◉Et sunt septem differentie. Prima differentia de proemio; secunda quod lux transit per diaffona corpora secundum verticationes linearum rectarum et reflectitur cum occurrerit corpori cuius diaffonitas fuerit diversa diaffonitati corporis in quo existit; tertia de qualitate reflexionis luminum in diaffonis corporibus; quarta differentia quod quicquid comprehenditur a visu ultra diaffona corpora quorum diaffonitas differt a diaffonitate corporis in quo visus existit cum fuerit declinis a perpendicularibus existentibus super superficies eorum, comprehenditur secundum reflexionem; quinta de fantasmatibus; sexta quomodo visus comprehendit visibilia secundum reflexionem; septima de fallaciis visus que accidunt ex reflexione.

◉There are seven chapters [in this book]. The first chapter [consists] of an introduction; the second [establishes] that light passes through transparent bodies along straight lines and is refracted⁑ when it encounters a body whose transparency is different from the transparency of the body in which it [originally] lies;⁑ the third [deals with] how light is refracted in transparent bodies; the fourth chapter [shows] that whatever is perceived by the visual faculty through transparent bodies whose transparency differs from the transparency of the body in which the visual faculty lies is perceived by means of refraction when [the visual faculty] lies to the side of the normals dropped [from the object’s surface] to the interface between the two transparent bodies; the fifth [deals] with the images [yielded by refraction];⁑ the sixth [explains] how the visual faculty perceives visible objects⁑ by means of refraction; [and] the seventh [is concerned] with the visual misperceptions that arise from refraction.

◉Prima differentia

◉Chapter One

1.1 ◉ Predictum est in proemio quarti tractatus huius libri quoniam visus tribus modis comprehendit visibilia—videlicet, secundum rectitudinem, et secundum conversionem a tersis corporibus, et secundum reflexionem ultra diaffona corpora que differunt in diaffonitate a diaffonitate aeris—et quod visus nichil comprehendit ex visibilibus nisi aliquo istorum trium modorum, et quod quolibet istorum modorum comprehendit visus visibilia et omnes res que sunt in visibilibus et omnibus modis visionis quorum distinctio declarata est in ultima differentia secundi tractatus.

1.1 ◉ It was pointed out in the introduction to the fourth book of this treatise [in Smith, Alhacen on the Principles, 295] that the visual faculty perceives visible objects in three ways—i.e., directly, or by reflection from polished bodies, or by refraction through transparent bodies that differ in transparency from the transparency of the air [in which the eye lies]—and [it was pointed out] that the visual faculty perceives nothing about visible objects except in one of these three ways; and [it was also pointed out] that in any one or all of these [three] ways the visual faculty perceives visible objects [as a whole] as well as all the properties of visible objects that were examined in detail in the last chapter of the second book.⁑

1.2 ◉ In precedentibus autem tractatibus declaratum est qualiter visus comprehendit visibilia secundum rectitudinem et secundum conversionem, et ostendimus diversitatem comprehensionis visus ad visibilia secundum utrumque istorum modorum. Remanet ergo declarare qualiter visus comprehendit visibilia secundum reflexionem ultra corpora diaffona. Nos autem in tractatu isto solummodo de reflexione tractabimus; et manifestabimus formam reflexionis, et distinguemus eius modos, et dividemus proprietates eius, et declarabimus quomodo accidit visui deceptio in huiusmodi visione. Et primo proponemus quedam fundamenta que certificant quicquid dependet ab hac re.

1.2 ◉ In the previous books, moreover, it was shown how the visual faculty perceives visible objects directly and by means of reflection, and we explained the various ways in which the visual faculty perceives visible objects according to each of those modes. It therefore remains to show how the visual faculty perceives visible objects by means of refraction through transparent bodies. And [so] in this book we will deal only with refraction; and we will clarify the formal nature of refraction, distinguish its modes, parse its specific characteristics, and explain how misperception arises in the visual faculty in this sort of vision. But first we will lay some foundations that confirm what will be entailed in this study.

◉Secunda differentia

◉Chapter Two

Quod lumen transit per diaffona corpora, et extenditur in eis secundum lineas rectas, et reflectitur cum occurrerit corpori diaffono differenti in diaffonitate a diaffonitate corporis in quo existit

2.1 ◉ Quoniam lumen quidem transit in aera et extenditur secundum rectas lineas declaratum est in tractatu primo huius libri. Aer autem est unum de corporibus diaffonis; aqua autem, et vitrum, et diaffoni lapides lumen transit per ipsa, et extenditur secundum lineas rectas. Hoc autem comprehenditur per experientiam.

2.1 ◉ That light in fact passes through air and extends [through it] along straight lines was demonstrated in the first book of this treatise.⁑ Air, however, is [only] one among [several] transparent bodies; light also passes through water, glass, and transparent stones, and it extends [through them] in straight lines. Indeed, this is observable by means of experiment.

2.2 ◉ Si quis ergo experiri voluerit, accipiet laminam ex ere rotundam cuius diameter non est minus uno cubito, et sit spissitudo eius aliquantulum fortis. Et habeat horas rotundas perpendiculares super superficiem eius, et sit altitudo horarum eius non minor latitudine duorum digitorum. In medio autem dorsi lamine sit aliquod corpus parvum columpnale rotundum cuius longitudo non minor latitudine trium digitorum, et sit perpendiculare super superficiem lamine. Et ponamus hoc instrumentum in retornativo in quo tornatorii retornant instrumenta cupri, et ponamus alterum dentem tornatorii in medio lamine et reliquum in medio extremitatis corporis quod est in dorso lamine. Et radamus revolvendo hoc instrumentum abrasione vera quousque verificetur rotunditas horarum suarum intus et extra, et adequetur superficies interior et exterior, et fiant due superficies equidistantes. Et abrademus etiam corpus quod est in dorso donec fiat rotundum.

2.2 ◉ Accordingly, if one wants to conduct [such] an experiment, he will take a round plate of bronze [henceforth referred to as the »register plate«], whose diameter is no less than a cubit, and it should be fairly thick [so as to remain rigid]. It should also have a round rim [attached] upright to its surface [around its perimeter], and its rim should be no less than two digits high [as illustrated in the bottom diagram of figure 7.2.1, p. 147].⁑ A small, round, cylindrical body no less than three digits long should be [attached] to the back side of the register plate at its center, and it should stand upright on the register plate’s surface [as illustrated in figure 7.2.2, p. 148]. We should insert this [entire] apparatus in a lathe, whose turning mechanism [is designed to] round off copper implements, and we should place one of the teeth of the turning mechanism at the center of the register plate [on its inner surface] and the other at the center of the outer end of the body attached to the back of the register plate. We should then even the [entire] apparatus out by turning it with a precise grinding until its rim is properly rounded both inside and out and the inner and outer surfaces [of the register plate] are smoothed out to form two parallel surfaces. We will also grind the body [attached] at the back [of the register plate] until it becomes [appropriately] round.

2.3 ◉ Cum ergo instrumentum hoc fuerit perfectum per abrasionem, signemus in superficie eius interiori duos diametros secantes se perpendiculariter et sic transeuntes per centrum eius. Deinde signemus punctum in basi hore instrumenti cuius distantia ab extremitate alterius duorum diametrorum secantium se est latitudo unius digiti. Deinde extrahamus ex isto puncto tertium diametrum transeuntem per centrum lamine quod extenditur in tota superficie eius. Deinde extrahemus a duobus extremis huius diametri duas lineas in superficie hore instrumenti perpendiculares super superficiem lamine. Deinde dividemus ex altera istarum duarum linearum tres lineas parvas equales quarum prima sequetur superficiem lamine, et longitudo cuiuslibet earum sit cum quantitate medietatis grani ordeacii. Fient igitur super lineam perpendicularem tria puncta que sunt fines istarum linearum.

2.3 ◉ When this apparatus is completely finished by grinding and polishing, we should mark off two diameters intersecting one another orthogonally on the inside surface [of the register plate] and therefore passing through its center [as represented by FG and BE in the top diagram of figure 7.2.1, p. 147]. We should subsequently mark a point at the base of the [inside wall of the] apparatus’s rim one digit to the side of the endpoint of either of the two intersecting diameters. Next we should draw a third diameter from this point passing through the center of the register plate and extending along its entire surface [as represented by AD in the top diagram of figure 7.2.1]. From the two endpoints of this [third] diameter we will draw two lines on the [inner] surface of the apparatus’s rim perpendicular to the surface of the register plate [i.e., FK and GL in figure 7.2.1]. On either of these two [perpendicular] lines, finally, we will mark off three short lines of the same length [end to end], the first of which will abut the surface of the register plate, and each of them should be half a grain of barley long.⁑ Three points will therefore be set on the perpendicular line, [and] these [points] constitute the endpoints of those [three short] lines.

2.4 ◉ Et deinde reducamus hoc instrumentum ad tornatorium, et signemus in ipso tres circulos equidistantes transeuntes per tria puncta que sunt super lineam perpendicularem super extremitatem diametri. Secetur igitur alia extremitas que est perpendicularis super aliam extremitatem huius diametri per istos tres circulos, et fient in ipsa tria puncta. Et fient in unoquoque trium circulorum duo puncta opposita que sunt extrema alicuius diametri ex eius diametris.

2.4 ◉ We should then reinsert this apparatus into the lathe, and we should mark off three parallel circles in it [on the inner wall of the rim] passing through the three [previously set] points that lie on the line perpendicular to the endpoint of the diameter [on the register plate’s face, as illustrated in the bottom diagram of figure 7.2.1]. The opposite line [on the inner wall of the ring] that is perpendicular to the other endpoint of this diameter should therefore be cut by those three circles, and they will form three points on that same [perpendicular]. In each of the three circles, moreover, two opposed points will be marked out at the endpoints of a diameter among [all] the diameters of that circle.

2.5 ◉ Deinde dividamus medium circulum ex istis tribus circulis per trecentas sexaginta partes, et si possibile fuerit per minuta. Deinde perforemus in hora instrumenti foramen rotundum cuius centrum sit medius punctus trium punctorum que sunt super alteram duarum linearum perpendicularium super extremitatem diametri lamine, et sit medietas diametri eius in quantitate distantie que est inter circulos. Perveniet igitur circumferentia foraminis inter duos circulos equidistantes qui sunt in extremitatibus.

2.5 ◉ We should then divide the middle of the three circles into 360 parts and into minutes, if that is possible. Next we should bore a round hole in the rim of the apparatus with its center being the middle point of the three points that lie on either of the two lines that are perpendicular to the endpoint of the plate’s diameter, and its radius should be the distance between the circles [i.e., half a grain of barley]. Hence, the perimeter of the hole will extend between the two outer parallel circles [that flank the middle one passing through the hole’s center].⁑

2.6 ◉ Postea accipiemus laminam subtilem quadratam aliquantule spissitudinis cuius longitudo sit in quantitate altitudinis hore instrumenti et cuius latitudo sit prope hoc. Et adequetur superficies eius quantum potest, et adequetur spissitudo eius etiam que sequetur alteram extremitatem eius quousque differentia communis inter superficiem faciei eius et inter superficiem spissitudinis eius fiet linea recta, quam lineam dividamus in duo equalia a cuius medio extrahamus lineam rectam in superficie faciei eius perpendicularem super illam rectam lineam que est communis differentia.

2.6 ◉ Afterwards we will take a small, moderately thick, square panel, whose length is equal to the height of the apparatus’s rim and whose width is around the same as that. Its surface should be ground as flat as it can be, and [the bottom surface along] its depth at either of its ends should be ground flat until the common section [that forms the edge] between the surface of its face and the [bottom] surface of its depth will form a straight line; we should bisect that line, and from the midpoint [so determined] we should draw a straight line on its face perpendicular to that straight line forming the common section [of the face and the bottom surface].

2.7 ◉ Deinde dividamus ex hac linea perpendiculari ex parte extremitatis que est super communem differentiam tres lineas equales inter se et equales unicuique parvarum linearum que distincte sunt super perpendicularem lineam in hora lamine. Fient igitur super lineam perpendicularem in facie lamine parve tria puncta. Deinde perforabimus hanc parvam lineam foramine rotundo cuius centrum sit medius punctus punctorum que distingunt lineas que sunt in ea, et sit medietas diametri eius equalis alicui uni linearum parvarum. Erit ergo hoc foramen equale foramini quod est in hora instrumenti.

2.7 ◉ From the endpoint of that perpendicular line on the side of the common section [at the bottom] we should mark off three lines equal to one another and equal to each of the short lines that were marked off on the perpendicular line on the [inner wall of the] register plate’s rim [at the endpoint of the third diameter]. Three points will thus be marked on the perpendicular line [drawn] on the face of the panel. Then we will drill a round hole through the short line [bounded by the three points just marked on the face of the panel] with its center at the middle point of the three points marked off by the [three short] lines [drawn] on it, and its radius should be equal to [the length] of any of the [three] short lines [i.e., half a grain of barley]. This hole will therefore be the same size as the hole in the rim of the apparatus.⁑

2.8 ◉ Deinde signabimus super diametrum lamine super cuius extremitates sunt due linee perpendiculares punctum in medio linee que est inter centrum lamine et extremitatem diametri que est in parte foraminis, et faciamus transire super hoc punctum lineam perpendicularem super diametrum. Deinde ponamus basim lamine parve super hanc lineam quousque differentia communis que est in parva lamina superponatur huic linee perpendiculari super diametrum, et erit punctus qui dividit differentiam communem que est in parva lamina in duo equalia superpositus super punctum signatum in diametro lamine.

2.8 ◉ On the diameter of the register plate at whose endpoints the two perpendicular lines [on the rim] lie we will then mark a point at the center of the line [forming the radius] between the register plate’s center and the endpoint of the diameter on the side of the hole [in the rim], and we should pass a line through this point perpendicular to the diameter. We should next place the base of the small panel on this line such that the common section [of the face and the bottom] of the small panel is flush with this line [drawn] perpendicular to the diameter and [so that] the point that bisects the common section of the small panel will coincide with the [mid]point marked on the register plate’s diameter.

2.9 ◉ Hoc autem toto facto, applicetur parva lamina cum maiori completa applicatione et consolidatione. Tunc ergo foramen quod est in parva lamina erit oppositum foramini quod est in hora instrumenti, et erit linea recta intellecta que copulat centra duorum foraminum in superficie circuli medii trium circulorum qui sunt in interiori hore instrumenti, et erit equidistans diametro lamine, et erit lamina parva que applicabitur puncto quasi hore astrolabii.

2.9 ◉ When all this is done, the small panel should be firmly and immovably attached [to the surface of the register plate]. Accordingly, the hole in the small panel will face the hole in the rim of the apparatus directly, the imaginary straight line that connects the centers of the two holes will lie in the plane of the middle of the three circles that are [inscribed] on the inner wall of the apparatus’s rim and will be parallel to the diameter on the register plate, and the small panel that will be attached at [that] point will be like the alidade of an astrolabe.⁑

2.10 ◉ Hoc autem completo, secetur de hora instrumenti quarta que est que sequitur quartam in qua est foramen ex quatuor quartis distinctis per duos primos diametros se perpendiculariter secantes, quarta propinqua tornatorio extra quod sunt hore, et adequetur locus sectionis donec fiat unum cum superficie lamine.

2.10 ◉ When this [step] is complete, the quadrant on the apparatus’s rim next to the quadrant in which the hole [in the rim] lies should be excised from among the four quadrants defined by the two initial diameters that intersect orthogonally, that quadrant being next to the turning mechanism outside of which the rim lies, and the [bottom] edge along which the cut is made should be smoothed until it it is flush with the surface of the register plate.⁑

2.11 ◉ Deinde accipiamus regulam eris cuius longitudo non sit minor sed maior uno cubito, et sit quadrate figure quam circumdent quatuor superficies equales in latitudine duorum digitorum, et adequentur superficies eius quantum possunt donec fiant equales et habentes angulos rectos. Deinde perforetur in medio alicuius superficiei eius foramen rotundum cuius amplitudo sit tanta quanta possit recipere corpus quod est in dorso instrumenti quod revolvatur in ipso non levi revolutione sed difficili, et sit foramen perpendiculare super superficiem regule et transiens in regulam ad aliam partem.

2.11 ◉ We should then take a bronze strip that is not less than but more than a cubit long [i.e., longer than the diameter of the register plate], it should be rectangular in shape so that the four flat surfaces [along its length] form a square two digits [on each] side [at the two ends], and its surfaces should be planed down as much as possible so as to be flat and form right angles. A round hole should then be drilled through the midpoint of one of its [lengthwise] surfaces, it should be just large enough to accommodate the [cylindrical] body that is [attached] at the back of the apparatus such that [the apparatus] may be turned in it not loosely but tightly, and the hole should be perpendicular to the surface of the strip and should pass through the strip to the other side.

2.12 ◉ Deinde ponamus instrumentum super regulam, et mittamus corpus quod est in dorso instrumenti in foramine quod est in medio regule donec superponatur superficies instrumenti superficiei regule. Hoc autem facto, secetur illud quod superfluit ex extremitatibus regule super diametrum lamine, nam regula longior est quam diameter lamine, quia sic posuimus eam. Cum ergo secaverimus duas superfluitates ex duabus extremitatibus regule, reducemus has duas superfluitates, et ponemus illas super duas extremitates regule ita quod ponemus duas extremitates superfluitatum super duas extremitates illius quod remansit de regula. Et applicabimus superficiem extremitatum cum superficie dorsi instrumenti, et erit illud quod ponetur ex utraque duarum superfluitatum super residuum regule equale latitudini unius digiti. Hac autem positione considerate, eminebuntur due superfluitates super duas extremitates regule, et si perforatum fuerit illud quod superfluit ex corpore, et missum fuerit in foramine eius stilus cupreus qui ipsum prohibeat exire erit melius. Hoc autem perfecto, perficietur instrumentum, et hec est forma dorsi instrumenti.

2.12 ◉ We should then attach the apparatus to the strip and insert the [cylindrical] body at the back of the apparatus into the hole in the middle of the strip until the [back] surface of the apparatus nests against the surface of the strip. Once this is done, the excess at the ends of the strip along the diameter of the register plate should be cut off, given that the strip is longer than the diameter of the plate because we constructed it that way. Thus, when we cut the two excess pieces from the two ends of the strip, we will remove these two excess pieces, and we will put them upon the two ends of the strip so that we will place the two ends of the excess pieces on the two ends of what remains of the strip. We will then attach the surface of the ends in the plane of the back of the apparatus, and the portion of the two excess pieces that will be attached to the remainder of the strip will amount to one digit [of overlap]. When this arrangement is in place, the two excess pieces will protrude beyond the two ends of [the long, central portion of] the strip [inserted into the axle], and it will be preferable if the overlapping portion of the excess piece is perforated and a copper pin inserted into the hole so as to keep it from sliding off. When this is done, the apparatus will be finished, and this is how the back of the apparatus takes form.⁑

2.13 ◉ Deinde accipiat experimentator regulam cupream parve latitudinis cuius latitudo sit duplum diametri foraminis quod est in hora instrumenti, et cuius spissitudo sit equalis diametro foraminis, et cuius longitudo non sit minor medietate cubiti. Et verificabitur ista regula donec fiat valde recta et vera, et fiant superficies eius equales et equidistantes. Deinde oblique secabimus alteram latitudinem eius quousque finis longitudinis eius contineat cum fine latitudinis eius angulum acutum ut possit homo declinare et movere eam quocumque voluerit, et ponet latitudinem eius ex alia extremitate perpendicularem super finem longitudinis eius. Deinde dividemus hanc latitudinem in duo equalia, et extrahemus a loco divisionis lineam in superficie faciei regule que extenditur in longitudine eius, et erit perpendicularis super latitudinem eius.

2.13 ◉ The experimenter should then take a thin copper ruler whose width should be twice the diameter of the hole in the rim of the apparatus [i.e., two grains of barley], whose thickness should be equal to the diameter of the hole [i.e., one grain of barley], and whose length should be no less than half a cubit. That ruler will be evened out [on all four of its faces] until it is perfectly straight and true and its surfaces are rendered plane and parallel. Then we will cut it slantwise at one of its ends along the width until the edge along [one of] its lengths forms an acute angle with its [newly cut] edge along the width so that a person can incline it and move it however he wishes [by pivoting it about its sharpened edge], and [the experimeter] will set its width at the other end perpendicular to its longitudinal edge. We will then bisect this widthwise edge, and from the point of bisection we will draw a line on the face of the ruler that extends its [entire] length, and [this line] will be perpendicular to the widthwise edge.⁑

2.14 ◉ Cum ergo hec regula fuerit superposita superficiei lamine, erit superficies eius superior in superficie circuli medii trium circulorum signatorum in interiori hore instrumenti, nam spissitudo huius regule est equalis diametro foraminis, et diameter foraminis equalis perpendiculari exeunti de centro foraminis quod est in hora instrumenti ad superficiem lamine, quia diameter foraminis est equalis duabus lineis trium linearum parvarum que distincte sunt de linea perpendiculari in interiori hore instrumenti. Cum ergo hec regula fuerit erecta super horam ipsius et fuerit superficies latitudinis eius super superficiem lamine, tunc linea descripta in medio eius erit in superficie medii circuli predicti, quia perpendicularis que egreditur a quolibet puncto huius linee ad finem longitudinis regule est equalis perpendiculari que egreditur a centro foraminis ad superficiem lamine, nam utraque istarum perpendicularium est equalis diametro foraminis.

2.14 ◉ Hence, when this ruler is applied [with its back face] on the surface of the register plate, its top face will lie in the plane of the middle circle of the three circles inscribed on the inner wall of the apparatus’s rim, for the thickness of this ruler is equal to the diameter of the hole [i.e., one grain of barley], and the diameter of the hole is equal to the perpendicular dropped from the center of the hole in [the inner wall of] the apparatus’s rim to the register plate’s surface, since the diameter of the hole is equal to two of the three short lines that were marked off on the perpendicular line [drawn] on the inner wall of the apparatus’s rim. Thus, when this ruler is stood on its edge with the surface of its edge applied to the surface of the register plate, the line drawn lengthwise through its middle will lie in the plane of the aforementioned middle circle because the perpendicular dropped from any point on this [mid]line to the edge along the length of the ruler [applied to the register plate’s surface] is equal to the perpendicular dropped from the center of the hole to the register plate’s surface, for both of those perpendiculars are equal to the diameter of the hole.

2.15 ◉ Cum ergo experimentator voluerit experiri transitum luminis in aqua per hoc instrumentum, accipiet vas rectarum horarum, ut cadum cupri, aut ollam figuli, aut consimile. Et sit altitudo horarum eius non minor medietate cubiti, et sit diameter circumferentie eius non minor diametro instrumenti. Et adequentur hore eius donec superficies que transit per horas eius sit superficies equalis, et ponamus in fundamento eius corpus diversarum partium aut diversorum colorum, ut anulus aut argentum depictum, aut depingatur in fundamento aque pictura manifesta.

2.15 ◉ Therefore, when the experimenter wants to test the passage of light into water empirically with this apparatus, he will take a vessel, such as a copper pot, a turner’s pottery jar, or the like, whose rim stands perpendicular [to its bottom]. The height of its rim should be no less than half a cubit and the diameter of its circumference no less than the diameter of the apparatus. [The lip of] its rim should be smoothed out until the plane passing through [the lip] of its rim is even, and we should place an object [consisting] of different segments or colors, such as a ring or a painted silver object, on its bottom, or a clear picture should be depicted at the bottom of the water [with which the vessel is to be filled].

2.16 ◉ Deinde fundatur in vas aqua clara donec impleatur, et expectetur donec motus eius quiescat. Cum ergo motus eius quieverit, erigatur aspiciens aut sedeat erectus, et aspiciat ad vas, et apponat visum suum corpori quod est in fundo aque aut picture que est in fundo aque donec linea inter visum et medium illius corporis aut illius picture sit perpendicularis super superficiem aque quoad sensum, et aspiciat corpus quod est in fundo aque aut picturam. Tunc inveniet illud eo modo quo est, et inveniet ordinationem suarum partium inter se adinvicem eo modo quo ordinarentur si aspiceret illud cum vas esset vacuum. Hoc autem declarato, certificabitur quod illud quod comprehenditur in fundo aque, cum aspexerit illud eadem positione qua aspexit corpus quod est in fundo aque et picturam, comprehenditur secundum ordinationem suarum partium.

2.16 ◉ Clear water should then be poured into the vessel until it is full, and [the experimenter] should wait until the water is perfectly calm. Accordingly, when its motion subsides, the viewer should stand straight up or sit upright and look into the vessel, directing his line of sight on the object lying at the bottom of the water or the picture lying at the bottom of the water until the line between [his] center of sight and the midpoint of that object or that picture is perpendicular to the water’s surface as far as can be empirically determined, and he should look at the object or picture at the bottom of the water. He will therefore find [that] it [appears] as it actually is, and he will find the arrangement of its parts among each other as they would be arranged if he looked at it when the vessel was empty. Having reached this determination, he will ascertain that, when he looks at anything according to the same vantage from which he looked at the object or picture at the bottom of the water, [whatever] is perceived at the bottom of the water is perceived according to the [true] arrangement of its parts [i.e., without any apparent distortion].⁑

2.17 ◉ Hoc autem certificato, si quis voluerit experiri transitum lucis, eligat locum super quem oritur lux solis in quo ponat vas, et preservet se ut superficies circumferentie vasis sit equidistans orizonti. Hoc autem potest observari hoc modo quod sit circumferentia superficiei aque equidistans circumferentie vasis: et si intus in vase aut prope circumferentiam eius fuerit signatus circulus equidistans circumferentie vasis, erit melius ad hoc quod circumferentia superficiei aque comparetur ad circumferentiam circuli.

2.17 ◉ With this point established, if one wants to test the passage of light empirically, he should choose a place upon which sunlight shines and put the vessel there, and he should make sure that the plane of the vessel’s [lip around its upper] circumference is parallel to the horizon. This can be done as follows to ensure that the plane formed by the water line is parallel to the [plane] of the vessel’s [lip]: if a circle is drawn inside the vessel or near its circumference parallel to the vessel’s lip, it will be best for that purpose that the circumference of the water’s surface be matched to the circumference of the circle.⁑

2.18 ◉ Deinde experimentor debet imponere instrumentum rotundum intra hoc vas ita quod due regule parve posite super duo extrema regule maioris superponantur hore vasis ex utraque parte. Tunc medietas instrumenti cum regula extensa in longitudine instrumenti erunt intra vas. Deinde addatur aqua aut diminuatur de ea donec fiat in superficie aque unum cum centro instrumenti, et sit aqua clara. Deinde revolvetur instrumentum in circuitu vasis donec obumbretur illud quod est intra aquam ex horis eius ab illo quod est supra aquam ex horis eius. Tunc teneatur regula altera manuum et revolvatur instrumentum reliqua manu super se in circuitu centri eius donec foramen quod est in hora instrumenti sit oppositum corpori solis, et transeat lumen solis in foramen, et perveniat ad alterum foramen, et transeat per aliud foramen. Cum ergo pertransiverit forma in duobus foraminibus, perveniet ad fundum aque. Tunc experimentator preservabit quod situs lucis in regula de secundo foramine sit situs equalis.

2.18 ◉ Then the experimenter should insert the round apparatus into this vessel so that the two small [excess] pieces applied to the two endpoints of the longer strip [to which the apparatus is attached by its axle] are hung from the rim of the vessel on each side. The midpoint of the apparatus, along with the strip that extends [behind and] across the apparatus [to which the back of the register plate is attached], will thus lie inside the vessel [below its lip]. Water should then be added or removed from it until the surface of the water reaches the center of the apparatus, and the water should be clear. Then the apparatus will be moved around [the lip of] the vessel until the portion of the [apparatus’s] rim that lies in the water is shaded by the portion of the rim that lies above the water. The [thin copper] ruler should therefore be held with one hand while the apparatus is rotated around [the axle at] its center with the remaining hand until the hole in the apparatus’s rim faces the sun [so that] the sunlight passes through the hole, reaches the other hole [in the small panel between it and the water], and passes through [that] other hole. Thus, when the [sunlight’s] form passes through the two holes, it will reach all the way down to the water. The experimenter will then make sure that the light [cast] on the ruler from the second hole is even [throughout].⁑

2.19 ◉ Hoc ergo situ preservato et luce preveniente ad superficiem aque, auferet experimentator manus suas ab instrumento, et stet erectus vel sedeat erectus, et inspiciat ad fundum aque ex quarta cuius hore sunt scisse, et preservet positionem quam preservaverat cum aspexerit corpus quod erat in fundo aque ut sit certus quod illud quod videt est secundum quod est. Tunc ergo cum intuebitur illud quod est intra aquam de hora instrumenti, inveniet lumen pertransiens ex duobus foraminibus super anterius hore instrumenti quod est intra aquam.

2.19 ◉ Now that the apparatus is set up this way with the light reaching the water’s surface, the experimenter should remove his hands from the apparatus and stand or sit upright, and he should look to the bottom of the water through the quadrant excised from the [apparatus’s] rim while maintaining the position he assumed when he looked at the object that lay at the bottom of the water in order to be certain that what he sees is according to its actual situation [in terms of location and the arrangement of parts]. Thus, when he will examine the portion of the apparatus’s rim that lies under water, he will find the light that passes through the two holes [shining] on the bottom rim of the apparatus that lies under water.

2.20 ◉ Et inveniet lumen inter duos circulos equidistantes extremos de tribus circulis signatis in anteriori parte hore instrumenti, aut addetur super distantiam que est inter circulos modicum, et erit additio eius ex duobus lateribus circulorum equalis. Sequitur ergo ex positione quod punctus qui est in medio luminis apparentis intra aquam quod est super interiorem partem hore instrumenti sit per medium circulum trium circulorum equidistantium qui sunt in interiori parte hore instrumenti. Et hoc lumen quod est intra aquam erit manifestius, quia hora superior instrumenti que circumdat superius foramen obumbrat interiorem partem hore instrumenti que circumdat lumen quod est in interiori parte hore instrumenti, et sic in illo loco non erit ex interiori parte hore instrumenti aliquid de lumine solis nisi lumen quod exit ex duobus foraminibus.

2.20 ◉ He will also find that the light [falls] between the two parallel outer circles of the three circles inscribed on the inner wall of the apparatus’s rim, or it will exceed the distance between [those] circles somewhat, but its excess will be equivalent on both sides of the circles. It therefore follows from [this] situation that the midpoint of the light appearing in the water on the inner wall of the apparatus’s rim lies on the middle circle of the three parallel circles [incised] on the inner wall of the apparatus’s rim. Moreover, the light that lies under water will be quite apparent because the upper portion of the apparatus’s rim that contains the upper hole shades the lower portion of the apparatus’s rim that contains the light on the inner wall of the apparatus’s rim, and so in that place there will be no sunlight on the inner wall of the apparatus’s rim other than the light that shines from the two holes.⁑

2.21 ◉ Deinde experimentator accipiet lignum minutum, sicut acum, et applicet eam in exteriori parte superioris foraminis quod est in hora instrumenti, et preservet se quod acus transeat per medium foraminis. Deinde aspiciat supra vas, et preservet positionem quam prius mensuravit. Tunc videbit umbram acus in medio lucis. Deinde incurret acum, attrahendo ipsam donec extremitas eius sit in medio foraminis, et intueatur lumen quod est intra aquam et quod est in superficie aque. Tunc inveniet umbram extremitatis acus in medio lucis que est intra aquam et in medio lucis que est in superficie aque.

2.21 ◉ Then the experimenter will take a fine wooden [stylus shaped] like a needle, and he should place it outside the upper hole in the apparatus’s rim, and he should ensure that the stylus⁑ is in line with the center of the hole. He should then look into the vessel from above, and he should maintain the position that he previously determined [by looking into the water along the perpendicular]. He will therefore see the shadow of the stylus in the middle of the [circle of] light [cast on the apparatus’s rim under water]. He should then move the stylus, drawing it inward [across the hole] until its [sharpened] endpoint lies at the center of the hole, and he should look at the light that is inside the water as well as the light on the water’s surface. He will thus find the shadow of the stylus’s [sharpened] endpoint at the middle of the light under water and at the middle of the light on the water’s surface.

2.22 ◉ Deinde mutet positionem acus, et ponat extremitatem eius etiam apud medium foraminis, et intueatur umbram. Tunc inveniet umbram extremitatis acus apud medium lucis. Deinde elevet acum, et inveniet lucem redeuntem ad suum statum intra aquam et in superficie aque. Deinde applicet acum in latere foraminis, et ponat eam cordam in foramine non diametrum, et intueatur lumen quod est intra aquam et in superficie aque. Tunc inveniet in utroque illorum umbram que est corda. Deinde elevet acum. Tunc inveniet lumen rediens ad suum locum, et si mutaverit situm acus in lateribus foraminis, inveniet umbram semper in latere luminis.

2.22 ◉ Next he should shift the position of the stylus [so as to bring it to the hole in the panel], and he should again place its [sharpened] endpoint at the middle of the hole and look at its shadow.⁑ Accordingly, he will find the shadow of the stylus’s [sharpened] endpoint at the middle of the light [on the rim and on the water’s surface]. He should then remove the stylus, and he will find that the light returns to its [original] situation in the water and on the water’s surface [i.e., replacing the shadow cast by the needle]. Then he should pose the stylus to the side of the hole, and he should place it along a chord, not a diameter of the hole, and he should look at the light inside the water and on the water’s surface. He will therefore find that the shadow at both of those [locations] forms a chord [in the circle of light]. Finally, he should remove the stylus. As a result, he will find that the light returns to its [original] place, and if he moves the stylus to the sides of the hole, he will always find the shadow at the side of the [circles of] light [cast on the water’s surface and on the rim under water].

2.23 ◉ Declarabitur ergo ex hac experientia quod punctus qui est in medio lucis que est intra aquam, que est circumferentia medii circuli, non exivit lux ad illum nisi ex puncto qui est medium lucis que est in superficie aque, et quod punctus qui est medium lucis que est in superficie aque non exibit lux ad ipsum nisi ex puncto quod est centrum foraminis superioris, et transit per punctum quod est centrum foraminis inferioris, scilicet foraminis quod est in horis, nam si non transisset per centrum foraminis inferioris, non manifestaretur medium lucis que est in superficie aque, cum acus esset in medio foraminis inferioris, sed non manifestaretur de luce que est in superficie aque nisi locus alius a medio eius.

2.23 ◉ From this experiment it will therefore be manifest that the [light at the] point in the center of the light inside the water, which lies on the circumference of the middle circle [inscribed on the rim’s inner wall], has reached that point only from the point in the middle of the light on the water’s surface, [and it is also clear] that the point at the middle of the light on the water’s surface will reach it only from the point at the center of the upper hole, and it passes through the point at the center of the lower hole, i.e., the hole that is in the panel, for if it did not pass through the center of the lower hole, the center of the light on the water’s surface would not be visible when the [point of the] stylus was in the middle of the lower hole, but instead some light other than that at the center [of the hole] would be visible at [the midpoint of the light on] the water’s surface.

2.24 ◉ Lux ergo que pervenit ad punctum quod est centrum lucis que est in superficie aque et lux que extenditur in aere non extenditur nisi secundum lineas rectas. Lux ergo que transit per centra duorum foraminum extenditur secundum rectitudinem linee transeuntis per centra duorum foraminum. Hec autem lux est illa que pervenit ad medium lucis que est in superficie aque. Punctus ergo qui est in medio lucis que est in superficie aque est in linea recta transeunte per centra duorum foraminum, et hec linea est in superficie medii circuli de tribus circulis signatis in interiori parte hore instrumenti, et est illi diameter, quia hec linea est equidistans diametro circuli qui est in superficie lamine. Cum ergo punctus qui est in medio lucis que est in superficie aque fuerit super hanc lineam, tunc iste punctus est in superficie circuli medii predicti. Punctus autem qui est in medio lucis que est intra aquam est in circumferentia medii circuli; ergo hec duo puncta sunt in superficie medii circuli.

2.24 ◉ Consequently, the light that reaches the point at the center of the light on the water’s surface and the light that extends through the air extend only along straight lines. The light that passes through the centers of the two holes therefore extends along the straight line passing through the centers of the two holes. This, moreover, is the light that reaches the center of the light on the water’s surface. Consequently, the point at the center of the light on the water’s surface lies on the straight line passing through the centers of the two holes, and this line lies in the plane of the middle circle among the three circles incised on the inner wall of the apparatus’s rim, and this is a diameter of it because this line is parallel to the diameter of the circle on the surface of the register plate. Since, therefore, the point at the center of the light on the water’s surface lies on this line, that point lies in the plane of the aforesaid middle circle. Moreover, the point at the center of the light [cast on the apparatus’s rim] inside the water lies on the circumference of the middle circle, so these two points lie in the plane of the middle circle.⁑

2.25 ◉ Si ergo lux que est in superficie aque latuerit et non fuerit bene manifesta, tunc experimentator mittet illam regulam in aquam, et applicet horam eius in superficie lamine, et ponat superficiem in qua signata est linea sequentem superficiem aque, et moveat illam donec superficies eius fiat cum superficie aque. Cum ergo superficies regule fuerit cum superficie aque, et regula fuerit erecta super horam eius, tunc linea que est in superficie ipsius erit in superficie circuli medii que transit per centra duorum foraminum. Hac autem positione preservata, apparebit lux que est in superficie aque super superficiem regule, et inveniet medium lucis super lineam que est in medio regule. Et si acus fuerit posita super medium superioris foraminis, tunc linea que est in medio regule obumbrabitur, et si extremitas acus fuerit posita super centrum foraminis, apparebit umbra extremitatis acus in medio lucis que est super regulam. Et si acus fuerit ablata, redibit lux sicut erat.

2.25 ◉ If the light on the water’s surface is faint and not clearly visible, then the experimenter will place the ruler in the water, and he should apply its edge to the surface of the register plate and pose the face on which the [mid]line is inscribed up to the waterline and adjust it until its surface is flush with the water’s surface. Hence, when the face of the ruler is flush with the surface of the water, and when the ruler is stood upon its edge, the [mid]line on its surface will lie in the plane of the middle circle that passes through the centers of the two holes. When this arrangement is in place, the light [shining] on the water’s surface will appear on the ruler’s surface, and [the experimenter] will find the center of the [resulting] light on the line through the middle of the ruler. And if the stylus is placed across the middle of the upper hole, the line that lies at the middle of the ruler will be darkened by its shadow, and if the [sharpened] endpoint of the stylus is placed at the center of the hole, the shadow of the stylus’s [sharpened] endpoint will appear in the center of the light cast on the ruler. Also, if the stylus is removed, the light will return as it was.

2.26 ◉ Cum hac ergo regula apparebit lux que est in superficie aque apparitione manifesta, et manifestabitur quod est super lineam transeuntem per centra duorum foraminum. Et iam posueramus superficiem aque apud centrum lamine. Cum ergo superficies regule cum superficie aque fuerit, erit superficies regule transiens per centrum lamine, et tunc erit remotio centri lucis a centro lamine equalis medietati latitudinis regule, que est equalis perpendiculari cadenti a centro foraminis super superficiem lamine. Et sic erit centrum lucis que est in superficie regule centrum circuli medii.

2.26 ◉ Therefore, since the light [shining] on the water’s surface will appear clearly on the ruler, it will also be evident that it lies on the line passing through the centers of the two holes. But earlier we brought the water up to the center of the register plate. Consequently, since the surface of the ruler coincides with the water’s surface, the surface of the ruler will pass through the center of the register plate, and so the distance of the center of the light [shining on the ruler’s surface] from the center of the register plate will be equal to half the width of the ruler, and that is equal to the perpendicular dropped from the center of the hole to the surface of the register plate [i.e., one grain of barley]. The center of the light on the ruler’s face will therefore lie at the center of the middle circle.

2.27 ◉ Deinde oportet experimentatorem auferre regulam subtilem, et mittere eam iterum in aquam, et applicare superficiem latitudinis eius cum superficie lamine, et ponere angulum eius acutum apud centrum lucis que est intra aquam, scilicet angulus qui est in superficie eius superiori. Deinde moveat regulam donec acuitas eius inferior, que est in superiori lamine, transeat per centrum lamine, et sic acuitas eius superior transibit per centrum circuli medii. Punctus ergo ex linea superiori regule qui est in superficie aque est centrum circuli medii; est ergo centrum lucis que est in superficie aque, et erit longitudo eius diameter ex diametris medii circuli.

2.27 ◉ The experimenter must then remove the thin ruler, place it again in the water, apply it to the surface of the register plate along its wider face, and put its acute angle, i.e., the angle at its top end, at the center of the light that lies in the water. He should then adjust the ruler [in place] until its lower edge, which lies on the upper part of the register plate [above the water], passes through the center of the register plate, and so its top edge will pass through the center of the middle circle. Therefore, the point on the upper edge of the ruler at the surface of the water [where it meets that surface] is [coincident with] the center of the middle circle, so it is [coincident with] the center of the light [shining] on the water’s surface, and [the line along its edge extending] its [full] length will form one of the diameters of the middle circle.

2.28 ◉ Hac autem ratione preservata, accipiat experimentator acum longam, et mittet eam in aquam, et ponat capud suum in puncto ultimitatis regule, et intueatur lucem que est intra aquam. Tunc inveniet umbram acus secantem lucem, et inveniet umbram capitis acus apud cornu regule que est apud medium lucis. Deinde mutet positionem acus, et capud eius sit loco eius ex fine regule. Tunc mutabitur situs umbre ex luce que est intra aquam, et erit umbra capitis acus inseperabilis a medio lucis. Deinde auferat acum, et tunc redibit lux ad suum locum. Deinde mittat acum in aquam iterum, et ponat capud eius in alio puncto finis regule, et intueatur umbram. Tunc inveniet secantem lucem que est intra aquam, et inveniet umbram capitis acus in medio lucis. Deinde mutet positionem acus super multitudinem punctorum ex acuitate regule, et inveniet umbram capitis eius semper in medio lucis.

2.28 ◉ Now when [the apparatus] is set up in this way, the experimenter should take a long needle,⁑ place it in the water, put its point at the top point [on the edge] of the ruler [where that edge meets the water’s surface], and look at the light inside the water. Accordingly, he will find that the shadow of the needle cuts the light, and he will find that the shadow of the needle’s point [which he holds] at the vertex [on the edge] of the ruler lies at the center of the light [cast on the rim inside the water]. He should then change the position of the needle, but its point should [still] lie on the edge of the ruler. Accordingly, the location of the shadow in the light at the bottom of the water will change, but the shadow of the point will be inseparable from the middle of the light. Then he should remove the needle, and the light will return to its [original] location. He should then put the needle back into the water and place its point at another point on the ruler’s edge, and he should look at the shadow. He will therefore find that it cuts the light inside the water, and he will find the shadow of the needle’s point at the center of that light. Then he should shift the location of the needle to several points on the ruler’s edge, and he will invariably find the shadow of its point at the center of the light [in the water].

2.29 ◉ Declarabitur ergo ex hac experientia declaratione manifesta quod lux que est in puncto mediante lucem que est intra aquam, que est super circumferentiam medii circuli, est perveniens ad illum punctum a puncto quod est medium lucis que est in superficie aque. Et declarabitur cum hoc quod hec lux extenditur super lineam rectam que est finis regule, nam experientia eius per extremitatem acus ex diversis locis in fine regule ostendit illam transeuntem per omne punctum finis regule. Hac ergo via experimentabitur transitus lucis per corpus aque, ex qua declarabitur quod extensio lucis per corpus aque est secundum verticationes rectarum linearum.

2.29 ◉ It will thus be patently obvious from this experiment that the light at the centerpoint of the light in the water, which lies on the circumference of the middle circle, reaches that point from the point at the center of the light on the water’s surface. And it will be manifest on that account that this light extends along the straight line formed by the edge of the ruler, for testing it with the [sharpened] endpoint of the needle at various spots on the ruler’s edge shows that it passes through every point on the edge of the ruler. Accordingly, in this way the passage of light through the body of water will be empirically tested, from which it will be demonstrated that the light extends through the body of the water along straight lines.⁑

2.30 ◉ Deinde oportebit experimentatorem quod ponat super centrum lucis signum fixum cum scalpsione. Deinde quando fuerit experimentator intuens punctum quod est in medio lucis que est intra aquam, inveniet ipsum non equidistans duabus extremitatibus diametri lamine, scilicet extra duas lineas perpendiculares super extremitatem diametri lamine qui est intra aquam. Et inveniet declinationem eius ab ista linea ad partem in qua est sol, et inveniet inter punctum quod est centrum medii lucis et punctum quod est communis differentia linee perpendiculari super extremitatem diametri lamine et puncto medio quod est extremitas diametri medii circuli transeuntis per centrum foraminum, inveniet dico distantiam sensibilem.

2.30 ◉ The experimenter will then need to put a permanent mark with an incising tool at the center of the light [at the bottom of the rim inside the water]. Accordingly, when the experimenter looks at the point at the center of the light inside the water, he will find that it is not in line with the two endpoints of the diameter on the register plate, i.e., that it lies outside [the plane containing] the two lines [incised on the inner wall of the apparatus’s rim] perpendicular to the endpoint of the diameter on the register plate in the water. And he will find that it inclines away from that line [i.e., the aforementioned perpendicular on the inner wall of the rim] in the direction of the sun, and I maintain that he will find a noticeable discrepancy between the point that lies at the center of the light [on the bottom of the rim] and the point that forms the common section of the line [on the inner wall of the rim] perpendicular to the endpoint of the diameter on the register plate and the midpoint at the end of the diameter of the middle circle, which passes through the centers of the two holes.

2.31 ◉ Hoc declarato, oportet mittere regulam subtilem in aquam, et applicare eam cum superficie lamine, et ponere terminum regule super centrum lamine, et movere regulam quousque acuitas eius sit perpendicularis super superficiem aque quoad sensum. Tunc igitur inveniet centrum lucis que est intra aquam inter acuitatem regule et lineam perpendicularem super diametrum lamine. Declarabitur ergo ex hoc quod hec reflexio est ad partem perpendicularis exeuntis a loco reflexionis perpendicularis super superficiem aque. Cum ergo certus fuerit experimentator de hoc, oportebit eum signare apud extremitatem regule que est super circumferentiam medii circuli que est extremitas perpendicularis exeuntis a centro medii circuli perpendicularis super superficiem aque signum fixum, ut primum quod signatum est apud centrum lucis.

2.31 ◉ Having established this point, [the experimenter] should place the thin ruler in the water, apply it to the surface of the register plate, pose the endpoint of the ruler at the center of the register plate, and adjust the ruler until its acute angle lies [in line with the] perpendicular to the water’s surface as far as can be empirically determined. He will therefore find that the center of the light [on the rim] inside the water lies between the [point of the] ruler’s acute angle and the line [drawn on the apparatus’s rim] perpendicular to the diameter on the register plate [at its endpoint below the water]. It will therefore be manifest from this that this refraction occurs toward the normal dropped perpendicular to the water’s surface from the point of refraction. Accordingly, when the experimenter has ascertained this point, he will need to put a permanent mark at the endpoint of the ruler, which lies on the perimeter of the middle circle at the endpoint of the normal dropped from the center of the middle circle at the water’s surface, just as he earlier marked the point at the center of the light [cast on the bottom of the rim].

2.32 ◉ Et iam declaratum est quod lux que pervenit ad punctum quod est centrum lucis que est intra aquam est lux extensa secundum rectitudinem linee continuantis duo centra foraminum, et hec linea pervenit ad centrum medii circuli equidistantis superficiei lamine, et est illi diameter. Si hec linea fuerit extensa in ymaginatione secundum rectitudinem intra aquam donec perveniat ad horam lamine, tunc igitur erit equidistans diametro lamine, et perveniet ad lineam perpendicularem in interiori parte hore lamine. Et cum centrum lucis que est intra aquam non est super perpendicularem lineam hore lamine, tunc lux que extenditur a medio lucis que est in superficie aque ad medium lucis que est intra aquam non extenditur secundum rectitudinem linee transeuntis per centra duorum foraminum, sed refertur.

2.32 ◉ Now it has already been demonstrated that the light reaching the point at the center of the light inside the water is the [same] light that extends along the straight line connecting the two centers of the holes, and this line reaches the center of the middle circle that is parallel to the surface of the register plate and forms a diameter on it. If this line is imagined to continue in a straight line below the water until it reaches the rim [on the circumference] of the register plate, then it will be parallel to the register plate’s diameter, and it will reach the perpendicular line on the inner wall of the register plate’s rim. But since the center of the light in the water does not lie upon the line [drawn] perpendicular on the [inner wall of the] register plate’s rim, the light that extends from the center of the light on the water’s surface to the center of the light inside the water does not extend along the straight line passing through the centers of the two holes; instead, it is diverted.⁑

2.33 ◉ Declaratum est autem quoniam hec lux extenditur recte a medio lucis que est in superficie aque ad medium lucis que est intra aquam. Ergo reflexio huius lucis est apud superficiem aque. Et iam declaratum est quoniam hec lux transit per centra duorum foraminum et in medio lucis que est in superficie aque quod est centrum circuli medii equidistantis superficiei lamine et medio lucis que est intra aquam quod est in circumferentia medii circuli, ex quo patet quod lumen perveniens ad centrum lucis que est intra aquam, dum extenditur in aere et postquam reflectitur intra aquam, est in eadem superficie equali, scilicet in superficie circuli medii trium circulorum qui sunt in interiori parte hore instrumenti.

2.33 ◉ It has also been demonstrated that this light extends [in a] straight [line] from the center of the light on the water’s surface to the center of the light in the water. Consequently, the refraction of this light occurs at the water’s surface. It has just been shown, as well, that this light passes through the centers of the two holes and through the midpoint of the light on the water’s surface, which constitutes the center of the middle circle parallel to the surface of the register plate, and through the midpoint of the light in the water that lies on the circumference of the middle circle; from this it is evident that, when it extends through the air and is subsequently refracted in the water, the light reaching the center of the light in the water lies in the same plane, i.e., in the plane of the middle circle of the three circles that are [inscribed] on the inner wall of the apparatus’s rim.

2.34 ◉ Et hec reflexio invenitur quando linea transiens per centrum foraminum fuerit declinis super superficiem aque, non perpendicularis, et numquam erit hec linea perpendicularis super superficiem aque in hora transitus lucis solis nisi quando sol fuerit in verticatione capitis. Et hoc erit in aliquibus locis et non in omnibus, in quibusdam temporibus, non in omnibus, neque sol transit per verticationem capitis habitantium in pluribus locis habitationis, et in istis locis distinguetur hec experimentatio in omni tempore; illi autem super quorum cenit transit sol, si voluerint hoc experiri, cavebunt tempus in quo sol transit per capita eorum.

2.34 ◉ But this refraction is observed when the line passing through the center of [each of] the holes is inclined, not perpendicular to the water’s surface, and this line will never be perpendicular to the water’s surface at the time of the sun’s transit [to its highest point in the sky] unless the sun reaches the [viewer’s] zenith. This, however, will be [the case] in some places, not in all places, and at certain times, not at all times, and [so] the sun does not pass through the zenith of those who live in many locations [on earth], but in those places [where it does] this experiment will be feasible at any time [the sun transits the zenith]. If those through whose zenith the sun passes do want to test [the perpendicular passage of light through water], they will be careful [to note] the time at which the sun passes through their zenith.⁑

2.35 ◉ Item accipiat experimentator frusta vitri clari quorum figure sunt cubice, et sit longitudo uniuscuiusque eorum dupla diametri foraminis quod est in hora instrumenti. Et adequentur superficies eorum vehementer per confricationem quousque superficies eius sint equales et equidistantes et latera eius sint recta. Deinde poliantur. Hoc autem completo, signetur in medio lamine linea recta transiens per centrum eius, et sit perpendicularis super diametrum eius super cuius extrema sunt due linee perpendiculares in interiori parte instrumenti, et transeat in utramque partem. Et signetur hec linea ferro ut descendat in corpus lamine, et remaneat ibi.

2.35 ◉ Now the experimenter should take some pieces of clear glass that are cubical in shape, and each [side] of them should be twice as long as the diameter of the hole in the apparatus’s rim [i.e., two grains of barley]. Their faces should be smoothed down by vigorous rubbing until their surfaces are flat and parallel and their edges are straight. They should then be polished. When this is done, a straight line should be inscribed on the face of the register plate through its center; it should be perpendicular to the diameter on it at whose endpoints the two perpendicular lines [drawn] on the inner wall of the apparatus fall, and it should extend on both sides [of that diameter]. This line should be incised with a steel [instrument] so that it cuts into the body of the register plate and remains there [permanently].

2.36 ◉ Deinde ponat unum vitrorum cubicorum super superficiem lamine, et applicet unum latus suorum laterum cum hac perpendiculari, et ponat medium lateris vitri vere super centrum lamine, et ponat corpus vitri ex parte foraminum. Est ergo diametrum lamine super cuius extrema sunt due linee perpendiculares transiens per medium superficiei vitri superposite lamine. Hac positione preservata, applicetur vitrum applicatione fixa per inglutum tali modo quod possit evelli.

2.36 ◉ [The experimenter] should then place one of the glass cubes upon the register plate’s surface and apply one of its edges to this perpendicular [just drawn], and he should position the midpoint of the glass cube’s [bottom] edge right at the center of the register plate while posing the body of the glass on the side of the [two] holes [on the upper side of the apparatus above the waterline]. Hence, the diameter on the register plate at whose endpoints the two perpendicular lines [drawn on the inner wall of the apparatus’s rim] fall passes through the middle of the face of the glass [cube] applied to the plate. When it is set up in this way, the glass [cube] should be attached solidly [to the plate] with adhesive in such a way that it can [nonetheless] be removed.⁑

2.37 ◉ Deinde accipiatur secundum vitrum, et ponatur ultra primum, scilicet ex parte foraminum, et applicetur aliqua superficierum eius superficiei primi vitri. Hoc preservato, applicetur secundum vitrum lamine applicatione fixa. Deinde accipiatur tertium vitrum, et applicetur secundo vitro, et adequetur superficies eius cum duabus superficiebus laterum secundi vitri, et applicetur lamine. Et sic fiat de pluribus vitris quousque perveniant intra ad horam perpendicularium super superficiem instrumenti, aut prope.

2.37 ◉ [The experimenter] should then take a second glass [cube] and place it in front of the first one, i.e., on the side of the [two upper] holes, and he should nest one of its faces against the face of the first glass [cube]. When this is properly set up, he should attach the second glass [cube] solidly to the face of the plate [so that it can nonetheless be removed]. He should then take a third glass [cube], nest it against the second glass [cube], align its surfaces [on each side] with the two surfaces on the sides of the second glass [cube], and affix it to the plate. He should do the same with more of the glass [cubes] until they reach [successively] to the panel [with their upright faces] perpendicular to the surface of the [register plate of the] apparatus, or [until they] nearly [reach that panel].

2.38 ◉ Cum ergo intra fuerint applicata superficiei lamine secundum positionem predictam, erit diameter lamine super cuius extremitates sunt due linee perpendiculares in extremitate instrumenti transiens per mediam superficiem vitrorum superpositorum lamine. Altitudo autem istorum vitrorum in latitudine est dupla diametri foraminis, sed diameter foraminis est equalis perpendiculari exeunti a centro foraminis super superficiem lamine et super diametrum eius. Ergo unaqueque perpendicularium exeuntium a centris superficierum vitrorum, scilicet superficierum perpendicularium super superficiem lamine secantium diametrum opposite duobus foraminibus, est equalis perpendiculari exeunti a centro foraminis super superficiem lamine et super diametrum lamine. Et erunt perpendiculares exeuntes a centris superficierum vitrorum ad superficiem lamine cadentes super diametrum lamine super cuius extremitates est perpendicularis egrediens a centro foraminis. Linea ergo que transit per centra duorum foraminum, si extendatur in ymaginatione secundum rectitudinem, transibit per centra superficierum vitrorum, scilicet superficierum perpendicularium super superficiem lamine opposite duobus foraminibus.

2.38 ◉ Therefore, when they are attached [in a line] to the surface of the register plate according to the aforesaid disposition, the diameter of the register plate upon whose endpoints the two perpendicular lines [drawn on the inner wall of the rim] fall will pass through the middle of the [bottom] surface of the glass [cubes] applied to the register plate. Moreover, the height of those glass cubes [above the register plate] is twice the diameter of the hole [in the panel], and the diameter of the hole [in the panel] is equal to the perpendicular dropped from the center of the hole to the surface of the register plate and [intersecting that surface] on the register plate’s diameter. Thus, each of the perpendiculars dropped from the centers of the faces of the glass [cubes], i.e., the faces upright on the register plate’s surface and cutting the diameter [on that surface] aligned with the two holes, is equal to the perpendicular dropped from the center of the hole to the surface of the register plate and to the register plate’s diameter. In addition, the perpendiculars dropped from the centers of the faces of the glass [cubes] to the surface of the register plate will fall on the diameter of the register plate at whose endpoints the perpendicular dropped from the center of [each] hole [in the apparatus’s rim and the panel] falls. Hence, if it is imagined to extend rectilinearly, the line passing through the centers of the two holes will pass through the centers of the faces of the glass [cubes], i.e., the faces that are perpendicular to the surface of the register plate and directed toward the two holes.

2.39 ◉ Deinde experimentator accipiat regulam subtilem predictam, et erigat illam super horam ipsius in superficie lamine, et ponat faciem eius in qua signata est linea ex parte primi vitri quod est super centrum lamine. Et ponat regulam prope vitrum, et ponat finem longitudinis regule secantem diametrum lamine perpendiculariter. Hoc ergo preservato, applicet regulam lamine applicatione fixa ita quod possit separari. Hac autem positione preservata in regula, tunc linea que est in superficie regule erit in superficie medii circuli ex tribus circulis signatis in interiori parte hore instrumenti, et erit linea recta transiens per centra duorum foraminum et per media superficierum vitrorum secans lineam que est in regula.

2.39 ◉ The experimenter should then take the thin ruler described before, stand it edgewise on the surface of the register plate, and pose the [wider] face on which the line was drawn toward the first glass [cube] at the center of the register plate. He should place the ruler near the glass [cube] and situate the lengthwise [bottom] edge of the ruler so that it intersects the diameter on the register plate orthogonally. Given this disposition, then, he should fasten the ruler firmly to the register plate so that it can [nonetheless] be removed from it. When the ruler is set up in this way, the line on the ruler’s face will lie in the plane of the middle circle among the three circles inscribed on the inner wall of the apparatus’s rim, and the straight line passing through the centers of the two holes, as well as through the centers of the faces of the glass [cubes], will intersect the line on the ruler.

2.40 ◉ Hoc toto completo, ponatur instrumentum in vas predictum. Sit autem vas vacuum aqua, et ponat vas in sole, et moveat instrumentum quousque lux solis transeat per duo foramina, et erit lux apud secundum foramen equalis, scilicet quod sit super omnia foramina, et si excesserit super foramen, erit vitrum continens foramen. Tunc igitur intueatur experimentator superficiem regule oppositam vitro, et inveniet lucem exeuntem a duobus foraminibus super superficiem regule, et inveniet illud quod circumdat lucem ex superficie regule obumbratum umbra hore instrumenti, et inveniet centrum lucis super lineam que est in superficie regule.

2.40 ◉ When all this is finally arranged, [the experimenter] should place the apparatus in the vessel described earlier. The vessel should be empty of water, though, and he should place the vessel in sunlight and adjust the apparatus until the sunlight passes through the two holes, and [so] the light at the second hole will be equivalent to that which lies at all the holes, and if it exceeds the [size of the] hole, the glass [cube can accommodate the excess because it] will embrace the hole [insofar as it extends beyond the hole on all sides]. The experimenter should therefore look at the surface of the ruler facing the [first] glass [cube], and he will find that the light passing through the two holes is [cast] on the ruler’s surface; he will also find that the shadow circumscribing the light on the ruler’s surface is the shadow [cast] by the [solid portion of the] apparatus’s rim [surrounding the hole], and he will find that the center of the light [on the ruler] lies on the [mid]line [drawn] on the ruler’s surface.⁑

2.41 ◉ Hoc ergo declarato, accipiat festucam subtilem, ut acum, et ponat illam super superius foramen, et ponat extremitatem perpendicularis super centrum foraminis, et intueatur lucem que est super regulam. Tunc inveniet umbram extremitatis festuce super centrum lucis, et inveniet illam super lineam que est in superficie regule. Tunc igitur accipiet experimentator pennam intinctam incausto, et signet super extremitatem umbre que est in medio lucis que est super regulam punctum. Ergo erit iste punctus super lineam que est in superficie regule.

2.41 ◉ When he has made this determination, [the experimenter] should take the thin stylus [shaped] like a needle,⁑ place it at the upper hole, pose its [sharpened] endpoint perpendicular to the center of the hole, and look at the light on the ruler. He will then find the shadow of the stylus’s [sharpened] endpoint at the center of the light [on the ruler], and he will find it on the line on the ruler’s face. The experimenter will take a pen dipped in ink, and he should mark the point where the shadow of the [stylus’s sharpened] endpoint [falls] on the midpoint of the light on the ruler. Accordingly, that point will lie on the [mid]line on the ruler’s face.

2.42 ◉ Deinde auferat acum a superiori foramine, et ponat ipsam super inferius foramen, scilicet quod est in hora, et ponat extremitatem acus super centrum foraminis, et intueatur lucem que est super regulam. Tunc inveniet umbram extremitatis acus super punctum quod est in superficie regule. Deinde auferat acum, et redibit lux ad suum locum. Declarabitur ergo ex hac experimentatione quod lux que est super punctum quod est in superficie regule est lux que transit per centra duorum foraminum.

2.42 ◉ Then he should remove the stylus from the upper hole and place it at the lower hole, i.e., the one in the panel, and he should place the [sharpened] endpoint of the stylus at the center of the hole and look at the light on the ruler. He will thus find the shadow of the stylus’s [sharpened] endpoint on the point [previously marked] on the ruler’s face. Then he should remove the stylus, and the light will return to its [original] place. It will therefore be manifest from this experiment that the light at the point [marked] on the ruler’s face is the light passing through the centers of the two holes.

2.43 ◉ Deinde accipiat experimentator calamum tinctum incausto, et signet punctum in vero medio superficiei vitri ex parte regule. Si vero non comprehendit medium vitri quoad sensum, signet in ipso duos diametros secantes se, et locus sectionis est medium superficiei vitri. Hoc autem facto, intueatur lucem que est super regulam, et inveniet umbram puncti que est in medio vitri super punctum quod est superficiei regule. Declarabitur ergo ex hoc quod lux que transit per duo centra duorum foraminum transit per punctum quod est in medio vitri.

2.43 ◉ The experimenter should then take a pen dipped in ink and mark the point right at the center of the face of the glass [cube] on the side of the ruler. If, however, he cannot determine the midpoint of the [face on the] glass [cube] empirically, he should draw the two diagonals that intersect one another [on that face], and the point of intersection constitutes the center of the glass [cube’s] face. Having done this, he should look at the light on the ruler, and he will find the shadow of the point at the center of the glass [cube’s face cast] on the point [previously marked] on the ruler’s face. It will thus be manifest from this that the light passing through the two centers of the two holes passes through the point at the center of the glass [cube’s face].

2.44 ◉ Hoc ergo declarato, oportet experimentatorem vitrum evellere et componere instrumentum secundo, et moveat ipsum quousque lux transeat per duo foramina. Deinde intueatur superficiem regule que est centrum lucis et lucem pervenientem ad centrum lucis que est in superficie regule, et est lux que transit per centra duorum foraminum. Declarabitur ergo ex hoc quod lux que transit per centra duorum foraminum transit etiam per punctum quod est in medio superficiei secundi vitri, et situs lucis transeuntis per centra duorum foraminum de superficiebus vitrorum in prima experimentatione, et cum hec lux transit per punctum qui est in medio vitri secundi, tunc lux que transit per centra duorum foraminum in prima experimentatione transit etiam per punctum quod est in medio vitri secundi.

2.44 ◉ Once this has been determined, the experimenter should remove the [first] glass [cube] and set up the apparatus with the second one, and he should adjust it until the light passes through the two holes. He should then look at the [point marked on the] ruler’s face that constitutes the center of the light and at the light [actually] reaching the center of the light on the ruler’s face, and this is the light that passes through the centers of the two holes. It will therefore be manifest from this that the light passing through the centers of the two holes also passes through the point at the center of the face of the second glass [cube], and [this is] the [same] location as that of the light passing through the centers of the two holes to the faces of the glass [cubes] in the first experiment, and since this light passes through the point at the center of the [face of the] second glass [cube], it follows that the light passing through the centers of the two holes in the first experiment also passes through the point at the center of [the face of] the second glass [cube].

2.45 ◉ Deinde oportet experimentatorem evellere secundum vitrum et experiri tertium, et sic de ceteris usque ad ultimum. Patebit ergo experimentatione hac quod lux que transit per centra duorum foraminum perveniens ad superficiem regule transit etiam per centra superficierum vitrorum omnium positorum super superficiem lamine. Manifestum est ergo quoniam sunt in rectitudine linee recte transeuntis per centra duorum foraminum, et lux que transit per centra duorum foraminum in experimentatione omnium vitrorum extenditur in rectitudine linee continuantis centra duorum foraminum.

2.45 ◉ The experimenter should then remove the second glass [cube] and test the third, and so on to the last one. It will therefore be obvious from this experiment that the light passing through the centers of the two holes and reaching the ruler’s surface also passes through the centers of the faces of all the glass [cubes] placed on the register plate’s surface. It is therefore evident that [all those centerpoints] lie on the straight line passing through the centers of the two holes, and, when all the glass [cubes] are tested, the light passing through the centers of the two holes extends [through the glass] in a straight line connecting the centers of the two holes.

2.46 ◉ Manifestum est ergo quoniam lux que extenditur per lineam rectam transeuntem per centra duorum foraminum transit etiam per centra superficierum vitrorum, ex quibus patet quod lux transit in corpus vitri in quo extenditur, postquam transit secundum lineas rectas, et quod lux que transit per centra duorum foraminum extenditur etiam in corpus vitri secundum rectitudinem linee per quam extendebatur in aere antequam pertransiret vitrum. Et illa linea per quam extenditur lux in aere est perpendicularis super superficiem vitri oppositam foramini, nam linea que transit per centra duorum foraminum est equidistans diametro lamine perpendicularis super primam superficiem superficierum vitrorum, quia est perpendicularis super differentiam communem inter superficiem vitri et superficiem lamine.

2.46 ◉ So it is clear that the light extending along the straight line passing through the centers of the two holes also passes through the centers of the faces of [all] the glass [cubes], from which it is obvious that the light passes into the body of the glass through which it extends and then continues on along straight lines and that the light passing through the centers of the two holes also extends through the body of the glass along the [same] straight line according to which it extended in the air before it passed into the glass. Moreover, the line along which the light extends through the air is perpendicular to the surface of the glass [cube] facing the hole, for the line passing through the centers of the two holes is parallel to the diameter on the register plate that is perpendicular to the first face of the glass [cubes], since it is perpendicular to the common section of the glass [cube’s] face and the surface of the register plate.⁑

2.47 ◉ Item accipiat experimentator medietatem spere vitree munde clare aut cristalline cuius semidiameter sit minor distantia que est inter hora et centrum lamine, et inveniat centrum basis eius super quod signet lineam subtilem cum incausto. Postea separet ex hac linea ex parte centri basis, quod est centrum spere, lineam equalem diametro foraminis quod est in hora instrumenti. Erit ergo hec linea equalis linee que est inter centrum foraminis quod est in hora instrumenti et superficiem lamine que est perpendicularis super superficiem lamine. Deinde statuamus super extremitatem linee separate a diametro lineam perpendicularem, et extrahamus illam in utramque partem.

2.47 ◉ Next the experimenter should take a hemisphere of clear, white glass or crystal, whose radius is less than the distance between the panel [containing the second hole] and the register plate’s center, and he should find the center of its base [circle] and draw a fine line through it with ink. Then to the side of the base’s center, which constitutes the center of the sphere [out of which the hemisphere is formed], he should mark off a line-segment [from that centerpoint] equal to the diameter of the hole in the apparatus’s rim [i.e., one grain of barley]. This line will therefore be equal to the line dropped perpendicular to the register plate’s surface from the center of the hole in the apparatus’s rim [i.e., one grain of barley]. At the endpoint of the line [just] marked off on the diameter we should erect a perpendicular line, and we should extend it on both sides [to the circumference of the base-circle of the sphere].

2.48 ◉ Deinde secemus vitrum super hanc lineam in confrictorio vel tornatorio donec locus sectionis fiat superficies equalis et perpendicularis super superficiem basis semicirculi, et mensuremus angulum qui est inter duas superficies per angulum rectum factum ex cupro donec verificetur superficies ista. Et tunc differentia communis huic superficiei et superficiei basis spere erit linea recta, et linea copulans centrum spere cum hac linea erit perpendicularis super superficiem factam. Postea sumatur in medio huius linee que est communis differentia particula parva que erit signum medii eius.

2.48 ◉ We should then cut the glass [hemisphere] along this line in a grinding machine or a lathe until the surface of the section cut off is flat and perpendicular to the surface of the semicircular [remnant at the] base, and we should measure the angle between the two surfaces with a right angle formed from copper until that surface is properly aligned. Consequently, the common section of this surface and the surface at the base of the [newly-cut quarter-]sphere will be a straight line, and the line joining the center of the sphere [encompassing the newly-cut quarter-sphere] and this line will be perpendicular to the surface [just] formed.⁑ Afterwards a tiny spot should be made at the middle of this line that forms the common section, and it will mark its midpoint.

2.49 ◉ Hoc completo, poliatur vitrum vehementissime, et ponatur super superficiem lamine et gibbositas eius ex parte foraminum, et sit pars facta in vitro super superficiem lamine. Et superponatur linea recta que est differentia communis duabus superficiebus equalibus que sunt in vitro super lineam scilicet signatam in lamina secantem diametrum perpendiculariter, et ponatur medium linee super centrum lamine. Hac ergo positione preservata, applicetur vitrum lamine applicatione fixa.

2.49 ◉ When this is finished, the glass should be vigorously polished and placed on the surface of the register plate with its convex side facing the [two] holes, and the section of the glass [quarter-sphere forming its semicircular base] should lie on the register plate’s surface. In addition, the straight line forming the common section of the two flat surfaces of the glass [quarter-sphere] should be placed on the line [previously] drawn on the register plate to cut its diameter orthogonally [through its center], and the middle of that line [formed by the two flat faces of the quarter-sphere] should be placed at the register plate’s center. When things are so arranged, the glass [quarter-sphere] should be firmly attached to the plate [so that it can nonetheless be removed].⁑

2.50 ◉ Deinde ponamus regulam subtilem super superficiem instrumenti, sicut ponebamus in experimentatione vitrorum cubicorum, et ponat superficiem regule in qua est linea ex parte vitri et prope illum. Deinde ponat instrumentum in predictum vas, et ponat vas in sole vacuum aqua, et moveat instrumentum donec lux solis transeat per duo foramina, et sit situs lucis de secundo foramine situs mediocris. Et intueatur regulam, et inveniet lucem transeuntem per duo foramina super superficiem regule. Deinde applicet stilum cum superiori foramina, et ponat extremitatem stili super centrum foraminis, et intueatur lucem que est in regula. Tunc inveniet umbram extremitatis stili apud centrum lucis. Deinde auferat stilum, et redibit lux ad suum locum.

2.50 ◉ We should next apply the thin ruler to the surface of the [register plate in the] apparatus, just as we applied it during the experiment with the glass cubes, and [the experimenter] should pose the face of the ruler on which the line is [drawn] toward [the upright face of] the glass [quarter-sphere] and near it. He should then insert the apparatus in the aforementioned vessel, place the vessel empty of water in sun[light], and adjust the apparatus until the sunlight passes through the two holes, and the light from the second hole should be disposed as usual [i.e., with its axis in line with the two holes].⁑ [The experimenter] should look at the ruler, and he will find the light passing through the two holes [cast] on the ruler’s surface. He should then put the stylus up to the top hole and place the [sharpened] endpoint of the stylus at the hole’s center, and he should look at the light on the ruler. Accordingly, he will find the shadow of the stylus’s [sharpened] endpoint at the center of the light [on the ruler]. Then he should remove the stylus, and the light will return to its [original] place.

2.51 ◉ Postea applicet stilum ad secundum foramen, et ponat extremitatem eius apud centrum secundum, et intueatur lucem que est in regula. Tunc inveniet umbram extremitatis stili apud centrum lucis. Postea ponat extremitatem stili apud centrum basis vitri quod est centrum spere, et intueatur lucem que est super regulam, et inveniet umbram extremitatis regule super centrum lucis. Deinde ponat stilum in medio lucis que est super convexum vitri oppositi foramini secundo quod est prope illum, et intueatur lucem que est super regulam, et inveniet umbram extremitatis stili apud centrum lucis, ex quo patet quod lux que transit per centra duorum foraminum transit etiam per centrum basis vitri et per medium superficiei lucis que est in convexo vitri.

2.51 ◉ Next he should put the stylus up to the second hole and place its [sharpened] endpoint at the center of [that] second [hole], and he should look at the light on the ruler. He will thus find the shadow of the stylus’s [sharpened] endpoint at the center of the light [on the ruler]. He should then put the [sharpened] endpoint of the stylus at the center of the [upright] face of the glass [quarter-sphere], which constitutes the center of the sphere [encompassing the quarter-sphere], and he should look at the light on the ruler, and he will find the shadow of the stylus’s [sharpened] endpoint at the center of the light [on the ruler].⁑ Finally, he should place the [point of the] stylus at the middle of the light that is [cast] on the convex surface of the glass [quarter-sphere] facing the second hole near it, and he should look at the light on the ruler, and he will find the shadow of the stylus’s [sharpened] endpoint in the middle of the light [on the ruler], from which it is evident that the light passing through the centers of the two holes also passes through the centerpoint [of the sphere] on the face of the glass [quarter-sphere] and through the midpoint of the spot of light [cast] on the convex surface of the glass [quarter-sphere].

2.52 ◉ Manifestum est igitur quod lux que transit in corpus vitri extenditur secundum rectitudinem linee transeuntis per centra duorum foraminum. Hec autem linea est diameter spere vitree, nam perpendicularis exiens a centro basis vitri ad laminam est equalis diametro foraminis. Diameter autem foraminis est equalis perpendiculari exeunti a centro foraminis ad superficiem lamine. Ergo perpendicularis exiens a centro basis vitri super superficiem lamine est equalis perpendiculari exeunti a centro foraminis ad superficiem lamine, et hee due perpendiculares cadunt super diametrum lamine.

2.52 ◉ It is therefore obvious that the light passing through the body of the glass extends along the straight line passing through the centers of the two holes. This line, moreover, forms a diameter of the sphere [encompassing the] glass [quarter-sphere], for the perpendicular dropped from the center [of that sphere] on the [upright] face of the glass [quarter-sphere] to the register plate is equal to the diameter of the hole. The diameter of the hole, in turn, is equal to the perpendicular dropped from the center of the hole to the surface of the register plate. Thus, the perpendicular dropped from the center [of the sphere encompassing the quarter-sphere] on the [upright] face of the glass [quarter-sphere] to the surface of the register plate is equal to the perpendicular dropped from the center of the hole to the surface of the register plate, and these two perpendiculars fall on the register plate’s diameter.

2.53 ◉ Linea ergo que transit per centra duorum foraminum, si fuerit extensa in rectitudine, perveniet ad centrum spere vitree. Erit ergo diameter huius spere; est ergo perpendicularis super superficiem huius spere. Experimentatione autem cubicorum vitrorum patuit quod lux que extenditur in corpus vitri est in rectitudine linee per quam extendebatur in aere, et linea per quam extendebatur in aere erat illic perpendicularis super superficiem vitri.

2.53 ◉ If it is extended in a straight line, then, the line passing through the centers of the two holes will reach the center of the sphere [encompassing the] glass [quarter-sphere]. It will thus constitute a diameter of this sphere, so it is perpendicular to the surface of this sphere. Furthermore, it was shown in the experiment with the glass cubes that the light extending through the body of the glass follows the straight line along which it extended in the air, and the line along which it extended through the air was the one that was perpendicular to the [upright] face of the glass [quarter-sphere].

2.54 ◉ Et oportet experimentatorem auferre regulam subtilem applicatam ad superficiem lamine, et ponat instrumentum secundo, et moveat ipsum quousque lux transeat per duo foramina, et intueatur horam instrumenti que est intra vas. Et inveniet lucem super horam instrumenti, et inveniet centrum lucis in puncto que est differentia communis inter circumferentiam circuli medii et lineam perpendicularem in hora instrumenti quod est extremitas diametri circuli medii transeuntis per centra duorum foraminum. Et lux que extenditur per hanc lineam erit differentia communis perveniens ad centrum spere vitree. Centrum ergo lucis que est in hora instrumenti, et centrum spere vitree, et centra duorum foraminum sunt in eadem linea recta, ex quo patet quoniam lux que transit in corpus vitri perveniens ad centrum eius, cum extrahitur in aere, extenditur in rectitudine linee per quam extendebatur in corpus vitri.

2.54 ◉ The experimenter should then remove the thin ruler that was attached to the register plate’s surface, insert the apparatus back in the vessel, adjust it until the light passes through the two holes, and look at the rim of the apparatus that lies inside the vessel. He will find that the light [appears] on the apparatus’s rim, and he will find the center of the light [cast on the apparatus’s rim] at the point that forms the common section of the middle circle’s perimeter and the perpendicular line on the apparatus’s rim at the endpoint of the middle circle’s diameter, which passes through the centers of the two holes. Also, the [axial] light that extends along this line will form a common section reaching to the center of the glass sphere [encompassing the quarter-sphere]. Thus, the center of the light on the apparatus’s rim, the center of the sphere [encompassing the] glass [quarter-sphere], and the centers of the two holes lie on the same straight line, from which it is evident that, when it extends through the air, the light passing into the body of the glass and reaching its centerpoint extends along the [same] straight line according to which it extended through the body of the glass.

2.55 ◉ Hec autem linea est perpendicularis super superficiem basis vitri que est equidistans diametro lamine qui est perpendicularis super superficiem basis vitri, quia est perpendicularis super lineam rectam que est differentia communis duabus superficiebus vitri equalibus, quarum altera est superposita superficiei lamine et reliqua erecta super superficiem lamine. Linea ergo transiens per centra duorum foraminum et per centrum spere vitree est perpendicularis super superficiem vitri; est ergo perpendicularis super superficiem aeris qui tangit hanc superficiem. Et si experimentator effuderit aquam in vas, remanente vitro in sua positione, et posuerit aquam supra centrum vitri, et inspexerit lucem que est in hora instrumenti, inveniet centrum lucis super extremitatem diametri medii circuli.

2.55 ◉ This line, moreover, is perpendicular to the plane of the [upright] face of the glass [quarter-sphere], and it is parallel to the diameter of the register plate, which is perpendicular to the plane of the [upright] face of the glass [quarter-sphere], since it is perpendicular to the straight line that forms the common section of the two plane surfaces of the glass [quarter-sphere], one of them being applied to the surface of the register plate and the other standing upright upon the surface of the register plate. Hence, the line passing through the centers of the two holes and through the center of the sphere [encompassing the] glass [quarter-sphere] is perpendicular to the [upright] face of the glass [quarter-sphere], so it is perpendicular to the surface of the air that is in contact with this face. And if the experimenter fills the vessel with water while leaving the glass in place, and if he brings the water up beyond the center of the glass [quarter-sphere] and looks at the light [cast] on the apparatus’s rim, he will find the center of the light at the endpoint of the middle circle’s diameter.⁑

2.56 ◉ Et si evulserit vitrum et posuerit illud in lamina econtra huic ordinationi, scilicet quod superficies equalis sit ex parte foraminum et convexitas vitri sit ex parte interioris vasis, et superposuerit lineam rectam que est in vitro que est differentia communis duabus suis superficiebus equalibus super lineam rectam que est in lamina secantem perpendiculariter diametrum lamine, et posuerit medium huius linee, scilicet que est in vitro, super centrum lamine, et inspexerit lucem sicut fecit in prima positione, inveniet lucem cadentem super horam instrumenti, et inveniet centrum lucis super punctum quod est differentia communis circumferentie medii circuli et linee stanti in hora instrumenti, ex quibus declarabitur quod lux que transit per centra duorum foraminum transit etiam in corpus vitri secundum rectitudinem linee per quam extendebatur in aere, et postquam egreditur corpus vitri, extenditur etiam in aere secundum rectitudinem linee per quam extendebatur in vitro.

2.56 ◉ Furthermore, if [the experimenter] pulls the glass [quarter-sphere] off [the register plate’s surface] and [re]applies it to the register plate with the opposite orientation, i.e., so that the flat surface faces the holes and the convexity of the glass faces the inside portion of the vessel [which has been emptied of water], and if he applies the straight line that forms the common section of the two plane surfaces of the glass [quarter-sphere] along the straight line on the face of the register plate that cuts the register plate’s diameter orthogonally and puts the center of this line, i.e., the line [forming the common section of the two plane faces] on the glass [quarter-sphere], at the register plate’s center, then when he looks at the light as he did in the first situation, he will find that the light falls on the apparatus’s rim, and he will find the center of the light at the point that forms the common section of the middle circle’s circumference and the line standing [perpendicular] on the [inner wall of the] apparatus’s rim, from which it will be manifest that the light passing through the centers of the two holes also passes through the body of the glass along the [same] straight line along which it extended in the air, and [it will also be manifest that] after it exits the body of the glass, it extends along the [same] straight line through the air [below the glass] along which it extended through the glass.

2.57 ◉ Et linea que transit per centra duorum foraminum est in hac positione etiam perpendicularis super superficiem vitri oppositam foramini, scilicet superficiem que est basis semispere, et hec linea est etiam perpendicularis super superficiem convexam, nam in hac positione etiam est diameter spere. Est ergo perpendicularis super superficiem spere; est ergo perpendicularis super superficiem aeris continentis superficiem spere. Et si experimentator infuderit aquam in vas, et relinquerit vitrum in sua positione, et posuerit aquam infra centrum vitri, et aspexerit lucem que est in hora instrumenti, inveniet centrum lucis in extremitate diametri medii circuli.

2.57 ◉ In addition, the line passing through the centers of the two holes in this situation is also perpendicular to the surface of the glass [quarter-sphere] on the other side of the holes, i.e., the face that forms the [upright face that constitutes slightly more than half the] base of the hemisphere, and this line is also perpendicular to the convex surface [of the glass], for in this situation it also forms the diameter of the sphere [encompassing the glass quarter-sphere]. It is therefore perpendicular to the surface of the sphere, so it is perpendicular to the surface of the air enclosing the sphere’s surface. And if the experimenter fills the vessel with water and leaves the glass in place, and if he brings the water up beneath the center of the sphere [on the upright face of the glass quarter-sphere] and looks at the light [cast] on the apparatus’s rim, he will find the center of the light [cast on the rim] at the endpoint of the middle circle’s diameter.⁑

2.58 ◉ Ex hiis igitur experimentationibus que fiunt per cubicum et spericum vitrum patet quod, si lux occurrerit corpori diaffono diverse diaffonitatis a corpore in quo est, et linea per quam extenditur fuerit perpendicularis super superficiem secundi corporis, tunc lux extenditur in secundo corpore in rectitudine linee per quam extendebatur in corpore primo, nec differt si secundum corpus fuerit grossius primo aut subtilius.

2.58 ◉ Hence, from these experiments conducted with the glass cubes and the glass [quarter-]sphere it is evident that, if light encounters a transparent body of different transparency from that of the body in which it lies, and if the line along which it extends is perpendicular to the surface of the second body, the light extends through the second body in the [same] straight line along which it extended in the first body, and it makes no difference whether the second body is denser or rarer than the first.⁑

2.59 ◉ Item oportet experimentatorem evellere vitrum, et revertere illud ad laminam, et ponere medium linee recte que est in ea super centrum lamine, et ponere superficiem equalem ex parte duorum foraminum et lineam que est in vitro que est differentia communis duabus suis superficiebus obliquam super diametrum lamine qualibet obliquatione, et ponere obliquationem diametri lamine super hanc lineam ad illam partem ad quam declinabat apud experimentationem aque. Necesse est igitur ut perpendicularis que egreditur a centro vitri que est super superficiem vitri perpendiculariter que extenditur in corpus vitri obliqua sit a linea transeunte per centra duorum foraminum ad partem in qua sunt duo foramina. Et applicet experimentator vitrum secundum hunc situm applicatione fixa, et ponat instrumentum in vas et vas in sole, et moveat instrumentum donec lux transeat per duo foramina, et intueatur lucem que est intra vas.

2.59 ◉ Now the experimenter should remove the glass [quarter-sphere] and put it back on the register plate, placing the middle of the straight line on it [that forms the common section of the two flat faces] at the center of the register plate and posing the [upright] flat face [of the quarter-sphere] toward the two holes with the line on the glass [quarter-sphere] forming the common section of its two [plane] faces inclined to some extent to the register plate’s diameter, and he should slant it with respect to the register plate’s diameter in the same direction it slanted in the experiment with water. The normal dropped from the center of the [sphere encompassing the] glass [quarter-sphere] and passing orthogonally through the [plane] surface of the glass into the body of the glass must therefore be oblique to the line passing through the centers of the two holes on the side of the two holes. The experimenter should attach the glass firmly [to the register plate] according to this disposition, insert the apparatus in the [evacuated] vessel and the vessel in sunlight, adjust the apparatus until the light passes through the two holes, and look at the light inside the vessel.

2.60 ◉ Tunc inveniet illam in interiori hore instrumenti, et inveniet centrum lucis in circumferentia medii circuli sed extra punctum qui est differentia communis circumferentie circuli medii et linee stanti in hora instrumenti, et declinatio eius erit ad partem in qua est sol. Erit ergo ad partem perpendicularis exeuntis a loco reflexionis, et hec lux extenditur in aere in rectitudine linee transeuntis per centra duorum foraminum, et hec linea in hoc situ pervenit ad centrum spere vitree, et est obliqua super superficiem vitri equalem.

2.60 ◉ Accordingly, he will find it [cast] on the inner wall of the apparatus’s rim and will find the center of the light on the circumference of the middle circle but outside the point that forms the common section of the middle circle’s circumference and the line standing [perpendicular] on the rim’s [inner] wall, and its inclination will be toward the sun. It will therefore incline toward the normal dropped from the point of refraction, and this light extends through the air along the straight line passing through the centers of the two holes, and in this situation this line reaches the center of the sphere [encompassing the] glass [quarter-sphere] but is oblique to the flat surface of the glass [facing the holes].

2.61 ◉ Huius autem lucis terminatio extensionis in vitro est a centro vitri; extenditur ergo in corpus vitri secundum lineam rectam exeuntem a centro spere; ergo est illi diameter. Hec igitur lux extenditur in corpus vitri secundum verticationem diametri alicuius eius; cum ergo pervenerit ad superficiem eius spericam, erit perpendicularis super illam, et cum extrahetur in aere, erit perpendicularis super aerem continentem superficiem spericam.

2.61 ◉ Moreover, the endpoint of this light extending into the glass is at the center of the [sphere encompassing the] glass [quarter-sphere], so it extends through the body of the glass along a straight line passing from the center of the sphere, [and] so it forms a diameter of that [sphere]. Hence, this light extends in the body of the glass straight along one of its diameters; so when it reaches its [convex] spherical surface, it will be perpendicular to it, and when it will extend into the air, it will be perpendicular to the air enveloping the [convex] spherical surface.

2.62 ◉ Non ergo reflectitur in aere, nec extenditur recte. Ergo reflectitur, sed non in corpus vitri, nec in convexo eius, neque in primo aere, neque in secundo. Ergo reflectitur apud centrum vitri, et hec lux est obliqua super superficiem vitri equalem in qua est centrum vitri, ex quibus patet quod, cum lux extenditur in aere, et transit in vitrum, et fuerit obliqua super superficiem vitri, reflectitur et non transibit recte. Et reflexio eius erit ad partem in qua est perpendicularis exiens a loco reflexionis, et corpus vitri grossius est corpore aeris.

2.62 ◉ It is therefore not refracted in the air, nor does it continue along the straight line [it followed through the air to reach the upright flat face of the quarter-sphere]. So it is refracted, but not in the body of the glass, nor at its convex surface, nor in the first [part of the] air [from which it passes into the glass], nor in the second [part of the air into which it passes from the glass]. Consequently, it is refracted at the center of the [sphere encompassing the] glass [quarter-sphere], and this light is oblique to the flat face of the glass on which the centerpoint lies, from which it is evident that, when the light extends in air and passes into the glass, and when it is oblique to the glass’s surface, it is refracted [there] rather than passing straight [through].⁑ Also, its refraction will be in the direction of the normal dropped from the point of refraction, and the body of glass is denser than the body of air.

2.63 ◉ Manifestum est igitur ex hac experimentatione et prima de reflexione lucis ab aere ad aquam, luce existente obliqua super superficiem aque, quoniam cum lux fuerit extensa in corpore subtiliori et occurrerit grossius corpus, et fuerit obliqua super grossius corpus, reflectetur ab ipso, et erit reflexio eius ad partem in qua est linea exiens a loco reflexionis que est perpendicularis super superficiem corporis grossioris.

2.63 ◉ From this experiment, as well as the first one dealing with the refraction of light from air to water when the light struck the water’s surface obliquely, it is manifest that, if light extends through a rarer body and encounters a denser body, and if it is oblique to [the surface of] the denser body, it will be refracted at that [surface], and its refraction will be toward the line dropped from the point of refraction normal to the surface of the denser body.⁑

2.64 ◉ Item oportet experimentatorem evellere vitrum et ponere ipsum econtra, scilicet quod superficies convexa sit ex parte foraminum, et ponat medium differentie communis que est in vitro super centrum lamine, et ponat differentiam communem obliquam super diametrum lamine, et applicet vitrum applicatione fixa. Et extrahat a centro lamine lineam in superficie lamine perpendicularem super differentiam communem que est in vitro. Erit ergo hec linea perpendicularis super superficiem vitri, nam superficies vitri equalis est perpendicularis super superficiem lamine.

2.64 ◉ The experimenter should now remove the glass [quarter-sphere] and replace it in the opposite way, i.e., so that its convex surface faces the holes, and he should place the midpoint of the common section of the [two plane faces of the] glass at the center of the register plate, pose the common section at a slant to the diameter of the register plate, and affix the glass firmly [to the plate’s surface]. He should then draw the line on the register plate’s surface that is normal to the common section of the [two plane faces of the] glass at the [register plate’s] center. This line will thus be normal to the [upright flat] face of the glass, for the [upright] flat face of the glass is [erected] perpendicular to the surface of the register plate.

2.65 ◉ Deinde experimentator ponat instrumentum in vase, vase existente sine aqua, et moveat instrumentum quousque lux transeat per duo foramina, et intueatur lucem que est intra vas. Tunc inveniet illam in interiori hore instrumenti, et inveniet centrum lucis in circumferentia medii circuli et extra punctum qui est differentia communis circumferentie medii circuli et linee perpendiculari in hora instrumenti, qui est extremitas diametri medii circuli. Et inveniet declinationem eius ad contrariam partem illi in qua est perpendicularis.

2.65 ◉ Then the experimenter should insert the apparatus in the vessel, which remains without water, and adjust the apparatus until the light passes through the two holes, and he should look at the light inside the vessel. Accordingly, he will find it on the inner wall of the apparatus’s rim, and he will find the center of the light on the circumference of the middle circle but outside the point that forms the common section of the middle circle’s circumference and the line [drawn] perpendicular on the [inner wall of the] apparatus’s rim, which lies at the endpoint of the middle circle’s diameter. He will also find that it inclines away from the normal.

2.66 ◉ Hec autem lux extenditur in vitro secundum rectitudinem linee transeuntis per centra duorum foraminum, quia hec linea est diameter vitri in hac etiam positione, quia transit per centrum vitri. In hac igitur positione reflexio lucis est etiam apud centrum vitri, et hec lux est obliqua super superficiem vitri equalem et super superficiem aeris contingentem vitrum, ex quibus patet quod, cum lux extenditur in vitro, et egreditur ad aerem, et fuerit obliqua super superficiem aeris, reflectetur, et reflexio eius erit in superficie circuli medii et ad partem contrariam illi in qua est linea exiens a loco reflexionis que est perpendicularis super superficiem aeris.

2.66 ◉ Moreover, this light extends through the glass along the straight line passing through the centers of the two holes because this line also forms the diameter of the [sphere encompassing the] glass [quarter-sphere] in this situtation, since it passes through the center of the [sphere encompassing the] glass [quarter-sphere]. In this situation, then, the refraction of the light also occurs at the center of the [sphere encompassing the] glass [quarter-sphere], and this light is oblique to the [upright] flat face of the glass as well as to the surface of the air in contact with the glass, from which it is clear that, when the light passes through the glass and exits into the air [below it], and when it is oblique to the air’s surface, it will be refracted, and its refraction will be in the plane of the middle circle but in the direction away from the line dropped normal to the air’s surface from the point of refraction.⁑

2.67 ◉ Et si experimentator effuderit aquam in vas, existente vitro in sua positione, et posuerit aquam super centrum vitri, et aspexerit lucem que est intra vas, inveniet lucem in interiori parte hore instrumenti. Et inveniet centrum lucis in circumferentia medii circuli, et inveniet illud extra extremitatem diametri medii circuli obliquum ad partem contrariam illi super quam cadit perpendicularis. Et inveniet distantiam centri lucis ab extremitate diametri medii circuli minorem distantia centri lucis ab hoc puncto in experientia egressus lucis a vitro ad aerem, quia aer est subtilior aqua, aqua autem est subtilior vitro.

2.67 ◉ Now if the experimenter pours water into the vessel while the glass remains in its position, if he brings the water up above the center of the [sphere encompassing the] glass [quarter-sphere], and if he looks at the light in the vessel, he will find the light [cast] on the inner wall of the apparatus’s rim. He will also find the center of the light on the circumference of the middle circle, and he will find [that] it [lies] outside the endpoint of the middle circle’s diameter, inclined in the direction away from the normal dropped [from the point of refraction]. He will also find that the distance between the center of the light and the endpoint of the middle circle’s diameter is less than the distance between the center of the light and that point in the experiment where the light passes from glass into air because air is rarer than water, whereas water is rarer than glass.

2.68 ◉ Ex hac ergo experimentatione et predicta patet quoniam, quando lux extenditur in corpore grossiori, et occurrerit corpori subtiliori, et fuerit obliqua super superficiem corporis subtilioris, reflectetur et non transibit recte, et reflexio eius erit ad partem contrariam illi in qua est perpendicularis exiens a loco reflexionis que est perpendicularis super superficiem corporis subtilioris. Et tantum declinabit a perpendiculari quanto corpus erit subtilius.

2.68 ◉ It is thus evident from this experiment and the aforementioned one that, when light extends through a denser body and encounters a rarer body, and when it is oblique to the surface of the rarer body, it will be refracted and will not pass straight through, and its refraction will be in the direction away from the normal dropped from the point of refraction to the surface of the rarer body. In addition, the more it will incline away from the perpendicular, the rarer [that] body will be.⁑

2.69 ◉ Item oportet experimentatorem evellere vitrum et ponere etiam ipsum in superficie lamine, et superponat lineam rectam que est in eo super lineam rectam que est in lamina, et ponat superficiem eius convexam ex parte duorum foraminum et lineam rectam que est in vitro extra centrum lamine. Et coniungat vitrum bene, et ponat regulam subtilem super superficiem lamine, et erigat eam super horam eius, et ponat superficiem eius in qua signatur linea ex parte vitri, et terminus eius secet diametrum lamine perpendiculariter, et applicet regulam hoc modo. Sic ergo linea que transit per centra duorum foraminum non transit per centrum spere sed per alium punctum superficiei vitri equalis, et erit obliqua super spericam superficiem.

2.69 ◉ Next the experimenter should remove the glass and place it again on the register plate’s surface, he should apply the straight line on it [formed by the common section] upon the straight line on the register plate [that passes through the center perpendicular to the diameter], and he should pose its convex surface toward the two holes with the straight line on the glass [quarter-sphere] eccentric to the register plate’s center [i.e., so that the centerpoint of the quarter-sphere’s common section at the bottom edge lies to either side of the register plate’s center]. He should then attach the glass firmly [to the register plate’s surface] and pose the thin ruler on its edge upon the register plate’s surface [so that] the line marked on its surface faces the [flat face of the] glass [quarter-sphere] and [so that] its edge cuts the register plate’s diameter orthogonally, and he should attach the ruler [to the register plate] accordingly. Thus, the line passing through the centers of the two holes does not pass through the center of the sphere [encompassing the glass quarter-sphere] but through some other point on the flat face of the glass [quarter-sphere], and [so] it will be oblique to the [convex] spherical surface.

2.70 ◉ Deinde oportet experimentatorem ponere instrumentum in vase et vas in sole, et moveat instrumentum quousque lux transeat per duo foramina, et intueatur superficiem regule. Tunc inveniet lucem super superficiem regule, et centrum eius super lineam que est in superficie regule, et centrum lucis extra rectitudinem linee que transit per centra duorum foraminum. Et inveniet declinationem eius ad partem in qua est centrum vitri, et inveniet lineam que transit per centra duorum foraminum perpendicularem super superficiem vitri equalem, est enim equidistans diametro, et diameter lamine est perpendicularis super superficiem vitri equalem. Et si lux transiret per duo centra foraminum et extenderetur secundum rectitudinem ad superficiem equalem, tunc extenderetur in rectitudine in aere. Sed cum centrum lucis que est in regula non est in rectitudine huius linee, ergo lux non extenditur in rectitudine ipsius ad superficiem equalem. Et lux in corpore vitri extenditur recte. Ergo lux que extenditur in corpore vitri non est in rectitudine linee que transit per centra duorum foraminum.

2.70 ◉ The experimenter should then insert the apparatus in the vessel and the vessel in sunlight, adjust the apparatus until the light passes through the two holes, and look at the ruler’s surface. He will therefore find the light [cast] on the ruler’s surface, and its midpoint [will lie] on the line marked on the ruler’s surface, but the center of the light [will] lie outside the straight line passing through the centers of the two holes. He will also find that it inclines toward the center of [the sphere encompassing] the glass [quarter-sphere], and he will find that the line passing through the centers of the two holes is perpendicular to the flat face of the glass [quarter-sphere], for it is parallel to the diameter [of the register plate], and the diameter of the register plate is perpendicular to the flat face of the glass [quarter-sphere]. But if the light were to pass through the two centers of the holes and extend straight to the flat face, then it would extend [through that face] in [the same] straight line [along which it passed] through the air. However, since the center of the light on the ruler does not lie directly on this line, the light does not extend along the same straight line to the flat face. But the light does extend straight through the body of the glass. Hence, the light that extends through the body of the glass does not lie on the [same] straight line that passes through the centers of the two holes.

2.71 ◉ Ergo est reflexa, sed non in aere neque in corpore vitri; ergo reflectitur apud spericam superficiem vitri, et est obliqua super superficiem spericam, quia linea que transit per duo centra foraminum non transit per centrum vitri, et hec lux, cum egreditur a superficie vitri equali, reflectitur. Sed cum regula subtilis fuerit valde propinqua superficiei vitri, tunc declinatio centri lucis que est in regula a rectitudine linee que extenditur in corpore vitri non latebit in tantum quod possit occultare reflexionem lucis in corpore vitri aut partem eius. Et hec reflexio erit ad partem in qua est centrum vitri; ergo est ad perpendicularem exeuntem a loco reflexionis perpendiculariter super superficiem vitri spericam, quia linea exiens a centro vitri ad punctum reflexionis est perpendicularis exiens a loco reflexionis super superficiem spericam.

2.71 ◉ It is refracted, then, but not in the air nor in the body of the glass, so it is refracted at the spherical surface of the glass [quarter-sphere], and it is oblique to the spherical surface because the line passing through the two centers of the holes does not pass through the center of the [sphere encompassing the] glass [quarter-sphere], and [so] when it exits the flat face of the glass, this light is refracted. Since the thin ruler is quite near the [flat] face of the glass [quarter-sphere], though, the inclination of the center of the light on the ruler away from the straight line [of light] that extends through the glass’s body will not be noticeable insofar as it might [otherwise] interfere with the refraction of the light in the body of the glass or [a significant] part of that refraction. And this refraction will incline toward the center of the [sphere encompassing the] glass [quarter-sphere], so it inclines toward the normal dropped from the point of refraction perpendicular to the spherical surface of the glass [quarter-sphere] because the line dropped from the center of the [sphere encompassing the] glass [quarter-sphere] to the point of refraction is the normal dropped from the point of refraction to the spherical surface [of the glass].⁑

2.72 ◉ Deinde oportet experimentatorem evellere vitrum et ponere econtra huic compositioni, scilicet quod ponat superficiem vitri equalem ex parte duorum foraminum, et ponat differentiam communem duabus superficiebus equalibus vitri super lineam secantem diametrum lamine perpendiculariter, et ponat medium differentie communis extra centrum lamine. Vitro autem coniuncto hoc modo, linea que transit per centra duorum foraminum non transit per centrum vitri, sed perveniet ad punctum de superficie eius equali in qua est centrum eius extra punctum centri, et erit perpendicularis super superficiem equalem, sicut predictum est. Et cum linea que transit per centra duorum foraminum extensa fuerit recte in ymaginatione, perveniet ad punctum quod est extremitas diametri circuli medii.

2.72 ◉ The experimenter should then remove the glass and replace it in the opposite way, i.e., he should pose its flat surface toward the two holes and place the common section of the two plane surfaces of the glass on the line intersecting the register plate’s diameter orthogonally, and he should place the midpoint of the common section outside the register plate’s center. With the glass attached in this way, the line passing through the centers of the two holes does not pass through the center of the [sphere containing the] glass [quarter-sphere]; rather, it will reach its flat face, upon which the centerpoint lies, at some point beyond its centerpoint, but it will be perpendicular to the flat face, as was pointed out earlier. When the line passing through the centers of the two holes is extended straight on in the imagination, moreover, it will reach the point that lies at the endpoint of the middle circle.

2.73 ◉ Et cum experimentator posuerit vitrum hoc modo, ponet instrumentum in vase et vas in sole, et moveat instrumentum donec lux transeat per duo foramina, et intueatur horam instrumenti, et inveniet lucem in interiori parte hore instrumenti. Et inveniet centrum lucis in circumferentia circuli medii et extra punctum quod est extremitas diametri circuli medii et declinans ad partem in qua est centrum spere vitree. Et linea que egreditur a centro huius spere in ymaginatione ad locum reflexionis est perpendicularis super superficiem huius spere; est ergo perpendicularis super superficiem aeris qui continet superficiem spere. Hec ergo reflexio est ad partem contrariam illi in qua est perpendicularis exiens a loco reflexionis super superficiem aeris continentis superficiem que extenditur in corpore aeris.

2.73 ◉ When the experimenter has placed the glass accordingly, he will insert the apparatus in the vessel and the vessel in sunlight, and he should adjust the apparatus until the light passes through the two holes and look at the rim of the apparatus, and he will find the light [cast] on the inner wall of the apparatus’s rim. He will also find the center of the light on the circumference of the middle circle but outside the point at the end of the middle circle’s diameter and inclining toward the center of the [sphere encompassing the] glass [quarter-sphere]. Furthermore, the line imagined to exit from the center of this sphere to the point of refraction is normal to the surface of this sphere, so it is perpendicular to the surface of the air that envelops the surface of the [quarter-sphere encompassed by this] sphere. This refraction thus occurs away from the normal dropped from the point of refraction to the surface of the air enveloping the surface [of the quarter-sphere] and extending through the air’s body.

2.74 ◉ Lux autem que transit per centra duorum foraminum transit in corpus vitri recte, quia est perpendicularis super superficiem vitri equalem oppositam duobus foraminibus, et perveniet ad convexum spere vitree. Et cum pervenit ad illam superficiem, non erit perpendicularis super illam, cum non sit diameter in spera, et omnis perpendicularis super superficiem spere est diameter illius aut secundum rectitudinem diametri illius. Sed lux que extenditur in corpore vitri hoc modo non est perpendicularis super superficiem aeris continentis convexum vitri, et hec lux invenitur reflexa. Ergo reflectitur apud convexum spere.

2.74 ◉ In addition, the light passing through the centers of the two holes passes straight through the body of the glass because it is perpendicular to the flat surface of the glass facing the two holes, and it will reach the convex surface of the glass [quarter-]sphere. When it reaches that surface, it will not be perpendicular to it, since it does not form a diameter in the sphere [encompassing the quarter-sphere], and every [line that is] perpendicular to the surface of a sphere forms a diameter on it or lies in a straight line with a diameter on it [if the line lies outside the sphere]. But the light that extends through the body of the glass in this way is not perpendicular to the surface of the air enveloping the convex surface of the glass, and [so] this light is found to be refracted. It is therefore refracted at the convex surface of the [quarter-]sphere.⁑

2.75 ◉ Et si experimentator effuderit aquam intra vas, vitro remanente in suo situ, et posuerit aquam infra centrum lamine, et aspexerit lucem que est in hora instrumenti, inveniet etiam lucem reflexam et ad partem in qua est centrum vitri. Erit ergo ad partem contrariam illi in qua est perpendicularis exiens a loco reflexionis que extenditur in corpore vitri a corpore aeris perpendicularis super concavitatem aeris continentis convexum vitri.

2.75 ◉ If the experimenter pours water into the vessel while the glass remains in place, and if he brings the water up below the center of the register plate and examines the light that is [cast] on the apparatus’s rim, he will find that the light is still refracted toward the center of the [sphere encompassing the] glass [quarter-sphere]. It will therefore occur in the direction away from the normal dropped from the point of refraction, and it extends into the body of the glass from the body of the air [along a line] perpendicular to the concave surface of the air enveloping the convex surface of the glass.⁑

2.76 ◉ Ex omnibus ergo hiis experimentationibus patet quod lux solis transit in omne corpus diaffonum secundum verticationes linearum rectarum, et si occurrerit corpori diaffono diverse diaffonitatis diaffonitati corporis in quo est, et linee per quas extenditur in primo corpore fuerint declinantes super superficiem secundi corporis, tunc lux reflectetur in secundo corpore in verticatione linearum rectarum aliarum a primis per quas extendebatur in primo corpore. Et si linee recte per quas extendebatur in primo corpore fuerint perpendiculares super superficiem secundi corporis, tunc lux extendetur in rectitudine eius, et non reflectetur.

2.76 ◉ From all these experiments, therefore, it is evident that sunlight passes through every transparent body along straight lines, and if it encounters a transparent body whose transparency differs from the transparency of the body in which it lies, and if the lines along which it extends through the first body are inclined to the surface of the second body, then the light will be refracted in the second body along straight lines other than the first ones along which it extended in the first body. But if the straight lines along which it extended in the first body are perpendicular to the surface of the second body, the light will extend directly into it and will not be refracted.

2.77 ◉ Et cum lux obliqua exiverit a corpore subtiliori ad grossius, reflectetur ad partem perpendicularis exeuntis a loco reflexionis perpendicularis super superficiem secundi corporis. Et cum lux obliqua fuerit extensa a grossiori ad subtilius, reflectetur ad partem contrariam perpendicularis exeuntis a loco reflexionis super superficiem secundi corporis. Cum ergo lux solis transit per omnia diaffona corpora secundum lineas rectas, ergo omnes luces extendentur in omnibus corporibus diaffonis, quia declaratum est in primo tractatu huius libri quod proprium est lucis semper extendi secundum lineas rectas, sive lux fuerit essentialis sive accidentalis, sive fortis sive debilis.

2.77 ◉ When the light passes obliquely from a rarer to a denser body, moreover, it will be refracted toward the normal dropped orthogonally from the point of refraction to the surface of the second body. And when the light passes obliquely from a denser into a rarer [body], it will be refracted away from the normal dropped orthogonally from the point of refraction to the surface of the second body. Accordingly, since sunlight passes through all transparent bodies along straight lines, all [other kinds of] light will extend through all transparent bodies [along straight lines] because it was shown in the first book of this treatise that it is a property of light always to extend along straight lines, whether the light is essential or accidental, or whether it is intense or faint.⁑

2.78 ◉ Preterea potest experimentator experiri luces accidentales in illo predicto instrumento et illis viis predictis in aliqua domo in quam intret lux diei per aliquod foramen alicuius quantitatis, si clauserit laminam, et posuerit instrumentum in oppositione foraminis, et aspexerit lucem que est intra aquam et ultra vitrum in hora instrumenti, et processerit per vias preostensas et in experimentatione lucis solis. Cum ergo experimentator expertus fuerit lucem accidentalem hiis predictis viis, inveniet lucem accidentalem transeuntem per corpus aque et per corpus vitri, et inveniet extensionem eius in vitro secundum verticationes linearum rectarum, et reflexionem si fuerit obliqua super superficiem corporis secundi, et rectam si fuerit perpendicularis super superficiem corporis secundi. In primo autem tractatu declaratum est quod lux omnis, sive essentialis aut accidentalis, fortis aut debilis, semper extenditur a quolibet puncto cuiuslibet corporis secundum lineam rectam.

2.78 ◉ Furthermore, the experimenter can test accidental light in the aforementioned apparatus according to the procedures described earlier if he encloses the [apparatus with its] register plate in a room into which daylight enters through a window of some [moderate] size, poses the apparatus toward the window, looks at the light [cast] on the apparatus’s rim inside the water or beyond the glass, and proceeds in the ways also shown in the experimentation dealing with sunlight.⁑ Accordingly, when the experimenter has tested the accidental light in these aforementioned ways, he will find that the accidental light passes through the body of water as well as through the body of glass, he will find that it extends through the glass along straight lines, and he will find that it is refracted if it is oblique to the surface of the second body or [continues] straight through if it is perpendicular to the surface of the second body. Furthermore, it was shown in the first book [of this treatise] that all light, whether essential or accidental, intense or faint, invariably extends in a straight line from any point on any [luminous or illuminated] body.

2.79 ◉ Ex istis ergo omnibus que declaravimus experientia et ratione patet quod omnis lux in omni corpore lucido essentialiter aut accidentaliter, fortiter aut debiliter extenditur a quolibet puncto illius per corpus diaffonum contingens illud corpus per omnem lineam rectam per quam poterit extendi, sive illud corpus contingens sit aer, aut aqua, aut lapis diaffonus. Et si luces extense per corpus contingens lucem que est principium eius occurrerint corpori diverse diaffonitatis ad diaffonitatem corporis in quo existit, si fuerint in lineis perpendicularibus super superficiem secundi corporis, extenduntur recte in secundo corpore. Et si fuerint in obliquis lineis super superficiem secundi corporis, reflectentur in secundo corpore, cum in secundo corpore extendentur in verticatione rectarum linearum aliarum a primis.

2.79 ◉ On the basis of everything we have shown by experiment and reason, then, it is clear that all light in any given luminous body, whether it does so essentially or accidentally, intensely or faintly, extends from any point on that body through a transparent body contiguous with that body by every straight line along which it can extend, no matter if that contiguous body is air, water, or a transparent stone. In addition, if the light extending through the contiguous body in which the light originates encounters a body of transparency differing from the transparency of the body in which it lies, it passes straight into the second body when it strikes the second body’s surface along perpendicular lines. If, on the other hand, it strikes the second body’s surface along oblique lines, it will be refracted in the second body because it will extend through the second body along straight lines other than the original ones [along which it traveled through the first body].

2.80 ◉ Et si lux fuerit reflexa, tunc linea per quam extendebatur lux in primo corpore et linea per quam reflectebatur in secundo erunt in eadem equali superficie, et quod reflexio eius, cum egressa fuerit a corpore subtiliori ad grossius, erit ad partem perpendicularis exeuntis a loco reflexionis super superficiem grossioris corporis. Et cum egressa fuerit a grossiori corpore ad subtilius, tunc reflexio eius erit ad partem contrariam illi in qua est perpendicularis exiens a loco reflexionis super superficiem subtilioris corporis.

2.80 ◉ If, moreover, the light is refracted, the line along which the light extended through the first body and the line along which it was refracted in the second body will lie in the same plane, and [it follows] that, when it passes from a rarer into a denser body, its refraction will occur in the direction of the normal dropped from the surface of the denser body at the point of refraction. But when it passes from a denser into a rarer body, its refraction will occur in the direction away from the normal passing through the surface of the rarer body at the point of refraction.

2.81 ◉ Quare autem reflectitur lux quando occurrit corpori diaffono diverse diaffonitatis hoc est quia transitus lucis per corpora diaffona fit per motum velocissimum, ut iam declaravimus in tractatu secundo. Luces ergo que extenduntur per corpora diaffona extenduntur motu veloci qui non patet sensui propter suam velocitatem. Preterea motus eorum in subtilibus corporibus, scilicet in illis que valde sunt diaffona, velocior est motu eorum in eis que sunt grossiora. in eis scilicet que minus sunt diaffona. Omne enim corpus diaffonum, cum lux transit in ipsum, resistit luci aliquantulum secundum quod habet de grossitie, nam in omni corpore naturali necesse est quod sit aliqua grossities, nam parve diaffonitatis non habet finem in ymaginatione, que est ymaginatio lucide diaffonitatis, et est quod omnia corpora naturalia perveniunt ad finem quem non possunt transire. Corpora ergo naturalia diaffona non possunt evadere aliquam grossitiem. Luces ergo, cum transeunt per corpora diaffona, transeunt secundum diaffonitatem que est in eis, et sic impediunt lucem secundum grossitiem que est in eis.

2.81 ◉ The reason light is refracted when it encounters [the surface of] a transparent body of different transparency is that the passage of light through transparent bodies occurs with an extraordinarily swift motion, as we showed earlier in the second book [of the treatise].⁑ Consequently, light extending through transparent bodies extends with a swift motion that cannot be perceived because of its speed. In addition, its motion in rare bodies, i.e., in those that are quite transparent, is quicker than its motion in those that are denser, i.e., in those that are less transparent. For when light passes through any transparent body, [that body] resists the light to some extent according to how dense it is because in every physical body there must be some density, since slight transparency has no limit in the imagination when it conceives of translucency, and what is [imaginable is] that all physical bodies [can] reach a limit [of rarity] they cannot transgress.⁑ As a result, transparent physical bodies cannot avoid some density. When it passes through transparent bodies, then, light travels [through them] according to the transparency that is in them, and so they impede the light according to the density that is in them.

2.82 ◉ Cum ergo lux transiverit per corpus diaffonum et occurrerit alii corpori diaffono grossiori primo, tunc corpus grossius resistet luci vehementius quam primum resistabat, et omne motum, cum movetur ad aliquam partem essentialiter aut accidentaliter, si occurrerit resistenti, necesse est ut motus eius transmutetur. Et si resistentia fuerit fortis, tunc motus ille reflectetur ad contrariam partem. Si vero debilis, non revertetur ad contrariam partem, nec poterit per illam procedere per quam inceperat, sed motus eius mutabitur.

2.82 ◉ Therefore, when light passes through a transparent body and encounters another transparent body that is denser than the first one, the denser body will resist the light more intensely than the first one resisted it, and when anything that moves in a given direction according to essential or accidental motion meets something to resist it, its motion must be altered.⁑ Furthermore, if the resistance is intense, that moving body will be reflected back in the opposite direction.⁑ If, on the other hand, [the resistance is] weak, the moving body will not be reflected back in the opposite direction, nor can it continue in the direction it originally followed, but its motion will be altered.

2.83 ◉ Omnium autem motorum naturalium que recte moventur per aliquod corpus passibile transitus super perpendicularem que est in superficie corporis in quo est transitus erit facilior. Et hoc videtur in corporibus naturalibus, si enim aliquis acceperit tabulam subtilem, et paxillaverit illam super aliquod foramen amplum, et steterit in oppositione tabule, et acceperit pilam ferream, et eicerit eam super tabulam fortiter, et preservaverit quod motus pile sit super lineam perpendicularem super superficiem tabule, tunc tabula cedet pile aut frangetur, si tabula fuerit subtilis et vis qua spera movetur fuerit fortis. Et si steterit in parte obliqua ab oppositione tabule et in illa eadem distantia in qua prius erat, et eicerit pilam super tabulam illam eadem vi qua prius eicerat, tunc spera labetur de tabula, si tabula non fuerit valde subtilis, nec movebitur ad illam partem ad quam primo movebatur, sed declinabit ad aliquam partem.

2.83 ◉ Now when physical bodies move straight through some permeable body, their passage will be easier along the normal to the surface of the body through which they pass. This [fact] is observed in [various cases of moving] physical bodies, for if someone takes a thin plank [of wood] and attaches it against a wide opening, then faces the plank, takes an iron ball, and hurls it forcefully at the plank while making sure that the ball travels along a path perpendicular to the plank’s surface, the plank will yield to the ball, or it will break, if the plank is thin [enough] and the force with which the ball moves is strong [enough]. But if he faces the plank obliquely at the same distance as before, and if he hurls the ball at that plank as forcefully as he did before, the ball will glance off the plank if the plank is not too thin, yet it will not move in the same direction it followed originally but will incline to a given side.

2.84 ◉ Et similiter si acceperit ensem, et posuerit coram se lignum, et percusserit eum ense ita quod ensis sit perpendicularis super superficiem ligni, tunc lignum secabitur magis. Et si ensis fuerit obliquus et percusserit lignum oblique, tunc lignum non secabitur omnino, sed forte secabitur in parte, aut forte ensis errabit deviando, et quanto magis ensis fuerit obliquus, tanto minus aget in lignum. Et alia multa sunt similia, ex quibus patet quod motus super perpendicularem est facilior et fortior et quod de obliquis motibus ille qui vicinior est perpendiculari est facilior remotiori.

2.84 ◉ Likewise, if he takes a sword, sets a log up in front of himself, and strikes it with the sword so that the sword [stroke] is perpendicular to the log’s surface, the log will be cut through to a considerable extent. On the other hand, if the sword is [swung] at an angle and strikes the log at a slant, the log will not be cut through entirely but may be partially cut, or the sword may deflect off [the log], and the more oblique the sword [stroke], the slighter its effect on the log. There are many other examples like these, from which it is evident that motion along the normal is easier and stronger and that among oblique motions the one that occurs [along a path inclining] more toward the normal is easier than [the one that occurs along a path inclining] farther away [from the normal].

2.85 ◉ Lux ergo, si occurrerit corpori diaffono grossiori illo corpore in quo existit, tunc impedietur ab illo ita quod non transibit in partem in qua movebatur, sed quia non fortiter resistit, non redibit in partem ad quam movebatur. Si ergo motus lucis transiverit super perpendicularem, transibit recte propter fortitudinem motus super perpendicularem, et si motus eius fuerit super lineam obliquam, tunc non poterit transire propter debilitatem motus. Accidet ergo quod declinetur ad partem in quam facilius movebitur quam in parte in quam movebatur. Sed facilior motuum est super perpendicularem, et quod vicinius est perpendiculari est facilius remotiori.

2.85 ◉ Hence, if light encounters a transparent body that is denser than the body in which it [originally] lies, it will be impeded by that body so as not to continue in the direction it [originally] followed, but since the [denser body] does not resist it [too] forcefully, it will not return back in the direction along which it [originally] moved. Consequently, if the light’s motion follows a perpendicular path, it will pass straight through on account of the force of [its] motion along the perpendicular, but if it moves along an oblique line, then it cannot pass [straight] through because of the weakness of its motion. It will happen, therefore, that it inclines toward the direction in which it will move more easily than the direction in which it [originally] moved [if it were to continue in that direction through the denser body]. The easiest of motions, however, is along the perpendicular, and motion nearer the perpendicular is easier than [motion] farther [from the perpendicular].

2.86 ◉ Et motus in corpore in quod transit, si fuerit obliquus super superficiem illius corporis, componitur ex motu in parte perpendicularis transeuntis in corpus in quo est motus et ex motu in parte linee que est perpendicularis super perpendicularem que transit in ipsum. Cum ergo lux fuerit mota in corpore diaffono grosso super lineam obliquam, tunc transitus eius in illo corpore diaffono erit per motum compositum ex duobus predictis motibus. Et quia grossities corporis resistit ei a verticatione quam intendebat, et resistentia eius non est valde fortis, ex quo sequeretur quod declinaret in partem ad quam facilius transiret, et motus super perpendicularem est facilimus motuum, necesse est ut lux que extenditur super lineam obliquam moveatur super perpendicularem exeuntem a puncto in quo lux occurrit superficiei corporis diaffoni grossi.

2.86 ◉ Furthermore, if it is oblique to the surface of the body through which it passes, the motion is composed of motion along the normal passing through the [surface of the] body in which the motion occurs and motion in the direction of the line that is perpendicular to the normal that passes through [the surface of] that body. Therefore, when light moves into a dense transparent body along an oblique line, its passage in that transparent body will occur with a motion composed of the two aforementioned motions. Since the density of the body resists that [motion] in the direction along which [the light] was incident, and since its resistance is not particularly intense, it would follow that [the light] would incline toward the direction in which it would pass more easily, and because motion along the normal is the easiest of motions, light that extends along an oblique line must move toward the normal dropped from the point at which the light strikes the surface of the dense transparent body.

2.87 ◉ Et quia motus eius est compositus duobus motibus, quorum alter est super lineam perpendicularem super superficiem corporis grossi et reliquus super lineam perpendicularem super perpendicularem hanc, et motus compositus qui est in ipso non omnino demittitur sed solummodo impeditur, necesse est ut lux declinet ad partem faciliorem parte ad quam prius movebatur, remanente in ipso motu composito. Sed pars facilior parte ad quam movebatur, remanente motu in ipso, est illa pars que est vicinior perpendiculari, unde lux que extenditur in corpore diaffono, si occurrerit corpori diaffono grossiori corpore in quo existit, reflectetur per lineam propinquiorem perpendiculari exeunti a puncto in quo occurrit corpori grossiori que extenditur in corpore grossiore quam linea per quam movebatur.

2.87 ◉ In addition, since its motion is composed of the two motions, one of them being along a line normal to the surface of the dense body and the other along a line perpendicular to this normal, and since the composite motion that is in it is in no way diminished but only interfered with, the light must incline in the direction [of motion that is] easier than [the motion that would result were it to continue] in the direction along which it [originally] moved if its composite motion stayed the same. But, given that its [overall composite] motion does stay the same, the direction of easier [motion] in comparison to that in which it originally moved is a direction closer to the normal, so if it strikes a transparent body denser than the body in which it [originally] lies, the light extending into the [denser] transparent body will be refracted along a line closer to the normal dropped from the point at which it strikes the denser body through which it extends than the line along which it moved [in the rarer body].⁑

2.88 ◉ Hec igitur est causa reflexionis splendorum in corporibus diaffonis que sunt grossiora corporibus diaffonis in quibus existunt, et ideo reflexio proprie inventa in lucibus obliquis. Cum ergo lux extenditur in corpore diaffono, et occurrerit corpori diaffono diverse diaffonitatis corporis in quo existit et grossioris, et fuerit obliqua super superficiem corporis diaffoni cui occurrit, reflectetur ad partem perpendicularis super superficiem corporis diaffoni extense in corpore grossiore.

2.88 ◉ This, then, is why radiant light refracts in transparent bodies that are denser than the transparent bodies in which it [originally] lies, and so refraction is found to be characteristic of oblique light. Hence, when light passes through a transparent body and encounters a transparent body of different transparency that is denser than the body in which it [originally] lies, and when it is oblique to the surface of the transparent body it encounters, it will be refracted toward the normal dropped to the denser body [at the point of refraction] on the surface of the [denser] transparent body.

2.89 ◉ Causa autem que facit reflexionem lucis a corpore grossiori ad corpus subtilius ad partem contrariam parti perpendicularis est quia, cum lux mota fuerit in corpore diaffono, repellet eam aliqua repulsione, et corpus grossius repellet eam maiori repulsione, sicut si lapis, cum movetur in aere, movetur facilius et velocius quam si movetur in aqua eo quod aqua repellit ipsum maiori repulsione quam aer. Cum ergo lux exiverit a corpore grossiori ad subtilius, tunc motus eius erit velocius, et cum lux fuerit obliqua super duas superficies corporis diaffoni que est differentia communis ambobus corporibus, tunc motus eius erit super lineam existentem inter perpendicularem exeuntem a principio motus eius et inter perpendicularem super lineam perpendicularem exeuntem etiam a principio motus. Resistentia ergo corporis grossioris erit a parte ad quam exit secunda perpendicularis. Cum ergo lux exiverit a corpore grossiori et pervenerit ad corpus subtilius, tunc resistentia corporis subtilioris luci que est in parte ad quam exit secunda perpendicularis erit minor prima resistentia, et sic motus lucis ad partem a qua resistebatur maior, et sic est de luce in corpore subtiliore ad contrariam partem parti perpendicularis.

2.89 ◉ The reason light refracts from a denser to a rarer body in the direction away from the normal is that, when light moves through a transparent body, [that body] pushes against it with some amount of repulsive force, and the denser body pushes against it with greater repulsive force, just as [happens] when a stone moving through air moves more easily and more swiftly than it moves in water because the water pushes against it with greater repulsive force than does air. Hence, when light passes from a denser into a rarer body, its motion will be faster [because it is less hindered], and when the light is oblique to the two [coincident] surfaces of the transparent body that form the interface of the two bodies, its motion will occur along a line that lies between the normal dropped from the initial point of its motion [into that interface] and the line perpendicular to the normal dropped from the initial point of motion. Consequently, the resistance of the denser body will be in the direction of the second perpendicular. Thus, when the light leaves the denser body and enters the rarer body, the resistance of the rarer body to the light posed in the direction of the second perpendicular will be less than the resistance in the first [body, which is denser], and so the motion of the light [will be] increased [after refraction] in the direction along which it was more forcefully resisted, and so it is that light [passing] into a rarer body [follows a path] away from the normal.⁑

◉Capitulum tertium

◉Chapter Three

De qualitate reflexionis lucis in corporibus diaffonis

3.1 ◉ In predicto capitulo declaratum est quod omnis lux que reflectitur a corpore diaffona ad aliud corpus diaffonum semper erit in una superficie equali. Lux ergo que reflectitur ab aere ad aquam est semper in eadem superficie equali; linea ergo recta que est per quam extenditur lux in aere et linea recta per quam reflectitur in aqua semper erunt in eadem superficie equali. Hec autem superficies apud inspectionem instrumenti predicti est medius circulus illis tribus predictis signatis in interiori parte hore instrumenti.

3.1 ◉ In the preceding chapter it was shown that all light [rays] refracted from one transparent body to another transparent body will always lie in a single plane [before and after refraction]. Consequently, [a] light [ray] refracted from air into water always lies in the same plane, so the straight line along which the light extends through the air and the straight line along which it is refracted into the water will always lie in the same plane. Furthermore, according to what what was observed in the apparatus described earlier, this plane is [the one passing through] the middle circle among the three previously-discussed [circles] inscribed on the inner wall of the apparatus’s rim.

3.2 ◉ Sed superficies interioris lamine est equidistans superficiei dorsi, cui superponitur superficies regule quadrate. Ergo superficies circuli medii est equidistans superficiei regule quadrate. Et superficies regule quadrate que est superposita dorso lamine est perpendicularis super alteram superficiem secantem superficiem superpositam dorso lamine, et hec superficies regule superponitur superficiei duarum differentiarum sibi applicatarum in duabus extremis regule. Sed superficies duarum differentiarum superponitur hore instrumenti.

3.2 ◉ But the inside surface of the register plate is parallel to its back surface, to which the surface of the rectangular strip [that holds the apparatus in the vessel] is attached [through its axle]. Therefore, the plane of the middle circle [on the rim’s inner wall] is parallel to the surface of the rectangular strip [that nests against the back surface of the register plate]. In addition, the surface of the rectangular strip nesting against the back of the register plate is perpendicular to either of the [two] surfaces cutting the surface that nests against the back of the register plate, and [either] of these surfaces of the strip is conjoined with the surface of the two excess pieces attached at the two ends of the strip. But the surface [at the ends] of the two excess pieces is flush with the rim of the apparatus.

3.3 ◉ Ergo superficies medii circuli est perpendicularis super superficiem transeuntem super horam instrumenti, et superficies transiens per horam instrumenti est equidistans orizonti apud experimentationem. Superficies ergo circuli medii est perpendicularis super superficiem orizontis. Cum ergo declaratum sit quod lux que est in aere et reflectitur in aqua est apud experimentationem in circumferentia medii circuli, manifestum est quoniam lux que extenditur in aere et reflectitur in aqua est semper in eadem superficie equali super superficiem orizontis.

3.3 ◉ Hence, the plane of the middle circle [on the rim’s inner wall] is perpendicular to the surface of the apparatus’s rim [i.e., to all its lines of longitude], and [every line of longitude on] the surface of the apparatus’s rim is parallel to the horizon according to [the way the] experiment [is set up]. The plane of the middle circle is therefore perpendicular to the plane of the horizon. Accordingly, since it has been shown experimentally that light extending through air and refracting in water does so within the perimeter [i.e., in the plane] of the middle circle, it is evident that the light extending through air and refracting in water always lies in the same plane [perpendicular] to the plane of the horizon.

3.4 ◉ Et etiam ymaginemur lineam a centro medii circuli ad centrum mundi. Sic ergo hec linea erit perpendicularis super superficiem aque, quia est diameter mundi. Sed hec linea est in superficie medii circuli; ergo est in superficie reflexionis. Ergo superficies reflexionis est perpendicularis super superficiem aque. Et iam declaratum est quod, cum lux reflectitur ex aere ad aquam, erit inter primam lineam per quam extenditur in aere, que est inter diametrum medii circuli, et inter perpendicularem exeuntem a centro medii circuli super superficiem aque. Et declaratum est etiam quoniam lux que est in puncto quod est centrum lucis que est intra aquam non pervenit ad ipsum nisi ex luce que extenditur a centro circuli medii. Lux ergo que reflectitur ex aere ad aquam reflectitur in superficie perpendiculari super superficiem aque, et reflexio eius erit ad partem perpendicularis exeuntis a loco reflexionis super superficiem aque, et non perveniet ad perpendicularem.

3.4 ◉ We may also imagine a line [extending] from the center of the middle circle to the center of the world. This line will thus be perpendicular to the water’s surface, since it is [a segment] of the diameter of the world.⁑ However, this line lies in the plane of the middle circle, so it lies in the plane of refraction. Consequently, the plane of refraction is perpendicular to the surface of the water. And it has already been shown that, when light is refracted from air into water, it will do so between [the continuation of] the original line along which it extends through the air, that line being along a diameter of the middle circle, and the normal dropped from the center of the middle circle to the water’s surface. It has also been shown that the light at the point constituting the center of the light under the water reaches it only from the light extending from the center of the middle circle. Consequently, the light that is refracted from the air into water is refracted in a plane perpendicular to the water’s surface, and it will be refracted toward the normal dropped to the water’s surface from the point of refraction but will not reach the normal [itself].⁑

3.5 ◉ Reflexio autem lucis ab aere ad vitrum hoc modo fit, declaratum enim est in experimentatione vitri quod, cum linea que transit per centra duorum foraminum, cum fuerit obliqua super superficiem vitri equalem et transiverit per centrum vitri, et superficies vitri equalis fuerit ex parte foraminum, tunc linea reflectetur apud centrum vitri, et reflexio eius erit in superficie circuli medii ad partem in qua est perpendicularis exiens a centro vitri super superficiem vitri equalem.

3.5 ◉ Refraction of light from air into glass occurs in this way too, for it was shown in the experiment with glass that, when the line [of light] passing through the centers of the two holes is oblique to the flat face of the glass and passes through the center of the [sphere encompassing the] glass [quarter-sphere], and when the flat face of the glass faces the holes, the line [of light] will be refracted at the center of the [sphere encompassing the] glass [quarter-sphere], and its refraction will occur in the plane of the middle circle [and will be] toward the normal dropped from the center of the [sphere encompassing the] glass [quarter-sphere] on the flat face of the glass.

3.6 ◉ Et declaratum est etiam quod, cum linea que transit per centra duorum foraminum fuerit obliqua super superficiem vitri spericam, et superficies sperica fuerit ex parte foraminum, tunc lux reflectetur in corpore vitri et apud superficiem vitri spericam. Et erit reflexio eius in superficie medii circuli et ad partem perpendicularis exeuntis a loco reflexionis super superficiem vitri spericam. Et superficies vitri equalis in qua est centrum circuli vitrei est perpendicularis super superficiem lamine; est ergo perpendicularis super superficiem medii circuli.

3.6 ◉ It has also been demonstrated that, when the line [of light] passing through the centers of the two holes is oblique to the spherical surface of the glass, and when the spherical surface faces the holes, the light will be refracted into the body of the glass at the spherical surface of the glass. Its refraction will also occur in the plane of the middle circle and toward the normal dropped from the point of refraction on the spherical surface of the glass. Furthermore, the flat face of the glass, upon which lies the center of the circle [forming the base] of the glass [hemisphere from which the quarter-sphere is formed], is perpendicular to the surface of the register plate, so it is perpendicular to the plane of the middle circle.

3.7 ◉ Superficies ergo medii circuli est perpendicularis super superficiem vitri equalem, et superficies circuli medii transit etiam per centrum spere vitree; in omnibus experimentationibus vitri ergo est perpendicularis super superficiem vitri spericam etiam. Lux ergo que extenditur in aere et reflectitur in corpore vitri, apud extensionem eius in aere et postquam reflectitur in vitro semper est in superficie perpendiculari super superficiem vitri, et semper reflexio eius erit ad partem perpendicularis exeuntis a loco reflexionis super superficiem vitri, sive superficies vitri fuerit equalis, sive sperica.

3.7 ◉ Hence, the plane of the middle circle is perpendicular to the flat face of the glass, and the plane of the middle circle also passes through the center of the sphere [encompassing the] glass [quarter-sphere], so in all the experiments with glass [the middle circle] is also perpendicular to the spherical surface of the glass. When it extends through air and refracts into the body of the glass, then, the light invariably lies in a plane perpendicular to the surface of the glass while it extends through the air and after it is refracted into the glass, and its refraction will invariably be toward the normal dropped from the point of refraction on the surface of the glass, whether the glass’s surface is flat or spherical.

3.8 ◉ Item declaratum est etiam quod linea que transit per duo centra foraminum, cum fuerit perpendicularis super superficiem vitri, et extensa fuerit in corpus vitri secundum rectitudinem, et superficies sperica fuerit ex parte foraminum, et fuerit hec linea—scilicet que transit per centra duorum foraminum—declinans super superficiem vitri equalem, et transiverit per centrum vitri et reflexa in corpore aeris contingentis superficiem vitri equalem et apud centrum vitri, tunc reflexio eius erit in superficie circuli medii et ad contrariam partem illi in qua est perpendicularis exiens a centro vitri super superficiem vitri equalem.

3.8 ◉ Likewise, it has been shown as well that, when the line [of light] passing through the two centers of the holes is perpendicular to the surface of the glass and extends straight through the body of the glass, then if the spherical surface faces the holes while this line [of light]—i.e., the one passing through the centers of the two holes—is oblique to the flat face of the glass but passes through the center of the [sphere encompassing the] glass [quarter-sphere] and is refracted in the body of the air in contact with the flat surface of the glass at the sphere’s center, its refraction will occur in the plane of the middle circle and away from the normal dropped from the [sphere’s] centerpoint on the flat face of the glass.

3.9 ◉ Et declaratum est etiam quod linea que transit per centra duorum foraminum, cum fuerit perpendicularis super superficiem vitri equalem, et si fuerit extensa in corpore vitri secundum rectitudinem, et superficies equalis fuerit ex parte foraminum, et hec linea—scilicet que transit per centra duorum foraminum—fuerit obliqua super superficiem vitri spericam et non transiens per centrum eius, et fuerit reflexa apud superficiem spericam in corpore aeris continentis superficiem spericam, tunc reflexio eius erit in superficie medii circuli et ad partem contrariam illi in qua est perpendicularis exiens a loco reflexionis super superficiem reflexionis. Et in hiis duobus sitibus superficies etiam medii circuli est perpendicularis super superficiem vitri equalem et spericam. Lux ergo que extenditur in corpore vitri et reflectitur in aere, dum extenditur in vitro et reflectitur in aere, semper est in superficie perpendiculari super superficiem aeris, et semper reflexio eius erit ad partem contrariam illi in qua est perpendicularis exiens a loco reflexionis super superficiem aeris.

3.9 ◉ It has also been shown that, when the line [of light] passing through the centers of the two holes is perpendicular to the flat face of the glass and passes through the body of the glass in a straight line, then if the flat face lies toward the holes while this line—i.e., the one passing through the centers of the two holes—is oblique to the spherical surface of the glass so as not to pass through its center, and if it is refracted at the spherical surface into the body of air enveloping the spherical surface, its refraction will occur in the plane of the middle circle and away from the normal dropped from the point of refraction on the surface at which refraction occurs. In these two cases, moreover, the plane of the middle circle is perpendicular to the flat surface as well as to the spherical surface of the glass. Therefore, when it extends through the glass and is refracted in the air, the light that extends through the body of the glass and is refracted in the air always lies in a plane perpendicular to the surface of the air [in contact with the glass], and its refraction will always be away from the normal dropped from the point of refraction on the surface of the air.

3.10 ◉ Ex omnibus ergo istis predeclaratis patet quod omnis lux reflexa a corpore diaffono ad aliud corpus semper reflectitur in superficie perpendiculari super superficiem secundi corporis, et si secundum corpus fuerit grossius primo, tunc reflexio eius erit ad partem perpendicularis exeuntis a loco reflexionis super superficiem secundi corporis, et non pervenit ad perpendicularem. Et si secundum corpus fuerit subtilius primo, reflexio erit ad partem contrariam illi in qua est perpendicularis exiens a loco reflexionis super superficiem secundi corporis secundum diversitatem figurarum superficierum corporum diaffonorum.

3.10 ◉ From everything that has already been shown, it is clear that all light that is refracted from one transparent body to another [transparent] body is invariably refracted in a plane perpendicular to the surface of the second body, and if the second body is denser than the first, its refraction will be toward the normal dropped from the point of refraction on the surface of the second body, but it will not reach the normal [itself]. If, on the other hand, the second body is rarer than the first, the refraction will occur away from the normal dropped from the point of refraction on the surface of the second body according to the different shapes of the surfaces of transparent bodies [i.e., according to whether the surfaces are plane or curved].

3.11 ◉ Et ex hiis etiam patet quod, cum lux reflectitur a corpore diaffono ad secundum corpus diaffonum et de secundo ad tertium, reflectetur etiam in superficie tertii si diaffonitas tertii differt a diaffonitate secundi. Si vero tertium fuerit grossius secundo, tunc reflexio lucis erit ad partem perpendicularis exeuntis a loco reflexionis super superficiem tertii. Si autem tertium fuerit subtilius secundo, tunc reflexio lucis erit ad partem contrariam illi in qua est perpendicularis; similiter si lux reflexa fuerit ad quartum corpus, et ad quintum, aut ad plura.

3.11 ◉ From these things it is also clear that, when light is refracted from one transparent body to a second transparent body and then from the second to a third, it will also be refracted at the surface of the third, if the transparency of the third [body] differs from the transparency of the second. But if the third [body] is denser than the second, the refraction of light will occur toward the normal dropped from the point of refraction on the surface of the third. If, on the other hand, the third [body] is rarer than the second, the refraction of the light will occur away from the normal, and the same [holds] if the light is refracted at a fourth body, or at a fifth, or at more.

3.12 ◉ Hoc autem quod declaravimus in hoc capitulo est qualiter omnes luces reflectuntur in corporibus diaffonis diverse diaffonitatis. Quare autem fit reflexio in superficie perpendiculari super superficiem corporis diaffoni hoc est quia linea per quam extenditur lux in primo corpore diaffono reflectitur ad partem perpendicularis in hac superficie, scilicet in qua est perpendicularis et prima linea, pars enim perpendicularis est in hac superficie. Ideo reflexio fit in superficie perpendiculari super superficiem corporis diaffoni.

3.12 ◉ Now what we have shown [so far] in this chapter is how all light is refracted in transparent bodies of varying transparency. The reason refraction occurs in a plane perpendicular to the surface of a transparent body is that the line along which the light extends through the first transparent body is refracted toward the normal within this plane, i.e., the plane containing the normal and the initial line [of incidence], for the normal [forming] part [of this couple] lies in this plane.⁑ Therefore, the refraction occurs in a plane perpendicular to the surface of the transparent body.

3.13 ◉ Quantitates autem angulorum reflexionis differunt secundum quantitates angulorum quos continet prima linea per quam extenditur lux in primo corpore et perpendicularis exiens a loco re flexionis super superficiem secundi corporis secundum diaffonitatem secundi corporis, nam quanto magis crescit angulus quem continet prima linea et perpendicularis, crescit angulus reflexionis, et quanto decreverit ille angulus, decrescit angulus reflexionis. Sed anguli reflexionum non observant eandem proportionem ad angulos quos continet prima linea cum perpendiculari; sed differunt hee proportiones in eodem corpore diaffono. Cum ergo prima linea per quam lux extenditur in primo corpore continuerit cum perpendiculari duos angulos inequales in duobus temporibus diversis aut in duobus locis diversis, tunc proportio anguli reflexionis que est ab angulo minori ad angulum minorem minor erit proportione anguli reflexionis anguli maioris ad angulum maiorem.

3.13 ◉ The sizes of the angles of refraction vary according to the sizes of the angles [of incidence] that the first line along which the light extends through the first body forms with the normal dropped from the point of refraction on the surface of the second body according to the transparency of the second body, for the larger the angle [of incidence] the first line forms with the normal becomes, the larger the angle of refraction becomes, whereas the smaller the angle [of incidence] becomes, the smaller the angle of refraction becomes. But the angles of refraction do not maintain the same ratio with the angles [of incidence] that the first line forms with the normal; rather these ratios vary in the same transparent body. Therefore, if the first line along which light extends through the first body forms two unequal angles with the normal at two different times or at two different locations, the ratio of the angle of refraction [resulting] from a smaller angle [of incidence] to [that] smaller angle [of incidence] will be less than the ratio of the angle of refraction [resulting] from a larger angle [of incidence] to the larger angle [of incidence].⁑

3.14 ◉ Cum ergo experimentator voluerit experiri istos angulos, dividat a circulo medio qui est in circumferentia instrumenti ex parte centri foraminis quod est in circumferentia instrumenti arcum decem partium ex illis partibus quibus medius circulus dividitur in 360. Deinde extrahamus a loco differentie lineam rectam perpendicularem super superficiem lamine, et copulemus extremitatem eius que est in lamina cum centro lamine per lineam rectam, et protrahamus ipsam in aliam partem.

3.14 ◉ So if the experimenter wants to investigate those angles empirically, he should measure off an arc on the middle circle [inscribed on the inner wall of the rim] at the outer edge of the apparatus consisting of ten of the 360 degrees into which the middle circle is subdivided [and he should measure that arc off] on the [right-hand] side of the center of the hole on the circumference of the apparatus. From the point [just] marked off we should then draw a straight line [on the inner wall of the apparatus’s rim] perpendicular to the surface of the register plate, connect the endpoint of this line on the register plate to the center of the register plate with a straight line, and extend this line to [the base of the apparatus’s rim on] the other side.

3.15 ◉ Deinde dividamus in circumferentia medii circuli etiam arcum sequentem primum, cuius quantitas sit 90, et signemus in extremitate huius arcus signum. Linea ergo que exit a centro medii circuli ad hoc signum erit perpendicularis super lineam exeuntem a centro medii circuli ad primum signum quod est in circumferentia medii circuli. Et erit arcus residuus qui est inter secundum signum et extremitatem diametri medii circuli qui transit per centra duorum foraminum 80 partium. Signemus igitur in extremitate huius diametri etiam signum.

3.15 ◉ Then we should measure off an arc of 90° on the periphery of the middle circle next to the first one, and we should mark the endpoint of this arc. Accordingly, the line extending from the center of the middle circle to this mark will be perpendicular to the line passing from the center of the middle circle to the first mark on the periphery of the middle circle. And the arc that remains between the second mark and the endpoint of the diameter of the middle circle that passes through the centers of the two holes will be 80°. We should therefore mark the endpoint of this diameter as well.⁑

3.16 ◉ Deinde ponamus instrumentum in vase, et preservemus ut circumferentia vasis sit equidistans orizonti, et incipiamus experiri ab hora ortus solis. Et infundamus in vas aquam claram quousque perveniat ad centrum lamine, et moveamus instrumentum donec prima linea signata in superficie lamine sit contingens superficiem aque. In hoc ergo statu linea que transit per centrum circuli medii equidistans est prime linee signate in superficie lamine cuius extremitas pervenit ad primum signum signatum in circumferentia circuli medii. Tanget etiam superficiem aque, locus enim harum duarum linearum non differt in respectu superficiei aque quoad sensum. Et hec linea continet cum linea exeunte a centro medii circuli ad secundum signum quod est in circumferentia medii circuli perpendicularem super superficiem aque angulum rectum, et diameter circuli medii qui transit per centra duorum foraminum continet cum hac perpendiculari exeunti a centro circuli medii super superficiem aque angulum cuius quantitas erit 80 partes, hunc enim angulum cordat arcus medii circuli qui est inter secundum et tertium signum. Arcus autem qui est inter centrum foraminis et primum signum, qui est decem partium, cordat angulum declinationis.

3.16 ◉ Next we should insert the apparatus in the vessel, and we should make sure that the lip of the vessel is parallel to the horizon, and we should begin the investigation at sunrise. We should fill the vessel with clear water until it reaches the center of the register plate, and we should adjust the apparatus until the first line marked on the register plate’s surface coincides with the water’s surface. In this situation, then, the line passing through the center of the middle circle is parallel to the first line marked off on the register plate’s surface, the endpoint of that line reaching the first mark on the periphery of the middle circle [which forms the plane of refraction]. It will also coincide with the water’s surface, for the location of these two lines does not differ with respect to the water’s surface as far as can be empirically determined. This line also forms a right angle with the line passing from the center of the middle circle to the second mark on the periphery of the middle circle, which is normal to the water’s surface, and the diameter of the middle circle passing through the centers of the two holes forms an angle that will be 80° with this normal dropped from the center of the middle circle on the water’s surface, for this angle subtends the arc on the middle circle between the second and third marks. The arc between the center of the hole and the first mark, which is 10°, subtends the angle of inclination.

3.17 ◉ Deinde oportet experimentatorem considerare solem et mutare instrumentum donec lux transeat per duo foramina, et tunc aspiciat lucem que est in hora instrumenti que est intra aquam, et signet super centrum lucis signum. Hoc ergo signum erit in circumferentia medii circuli. Deinde auferat instrumentum, et aspiciat tertium signum, quod est inter extremitatem medii circuli et inter secundum signum quod est extremitas perpendicularis exeuntis a centro medii circuli super superficiem aque. Ex hac ergo experimentatione patebit quod angulus reflexionis est ille quem cordat arcus qui est inter centrum lucis et tertium signum quod est extremitas linee transeuntis per centra duorum foraminum per quam extendebatur lux. Et ex numero partium huius arcus patebit quantitas anguli reflexionis et quantitas proportionis anguli reflexionis ad 80 partes que sunt angulus quem continet linea per quam extendebatur lux cum perpendiculari exeunti a puncto reflexionis super superficiem aque.

3.17 ◉ The experimenter should then take account of [the position of] the sun and adjust the apparatus until the light shines through the two holes, and he should thus observe the light on the rim of the apparatus under water and mark the center of that light. This mark will therefore lie on the periphery of the middle circle. Then he should remove the apparatus and look at the third mark, which lies between the endpoint [of the diameter coincident with the water’s surface] on the middle circle and the second mark at the endpoint of the normal dropped from the center of the middle circle on the water’s surface. On the basis of this experiment, then, it will be clear that the angle of refraction is the one subtended by the arc between the center of the light and the third mark at the endpoint of the line passing through the centers of the two holes along which the light extended [to the water]. According to the number of degrees in this arc, the size of the angle of refraction will be revealed, as will the size of the angle of refraction in proportion to the 80° that constitutes the angle that the line along which the light extended [to the water] forms with the normal dropped from the point of refraction on the water’s surface [i.e., the angle of incidence].⁑

3.18 ◉ Deinde oportet experimentatorem delere signum et lineam signatam in lamina et distinguere inter circumferentiam medii circuli ex parte centri foraminis quod est in hora instrumenti arcum cuius quantitas sit viginti partes. Et signet in extremitate eius signum, et extrahat ab hoc signo perpendicularem super superficiem lamine, et extrahat ab eius extremitate lineam ad centrum lamine. Et protrahamus illam in utramque partem, et dividamus etiam arcum sequentem illum cuius quantitas erat viginti in 90, et signemus in ipso signum. Et sit arcus qui est inter secundum signum et extremitatem linee transeuntis per centra duorum foraminum 70 partes, et signemus in extremitate huius linee signum.

3.18 ◉ Now the experimenter should erase the mark and the line drawn on the register plate and measure off an arc 20° in size on the periphery of the middle circle to the [right-hand] side of the center of the hole in the apparatus’s rim. He should mark the endpoint of this [arc] and from this mark draw a line [on the inner wall of the rim] perpendicular to the register plate’s surface, and from its endpoint he should draw a line to the center of the register plate. We should extend this line to the other side, cut the arc next to the arc that was [marked off at] 20° with an arc of 90°, and mark that point. The arc between this second mark and the end of the line passing through the centers of the two holes should be 70°, and we should mark the endpoint of this line.

3.19 ◉ Deinde ponamus instrumentum in vas et revolvamus illud quousque linea signata in lamina tangat superficem aque. Linea ergo que exit a centro circuli medii ad secundum signum erit perpendicularis super superficiem aque, ut predictum est, et linea que transit per centra duorum foraminum continet cum hac perpendiculari angulum 70 partium. Deinde experimentator consideret solem, et moveat instrumentum quousque lux pertranseat per duo foramina, et signemus super centrum lucis signum. Et auferat instrumentum, et inspiciat signa que sunt in circumferentia medii circuli, ex qua experimentatione habebit quantitatem anguli reflexionis et proportionem eius ad angulum quem continet linea per quam extenditur lux cum perpendiculari exeunte a loco reflexionis que est in hoc statu 70 partes.

3.19 ◉ Then we should insert the apparatus in the vessel and turn it until the line drawn on the register plate coincides with the water’s surface. Accordingly, the line dropped from the center of the middle circle to the second mark will be normal to the water’s surface, as was described earlier, and the line passing through the centers of the two holes forms an angle of 70° with this normal. The experimenter should then take account of the [position of the] sun and adjust the apparatus until the light shines through the two holes, and we should mark the center of the light [refracted to the apparatus’s rim under water. The experimenter] should remove the apparatus and look at the marks on the periphery of the middle circle, [and] from this experiment he will have the size of the angle of refraction as well as its size in relation to the angle that the line along which the light extends [to the water] forms with the normal dropped from the point of refraction, which in this case is 70°.⁑

3.20 ◉ Deinde experimentator auferat instrumentum et deleat signa et lineam que est in lamina, et dividat arcum ex parte foraminis cuius quantitas sit 30 partes. Et procedat ut in primis ablationibus, et sic habebit quantitatem anguli reflexionis et proportionem eius ad angulum quem continet linea per quam extendebatur lux cum perpendiculari exeunte a loco reflexionis, qui est in hoc situ 60 partes. Deinde dividamus arcum cuius quantitas sit 40 partes; deinde arcum cuius quantitas sit 50 partes; deinde 60; deinde 70; deinde 80; et consideret unumquemque istorum arcuum, et sic habebit quantitates angulorum reflexionis et angulorum declinationis quos cordant primi arcus distincti ex parte centri foraminis, et habebit proportionem angulorum reflexionis ad angulos quos continent prime linee per quas extendebatur lux cum perpendiculari que est in superficie aque, qui crescunt per decem. Et si experimentator voluerit quod anguli crescant per quinque, bene poterit facere, et si voluerit per minus quam quinque, bene poterit facere predicto ordine.

3.20 ◉ The experimenter should then remove the apparatus, erase the marks and the line on the register plate, and measure off an arc of 30° to the [right-hand] side of the hole. He should proceed as he did in the first [cases] when removing [the marks and lines], and so he will have the size of the angle of refraction as well as its relation to the angle that the line along which the light extended [to the water] forms with the normal dropped from the point of refraction, which is 60° in this case. Next we should measure off an arc of 40° [to the right-hand side of the hole], then an arc 50° in size, then 60°, then 70°, then 80°; and [the experimenter] should take account of each of those arcs, and so he will have the sizes of the angles of refraction as well as the [respective] angles of inclination subtended by the first arc marked off to the side of the hole’s center, and he will [also] have the ratio of the angles of refraction to the angles that the first lines along which the light extended [to the water] form with the normal to the water’s surface, these angles increasing in increments of ten degrees. Moreover, if the experimenter wants the angles to increase by [increments of] five [degrees], he can certainly make [them do so], and if he wants [them to increase] by [increments of] less than five [degrees], he can certainly make [them do so] in the order just described.⁑

3.21 ◉ Et cum experimentator voluerit experiri per vitrum, dividat arcus, et signet predicta signa, et superponat vitrum predictum superficiei lamine, et superponat differentiam eius communem linee signate in lamina. Et ponet superficiem vitri equalem ex parte foraminum, et applicet vitrum bene. Et ponat instrumentum in vase, et moveat ipsum quousque lux transeat per duo foramina, et signet super centrum lucis signum. Et auferat instrumentum, et intueatur arcus. Et deinde deleat signa, et dividat alios arcus, et signet alia signa, et inspiciat arcus prout aspexerit per aquam, et sic habebit quantitates reflexionum in transitu lucis de aere ad vitrum.

3.21 ◉ Furthermore, if the experimenter wants to do the experiment with glass, he should mark off the [appropriate] arc [of 10°] and make the marks described before, and he should apply the glass [quarter-sphere] described earlier to the surface of the register plate and place its common section on the line drawn on the register plate. He will also pose the flat face of the glass toward the holes and affix the glass firmly [to the register plate]. He should insert the apparatus in the vessel, adjust it until the light shines through the two holes, and mark the center of the light [cast on the rim opposite the holes]. Then he should remove the apparatus and look at the [resulting] arc. Subsequently he should erase the marks, measure off the other arcs [from 20° to 80°], mark the other points, and examine the arcs as he examined them for water, and so he will have the sizes of [the angles of] refraction in the passage of light from air to glass.⁑

3.22 ◉ Et si voluerit experiri reflexionem lucis de vitro ad aerem et ad aquam, applicet vitrum econtra primi situs, scilicet quod ponat convexum eius ex parte duorum foraminum, et ponat medium communis differentie que est in vitro super centrum lamine. Tunc ergo lux que transit per centra duorum foraminum pervenit recte ad centrum vitri et reflectitur apud illum de vitro ad aerem. Deinde dividat arcus successive, et mutet positionem vitri, et sic habebit angulos reflexionis particulares et proportionem eorum ad angulos quos continet prima linea per quam extenditur lux cum linea perpendiculari super superficiem contingentem superficiem vitri.

3.22 ◉ Now if he wishes to test the refraction of light from glass to air or to water, he should apply the glass in the opposite way from [that in] the first case, i.e., he should pose its convex surface toward the two holes, and he should place the midpoint of the common section on the glass at the center of the register plate. Accordingly, the light passing through the centers of the two holes reaches straight to the center of the [sphere encompassing the] glass [quarter-sphere] and is refracted at that point from the glass into the air. He should then mark off the arcs [from 10° to 80°] successively and change the position of the glass [accordingly], and thus he will have the angles of refraction specific [to those arcs] as well as their relation to the angles that the first line along which the light extends [to the glass] forms with the normal to the surface [of the air] in contact with the [flat] face of the glass.⁑

3.23 ◉ Et cum experimentator expertus fuerit hos duos predictos situs, videbit quoniam quantitates angulorum reflexionis de aere ad vitrum et de vitro ad aerem semper erunt equales, cum angulus quem continet linea per quam extenditur lux ad locum reflexionis cum linea perpendiculari, cum reflectatur de aere ad vitrum, equalis sit angulo quem continet linea per quam extenditur lux a loco reflexionis cum perpendiculari cum reflectitur a vitro.

3.23 ◉ And when the experimenter has put the two cases just discussed to the experimental test, he will see that the sizes of the angles of refraction from air to glass and from glass to air will always be equal if the angle that the line along which the light extends to the point of refraction forms with the normal, when [the light] is refracted from air into glass, is equal to the angle that the line along which the light extends from the point of refraction forms with the normal, when it is refracted from glass [into air].⁑

3.24 ◉ Et si quis voluerit experiri quantitates angulorum reflexionis qui sunt apud convexum vitri, dividat de circumferentia medii circuli ex parte centri foraminis quod est in hora instrumenti arcum cuius quantitas sit decem partium, et extrahat ab extremitate eius perpendicularem super superficiem lamine in superficie hore instrumenti, sicut prius fecerat. Deinde dividat ex hac linea incipiens a centro lamine lineam equalem semidiametro vitri, et ab extremitate huius linee extrahat perpendicularem super diametrum lamine super cuius extremitates sunt due linee perpendiculares in hora instrumenti, et protrahat hanc perpendicularem in utramque partem. Deinde superponat vitrum super superficiem lamine, et superponat differentiam eius communem predicte perpendiculari, et ponat medium differentie communis super punctum a quo extracta fuerit perpendicularis.

3.24 ◉ If one wants to investigate the sizes of the angles of refraction [resulting from the incidence of light] at the convex surface of the glass, he should measure off an arc of ten degrees on the periphery of the middle circle from the [right-hand] side of the center of the hole in the apparatus’s rim, and from its endpoint he should draw a line on the [inner] wall of the apparatus’s rim perpendicular to the register plate’s surface, just as he did earlier. Starting from the center of the register plate he should then measure off a distance [on the line drawn] from [the bottom endpoint of] that line [to the register plate’s center] equal to the radius of the [sphere encompassing the] glass [quarter-sphere], and from the endpoint of this line [which lies between the register plate’s center and the apparatus’s rim] he should draw a line perpendicular to the diameter of the register plate at whose endpoints lie the two perpendicular lines [drawn] on [the inner wall of] the apparatus’s rim, and he should extend this perpendicular to both sides. Then he should place the glass on the register plate’s surface and put its common section on the perpendicular just mentioned, and he should put the midpoint of that common section at the point from which the perpendicular [to the diameter] was drawn.

3.25 ◉ Et sic erit centrum vitri in superficie medii circuli, et linea que transit per centra duorum foraminum erit perpendicularis super superficiem vitri equalem, est enim equidistans diametro lamine, qui est perpendicularis super differentiam communem que est in vitro. Et centrum circuli medii erit in convexo vitri, nam linea que exit a centro circuli medii ad centrum lamine est equalis linee exeunti a centro vitri ad medium differentie communis, et utraque istarum linearum est perpendicularis super superficiem lamine. Ergo due linee sunt equales et equidistantes, et linea que copulat centrum vitri cum centro medii circuli est equalis linee que copulat centrum lamine et medium differentie communis que est in vitro. Hec autem linea posita fuit equalis semidiametro vitri; ergo linea equidistans ei est equalis semidiametro vitri. Centrum ergo medii circuli est in convexo vitri; linea ergo que transit per centra duorum foraminum, que transit per centrum medii circuli, tenet cum linea exeunti a centro vitri angulum equalem angulo qui est apud centrum lamine.

3.25 ◉ Hence, the center of [the sphere encompassing] the glass [quarter-sphere] will lie in the plane of the middle circle, and the line passing through the centers of the two holes will be perpendicular to the flat face of the glass because it is parallel to the diameter of the register plate, which is perpendicular to the common section on the glass. Moreover, the center of the middle circle will lie on the convex surface of the glass, for the line extending from the center of the middle circle to the center of the register plate is equal to the line dropped from the center of the [sphere encompassing the] glass [quarter-sphere] to the midpoint of the common section [i.e., one grain of barley], and both of those lines are perpendicular to the register plate’s surface. The two lines are thus equal and parallel, and the line joining the center of the [sphere encompassing the] glass [quarter-sphere] and the center of the middle circle is equal to the line joining the center of the register plate and the midpoint of the common section on the glass. This line, however, was constructed equal to the radius of the glass [quarter-sphere], so the line parallel to it is equal to the radius of the glass [quarter-sphere]. The center of the middle circle thus lies on the convex surface of the glass, so the line passing through the centers of the two holes, which [also] passes through the center of the middle circle, forms an angle with the line extending from the center of the [sphere encompassing the] glass [quarter-sphere] that is equal to the angle at the center of the register plate [formed by the respective parallels].

3.26 ◉ Extendantur ergo due linee in ymaginatione recte in utramque partem, scilicet diameter vitri predictus et linea que transit per centra duorum foraminum. Perveniet ergo ad circumferentiam medii circuli, sunt enim ambe in superficie medii circuli. Ergo due linee divident a circumferentia medii circuli ex utraque parte arcum cuius quantitas est decem partium, et extremitates linee que transit per centra duorum foraminum sunt note, altera enim earum est centrum foraminis, et altera punctus oppositus centro foraminis, et altera duarum extremitatum linee que transit per centrum vitri est extremitas arcus quam separaverat a circumferentia medii circuli, qui distat a centro foraminis decem partibus. Reliqua ergo extremitas linee que transit per centrum vitri distat a linea que transit per centra duorum foraminum decem partibus in parte opposita primo signo. Signemus igitur extremitatem huius diametri et extremitatem linee que transit per centra duorum foraminum, quamvis locus iste sit notus, quia est super lineam perpendicularem in hora instrumenti.

3.26 ◉ Accordingly the two lines, i.e., the aforementioned diameter in the glass [quarter-sphere] and the line passing through the centers of the two holes, may be imagined to extend straight on both sides. [Each] will therefore reach the periphery of the middle circle, for they both lie in the plane of the middle circle. As a result, the two lines will measure off an arc on both sides of the periphery of the middle circle that is ten degrees in size, and the endpoints of the line passing through the centers of the two holes are already determined, one of them being the center of the hole [in the apparatus’s rim], the other being the point opposite the center of the hole, and one of the two endpoints of the line passing through the center of the [sphere encompassing the] glass [quarter-sphere] is the endpoint of the arc that it had measured off on the periphery of the middle circle, and it lies ten degrees away from the center of the hole. The remaining endpoint of the line passing through the center of the [sphere encompassing the] glass [quarter-sphere] therefore lies ten degrees on the opposite side of the first line marked off, which passes through the centers of the two holes. So we should mark the endpoint of this diameter as well as the endpoint of the line passing through the centers of the two holes, even though this point is [already] determined because it lies on the line [drawn] perpendicular [to the register plate’s surface] on the [inner wall of the] apparatus’s rim.

3.27 ◉ Et intueatur experimentator signum, et inveniet illud remotius ab extremitate linee que transit per centrum vitri plus quam sit extremitas linee que transit per centra duorum foraminum. Hec ergo reflexio est ad partem contrariam perpendiculari a loco reflexionis, quia perpendicularis exiens a loco reflexionis est linea que transit per centrum vitri. Et arcus circumferentie medii circuli que est inter centrum lucis et extremitatem linee que transit per centra duorum foraminum est quantitas anguli reflexionis, angulus enim reflexionis est apud centrum circuli medii, lux enim extenditur super lineam transeuntem per centra duorum foraminum recte donec perveniat ad convexum vitri et spericum. Angulus ergo reflexionis erit apud centrum circuli medii, qui est super convexum vitri, et arcus qui est inter centrum lucis et extremitatem linee que transit per centra duorum foraminum est ille qui cordat angulum reflexionis qui est decem partium.

3.27 ◉ [Having inserted the apparatus into the vessel while allowing the sunlight to shine through the two holes, and having subsequently marked the point at which the center of the refracted light strikes the inner wall of the rim],⁑ the experimenter should examine the mark, and he will find that it lies farther away from the endpoint of the line passing through the center of the [sphere encompassing the] glass [quarter-sphere] than the endpoint of the line passing through the centers of the two holes does. This refraction is therefore in the direction away from the normal [dropped] from the point of refraction, since the normal dropped from the point of refraction is a line passing through the center of the [sphere encompassing the] glass [quarter-sphere]. Moreover, the arc on the periphery of the middle circle that lies between the center of the light and the endpoint of the line passing through the centers of the two holes is the size of the angle of refraction, for the angle of refraction lies [with its vertex] at the center of the middle circle because the light extends straight along the line passing through the centers of the two holes until it reaches the convex spherical surface of the glass. Consequently, the angle of refraction will lie [with its vertex] at the center of the middle circle, which lies on the convex surface of the glass, and the arc between the center of the light and the endpoint of the line passing through the centers of the two holes is the one subtending the angle of refraction that is [produced when the angle of inclination is] ten degrees.

3.28 ◉ Deinde oportet experimentatorem evellere vitrum et dividere incipiens a centro foraminis arcum cuius quantitas sit viginti, et procedat, ut prius. Et sic habebit quantitatem anguli reflexionis differentem a quantitate anguli qui est viginti. Et sic dividat alios arcus successive, et experiatur reflexiones eorum, sicut in primis, et habebit quantitates angulorum reflexionis qui sunt apud convexum vitri, et hee eedem quantitates sunt quantitates angulorum reflexionis lucis de aere ad vitrum, hoc enim declaratum est in predictis duabus experimentationibus. Sed reflexio de aere ad vitrum est ad partem perpendicularis, reflexio vero de vitro ad aerem est ad partem contrariam perpendiculari. Et si quis voluerit experiri vitrum et aquam etiam a convexo vitri et a superficie eius equali, habebit quantitates angulorum reflexionis de vitro ad aquam, aqua enim efficitur in loco aeris.

3.28 ◉ The experimenter should then remove the glass and measure off an arc 20° in size, starting from the center of the hole, and he should proceed as before. He will thus have an angle of refraction different in size from the size of the angle of [incidence of] 20°. He should measure off the other arcs accordingly in succession, and he should test their refractions, just as in the initial cases, and he will have the sizes of the angles of refraction at the convex surface of the glass, and these same angles will be the same size as the angles of refraction [for the passage] of light from air into glass, for this was shown in the two preceding experiments. But the refraction of air into glass is toward the normal, whereas the refraction from glass to air is away from the normal. And if one also wants to test [refraction between] glass and water at the convex surface of the glass [quarter-sphere] as well as at its flat face, he will have the angles of refraction for glass to water, for glass acts in place of the air.⁑

3.29 ◉ Et si quis voluerit experiri quantitates angulorum reflexionis apud concavum vitri, accipiat vitrum concavum concavitate columpnali in quantitate semicolumpne. Et sit figura universi vitri equidistantium superficierum, et longitudo eius sit maior diametro vitri sperici uno grano ordei, et latitudo eius sit similiter. Et sit spissitudo eius sicut duplum diametri foraminis quod est in hora instrumenti, et concavitas eius sit in uno suorum laterum. Et vas concavitatis columpnalis sit in superficie vitri quadrati, et longitudo columpne sit in spissitudine vitri. Et semidiameter basis columpne sit in quantitate semidiametri vitri sperici, et sint fines vitri linee recte verissime. Hoc autem instrumentum sic bene potest fieri super formam, ita quod forma fiat eadem doctrina predicta, et dissolvatur vitrum et infundatur super formam predictam.

3.29 ◉ If one wants to investigate the sizes of the angles of refraction [for light incident] on a concave glass surface, he should take a concave piece of glass whose hollow is semicylindrical in shape. The surfaces of the entire glass piece should be parallel, and it should be longer by one grain of barley than the radius of the glass [quarter-]sphere and wider by the same amount. It should be twice as thick as the diameter of the hole in the apparatus’s rim [i.e, two grains of barley], and its concavity should be on one of its sides [only]. The [semi]cylindrical hollow should lie in the plane of the square surface of the glass, and the length [along the axis of the semi]cylinder should lie along the height of the glass. The radius of the [semi]cylinder’s base should be the same as the radius of the glass [quarter-]sphere [used in earlier experiments], and the edges of the glass should form perfectly straight lines. This device can certainly be produced in a mold as long as that mold is formed according to the description just given and the glass is melted and poured into the mold just described.

3.30 ◉ Si ergo experimentator voluerit experiri reflexionem hoc instrumento, dividat de circumferentia medii circuli arcum cuius quantitas sit illa quam vult experiri, et extrahat ab extremitate arcus perpendicularem super superficiem lamine, ut predictum est. Et copulet extremitatem perpendicularis cum centro lamine linea recta quam protrahat in alteram partem, et dividat ex hac linea in altera parte, scilicet in qua sunt duo foramina, lineam equalem semidiametro basis columpne. Et extrahat ab extremitate eius perpendicularem super diametrum lamine, et protrahat illam in utramque partem. Deinde superponat vitrum lamine, et ponat dorsum concavitatis ex parte duorum foraminum. Et superponat duas superfluitates que superfluunt super diametrum columpne huic perpendiculari, et preservet quod sint distantie duarum extremitatum diametri basis concavitatis a puncto a quo exivit perpendicularis distantie equales. Erit ergo centrum basis concavitatis columpnalis super punctum a quo exivit perpendicularis et super punctum cuius distantia a centro lamine est in quantitate semidiametri basis concavitatis. Hoc situ preservato, applicet vitrum fixa applicatione.

3.30 ◉ Accordingly, if the experimenter wants to test refraction with this device, he should measure off the arc he wants to test on the periphery of the middle circle and draw a line [on the inner wall of the apparatus’s rim] from the endpoint of the arc that is perpendicular to the register plate’s surface, as was described earlier. He should then join the endpoint of the perpendicular and the center of the register plate with a straight line that he extends to the other side [of the register plate], and on one segment of that line, i.e, the one on the [other] side of the two holes, he should measure off a line [from the center of the register plate] equal to the radius of the base of the [semi]cylinder [hollowed out of the glass]. From its endpoint he should drop a perpendicular to the diameter on the register plate and extend it on both sides. Then he should put the glass piece on the register plate and pose the back of the hollow toward the two holes. He should apply the [surfaces of the] two end pieces extending beyond the diameter of the cylinder [across the base of the semicylindrical hollow] flush with this perpendicular and make certain that the two endpoints of the diameter at the hollow’s base are equidistant from the point at which the perpendicular was dropped. Consequently, the centerpoint [of the diameter] at the base of the [semi]cylindrical hollow will lie on the point from which the perpendicular was dropped and [thus] on the point whose distance from the centerpoint of the register plate is equal to the radius of the [semicylindrical] hollow’s base. When everything is set up this way, [the experimenter] should affix the glass firmly [to the register plate].

3.31 ◉ Et erit superficies medii circuli secans foramen columpnale et equidistans basi eius, nam basis eius in hac dispositione est in superficie lamine. Superficies ergo circuli medii facit in superficie columpnali concava semicirculum, et est diameter huius circuli medii equidistans diametro basis concavitatis. Erit ergo linea que egreditur a centro huius dimidii ad centrum basis concavitatis, que est perpendicularis super superficiem lamine, equalis perpendiculari exeunti a centro circuli medii perpendicularis super superficiem lamine. Et perpendicularis que exit a centro medii circuli ad centrum lamine est equalis semibasi columpne; ergo linea que exit a centro circuli medii ad centrum semicirculi qui sit in superficie columpne est equalis semidiametro huius dimidii. Centrum ergo circuli medii est in circumferentia semicirculi facti; est ergo in concavo columpne. Et quia terminus vitri superponitur linee perpendiculari super punctum lamine, erit diameter lamine perpendicularis super superficiem vitri equalem, nam superficies vitri equales sunt perpendiculares super se adinvicem. Erit ergo linea que transit per centra duorum foraminum perpendicularis super superficiem vitri equalem que est in parte convexa vitri, quia est equidistans diametro lamine, et hec superficies equalis vitri est ex parte foraminum.

3.31 ◉ And [so] the plane of the middle circle [on the apparatus’s rim] will cut the [semi]cylindrical hollow and will be parallel to its base because in this case its base lies on the surface of the register plate. Hence, the plane of the middle circle [on the apparatus’s rim] cuts a semicircle on the surface of the [semi]cylindrical hollow, and the diameter of this middle circle is parallel to the diameter [of the circle encompassing the semicircle] at the base of the hollow. The line dropped from the center of this semicircle to the center of the base of the hollow, which is perpendicular to the surface of the register plate, will thus be equal to the perpendicular dropped from the center of the middle circle [on the apparatus’s rim] to the surface of the register plate [i.e., both will be one grain of barley in length]. Moreover, the perpendicular dropped from the center of the semicircle [at the hollow’s base] to the center of the register plate is equal to the radius of the [semi]cylinder, so the line dropped from the center of the middle circle [on the apparatus’s rim] to the center of the semicircle on the surface of the [semi]cylinder is equal to the radius of this semicircle. As a result, the center of the middle circle lies on the circumference of the semicircle formed [by the middle circle cutting the hollow], and so it lies on the concave [surface of the semi]cylinder [forming the hollow]. And since the edge of the glass piece is flush with the line [drawn] perpendicular through a point on the register plate, the diameter of the register plate will be perpendicular to the flat face of the glass, for the flat faces of the glass are perpendicular to one another. Thus, the line passing through the centers of the two holes will be perpendicular to the flat face of the glass on the side of the glass’s convexity because it is parallel to the diameter on the register plate, and this flat surface of the glass faces the two holes.

3.32 ◉ In hoc situ ergo lux que extenditur per lineam que transit per centra duorum foraminum extenditur in corpore vitri recte donec perveniat ad concavum vitri. Et tunc reflectitur apud concavum vitri, cum non transeat per centrum circuli, qui est in concavo vitri, nec est perpendicularis super concavum vitri; ergo reflectitur in concavo vitri. Et hec linea occurret concavo vitri in uno puncto; ergo differentia communis huic linee et concavo vitri est centrum circuli medii. Ergo lux que extenditur per lineam que transit per centra duorum foraminum reflectitur apud centrum circuli medii; ergo arcus qui est inter centrum lucis et extremitatem linee que transit per centra duorum foraminum cordat angulum reflexionis.

3.32 ◉ In this situation, then, the light extending along the line that passes through the centers of the two holes extends straight through the body of the glass until it reaches the concave surface of the glass. It is then refracted at the concave surface of the glass because it cannot pass through the center of the [middle] circle, which lies on the concave surface of the glass, and it is not perpendicular to the concave surface of the glass, so it is refracted at the concave surface of the glass. This line, moreover, will intersect the concave surface of the glass at one point, so the common section of this line and the concave surface of the glass is the center of the middle circle [on the apparatus’s rim]. Accordingly, the light extending along the line that passes through the centers of the two holes is refracted at the center of the middle circle [on the apparatus’s rim], so the arc between the center of the [refracted] light [cast on the inner wall of the rim] and the endpoint of the line passing through the centers of the two holes subtends the angle of refraction.⁑

3.33 ◉ Hac igitur via posset quis experiri quantitates angulorum reflexionis qui fiunt in concavis vitri addendo in arcubus parum parum. Et hec reflexio est a vitro concavo ad aerem, et erunt anguli adquisiti hac reflexione idem illis qui fiunt ex aere ad vitrum in concavo vitri, declaratum est enim paulo ante quod angulus reflexionis a vitro ad aerem et ab aere ad vitrum est idem, cum angulus quem continet prima linea per quam extenditur lux et perpendicularis exiens a loco reflexionis sit idem angulus. Hac ergo via posset quis habere quantitates angulorum reflexionis de aere ad aquam, et de aere ad vitrum, et de vitro ad aerem, et de vitro ad aquam a superficie equali, et concava, et convexa.

3.33 ◉ This, then, is how one could investigate the sizes of the angles of refraction produced in concave glass surfaces by increasing the [size of the] arcs [on the rim] bit by bit [according to the increments chosen]. Moreover, this refraction occurs at the concave surface of the glass into air, and the angles obtained from this refraction will be the same as those produced [in refraction] from air to glass through the concave surface of the glass, for it was shown a bit earlier that the angle of refraction [when light passes] from glass to air and from air to glass is the same because the angle that the first line along which the light extends forms with the normal dropped from the point of refraction is the same angle [reciprocally in both cases]. Hence, in this way one could obtain the sizes of the angles of refraction from air to water, air to glass, glass to air, and glass to water at a plane, concave, and convex surface.⁑

3.34 ◉ Hiis ergo angulis experimentatis et proportionibus eorum notis, experimentator inveniet [1] quoslibet duos angulos quorum utrumque continet prima linea per quam extenditur lux et perpendicularis exiens a loco reflexionis super superficiem corporis diaffoni, inveniet dico in eisdem corporibus diaffonis. Et erunt duo anguli diversi, nam angulus reflexionis ab angulo maiori ex illis erit maior duobus angulis reflexionis ab angulo minori, et excessus anguli reflexionis super angulum reflexionis erit minor excessu anguli maioris quem continet prima linea cum perpendiculari super angulum minorem quem continet prima linea cum perpendiculari. [2] Et proportio anguli reflexionis ab angulo maiori ad angulum maiorem erit maior proportione anguli reflexionis ab angulo minore ad angulum minorem. [3] Et illud quod restat post angulum reflexionis de angulo maiori est maius illo quod remanet post angulum reflexionis de angulo minore, [4] et remotio anguli reflexionis cum lux exiverit de corpore subtiliori ad corpus grossius semper erit minor angulo quem continet linea per quam extenditur lux ad locum reflexionis cum perpendiculari exeunti a loco reflexionis. [5] Et si lux exiverit a corpore grossiori ad subtilius, tunc angulus reflexionis erit medietas coniuncti duorum angulorum. [6] Et si comparaveris angulos reflexionis qui sunt inter aliquod istorum corporum diaffonorum et aliud corpus grossius illis ad angulos reflexionis qui sunt inter illud corpus idem diaffonum subtilius et aliud corpus grossius primo grosso, invenies proportiones maiores angulorum reflexionis ad angulos quos continet prima linea et perpendicularis, qui sunt inter corpus subtilius et corpus grossius quod magis grossum est proportionibus angulorum reflexionis ad angulos quos continet prima linea cum perpendiculari, qui sunt inter idem corpus subtilius et corpus grossius quod minus est grossum, scilicet quoniam, si fuerint duo anguli equales quorum utrumlibet continet prima linea per quam extenditur lux et perpendicularis que exit a loco reflexionis, quorum alter est inter corpus subtilius et corpus grossius illo et alter inter illud idem corpus subtilius et corpus grossius primo grosso, tunc angulus reflexionis qui est in corpore grossiori erit maior angulo reflexionis qui est in corpore grossiori quod est minus grossum. [7] Et similiter, si reflexio fuerit a corpore grossiori ad corpus subtilius quod magis est subtile, maior erit angulo reflexionis qui est ab illo corpore eodem grossiori ad corpus subtilius quod est minus subtile. Hec ergo sunt omnia que pertinent ad qualitates reflexionis lucis in corporibus diaffonis.

3.34 ◉ Thus, I maintain that, when these angles are tested and their relationships taken into account, the experimenter will discover [that the following rules invariably apply].

◉Quartum capitulum

◉CHAPTER FOUR

Quoniam quicquid visus comprehendit ultra corpora diaffona que differunt in diaffonitate a corpore in quo est visus, cum fuerit obliquum a lineis perpendicularibus super superficiem corporis, est comprehensio secundum reflexionem

4.1 ◉ In predicto autem capitulo patuit quod lux transit de vitro ad aerem, et de aere ad vitrum, et de aere ad aquam, et cum transit de vitro ad aerem et ad aquam, constat quod transit de aqua ad aerem, aqua enim est subtilior vitro, cum fuerit clara. Et cum transit de aere ad vitrum, transibit de aqua ad vitrum, cum aqua sit grossior aere. Preterea patuit quod omnes luces accidentales et essentiales, et fortes et debiles transeunt per hec corpora diaffona hiis modis. Ergo omne corpus lucidum quacumque luce mittit lucem suam in omne corpus diaffonum, et si occurrerit aliud corpus diaffonum, transibit in alio corpore aut reflexe aut recte.

4.1 ◉ Now in the preceding chapter it was shown that light passes from glass to air, from air to glass, and from air to water, and since it passes from glass to air or to water, it is certainly the case that it passes from water to air, for, when it is clear, water is rarer than glass.⁑ And since it passes from air to glass, it will pass from water to glass because water is denser than air. It was further shown that all accidental or essential light, [whether] intense or faint, passes through these transparent bodies in these ways. Therefore, every body that is illuminated by any light whatever sends its light through every transparent body, and if it encounters another transparent body, it will pass into that other body either refracted or straight.

4.2 ◉ Et in primo declaratum est quod a quolibet puncto cuiuslibet corporis lucidi oritur lux per quamcumque lineam rectam que potest extendi ex illo puncto, ex quibus patet quod a quolibet puncto cuiuslibet corporis diaffoni contingentis aliquod corpus lucidum quacumque luce oritur lux per omnem lineam rectam que poterit extendi ex illo puncto, et transit in corpore diaffono tangenti illud punctum. Et si occurrerit aliud corpus diaffonum diverse diaffonitatis a diaffonitate corporis tangentis illud, transibit etiam in ipsum aut reflexe aut recte, sive primum corpus sit subtilius secundo, sive secundum sit subtilius primo.

4.2 ◉ In the first [book], moreover, it was demonstrated that light propagates from every point on any luminous body along any straight line that can be extended from that point, from which it obviously follows that light propagates from every point on any transparent body in contact with any body that is illuminated by any light whatever along every straight line that can be extended from that point, and it continues on through the transparent body in contact with that point. And if it encounters another transparent body, whose transparency is different from the transparency of the [original] body in contact with it, it will also pass through that body either refracted or straight, [no matter] whether the first body is rarer than the second, or whether the second is rarer than the first.

4.3 ◉ Et in primo etiam declaratum est quod ab omni corpore colorato lucido color oritur cum luce que est in ipso mixtus cum luce, et quod visus, cum comprehenderit lucem, comprehendit formam coloris mixtam sibi, ex quibus patet quod corpora colorata que sunt in aqua et ultra corpora diaffona que differunt in diaffonitate a diaffonitate aeris, cum in eis fuerit lux essentialis aut accidentalis, fortis aut debilis, tunc lux que est in eis oritur a quolibet puncto cum forma coloris que est in illo puncto, et transit lux mixta cum colore in corpore aque et in omni corpore diaffono contingente ipsa, et extenditur lux cum forma coloris in corpore aque et in omni corpore diaffono per lineas rectas donec perveniat ad superficiem aque aut illius corporis diaffoni.

4.3 ◉ It was also demonstrated in the first [book] that from every illuminated, colored body, color mingled with light is propagated along with the light in it, and [it was demonstrated] that, when it perceives the light, the visual faculty perceives the form of the color mingled with it. From these points it is evident that, if there is essential or accidental light, [whether] intense or faint, in colored bodies lying in water or beyond transparent bodies that differ in transparency from the transparency of air, light propagates from every point on them along with the form of the color at that point, and the light passes mingled with color through the body of the water as well as through every transparent body in contact with it, and the light coupled with the form of the color extends through the body of the water and through every [other] transparent body along straight lines until it reaches the surface of the water or [the surface] of that [other] transparent body.

4.4 ◉ Et cum fuerit aer aut aliud corpus diaffonum tangens aquam, tunc in illud corpus diaffonum transibit lux cum forma mixta sibi in aere aut in alio corpore diaffono per lineas rectas, et hee linee secunde in maiori parte secabunt primas lineas per quas extendebantur, et quedam earum erunt in rectitudine primarum linearum. Et omnia corpora que sunt in aqua et ultra diaffona corpora que differunt a diaffonitate aeris, cum fuerint in loco lucido, scilicet cum lux orta fuerit super aquam in qua sunt, tunc lux perveniet ad ipsa, manifestum est enim quod omnis lux transit in omne corpus diaffonum.

4.4 ◉ Furthermore, if air or another transparent body is in contact with the water, the light will pass through that transparent body with the form [of color] mingled with it in the air or in the other transparent body along straight lines, and for the most part these second [straight] lines will cut [i.e., form angles with] the first lines along which [the light and color-forms] extended [through the first body], but some of them will be right in line with the first lines. Also, when any objects lie in water or beyond [other] transparent bodies that differ in transparency from the air, and when they lie in a lighted spot, i.e., when light is propagated into the water in which they lie, the light will reach them, for it is obvious that all light passes through every transparent body.

4.5 ◉ Ergo omne corpus in aqua existens aut in alio corpore diaffono, cum super aquam illam aut super illud corpus diaffonum orta fuerit lux, illud corpus erit lucidum, et a quolibet puncto ipsius orietur forma lucis que est in ipso cum forma coloris, et extenditur in universo illius aque aut illius corporis diaffoni per omnem lineam rectam que poterit extendi ab ipso puncto donec perveniat lux cum forma coloris qui est in illo puncto ad superficiem aque aut ad superficiem illius corporis diaffoni.

4.5 ◉ Therefore, when light shines on water or on [another] transparent body, every body lying in the water or in the other transparent body will be illuminated, and from any point on it the form of light along with the form of the color in it will be propagated, and it extends throughout that water or that [other] transparent body along every straight line that can be projected from that point until the light at that point reaches the surface of the water or the surface of that [other] transparent body along with the form of the color.

4.6 ◉ Sed non potest extrahi ab eodem puncto alicuius superficiei ad eandem superficiem linea perpendicularis nisi una. Ergo a quolibet puncto cuiuslibet corporis colorati lucidi existentis in corpore diaffono oritur forma lucis cum forma coloris in universo corporis diaffoni in quo existit secundum lineas rectas, et pervenit forma ad universum oppositum de superficie corporis diaffoni. Et una illarum linearum erit perpendicularis super superficiem corporis diaffoni et super superficiem continuam cum superficie corporis diaffoni; relique autem linee erunt oblique super superficiem corporis diaffoni.

4.6 ◉ But only one perpendicular line can be dropped from a given point on any surface to that same surface. From any point on any luminous colored body lying in a transparent body, then, the form of light along with the form of color is propagated along straight lines throughout the transparent body in which it lies, and the form [of that light and color] reaches the entire facing surface of the transparent body. But [only] one of those lines [along which the form extends] will be perpendicular to the surface of the transparent body as well as to the surface [of another transparent body] in contact with the surface of the transparent body; the rest of the lines will be oblique to the surface of the transparent body.

4.7 ◉ Sed in precedenti capitulo declaratum est quoniam lux, cum extenditur in corpore diaffono et occurrerit alii corpori diaffono diverso a diaffonitate primi corporis, et linea per quam extensa est lux in primo corpore fuerit perpendicularis super superficiem secundi corporis, tunc lux extendetur in rectitudine eius in secundo corpore. Et si linea per quam extenditur lux fuerit obliqua super superficiem secundi corporis, tunc lux reflectitur. Et cuiuslibet puncti cuiuslibet corporis colorati et lucidi existentis in corpore diaffono forma lucis et coloris extenditur in universo corpore diaffono, et pervenit opposite ad superficiem corporis diaffoni.

4.7 ◉ In the previous chapter, however, it was shown that, when light extends through a transparent body and encounters another transparent body whose transparency is different from the transparency of the first body, and when the line along which the light extends through the first body is perpendicular to the surface of the second body, the light will extend straight through the second body. On the other hand, if the line along which the light extends [through the first body] is oblique to the surface of the second body, the light is refracted. And [so] from any point on any colored and illuminated body lying in a transparent body the form of the light and color [at that point] extends throughout the transparent body, and it reaches the facing surface of the transparent body.

4.8 ◉ Et si fuerit aliud corpus diaffonum contingens illud corpus diaffonum, et fuerit alterius diaffonitatis, tunc forma que pervenit ad superficiem illius corporis diaffoni transit in corpus ipsum contingens, et omnes erunt reflexe, preter quam forma que est in perpendiculari, extenditur enim secundum rectitudinem in corpore contingente. Et si forte perpendicularis ceciderit super punctum superficiei continue cum superficie corporis qui non est in ipso corpore diaffono, tunc illa forma delebitur, et tunc omnes forme que transeunt in corpus contingens erunt reflexe.

4.8 ◉ If, moreover, another transparent body is in contact with that transparent body, and if it is of a different transparency, then the form that reaches the surface of that [first] transparent body passes into the body in contact with it, and all [such forms], except the form that passes along the perpendicular, will be refracted because [that particular form] extends straight into the [transparent] body in contact [with the original transparent body]. But if by chance the [form passing along the] perpendicular reaches a point on the surface in contact with the [original transparent] body’s surface that is not on that transparent body, that form will be blocked, and so all the forms that pass into the [transparent] body in contact [with the original transparent body] will be refracted.⁑

4.9 ◉ Ergo forme omnium visibilium que sunt in aqua, et in celo, et in omnibus corporibus diaffonis contingentibus aerem que differunt a diaffonitate aeris, extenduntur in universo aere opposito secundum lineas rectas, et ille linee que fuerint ex istis lineis declinate per quas extenduntur forme super superficiem aeris contingentis superficiem corporis diaffoni, forma que extenditur per illas erit reflexa, et que fuerint ex illis perpendiculares super superficiem aeris contingentis superficiem corporis diaffoni, forma que extenditur per illas erit secundum rectitudinem ipsarum.

4.9 ◉ Consequently, the forms of all visible objects that lie in water, in the heavens, and in all [other] transparent bodies that are in contact with the air and that differ in transparency from the air extend throughout the facing air along straight lines, and a form extending along oblique lines among [all] the lines according to which the forms pass to the surface of the air in contact with the surface of the transparent body will be refracted, whereas a form extending to the surface of the air in contact with the surface of the transparent body along perpendicular lines among all these lines will be [propagated] straight through [that surface].

4.10 ◉ Et cum iam declaratum sit quod a quolibet puncto cuiuslibet corporis colorati et lucidi extenditur forma lucis et coloris in universo corpore diaffono, et pervenit ad superficiem eius, et reflectitur a superficie eius, ergo forma que extenditur ab uno puncto ad superficiem corporis diaffoni erit continua coniuncta. Et cum forma fuerit continua, et superficies corporis diaffoni fuerit continua coniuncta, et forma fuerit reflexa in alio corpore diaffono, tunc reflectetur continua. Et cum forma reflexa fuerit continua et occurrerit corpus densum, tunc forma perveniet ad illud corpus diaffonum, et sic locus corporis diaffoni per quem extenditur forma puncti quod est in primo corpore que reflectitur a superficie primi corporis ad illum locum, cum fuerit lucidus coloratus, mittit formam lucis et coloris a quolibet puncto ipsius per omnem lineam rectam que poterit extendi ex illo puncto.

4.10 ◉ And since it has just been shown that the form of light and color extends throughout [any] transparent body from any point on any colored and luminous body [in it] and [that it thereby] reaches its surface and is refracted at its surface, the form that extends from a single point to the surface of the transparent body will be continuous and coherent. And since the form is continuous, and since the surface of the transparent body is continuous and coherent, then, when the form is refracted into another transparent body, it will be refracted [as a] continuous [whole]. And since the refracted form is continuous and encounters a dense body [i.e., the second transparent medium], the form will reach that transparent body, and so the spot on the [denser] transparent body through which the form of the point in the first body extends is the spot at which the form is refracted from the first transparent body, when [that spot] is illuminated and colored, and it sends the form of [its] light and color to any point in that second body along every straight line that can be extended from that point.⁑

4.11 ◉ Accidit ergo ex hoc quod sint linee reflexe ad illum locum ex lineis per quas extenditur forma illius loci. Et iam extendebatur forma cuiuslibet puncti illius loci per unam illarum linearum reflexarum. Forma ergo illius loci ex corpore denso colorato lucido erit in loco ex superficie corporis diaffoni apud quem reflectitur forma unius puncti extensi ad illum locum superficiei corporis diaffoni que reflectitur ad eundum locum corporis densi, ex quo sequitur quod forma loci corporis densi que extenditur ad illum locum corporis diaffoni reflectitur super easdem lineas extensas ab uno puncto ad illum locum corporis diaffoni.

4.11 ◉ It therefore follows from this that among the lines along which the form of that spot extends [through the second medium] there are refracted ones at that spot, and the form of every point on that spot has already extended along one of those refracted lines. The form of that spot on the dense body, which is luminous and colored, will therefore lie at that spot on the surface of the transparent body where the form of a single point extending to that spot on the surface of the transparent body is refracted to that same spot on the surface of the dense body, from which it follows that the form of the spot on the dense body that extends to that point on the transparent body is refracted along the same lines extending from the single point to that spot on the transparent body.⁑

4.12 ◉ Et cum forma loci corporis diaffoni fuerit reflexa super illas easdem lineas, tunc perveniet ad illud idem punctum, ex quo declaratur quod, si ymaginatus fueris piramidem extensam a quolibet puncto aeris secundum lineas rectas, et piramis fuerit coniuncta continua, et pervenerit illa piramis ad superficiem corporis diaffoni diverse diaffonitatis ab aere, et ymaginatus fueris omnem lineam rectam que possit extendi ex illa piramide reflecti apud superficiem corporis diaffoni in loco quem exigit eius declinatio, et si aliqua fuerit perpendicularis, extendetur recte. Tunc efficitur ex hoc corpus continuum reflexum in corpore diaffono quod differt a diaffonitate aeris. Et cum hoc corpus reflexum pervenerit ad corpus densum, tunc illud corpus densum, si fuerit coloratum et lucidum, mittit formam lucis et coloris que sunt in ipso in hoc corpore reflexo ymaginato per quamlibet lineam rectam que poterit extendi in hoc corpore reflexo a linea extensa in corpore piramidis a puncto qui est in aere, nam omne corpus coloratum lucidum proprie mittit formam suam a quolibet puncto ipsius per omnem lineam rectam que poterit extendi ab illo puncto.

4.12 ◉ And when the form at the spot on the transparent body is refracted along those same lines, it will reach the same point, from which it is clear that, if you imagine a cone extending from any point in the air along straight lines, and if the cone is continuous and coherent, and if that cone reaches the surface of a transparent body of different transparency than the air, and if you imagine that every straight line that can be extended within that cone is refracted at the surface of the transparent body at a point that requires it to be deflected, and if one of them is perpendicular, it will continue straight on. This [cone] thus forms a continuous body refracted in the transparent body that differs in transparency from the air. And when this refracted body reaches an opaque body, then, if that opaque body is colored and luminous, it sends the form of the light and color that are in it through this imaginary refracted body along every straight line that can be extended in this refracted body by means of a line extending in the body of the cone to a point in the air, for every colored and illuminated body by its nature sends its form from every point on it along every straight line that can be extended from that point.

4.13 ◉ Erit ergo forma puncti illius loci corporis densi extensa per quamlibet linearum reflexarum ad illum locum corporis densi. Perveniet ergo forma illius a corpore denso colorato lucido ad locum superficiei corporis diaffoni in quem reflectuntur ille linee, et cum pervenerit forma ad illum locum superficiei corporis diaffoni, necessario reflectetur per easdem lineas extensas ad illum locum ab uno puncto qui est in aere. Et cum centrum visus fuerit in illo puncto qui est in aere, tunc forma que est forma loci colorati corporis densi quod est in corpore diaffono quod differt a diaffonitate aeris et est de numero illarum linearum per quas extenditur forma ad centrum visus, tunc forma que extenditur per illam lineam pervenit ad centrum visus recte. Forme autem que extenduntur per omnes alias lineas que constituunt piramidem extensam a centro visus erunt reflexe, non recte.

4.13 ◉ Accordingly, the form of the point at that spot on the opaque body will extend along any one of the refracted lines at that point on the opaque body. So the form of that point on the opaque, illuminated, colored body will reach a spot on the surface of the transparent body at which those lines are refracted, and when the form reaches that spot on the surface of the transparent body, it will necessarily be refracted along the same lines extending to that spot from one point in the air. And if a center of sight lies at that point in the air, the form of the spot on the opaque, colored body that lies in the transparent body that differs in transparency from the air and that extends [orthogonally] along one of those lines according to which the form extends to the center of sight, the form extending along that line reaches straight to the center of sight. On the other hand, the forms that extend along all the other lines forming the cone that extends from the center of sight will be refracted, not straight.⁑

4.14 ◉ Et in primo tractatu declaratum est quod aer recipit formam visibilium et reddit illam omni corpori opposito et quod aer deferens formam, cum tetigerit visum, transibit forma que est in ipso corpus visus, et sic visus comprehendit visibilia que aer reddit visui. Ex omnibus ergo istis patet quod forma omnis corporis colorati lucidi existentis in corpore diaffono diverse diaffonitatis aeris extenditur in corpore diaffono in quo existit, et reflectitur in aere, et extenditur in aere secundum lineas rectas, et quod quedam linearum rectarum per quas forma reflectitur in aere coniunguntur apud idem punctum aeris. Et cum centrum visus fuerit apud illum punctum, tunc visus comprehendet illud visum secundum reflexionem, et si aliquid ipsius comprehenditur recte, non erit nisi unum punctum tantum. Hoc ergo modo comprehendit visus res que sunt in aqua, et in celo, et omnia visibilia que sunt ultra corpora diaffona que differunt a diaffonitate aeris.

4.14 ◉ In addition, it was shown in the first book that air receives the form of visible objects and transmits it to every facing body and that, when the air transmitting that form is in contact with the eye, the form that is in it will pass into the body of the eye, and so the visual faculty perceives the [forms of the] visible objects that the air transmits to the eye. From all of these things, therefore, it is evident that the form of every colored and illuminated body lying in a body whose transparency is different from the transparency of the air extends through the transparent body in which it lies, is [then] refracted in the air, and [subsequently] continues through the air in straight lines, and [it is also evident] that some of the straight lines along which the form is refracted in the air converge at the same point in the air. And [so] when the center of sight is at that point, the visual faculty will perceive that visible object according to refraction, and if any of [the points on the form] is perceived by direct vision, it will only be a single point. This, then, is how the visual faculty perceives things that lie in water and in the heavens, and [it is how it perceives] every visible object that lies behind transparent bodies that differ in transparency from the air.

4.15 ◉ Quoniam autem hoc verum sit sic poterit experimentari. Accipiat ergo experimentator predictum instrumentum, et ponat ipsum in vase, et ponat vas in loco lucido quacumque luce ita quod lux perveniat ad interius vasis. Et infundat intra vas aquam quousque perveniat ad centrum lamine. Deinde diminuat foramina cum cera ita quod non remaneat de foramine nisi modicum in medio eorum, et mittat in duobus foraminibus unum calamum ita quod spatium quod est inter duo foramina sit determinatum. Deinde moveat instrumentum donec diameter lamine super cuius extremitates sunt due linee perpendiculares in hora instrumenti sit perpendicularis super superficiem aque. Deinde accipiat stilum subtilem album, et mittat eum in vas, et eius extremitatem ponat in puncto circuli medii, qui est differentia communis circumferentie circuli medii et linee perpendiculari in hora instrumenti qui est extremitas diametri circuli medii qui transit per centra duorum foraminum. Deinde ponat experimentator alterum visum super superius foramen, et claudat reliquum, et intueatur horam instrumenti quod est intra aquam, tunc enim videbit extremitatem stili.

4.15 ◉ That this is in fact the case can be determined experimentally. Accordingly, the experimenter should take the apparatus discussed earlier, insert it in the vessel, and set the vessel up in a location illuminated by some light so that the light reaches inside the vessel. He should then pour water into the vessel until it reaches the center of the register plate. Next he should constrict the holes with wax so that only a narrow opening at the center of the holes is left, and he should push a [hollow] reed through the two holes so that the space between the two holes is determined [i.e., so that the axial line of sight through the hollow reed coincides with the line passing through the centers of the two holes]. He should then adjust the apparatus until the diameter on the register plate at whose endpoints the two perpendicular lines [drawn] on the [inner wall of the] apparatus’s rim lie is perpendicular to the surface of the water. He should then take a thin white stylus, put it into the vessel, and place its endpoint at the point on the middle circle [inscribed on the inner wall of the rim] that constitutes the common section of the middle circle’s circumference and the perpendicular line on [the inner wall of the] apparatus’s rim at the endpoint of the diameter of the middle circle, which passes through the centers of the two holes. After that the experimenter should place one of his eyes at the top hole, close the other, and look at the rim of the apparatus under water, for he will then see the endpoint of the stylus.

4.16 ◉ Et declarabitur ergo ex hac experimentatione quod comprehensio eius ad extremitatem stili est secundum rectitudinem perpendicularis egredientis ab extremitate stili super superficiem aque, nam linea que transit per centra duorum foraminum in qua est centrum visus et extremitas stili, ex cuius verticatione comprehendit visus extremitatem stili, est perpendicularis super superficiem aque. In primo autem patuit quod nichil comprehendit visus nisi secundum rectitudinem linearum que extenduntur per centrum visus. Visus ergo comprehendit extremitatem stili a verticatione linee que transit per centra duorum foraminum, et hec linea extenditur ad extremitatem stili recte, et est perpendicularis super superficiem aque.

4.16 ◉ It will therefore be manifest from this experiment that he perceives the endpoint of the stylus along the normal dropped from the endpoint of the stylus to the water’s surface, for the line passing through the centers of the two holes, upon which the center of sight and the endpoint of the stylus lie and along which the center of sight perceives the stylus’s endpoint, is perpendicular to the water’s surface. Moreover, in the first [book] it was shown that the visual faculty perceives nothing unless [it does so] along straight lines that extend through the center of sight. Therefore, the visual faculty perceives the endpoint of the stylus along the straight line that passes through the centers of the two holes, and this line extends straight to the endpoint of the stylus and is perpendicular to the water’s surface.

4.17 ◉ Deinde oportet experimentatorem declinare instrumentum donec linea que transit per centra duorum foraminum sit obliqua super superficiem aque. Et mittat stilum in aqua, et ponat extremitatem eius super primum punctum, scilicet super extremitatem diametri circuli medii qui transit per centra duorum foraminum, et ponat visum suum super superius foramen, et intueatur horam instrumenti que est intra aquam. Tunc enim non videbit extremitatem stili. Et deinde moveat stilum ad partem contrariam illi in qua est visus, et tunc non videbit extremitatem stili. Deinde moveat stilum ad partem in qua est visus, et moveat extremitatem stili per circumferentiam circuli medii suaviter et molliter, et intueatur horam instrumenti, tunc enim videbit extremitatem stili.

4.17 ◉ Now the experimenter should incline the apparatus [by turning it on its axis] until the line passing through the centers of the two holes is oblique to the water’s surface. He should then put the stylus in the water, place its endpoint at the first point, i.e., at the endpoint of the middle circle’s diameter that passes through the centers of the two holes, put his eye at the top hole, and look at the rim of the apparatus under water. In that case, he will in fact not see the endpoint of the stylus. Then he should move the stylus in the direction away from that in which the eye lies, and he will therefore not see the endpoint of the stylus. Subsequently he should move the stylus to the same side as the eye and shift the stylus’s endpoint on the circumference of the middle circle smoothly and gently, and he should [continue to] look at the rim of the apparatus [under water], in which case he will [eventually] see the endpoint of the stylus.⁑

4.18 ◉ Et tunc figat extremitatem stili in suo loco; deinde precipiat alii ut mittat in vas perpendicularem neque grossam nec gracilem, et ponat illam apud superficiem aque in oppositione secundi foraminis ut sit apud centrum circuli medii. Et intueatur experimentator interius vasis, et tunc non videbit extremitatem stili. Deinde precipiat auferre lignum, et tunc videbit extremitatem stili. Deinde figat extremitatem stili in suo loco, et elevet visum suum a foramina, et auferat instrumentum suum a vase, existente extremitate stili in suo loco, et intueatur locum in quo est extremitas stili. Tunc enim videbit inter ipsum et diametrum circuli medii distantiam sensibilem. Et si miserit regulam subtilem in aquam in hora experimentationis, et acumen eius fecerit transire per centrum lamine, et signaverit locum circuli medii qui est apud extremitatem regule signo, et abstulerit instrumentum, et aspexerit locum extremitatis stili, videbit locum extremitatis stili medium inter locum extremitatis regule et diametrum circuli medii.

4.18 ◉ Accordingly, he should fix the endpoint of the stylus in place [and] then direct someone else to put a [wooden rod of some kind or a needle] that is neither [too] thick nor [too] thin upright⁑ into the vessel and to place it at the water’s surface facing the second hole so that it lies at the center of the middle circle. The experimenter should then look into the vessel, and he will not see the endpoint of the stylus. Next he should direct [the other person] to remove the rod, and in that case he will see the endpoint of the stylus.⁑ Afterwards he should fix the endpoint of the stylus in place, raise his eye from the hole, remove his apparatus from the vessel, while leaving the endpoint of the stylus in its place, and look at where the endpoint of the stylus lies. He will then see that there is a noticeable discrepancy between that point and the [endpoint of the] diameter of the middle circle. And if at the time of the experiment he places the thin ruler on the rim under water and has its pointed edge pass through the center of the register plate, and if he marks the point on the middle circle at the endpoint of the ruler, removes the apparatus, and looks at where the endpoint of the stylus lies, he will see that the point at which the endpoint of the stylus lies is between the point marking the endpoint of the ruler and the diameter of the middle circle.⁑

4.19 ◉ Deinde oportet eum auferre instrumentum, et infundere aquam in vas, et applicare vitrum lamine, et ponere superficiem vitri equalem ex parte foraminum, et ponere differentiam communem que est in ipso super lineam secantem diametrum lamine perpendiculariter. Sic ergo erit linea que transit per centra duorum foraminum perpendicularis super superficiem vitri equalem et super superficiem eius convexam. Deinde ponat instrumentum in aqua, et mittat stilum in vas, et ponat extremitatem stili super extremitatem diametri circuli medii, et ponat visum suum super superius foramen, et intueatur horam instrumenti. Tunc videbit extremitatem stili, et si moverit extremitatem stili, et extraxerit illam a puncto quod est extremitas diametri medii circuli, non videbit extremitatem stili, ex quo patet quod extremitatem stili comprehendit recte, nam duo centra foraminum et extremitas diametri circuli medii sunt in eadem linea recta, et experimentator non comprehendit extremitatem stili in hoc statu, cum extremitas stili non fuerit super extremitatem diametri. Et si evulserit vitrum, et posuerit ipsum econtra, scilicet quod ponat convexum vitri ex parte duorum foraminum et differentiam eius communem super primum locum, et expertus fuerit extremitatem stili, videbit illam, cum fuerit in extremitate diametri circuli medii. Ideo in hoc situ etiam linea que transit per centra duorum foraminum ex cuius verticatione comprehendit visus extremitatem stili erit perpendicularis super superficiem vitri equalem et super superficiem eius convexam.

4.19 ◉ Next [the experimenter] should remove the apparatus and pour water into the vessel, and he should affix the glass [quarter-sphere] to the register, posing the flat face of the glass toward the holes and placing the common section [formed by the two flat faces] on it flush with the line cutting the diameter of the register plate orthogonally. Hence, the line passing through the centers of the two holes will be perpendicular to the flat face of the glass [quarter-sphere] as well as to its convex surface.⁑ [The experimenter] should then place the apparatus in the water, put the stylus in the vessel, pose the endpoint of the stylus at the endpoint of the middle circle’s diameter, put his eye up to the top hole, and look at the apparatus’s rim [under water]. Accordingly, he will see the stylus’s endpoint, and if he moves the stylus endpoint and shifts it from the point that marks the endpoint of the middle circle’s diameter, he will not see the stylus’s endpoint, from which it follows that he perceives the stylus’s endpoint straight on, for the two centers of the holes and the endpoint of the middle circle’s diameter lie on the same straight line, and in this situation, when the stylus’s endpoint does not lie on the diameter’s endpoint, the experimenter does not perceive the stylus’s endpoint. Furthermore, if he removes the glass [quarter-sphere] and places it in the opposite way, i.e., if he poses the convex surface of the glass toward the two holes with its common section [left] where it was in the first case, and if he does the test with the stylus’s endpoint, he will see it when it lies at the endpoint of the middle circle’s diameter. So in this case as well, the line passing through the centers of the two holes along which the visual faculty perceives the stylus’s endpoint will be perpendicular to the flat face of the glass [quarter-sphere] as well as to its convex surface.⁑

4.20 ◉ Deinde oportet experimentatorem evellere vitrum et extrahere a centro lamine lineam rectam in superficie lamine que contineat cum diametro lamine super cuius extremitates sunt due linee perpendiculares in hora instrumenti angulum obtusum, et extrahat illam donec perveniat ad horam instrumenti. Deinde extrahat a centro lamine lineam in superficie lamine que continet cum prima linea angulum rectum, et protrahat illam in utramque partem. Tunc hec linea continet cum diametro lamine angulum acutum, et diameter lamine erit obliquus super hanc lineam. Deinde superponat vitrum lamine, et ponat differentiam eius communem super lineam quam ultimo signavit in superficie lamine, et ponat superficiem vitri equalem ex parte duorum foraminum, et ponat medium differentie communis super centrum lamine.

4.20 ◉ Now the experimenter should remove the glass and draw a straight line on the surface of the register plate from the center of the register plate so as to form an obtuse angle with the diameter on the register plate at whose endpoints the two perpendicular lines [drawn] on the [inner wall of the] apparatus’s rim lie, and he should extend it until it reaches the apparatus’s rim. From the center of the register plate he should then draw a line on the register plate’s surface that forms a right angle with the first line, and he should extend it on both sides. This line will therefore form an acute angle with the diameter on the register plate, and the diameter on the register plate will be oblique to this line. Next he should apply the glass [quarter-sphere] to the register plate, put its common section flush with the line he drew last on the register plate, pose the flat face of the glass [quarter-sphere] toward the two holes, and place the midpoint of the common section at the register plate’s center.

4.21 ◉ Sic ergo erit centrum vitri super centrum circuli medii, ut prius declaratum est, et linea que transit per centra duorum foraminum transibit per centrum vitri. Et hec linea erit obliqua super superficiem vitri equalem, nam diameter lamine illi equidistans est obliquus super differentiam communem que est in vitro. Et hec linea etiam erit perpendicularis super superficem vitri convexam, quia transit per centrum eius.

4.21 ◉ Accordingly, the center of the [sphere encompassing the] glass [quarter-sphere] will lie at the center of the middle circle, as was shown before, and the line passing through the centers of the two holes will pass through the center of the [sphere encompassing the] glass [quarter-sphere]. This line will be oblique to the flat face of the glass, for the diameter on the register plate parallel to it is oblique to the common section on the glass. And yet this line will be perpendicular to the convex surface of the glass because it passes through its center.

4.22 ◉ Deinde extrahat experimentator ab extremitate linee quam primo signavit in lamina lineam perpendicularem in hora instrumenti, et ducat illam ad circumferentiam circuli medii, et hee linee sint nigre. Erit ergo punctus ad quem pervenit, cum ab illo extracta fuerit linea ad centrum circuli medii, quod est centrum vitri, perpendicularis super superficiem vitri equalem et superficiem vitri spericam. Super superficiem autem vitri equalem est perpendicularis, quia est equidistans prime linee signate in lamina super differentiam communem que est in vitro. Super spericam vero, quia transit per centrum eius. Punctus ergo ad quem pervenit linea extracta in hora instrumenti qui est super circumferentiam circuli medii est casus in quo cadit perpendicularis exiens a centro vitri super superficiem vitri planam.

4.22 ◉ The experimenter should next draw a perpendicular line on the [inner wall of the] apparatus’s rim from the endpoint of the line that he first drew on the register plate and extend it to the circumference of the middle circle, and these lines should be black. Accordingly, if a line is extended from the point [that this perpendicular on the rim] reaches [on the middle circle] to the center of the middle circle, which is the center of the [sphere encompassing the] glass [quarter-sphere], it will be normal to the flat face of the glass as well as to the [convex] spherical surface of the glass. On the one hand, it is normal to the flat face of the glass because it is parallel to the first line drawn on the register plate to the common section on the glass. On the other hand, [it is normal] to the [convex] spherical surface because it passes through its center [of curvature]. Consequently, the point at which the [perpendicular] line drawn on the [inner wall of the] apparatus’s rim intersects the circumference of the middle circle is where the normal dropped from the center of the [sphere encompassing the] glass [quarter-sphere] falls on the flat face of the glass.

4.23 ◉ Deinde oportet experimentatorem ponere instrumentum in vas et ponere extremitatem stili in puncto quod est extremitas diametri circuli medii, et ponat suum visum super superius foramen, et intueatur horam instrumenti. Tunc non videbit extremitatem stili. Deinde moveat stilum ad partem contrariam illi in qua est casus perpendicularis, et tunc etiam non videbit extremitatem stili. Deinde moveat stilum ad illam in qua est casus perpendicularis et per circumferentiam circuli medii, tunc enim, si motus fuerit suavis, videbit extremitatem stili. Et tunc figat extremitatem stili in suo loco in quo apparuit. Deinde precipiat alicui cooperire centrum vitri tenui subtili ligno, et tunc non videbit extremitatem stili, et si abstulerit coopertorium, videbit ipsum.

4.23 ◉ The experimenter should now insert the apparatus in the vessel and place the endpoint of the stylus at the endpoint of the middle circle’s diameter, and he should put his eye up to the top hole and look at the apparatus’s rim. In that case he will not see the stylus’s endpoint. He should then move the stylus on the opposite side of where the normal is dropped, and in that case he will still not see the stylus’s endpoint. Finally, he should move the stylus along the circumference of the middle circle on the side where the normal is dropped, for in that case, if the adjustment is smooth, he will [eventually] see the stylus’s endpoint. Accordingly, he should fix the endpoint of the stylus in the place where it appeared. Then he should direct someone to block the center of the [sphere encompassing the] glass [quarter-sphere] with a fine, thin rod, and in that case he will not see the endpoint of the stylus, whereas if he removes the block, he will see it.⁑

4.24 ◉ Ex hac ergo experimentatione patet quod, cum visus comprehendit extremitatem stili recte, est secundum reflexionem, et quod reflexio est a centro vitri, et quod forma reflexa est in superficie circuli medii, qui est perpendicularis super superficiem vitri equalem apud quam fit reflexio perpendicularis, ut prius declaratum est. Et si experimentator aspexerit locum extremitatis stili, inveniet ipsum inter casum perpendicularis et extremitatem diametri circuli medii, qui transit per centra duorum foraminum. Linea ergo que exit ab extremitate stili ad centrum vitri, cum extensa fuerit in illa recte in aere, et extensa fuerit cum illa in aere perpendicularis exiens a centro vitri super superficiem vitri equalem, erit media inter perpendicularem et lineam que transit per centra duorum foraminum. Et forma extremitatis stili, que extensa est ab extremitate stili ad centrum vitri, extensa est super hanc lineam et extensa est in rectitudine eius ad centrum vitri, hec enim linea est perpendicularis super superficiem vitri spericam, que est ex parte extremitatis.

4.24 ◉ From this experiment it is therefore evident that, when the visual faculty perceives the endpoint of the stylus [as if it lay] along a straight line [of sight], it is [perceived] according to refraction, and [it is also evident] that the refraction occurs at the center of the [sphere encompassing the] glass [quarter-sphere] and that the form is refracted in the plane of the middle circle, which is perpendicular to the flat face of the glass, where the refraction occurs [away] from the normal, as was previously demonstrated. And if the experimenter looks at the location of the stylus’s endpoint, he will find it between where the normal falls [on the apparatus’s rim] and the endpoint of the middle circle’s diameter, which passes through the centers of the two holes. Therefore, since it extends straight along that line through the air, and since the normal dropped from the center of the [sphere encompassing the] glass [quarter-sphere] on the flat face of the glass extends through the air along with it, the line dropped from the endpoint of the stylus to the center of the [sphere encompassing the] glass [quarter-sphere] will lie between the normal and the line passing through the centers of the two holes. And the form of the stylus’s endpoint, which passes from the endpoint of the stylus to the center of the [sphere encompassing the] glass [quarter-sphere], extends along this line and continues straight along it to the center of the [sphere encompassing the] glass [quarter-sphere], for this line is perpendicular to the [convex] spherical surface of the glass, which faces the [stylus’s] endpoint.

4.25 ◉ Deinde cum hec forma fuerit reflexa super lineam que transit per centra duorum foraminum, quia linee radiales que exeunt a visu in hoc situ non perveniunt ad vitrum preter lineam que transit per centra duorum foraminum, calamus enim qui extenditur inter duo foramina secat omnem lineam exeuntem a visu ad vitrum preter quam lineam que transit per centra duorum foraminum. Visus autem non comprehendit formas nisi ex verticationibus harum linearum tantum; ergo forme non extenduntur nisi recte; ergo visus non comprehendit hanc formam nisi ex verticatione huius linee perpendicularis. Ergo que extenditur recte in aere est perpendicularis super superficiem aeris contingentis superficiem vitri equalem. Ergo hec reflexio erit ad partem contrariam parti perpendicularis exeuntis a loco reflexionis super superficiem aeris, nam linea que transit per centra duorum foraminum magis distat a perpendiculari que extenditur in aere quam linea que exit ab extremitate stili ad centrum vitri que extenditur in aere. Et hec forma exit a vitro et reflectitur in aere, et aer est subtilior vitro, et hoc modo fuit reflexio forme de aqua ad aerem, visus enim comprehendit extremitatem stili in aqua ab isto loco, scilicet quia comprehendit extremitatem stili quando fuerit inter casum perpendicularis et extremitatem diametri circuli medii, qui transit per centra duorum foraminum. Et illa forma etiam exivit ab aqua et reflexa est in aere, et aer est subtilior aqua.

4.25 ◉ Then, since this form is refracted along the line passing through the centers of the two holes, [it follows] that, in this case, [of all] the radial lines extending from the center of sight only the line passing through the centers of the two holes reaches the glass, for the reed extending between the two holes cuts every line that passes out from the center of sight to the glass except the line passing through the centers of the two holes. But the visual faculty perceives forms only along such straight lines, so [visible] forms only extend [along] straight [lines], [and] so the visual faculty perceives this form only along this line [which is] perpendicular [to the surface of the glass]. Hence, [the form] that extends straight through the air is perpendicular to the surface of the air in contact with the flat face of the glass. This refraction, then, will be away from the normal dropped from the point of refraction on the surface of the air, for the line passing through the centers of the two holes lies farther from the normal that extends through the air than the line dropped from the stylus’s endpoint to the center of the [sphere encompassing the] glass [quarter-sphere] and continuing through the air. This form passes from the glass and is refracted in the air, and air is rarer than glass, and this is the way the form was refracted from water to air, for the visual faculty perceives the endpoint of the stylus at this location in water, i.e., it perceives the endpoint of the stylus when it lies between where the normal falls [on the apparatus’s rim] and the endpoint of the diameter of the middle circle, which passes through the centers of the two holes. And that form too passed from the water and was refracted in the air, and air is rarer than water.

4.26 ◉ Deinde oportet experimentatorem evellere vitrum et ponere ipsum super laminam econtra huius situs, scilicet quod ponat convexum eius ex parte duorum foraminum, et ponat differentiam eius communem super lineam equalem in superficie lamine in qua posuerat illam in predicto situ, et ponat medium communis differentie super centrum lamine. Et sic linea que transit per centra duorum foraminum erit obliqua super superficiem vitri equalem et perpendicularis super superficiem eius convexam. Et applicet vitrum hoc situ, et ponat instrumentum in vas, et ponat extremitatem stili super extremitatem diametri circuli medii, ut prius fecerat, et ponat visum suum super superius foramen, et intueatur horam instrumenti, non enim videbit tunc extremitatem stili. Deinde moveat stilum ad partem casus perpendicularis, et tunc non videbit extremitatem stili. Deinde moveat illud ad partem contrariam illi in quo est casus perpendicularis per circumferentiam circuli medii, et suaviter, tunc enim videbit extremitatem stili. Sic ergo linea recta que exit ab extremitate stili ad centrum vitri, cum fuerit extensa recte in corpore vitri, et extensa fuerit cum ipsa perpendicularis exiens a centro vitri super superficiem vitri, erit linea que transit per centra duorum foraminum media inter duas lineas. Et forma extremitatis stili que extenditur super hanc lineam, cum fuerit extensa ad centrum vitri, reflectebatur super lineam que transit per centra duorum foraminum. Erit ergo reflexio ista ad partem perpendicularis exeuntis a loco reflexionis super superficiem vitri, et hec forma exit ab aere, et reflectitur in vitro, et vitrum est grossius aere.

4.26 ◉ Now the experimenter should remove the glass and place it on the register plate in the opposite way, i.e., he should pose its convex surface toward the two holes and put its common section on the straight line on the register plate where he put it in the previous case, and he should put the midpoint of [that] common section on the center of the register plate. Thus, the line passing through the centers of the two holes will be oblique to the flat face of the glass and normal to its convex surface. He should then affix the glass in this position, insert the apparatus in the vessel, put the endpoint of the stylus at the endpoint of the middle circle’s diameter, just as he did before, and he should position his eye at the top hole and look at the apparatus’s rim [under water], for in that case he will not see the stylus’s endpoint. He should then move the stylus toward where the normal falls [on the rim], and in that case he will not see the stylus’s endpoint. Next he should move it away from where the normal falls on the periphery of the middle circle, and he should do so smoothly, in which case he will [eventually] see the stylus’s endpoint. It therefore follows that, when the straight line passing from the endpoint of the stylus to the center of the [sphere encompassing the] glass [quarter-sphere] is extended straight into the glass, and when the normal is dropped from the center of the [sphere encompassing the] glass [quarter-sphere] to the [flat] face of the glass, the line passing through the centers of the two holes will lie between [these] two lines. And when it reached the center of the [sphere encompassing the] glass [quarter-sphere], the form of the stylus’s endpoint extending along this line was refracted along the line that passes through the centers of the two holes. Hence, that refraction will be toward the normal dropped from the point of refraction on the surface of the glass, and this form passes from the air and is refracted in the glass, and glass is denser than air.⁑

4.27 ◉ Ex omnibus igitur istis experimentationibus patet quod visus comprehendit visibilia que sunt in aqua et ultra corpora diaffona que differunt a diaffonitate aeris secundum reflexionem preter quam illa que sunt super lineas perpendiculares super superficiem corporis diaffoni in quo existit, et quod reflexio formarum ipsorum est in superficiebus perpendicularibus super superficies corporum diaffonorum, omne enim quod experimentatum est per predictum instrumentum invenitur reflecti in superficie medii circuli, ex quo patuit quod est perpendicularis super superficies corporum diaffonorum et super superficies corporum contingentium superficies eorum. Ex hac ergo experimentatione declarabitur etiam quod forme que comprehenduntur a visu secundum reflexionem que exeunt a grossiori corpore ad subtilius reflectuntur ad partem contrariam illi in qua est perpendicularis exiens a loco reflexionis super superficiem corporis diaffoni, et que exeunt a subtiliori ad grossius reflectuntur ad partem in qua est perpendicularis predicta.

4.27 ◉ From all these experiments, then, it is obvious that the visual faculty perceives visible objects in water and beyond transparent bodies that differ in transparency from air according to refraction, except those that lie on lines that are perpendicular to the surface of the transparent body in which they lie, and [it is also obvious] that the refraction of their forms occurs in planes that are perpendicular to the surfaces of the transparent bodies, for everything tested with the aforementioned apparatus is found to be refracted in the plane of the middle circle, on the basis of which it was shown that [this plane] is perpendicular to the surfaces of the transparent bodies [in which the objects lie] as well as to the surfaces of the [transparent] bodies in contact with their surfaces. And from this series of tests it will also be manifest that, when they pass from a denser to a rarer [transparent] body, the forms that are perceived by the visual faculty according to refraction are refracted away from the normal dropped from the point of refraction on the surface of the transparent body, whereas those passing from a rarer to a denser [body] are refracted toward the aforesaid normal.

4.28 ◉ Stelle autem comprehenduntur etiam secundum reflexionem, nam corpus celi est subtilius corpore aeris, scilicet maioris diaffonitatis. Hoc autem potest experiri experimentatione que ostendet quod stelle comprehenduntur secundum reflexionem et ex qua patebit etiam quod corpus celi est magis diaffonum corpore aeris. Et cum quis hoc voluerit experiri, accipiat instrumentum de armillis, et ponat illud in loco eminenti in quo poterit apparere orizon orientalis, et ponat instrumenti armillarum suo modo proprio, scilicet quod ponat armillam que est in loco circuli meridiei in superficie circuli meridiei, et polus eius sit altior terra secundum altitudinem poli mundi super orizonta loci in quo ponitur instrumentum. Et in nocte preservet aliquam stellarum fixarum magnarum que transit per verticationem capitis illius loci aut prope, et preservet illam in ortu suo ab oriente. Stella autem orta, revolvat armillam que revolvitur in circuitu poli equinoctialis donec fiat equidistans stelle, et certificetur locus stelle ex armilla, et sic habebit longitudinem stelle a polo mundi. Et deinde preservet stellam quousque perveniat ad circulum meridiei, et moveat armillam quam prius moverat donec fiat equidistans stelle, et sic habebit longitudinem stelle a polo mundi, cum stella fuerit in verticatione capitis. Hoc autem facto, inveniet remotionem stelle a polo mundi apud ascensionem minorem remotione eius a polo mundi in hora existentie eius in verticatione capitis, ex quo patet quod visus comprehendit stellas reflexe, non recte.

4.28 ◉ Indeed, even the stars are perceived by means of refraction, for the body of the heavens is rarer than the body of air, i.e., of greater transparency.⁑ This can actually be empirically determined by an experiment that will show that the stars are perceived by means of refraction, from which it will also be evident that the body of the heavens is more transparent than the body of the air. And if one wants to conduct this experiment, he should take an armillary sphere, set it up on a high location from which the eastern horizon can be seen, and arrange the armillary apparatus in the way appropriate to it, i.e., he should pose the ring representing the meridian circle in the plane of the meridian circle, and its pole should be [pointing] above the earth according to the altitude of the celestial pole on the horizon at the place where the apparatus is posed.⁑ Then at night he should track one of the large fixed stars that passes through the zenith of his location, or nearly so, and he should track it at its rising in the east. When the star rises, he should turn the ring that rotates about the equinoctial pole until it is in line with the star, and he should determine the star’s location on the ring, and he will thus have the [angular] distance of the star from the celestial pole. Then he should track the star until it reaches the meridian circle and adjust the ring, as he adjusted it before, until it is in line with the star, and so he will have the [angular] distance of the star from the celestial pole when the star lies directly overhead. When this is done, he will find that the distance of the star from the celestial pole at its rising is less than its distance from the celestial pole when it lies at the zenith, from which it is clear that the visual faculty perceives stars by means of refraction, not by means of direct vision.⁑

4.29 ◉ Stella enim fixa semper movetur per eundem circulum de circulis equidistantibus equatori diei, et numquam exit ab ipsa ita quod appareat nisi in longissimo tempore. Et si stella comprehenderetur recte, tunc linee radiales extenderentur a visu ad stellas recte, et extenderentur forme stellarum per lineas radiales recte quousque pervenirent ad visum. Et si forma extenderetur a stella ad visum recte, tunc visus comprehenderet eam in suo loco, et sic inveniret distantiam stelle fixe a polo mundi in eadem nocte eandem. Sed distantia stelle mutatur in eadem nocte a polo mundi, ergo visus non recte comprehendit stellam. In celo autem non est corpus densum tersum, nec in aere, a quo possunt forme converti, et cum visus comprehendit stellam et non recte nec secundum conversionem, ergo est secundum reflexionem, cum hiis solis tribus modis comprehenduntur res a visu. Ex diversitate ergo distantie eiusdem stelle in eadem nocte a polo mundi patet procul dubio quod visus comprehendit stellas reflexe. Ergo corpus in quo sunt stelle fixe differt in diaffonitate ab aere.

4.29 ◉ [This is so] because the fixed star always moves on the same circle among circles parallel to the equator [which marks the division] of [equal night and] day, and to all appearances it never deviates from [that circle], unless [it does so] over an extremely long [period of] time.⁑ If the star were perceived by direct vision, then the lines of sight would extend straight from the center of sight to the stars, and the forms of the stars would extend along straight radial lines until they reached the center of sight. And [so], if the form extended straight from the star to the center of sight, the visual faculty would perceive it in its [actual] location, and so [the experimenter] would find the distance of the fixed star from the celestial pole to be the same during the same night. But the distance of the star from the celestial pole changes during the same night, and so the visual faculty does not perceive the star by direct vision. Moreover, neither in the heavens nor in the air is there a dense, polished body from which forms can be reflected, and since the faculty of sight perceives the star neither by direct vision nor by reflection, it does so by refraction because things are perceived by sight according to these three modes alone [i.e., by direct, reflected, or refracted vision].⁑ Consequently, on the basis of the variation in the same star’s distance from the celestial pole during the same night it is evident without the slightest doubt that the visual faculty perceives heavenly bodies by refraction. Therefore, the body in which the fixed stars lies differs in transparency from the air.⁑

4.30 ◉ Preterea potest experiri diaffonitas corporis celi per experimentationem lune, nam, cum equaveris locum lune in aliqua hora prope ortum eius, et post in nocte nota et in loco noto et verificaveris locum eius a polo mundi. Deinde posueris instrumentum horarum in illa nocte ante ortum lune, et scias altitudinem lune. Et preservaveris lunam usque ad ortum eius, et perveniat tempus ad minutum idem eiusdem hore quod habet luna, et preservaveris altitudinem lune quam habet in illa hora a verticatione capitis, et preservaveris quod instrumentum elevationis sit divisum per minuta et per minora minutis si possibile est. Tunc invenies distantiam lune a verticatione capitis in illa hora per instrumentum minorem spatio remotionis a verticatione capitis in illa hora per computationem. Ergo lux lune non extenditur per duo foramina instrumenti per quod sumpta est elevatio recte, tunc enim distantia eius a verticatione capitis esset eadem cum illa que inventa est per computationem, sed distantia inventa per instrumentum differt a distantia per computationem. Ergo lux lune non extenditur a celo ad aerem per lineas rectas; ergo secundum reflexionem. Ex hiis ergo experimentationibus patet quod comprehendit visus omnes stellas que sunt in celo reflexe. Ergo universum celum differt a diaffonitate aeris. Restat ergo declarare quod corpus celi differt in subtilitate ab aere.

4.30 ◉ The [relative] transparency of the heavens can be tested subsequently by an experiment with the moon, for [you can begin this experiment] if you compare the location of the moon at a certain time near its rising and if, later in the night [at a] determinate [time] and at a determinate location, you verify its position with regard to the [north] celestial pole [i.e., its angular distance from that pole. This can be done] if you set up an apparatus with sighting-holes before moonrise during that night, and you should [thereby] know the moon’s altitude [from the pole]. You then track the moon until its rising, and the time should come to the very minute of the very hour that the moon has [risen], and you should take into account the moon’s altitude from the zenith at that time and make sure that the apparatus for [measuring that] altitude is divided into minutes and less than minutes, if it is possible. You will then find that the distance of the moon from the zenith at that time according to the apparatus is less than its distance from the zenith at that time according to calculation. Hence, the moon’s light does not extend through the two [sighting] holes of the apparatus by means of which its elevation is supposed to be correctly found, for then its distance from the zenith would be the same as that which is arrived at by calculation, but the distance found by means of the apparatus differs from the distance [found] by calculation.⁑ Thus, the moon’s light does not extend from the heavens into the air along straight lines, so [it must do so] by refraction. From these experiments, therefore, it is evident that the visual faculty perceives all heavenly bodies by means of refraction. Therefore, the entire heavens differ in transparency from the air. It remains, then, to show [by geometry] that the body of the heavens differs in rarity from the air.

4.31 ◉ [PROPOSITIO 1] Et hoc declarabitur per experimentationem predictam. Sit ergo circulus meridiei in loco experimentationis circulus ABG [FIGURE 7.4.1, p. 440], et cenit capitis B, et polus mundi D, et centrum mundi E. Et continuemus B cum E, et sit locus visus Z, et circulus equidistans equinoctiali cuius distantia a polo mundi est illa in qua invenitur stella in hora certificationis distantie prime circulus HT. Et sit locus stelle in illa hora H, et sit circulus equidistans equinoctiali cuius distantia a polo est illa in qua invenitur stella in secunda hora circulus KB. Iste ergo circulus erit ille in quo requiescet stella secundum verticationem, nam, cum stella fuerit in verticatione capitis aut valde prope, tunc visus comprehendet illam recte, quia linea recta que transit per visum et per verticationem capitis est perpendicularis super concavum spere celi et perpendicularis super convexum aeris. Et cum sit perpendicularis super utrumque corpus, ergo visus comprehendet stellam que est super hanc lineam recte, sive hec duo corpora celi et aeris fuerint diverse diaffonitatis aut consimilis.

4.31 ◉ [PROPOSITION 1] This will be demonstrated for the experiment just discussed. Accordingly, let ABG [in figure 7.4.45, p. 179] be [a segment of] the circle of the meridian at the location of the experiment, and let B be the zenith, D the [north] celestial pole, E the center of the universe [and circle AF the equator]. Let us connect B and E, let Z [on the earth’s surface] be where the center of sight lies, and let HT be the circle parallel to the equator whose distance from the celestial pole is that at which the star is found at the time the first distance is determined [i.e. at the horizon]. Let H be where the star is at that time, and let KB be the circle parallel to the equator, whose distance from the pole is that at which the star is found at the second time [i.e., at the zenith]. That circle will thus be the one on which the star will lie on a line [passing] straight [through the interface between the heavens and the air], for when the star lies at the zenith, or very nearly so, the visual faculty will perceive it by direct vision because the straight line [EZB] that passes through the center of sight and the zenith is perpendicular to the concave surface of the heavenly sphere as well as perpendicular to the convex surface of the air. And since [that line of sight] is perpendicular to both bodies, the visual faculty will perceive the star on that line by direct vision, whether these two bodies of heaven and air are of different or of the same transparency.

4.32 ◉ Cum ergo stella fuerit in verticatione capitis aut prope, visus comprehendit illam in suo vero circulo equidistanti equinoctiali super quem movebatur ab initio noctis quousque pervenit ad circulum meridiei. Circulus ergo KB est ille in quo erat stella in experimentatione. Et sit circulus verticationis qui transit per stellam in hora experimentationis prima circulus BHK, et secet iste circulus circulum KB in puncto K. Et quia distantia stelle a polo mundi fuit in prima experimentatione minor quam in secunda, erit circulus HT propinquior polo circulo KB; ergo punctus H est propinquior cenit capitis quam punctus K.

4.32 ◉ Hence, when the star lies at the zenith, or very nearly so, the visual faculty perceives it on its actual circle, which is parallel to the equator and upon which it moved at the beginning of night until reaching the meridian circle. Circle KB is thus the one on which the star lay in the experimental determination [at that point]. Let BHK be the circle of the zenith that passes through the star at the first time in the experiment, and let that circle intersect circle KB at point K.⁑ Since the distance of the star from the celestial pole was less in the first than in the second [stage of the] experiment, circle HT [parallel to the equator] will lie closer to the pole than circle KB, so point H lies closer to the zenith than does point K.

4.33 ◉ Et continuemus duas lineas HZ, KZ. Quia ergo stella comprehenditur a visu in hora experimentationis prima in puncto H, et tunc erat in superficie circuli BHK verticalis, et stella erat in illa hora in circumferentia circuli KB, ergo stella erat in illa hora in puncto K, et comprehendebatur a visu in puncto H et per rectitudinem linee ZH, visus enim nichil comprehendit nisi per verticationes radialium linearum per quas forme perveniunt ad visum. Visus ergo comprehendit stellam in puncto H, quia forma pervenit ad illum in rectitudine linee HZ. Et cum visus comprehendit illam in rectitudine linee HZ, et linea recta que est inter stellam et visum est linea KZ, manifestum est ergo quod visus non comprehendit stellam que est in puncto K recte; ergo reflexe.

4.33 ◉ Let us join the two lines HZ and KZ. Accordingly, since the star is perceived by the visual faculty at point H at the first time of the experiment, and since it was thus in the plane of vertical circle BHK while at that time the star was [actually] on the circumference of circle KB, the star was at point K at that time, but it was perceived by the visual faculty at point H along straight line ZH, for the visual faculty perceives nothing unless [it does so] by means of straight radial lines along which forms reach the center of sight. Thus, the visual faculty perceives the star at point H because the form reaches it along straight line HZ. And [so], since the visual faculty perceives it along straight line HZ, and since the straight line between the star and the center of sight is [actually] KZ, it is obvious that the visual faculty does not perceive the star that is at point K by direct vision, so [it perceives] it by means of refraction.

4.34 ◉ Sit ergo locus reflexionis M, et continuemus KM, et protrahamus illam recte usque Z. Forma ergo stelle que pervenit ad Z, ex qua visus comprehendit stellam, extenditur a stella per lineam KM, et reflectitur per lineam MZ. Et non reflectuntur forme nisi cum occurrerit corpus diverse diaffonitatis a diaffonitate corporis in quo existerit. Ergo corpus in quo est stella, scilicet celum, est diaffonum differens a diaffonitate ab aere, et quod locus reflexionis est apud superficiem que transit inter duo corpora que differunt in diaffonitate. Punctus ergo M est punctus in concavitate celi. Et continuemus lineam inter E, M, et sit diameter spere celi. Erit ergo linea EM perpendicularis super superficiem celi concavam contingentem aerem et super superficiem aeris convexam. Et cum forma stelle que est in puncto K extenditur per lineam MK et reflectetur in aere per lineam MZ, patet quod hec reflexio est ad partem in qua est perpendicularis EM, que transit per punctum reflexionis, que est perpendicularis super superficiem aeris. Et cum reflexio in aere est ad partem perpendicularis exeuntis per locum reflexionis, ergo corpus aeris est grossius corpore celi.

4.34 ◉ So let M [in figure 7.4.45a, p. 179] be the point of refraction [within the plane of refraction defined by circle BHK in figure 7.4.45], and let us join KM and extend it straight to Z.⁑ Accordingly, the form of the star, which reaches Z, at which [point] the visual faculty perceives the star, extends from the star along line KM and is refracted along line MZ. Moreover, forms only refract when they encounter a body of different transparency from the transparency of the body within which they lie. Therefore, the body in which the star lies, i.e., the heavens, is of a transparency differing from the transparency of the air, and [it follows] that the point of refraction is at the interface passing between two bodies that differ in transparency. Point M is therefore a point on the concave surface of the heavens [contiguous with the convex surface of the atmosphere]. Let us draw a line between E and M, and let it be a [radial segment of a] diameter of the heavenly sphere. Line EM will thus be perpendicular to the concave surface of the heavens that is in contact with the air and [also] to the convex surface of the air. And since the form of the star at point K extends along line MK and will be refracted in the air along line MZ, it is clear that this refraction occurs toward normal EM, which passes through the point of refraction and is normal to the surface of the air. Since, therefore, the refraction in the air is toward the normal passing through the point of refraction, the body of the air is denser than the body of the heavens.⁑

4.35 ◉ Patet ergo quod hoc quod invenimus per experimentationem de apparitione stellarum signat demonstrative quod visus non comprehendit stellas nisi reflexe, et quod corpus aeris est grossius corpore celi, et quod corpus celi est subtilius corpore aeris. Ex hiis ergo omnibus patet quod omnia que comprehenduntur a visu ultra corpora diaffona quorum diaffonitas differt a diaffonitate aeris, si visus fuerit obliquus a perpendicularibus egredientibus ex ipsis super superficiem diaffonorum corporum in quibus existunt, comprehenduntur reflexe.

4.35 ◉ It is thus evident that what we discover about the appearance of the stars by experiment indicates conclusively that the visual faculty perceives the stars solely by means of refraction, that the body of the air is denser than the body of the heavens, and that the body of the heavens is rarer than the body of the air. From all these [experimental demonstrations], then, it is clear that all objects that are perceived by the visual faculty through transparent bodies whose transparency is different from the transparency of air are perceived by means of refraction if the center of sight lies to the side of the normals dropped from those objects to the surface of the transparent bodies in which they lie.

◉Quintum capitulum

◉Chapter Five

De ymagine

5.1 ◉ Ymago est forma rei visibilis quam visus comprehendit ultra diaffonum corpus quod differt in sui diaffonitate a diaffonitate aeris, cum visus fuerit obliquus a perpendicularibus exeuntibus ab illo visibili ad superficiem illius corporis diaffoni, nam forma quam comprehendit visus in corpore diaffono de re visa que est ultra ipsum corpus non est ipsa res visa, quoniam visus tunc non comprehendit rem visam in suo loco nec in sua forma sed in alio loco et in alio modo, scilicet reflexione, et cum hoc comprehendit illam rem in sui oppositione. Hec autem forma dicitur ymago. Hec autem comprehenditur ratione et experientia.

5.1 ◉ An image is the form of a visible object that the visual faculty perceives through a transparent body whose transparency differs from the transparency of air, when the center of sight lies to the side of the normals dropped from that visible object to the surface of that transparent body, for the form of the visible object that the visual faculty perceives through the transparent body is not the visible object itself because in this case the visual faculty does not perceive the visible object in its [actual] location nor in its [true] form but in another place and in another way, i.e., according to refraction, and yet it perceives that object in a directly facing position. Such a form is called an »image.«⁑ This [point], moreover, is understood on the basis of reason as well as empirical test.

5.2 ◉ Ratione vero quoniam ex predicto capitulo patet quod visum quod est in diaffono corpore diverse diaffonitatis ab aere comprehenditur a visu reflexe, cum visus fuerit declinis a perpendicularibus exeuntibus a re visa super superficiem corporis diaffoni. Et cum visus comprehenderit huiusmodi visum reflexe, nec est in oppositione eius, nec comprehendit ipsum recte, nec sentit ipsum comprehendere ipsum reflexe, patet quod ipse comprehendit ipsum extra suum locum.

5.2 ◉ By reason, on the one hand, it is evident from the preceding chapter that a visible object [lying] in a transparent body that differs in transparency from the air is perceived by the visual faculty according to refraction if the center of sight lies to the side of the normals dropped from the visible object to the transparent body’s surface. Since the visual faculty perceives the visible object in such a refracted way, [and since] the object does not face the eye directly, [and since] the visual faculty does not perceive it directly or sense that it is perceiving it according to refraction, it is obvious that it perceives the object outside of its [actual] location.

5.3 ◉ Per experientiam vero sic potest cognosci. Nam si aliquis acceperit vas habens horas erectas perpendiculares, in cuius medio posuerit aliquod visum manifestum, ut obulum aut denarium, et steterit a longe quousque videat rem visam in profundo vasis, deinde elongaverit se a re visa quousque non videat rem paulatim paulatim, tunc initio occultationis stet in suo loco, et precipiat alii infundere aquam in vas, ipso existente in suo loco, nec moveat visum nec mutet situm, tunc enim, cum aspexerit aquam que est in vase, videbit rem visam postquam non viderat eam. Et videbit eam in eius oppositione, ex quo patet quod forma quam videt in aqua non est in loco visi, nam forma quam vidit in aqua que est in vase non est in loco visi, nam si forma esset in loco visi, tunc visus comprehenderet rem visam, non existente aqua in vase. Visus enim in secundo statu comprehendit rem visam in sui oppositione ipsa non existente. Hoc igitur modo declarabitur utroque modo, ratione videlicet et experientia, quod ymago rei vise quam visus comprehendit reflexe non est in loco rei vise.

5.3 ◉ By empirical test, on the other hand, this can be understood as follows. If one takes a container with a rim that is perpendicular [to its bottom] and places some conspicuous visible object, such as an obolus or denarius,⁑ on the middle of its [bottom], and if he stands away from it so that he can see the visible object at the bottom of the container and then moves back from the visible object little by little until he cannot see the object [any longer], then if he stands at the point where the coin first disappears from view and directs someone else to pour water in the container while he stays in place without moving his line of sight or his position, then when he looks into the water in the container, he will see the visible object after not having seen it.⁑ He will also see it in a directly facing position, from which it is evident that the form he sees in the water is not where the visible object [actually] is, for the form he saw in the water in the container is not where the visible object [actually is] because, if the form were in the [actual] place [occupied by the] visible object, the visual faculty would perceive the visible object [there] when there was no water in the container. For in the second situation the visual faculty perceives the visible object in a directly facing position when it is not [actually there]. In this way, then, it will be demonstrated by both means, namely, reason and experiment, that the image of a visible object that the visual faculty perceives according to refraction does not lie where the visible object [itself] lies.

5.4 ◉ Deinde dico quod ymago cuiuslibet puncti quod visus comprehendit reflexe est in puncto quod est differentia communis linee per quam forma pervenit ad visum et perpendiculari exeunti ab illo puncto viso super superficiem diaffoni corporis. Hoc autem declarabitur per experientiam hoc modo. Accipiat aliquis circulum ligneum, cuius diameter non sit minor uno cubito, et adequet superficiem eius quantumcumque poterit. Et inveniat centrum eius, et extrahat in ipso diametros sese intersecantes quotcumque voluerit, et signentur ferro ut appareant, et impleat lineas illas corpore albo, ut cerusa mixta lacte nivio, et punctum centri sit nigrum. Hoc autem perfecto, accipiat vas amplum, ut pelvim, habens horas elevatas, et ponat vas in loco luminoso. Et infundat in vas aquam claram, et sit altitudo aque minor diametro circuli et maior semidiametro eius. Et mensuretur hoc ipso circulo quousque aqua transeat centrum circuli aliquibus digitis, duobus scilicet diametris aut pluribus signatis in ipso vase, scilicet quod sit aqua cooperiens aliquam partem utriusque diametri et quod remaneat altera pars extra aquam.

5.4 ◉ I say, then, that the image of any point that the visual faculty perceives according to refraction lies at the point that forms the common section of the line along which the form reaches the center of sight and the normal dropped from that visible point to the surface of the transparent body.⁑ This, moreover, will be proven by experiment in the following way. Let someone take a wooden disk whose diameter is no less than a cubit, and let him plane and smooth its surface as thoroughly as he can. He should then find its center, draw as many intersecting diameters through it as he wishes, score them with an iron [incising tool] so that they are [clearly] visible, and fill those [scored] lines with a white substance, such as white lead mixed with snow-white milk,⁑ and the centerpoint should be black. When this is done, he should take a wide vessel, such as a wash-basin, that has a high brim, and he should place the vessel in a lighted spot. Then he should pour clear water into the vessel, and the depth of the water should be less than the diameter of the [wooden] disk but greater than its radius. He should determine this with the disk itself until the water rises above the center of the disk by some [number of] digits, i.e., above the two or more diameters scored [on the wooden disk] in that vessel, so that the water covers some portion of each diameter while the other portion remains above water.

5.5 ◉ Et expectet donec aqua quiescat in vase, et tunc mittat circulum ligneum in vas, et erigat circulum super horam ipsius, et ponat superficiem ipsius in qua sunt linee signate ex parte visus; deinde moveat circulum donec aliquis diametrorum suorum sit perpendicularis super superficiem aque. Deinde demittat visum suum, et erigat vas quousque visus suus appropinquet equidistantie superficiei aque et extra horam vasis et supra superficiem aque in tantum ut possit videri centrum circuli, experientia enim secundum hunc modum erit manifestior.

5.5 ◉ He should wait until the water in the vessel becomes still, then insert the wooden disk in the vessel and stand the disk upright on its rim, and he should pose the surface on which the lines are scored toward his eye and then turn the disk until one of its diameters is perpendicular to the water’s surface. Afterwards he should lower his eye and set the vessel up so that his line of sight outside the rim of the vessel is nearly parallel to the water’s surface but high enough above the water’s surface that the center of the disk can be seen, for the experiment will be clearer [if it is conducted] in this way.

5.6 ◉ Hoc igitur facto, intueatur centrum circuli et diametrum circuli perpendicularem super superficiem aque, tunc enim inveniet centrum circuli in rectitudine diametri perpendicularis. Deinde intueatur diametrum circuli declinem, cuius pars est preminens aque, tunc enim inveniet ipsum incurvatum, cuius incurvatio erit apud superficiem aque. Et illa pars que est intra aquam continet cum illa que est extra aquam angulum obtusum. Et inveniet angulum ex parte diametri perpendicularis, et inveniet illud quod est intra aquam rectum continuum, ex quo patet quod forma puncti que est centrum circuli, scilicet forma quam visus comprehendit, non est apud centrum circuli, nam si esset apud centrum circuli, tunc esset in rectitudine diametri declinis, nam in rei veritate talem habet situm.

5.6 ◉ When this is all done, therefore, he should look at the center of the disk as well as the diameter on the disk that is perpendicular to the water’s surface, for in that case he will find that the disk’s center lies in a straight line with the perpendicular diameter. Then he should look at a slanted diameter that has a portion standing above the water, for in that case he will find that it is bent and that its bending will occur at the water’s surface. Furthermore, the portion that is under water will form an obtuse angle with the portion that is above water. He will also find that the angle faces the perpendicular diameter, and he will find that the portion under water continues in a straight line, from which it is clear that the form of the point at the center of the circle, i.e., the form that the visual faculty perceives, is not [actually] at the center of the disk, for if it were at the disk’s center, it would lie on the straight continuation of the slanted diameter because in actuality that is where it is.⁑

5.7 ◉ Cum ergo visus comprehendit hoc punctum extra rectitudinem diametri declinis, et angulus quem continent partes diametri declinis sequitur diametrum perpendicularem, tunc punctus qui est forma centri est elevatus a centro. Et quia visus comprehendit hoc punctum in rectitudine diametri perpendicularis super superficiem aque, erit hoc punctum, quod est forma puncti quod est in centro, extra centrum et elevatum a centro, et cum hoc est in rectitudine perpendicularis exeuntis a centro super superficiem aque. Et declarabitur ex incurvatione diametri declinis apud superficiem aque, et rectitudine eius quod est intra aquam ex diametro et continuatione eius quod omne punctum partis que est intra aquam ex diametro declini est elevatum a suo loco.

5.7 ◉ Since, therefore, the visual faculty perceives this point outside the straight continuation of the slanted diameter, and since the angle that the segments of the slanted diameter form faces the perpendicular diameter, the point that constitutes the form of the center lies above the [actual] center. Moreover, because the visual faculty perceives this point on the straight continuation of the diameter that is perpendicular to the water’s surface, this point, which constitutes the form of the point at the center, will lie outside the center and above the center, yet nonetheless it lies on the normal dropped straight from the [actual] center to the water’s surface. It will also be clear from the bending of the slanted diameter at the water’s surface, as well as from the straightness of the [portion] of that diameter and its continuation under water, that every point on the portion of the slanted diameter under water lies above its [actual] location.⁑

5.8 ◉ Deinde oportet quod experimentator revolvat circulum ligneum quousque diameter declinis fiat perpendicularis super superficiem aque et diameter qui erat perpendicularis fiat declinis. Deinde demittat visum suum, et intueatur centrum circuli, et tunc inveniet formam centri in rectitudine diametri qui nunc est perpendicularis super superficiem aque extra cuius rectitudinem erat forma centri quando erat declinis, et inveniet formam extra rectitudinem diametri qui est nunc declinis qui prius erat perpendicularis super superficiem aque. Et inveniet diametrum declinem incurvatum apud superficiem aque, et angulus incurvationis erit ex parte diametri declinis. Et si in circulo fuerint plures diametri, et revolverit experimentator circulum quousque unusquisque eorum fuerit perpendicularis successive super superficiem aque, et fuerit diameter qui sequitur illum diametrum declinis, et aliqua pars eius fuerit extra aquam, tunc inveniet formam puncti quod est centrum circuli semper in rectitudine diametri perpendicularis elevatam a rectitudine diametri declinis. Et semper inveniet illud quod est intra aquam rectum.

5.8 ◉ Next the experimenter should revolve the wooden disk until the slanted diameter becomes perpendicular to the water’s surface and the diameter that was perpendicular becomes slanted. He should then lower his eye and look at the disk’s center, and in that case he will find the form of the center on the straight continuation of the diameter that is now perpendicular to the water’s surface and outside of which the form of the center lay when the diameter was slanted, and he will find the form [of the center] outside the straight continuation of the diameter that is now slanted but was perpendicular to the water’s surface before. He will also find the slanted diameter bent at the water’s surface, and the angle of [its] bending will be on the side [away] from the diameter [that was formerly] slanted. Furthermore, if there are several diameters on the disk, and if the experimenter revolves the disk so that each of them in turn is perpendicular to the water’s surface, and if the diameter next to that diameter is slanted and part of it lies outside the water, then he will find that the form of the point at the center of the disk on the straight continuation of the perpendicular diameter always lies above the straight continuation of the slanted diameter. And he will always find that what lies under water is straight.

5.9 ◉ Ex omnibus igitur istis patet quod forma cuiuslibet puncti comprehensi a visu in corpore diaffono grossiore corpore aeris comprehenditur extra suum locum et elevatum a suo loco et in rectitudine perpendicularis exeuntis ab illo puncto super superficiem corporis diaffoni, cum linea que continuat centrum visus cum illo puncto non fuerit perpendicularis super superficiem corporis diaffoni. Omne autem punctum comprehenditur a visu in eius oppositione et in rectitudine linee recte per quam extenditur forma ad visum; puncta ergo que visus comprehendit reflexive comprehenduntur in eius oppositione et in rectitudine linee recte per quam forma pervenit ad visum.

5.9 ◉ From all these [observations] it is therefore evident that the form of any point perceived by the visual faculty in a transparent body denser than the body of air is perceived outside its [actual] location and above its [actual] location on a straight line with the normal dropped from that point to the surface of the transparent body, when the line connecting the center of sight with that point is not perpendicular to the transparent body’s surface. Furthermore, every point is perceived by the visual faculty in a directly facing position on the straight line along which the form extends to the center of sight, so points that the visual faculty perceives by means of refraction are perceived in a directly facing position on the straight line along which the form reaches the center of sight.

5.10 ◉ Hoc autem declarabitur per experimentationem comprehensionis rerum visibilium secundum reflexionem per illud instrumentum predictum, nam si experimentator clauserit secundum foramen quod est in instrumento, tunc non comprehendet rem visam quam comprehendebat secundum reflexionem, et cum clauserit secundum foramen, nichil aliud fecit nisi secare lineam rectam ymaginabilem que exit a centro visus ad locum reflexionis, ex quo patet quod forma que extenditur a visu in corpore diaffono in quo est res visa et reflectitur in corpore diaffono in quo est visus extenditur per lineam rectam que exit a centro visus ad locum reflexionis. Et omne punctum quod comprehenditur a visu in corpore diaffono magis grosso quam corpus sit aeris, si centrum visus fuerit extra perpendicularem exeuntem ab illo puncto super corpus diaffonum, comprehenditur in puncto quod est differentia communis linee super quam pervenit forma ad visum et perpendiculari exeunti a puncto viso super superficiem corporis diaffoni quod est ex parte visus.

5.10 ◉ This will be demonstrated by means of an experimental test of the perception of visible objects according to refraction based on the apparatus discussed earlier [in chapters 3 and 4], for if the experimenter closes off the second hole in the apparatus [i.e., the one in the panel attached to the register plate], he will not perceive a visible object that he perceived [earlier through the holes] according to refraction, but when he closes the second hole, he has done nothing other than cut the imaginary straight line extending from the center of sight to the point of refraction, from which it is clear that the form extending from what is seen in the transparent body in which the visible object lies and refracted in the transparent body in which the center of sight lies extends along the straight line that passes from the center of sight to the point of refraction. And every point perceived by the visual faculty in a transparent body denser than the body of air is perceived at the point that forms the common section of the line along which the form reaches the eye and the normal dropped from the visible point to the surface of the transparent body on the side of the center of sight, provided that the center of sight lies outside the normal dropped from that point to the transparent body.

5.11 ◉ Si autem experimentator voluerit experiri ymaginem rei vise cuius forma reflectitur a corpore subtiliore ad corpus grossius, accipiat frustrum vitri cuius superficies sint equate equidistantes habens in longitudine octo digitos, et in altitudine quatuor, et in spissitudine quatuor. Et accipiat circulum ligneum predictum, et signet in dorso eius cordam in longitudine decem digitorum, et dividat illam in duo equalia, et continuet locum divisionis cum centro circuli linea recta que transit in utramque partem. Hec ergo linea erit perpendicularis super primam lineam. Deinde continuet alteram extremitatem corde cum centro circuli linea recta que etiam transeat in utramque partem, et hii duo diametri sint signati ferro, quorum perpendicularem impleat corpore albo et aliud alterius modi corpore.

5.11 ◉ Now if the experimenter wants to test the image of a visible object whose form is refracted from a rarer body to a denser body, he should take a block of glass that has flat, parallel surfaces and is eight digits long, four digits high, and four digits wide. He should then take the wooden disk just described, mark off a chord ten digits long on its back side, bisect it, and connect the point of bisection with the center of the disk by a straight line extending [to the disk’s edge] on both sides [of that point]. This line will thus be perpendicular to the first line. Then he should connect either endpoint of the chord to the center of the circle with a straight line that also extends on both sides, and these two diameters should be scored [into the wood] with the iron [incising too], and the perpendicular should be filled with the white substance while the other [is filled] with some other kind of [colored] substance.

5.12 ◉ Deinde ponat vitrum longum super dorsum instrumenti circuli lignei, et superponat alteram extremitatem longitudinis eius medietati corde, et distinguat de vitro tres digitos, ex quibus duo erunt ex parte diametri declinis extra circulum, et remanebit de longitudine vitri unus digitus qui erit ultra diametrum perpendicularem supra cordam. Et sit corpus vitri ex parte centri, et applicet vitrum secundum hunc situm circulo ligneo applicatione fixa. Sic ergo diameter perpendicularis super cordam erit perpendicularis super extremitates vitri equidistantes, et alter diameter erit declinis super has duas superficies.

5.12 ◉ Next he should place the glass block lengthwise on the back of the circular wooden device [upon which the new lines have been incised] and apply one of its long edges on the middle of the [bisected] chord, and he should mark off three digits on the glass, two of which will lie beyond the slanted diameter outside the circle, and one digit of the glass’s length will remain [to project] beyond the diameter that is perpendicular to the chord. The glass block should lie on the side of [i.e., above] the center, and [the experimenter] should affix the glass firmly to the wooden disk according to this position. Thus, the diameter that is perpendicular to the chord will be perpendicular to the parallel edges of the glass, and the other diameter will be inclined to these two surfaces.⁑

5.13 ◉ Deinde oportet quod experimentator ponat horam circuli in qua est extremitas vitri eminens ex parte sui visus, et ponat alterum visum in differentia communi circumferentie et extremitati vitri, que est extremitas diametri declinis, et appropinquet suum visum vitro quantum potuerit ita quod non possit per illum videre ex superficie aliquid preter extremitatem diametri declinis. Reliquus autem visus sit in parte in qua est vitrum et circulus. Deinde cooperiat illud quod opponitur alteri visui ex superficie vitri cum bombace, quem applicet super aliquam partem vitri ita quod comprehendat diametrum declinem, qui est ultima linea per unum visum qui contingit vitrum, et non videat ultra hanc lineam, et videat lineam albam perpendicularem utroque visu.

5.13 ◉ The experimenter should then put the edge of the disk on which the projecting end of the glass lies toward his eye, placing one eye at the common section of the circumference and the [top] edge of the glass, which is the endpoint of the slanted diameter, and he should bring his eye as close as he can to the glass so that he cannot see any of its [top] surface except the endpoint of the slanted diameter. The remaining eye will lie on the side where the glass and the disk lie. Then he should block the part of the glass that faces the other eye with a piece of paper⁑ that he places over some portion of the glass so that he may [only] see the slanted diameter, which is the last line [viewed] by the one eye that is up close to the glass, and [so that] he may not see beyond this line but may see the white perpendicular line with both eyes.

5.14 ◉ Ipso autem existente in hoc situ, intueatur centrum circuli, et inveniet illud in rectitudine linee albe que est perpendicularis super superficiem vitri. Et intueatur diametrum declinem apud cuius extremitatem tenet visum suum, et tunc videbit eum incurvatum apud superficiem vitri que est ex parte centri, et inveniet angulum incurvationis ex parte circumferentie. Visus autem comprehendit partem huius diametri declinis que est sub vitro in rectitudine, nam visus tangit superficiem vitri, et diametri perpendicularis una pars est sub vitro, alia extra vitrum ex parte centri, alia extra vitrum ex parte extremitatis diametri.

5.14 ◉ When everything is in place this way, he should look at the disk’s center, and he will find that it lies on the straight continuation of the white line that is perpendicular to the surface of the glass. He should also look at the slanted diameter at whose endpoint he holds his eye, and in that case he will see it bent at the surface of the glass on the side of the center, and he will find that the angle of bending faces the circumference [of the wooden disk]. Moreover, the visual faculty perceives the part of this slanted diameter below the glass in a straight line, for the eye is right up against the surface of the glass, and one part of the perpendicular diameter lies below the glass, another beyond the [bottom edge of the] glass on the side of the center, and another beyond the glass on the side of the diameter’s endpoint.

5.15 ◉ Pars igitur que est sub vitro comprehenditur a visu extra vitrum secundum reflexionem, et pars que est ex parte extremitatis diametri comprehenditur a visu extra vitrum, qui est visus extra vitrum recte et sine reflexione. Pars autem que est ex parte centri comprehenditur ab utroque visu secundum reflexionem, nam linee que exeunt a centro visus contingentis vitrum et extenduntur in corpore vitri, quando perveniunt ad superficiem vitri que est ex parte extremitatis centri, omnes erunt declines super superficiem vitri. Pars ergo que est ex parte centri ex diametro perpendiculari comprehenditur a visu contingenti vitrum secundum reflexionem.

5.15 ◉ Consequently, the part below the glass is perceived by the visual faculty outside the glass according to refraction, and the part on the side of the diameter’s endpoint is perceived by the visual faculty outside the glass, and it is seen straight outside the glass without refraction. The part that lies on the side of the center is perceived by both eyes according to refraction because, when they reach the surface of the glass at the edge on the side of the center, all the lines extending from the center of sight that is right up against the glass and [then] continuing through the body of the glass will be oblique to the glass’s surface. Therefore, the part of the perpendicular diameter on the side of the center is perceived according to refraction by the eye right up against the glass.

5.16 ◉ Linee vero que exeunt a reliquo visu ad superiorem superficiem vitri erunt declines super superficiem vitri superiorem, et cum extenduntur ad aliam superficiem vitri que est ex parte centri erunt etiam declines. Reliquus ergo visus etiam comprehendit partem diametri perpendicularis que est ex parte centri duabus reflexionibus; partem autem que est sub vitro una sola reflexione, partem vero superiorem absque reflexione, et cum hoc toto uterque visus comprehendit hunc diametrum rectum. Et si experimentator cooperuerit alterum visum et aspexerit per visum qui est ex parte vitri, comprehendet perpendicularem rectum. Et si elevaverit visum suum a vitro et intuens fuerit diametrum perpendicularem ultra vitrum, comprehendet ipsum rectum cum hoc quod comprehendit ipsum secundum reflexionem.

5.16 ◉ On the other hand, the lines extending from the remaining eye to the top surface of the glass will be oblique to the top surface of the glass, and when they continue to the other surface of the glass that lies on the side of the center, they will also be slanted. Thus, the remaining eye also perceives the part of the perpendicular diameter on the side of the center according to two refractions, the part below the glass by one single refraction and the upper part without refraction, and despite all this both eyes perceive this diameter straight. And if the experimenter covers one or the other eye and looks with the eye that lies on the side of the glass, he will perceive the perpendicular [diameter as] straight. And if he raises his eye from the glass and looks at the perpendicular diameter behind the glass, he will perceive it as straight despite his perceiving it according to refraction.⁑

5.17 ◉ Causa autem huius est quoniam omne punctum diametri perpendicularis, quando comprehenditur a visu secundum reflexionem, comprehenditur non suo loco, sed tamen comprehendit ipsum in loco qui est in rectitudine perpendicularis que exit ab illo super superficiem vitri, et iste diameter est perpendicularis que exit a quolibet puncto eius ad superficiem vitri, et nullum punctum comprehenditur reflexive nisi super ipsum. Cum igitur visus comprehendit hunc diametrum rectum et comprehendit formam centri in rectitudinem huius diametri, forma centri quam visus comprehendit ultra vitrum, quando visus tangit vitrum, est in rectitudine perpendicularis exeuntis a centro super superficiem vitri.

5.17 ◉ The reason for this is that, when it is perceived by the visual faculty according to refraction, no point on the perpendicular diameter is perceived in its [actual] location, but [the visual faculty] still perceives it in a location on the normal that extends straight from it to the surface of the glass, and that [perpendicular] diameter constitutes a normal dropped from any point on it to the surface of the glass, and no point on it is perceived according to refraction except on that diameter. Therefore, since the visual faculty perceives this diameter as straight and perceives the form of the center on the straight continuation of this diameter, the form of the center that the visual faculty perceives through the glass lies in a straight line with the normal dropped from the center to the surface of the glass, when the eye is right up against the glass.

5.18 ◉ Et cum comprehenderit diametrum declinem incurvatum, comprehendit partem eius que exit a vitro, que est ex parte centri, non in suo loco. Punctus centri non comprehenditur a visu nisi preter suum locum, et cum angulus incurvationis fuerit ex parte circumferentie, tunc punctum quod est forma centri est sub centro, ex quo patet quod ymago cuiuslibet puncti comprehensi a visu ultra corpus diaffonum subtilius corpore diaffono quod est in parte visus est in rectitudine linee que exit ab illo puncto perpendiculariter super superficiem corporis diaffoni quod est in parte visus, et est remotior a superficie corporis diaffoni quod est in parte visus quam ipsum punctum. Et omne punctum comprehensum a visu est in rectitudine linee per quam pervenit forma ad visum, et ymago cuiuslibet puncti comprehensi a visu ultra corpus diaffonum subtilius corpore diaffono quod est in parte visus est in differentia communi linee per quam forma pervenit ad visum et perpendiculari que exit a puncto viso super superficiem corporis diaffoni quod est in parte visus.

5.18 ◉ Moreover, when [the visual faculty] perceives the slanted diameter as bent, it does not perceive the portion of it that passes out of the glass on the side of the center in its [actual] place. [Thus] the centerpoint is only perceived by the visual faculty [somewhere] other than in its [actual] place, and since the angle of bending faces the circumference [of the wooden disk], the point that constitutes the form of the center lies below the [actual] center, from which it is clear that the image of any point perceived by the visual faculty through a transparent body rarer than the transparent body on the side of the eye lies on the straight line dropped orthogonally from that point to the surface of the transparent body on the side of the eye, and it lies farther from the surface of the transparent body on the side of the eye than the [actual] point itself. And every point perceived by the visual faculty lies on the straight line along which the form reaches the center of sight, and the image of any point perceived by the visual faculty through a transparent body that is rarer than the transparent body on the side of the eye lies at the common section of the line along which the form reaches the eye and the normal dropped from the visible point to the surface of the transparent body on the side of the eye.

5.19 ◉ Ex omnibus ergo istis declaratis in hoc capitulo patet quod ymago cuiuslibet puncti cuiuslibet visi comprehensi a visu ultra corpus diaffonum diverse diaffonitatis a diaffonitate corporis quod est in parte visus, cum visus fuerit declinis a perpendicularibus exeuntibus ab illa re visa super superficiem corporis diaffoni quod est in parte visus, est in differentia communi linee per quam forma illius puncti pervenit ad visum et perpendicularis que exit ab illo puncto super superficiem corporis diaffoni quod est in parte visus, sive corpus diaffonum quod est in parte visus sit subtilius corpore diaffono quod est in parte rei vise aut grossius.

5.19 ◉ Hence, from everything that has been shown in this chapter it is evident that, when the eye lies to the side of the normals dropped from the visible object to the surface of the transparent body on the side of the eye, the image of any point on any visible object perceived by the visual faculty through a transparent body differing in transparency from the transparency of the body on the side of the eye lies at the common section of the line along which the form of that point reaches the eye and the normal dropped from that point to the surface of the transparent body on the side of the eye, whether the transparent body on the side of the eye is rarer or denser than the transparent body on the side of the visible object.

5.20 ◉ Quare autem visus comprehendit rem visam in loco ymaginis et quare ymago est in loco sectionis inter lineam per quam forma pervenit ad visum et inter perpendicularem que exit a puncto viso ad superficiem corporis diaffoni postea dicetur. Quod autem visus comprehendit formam puncti visi quam comprehendit reflexive etiam in rectitudine linee per quam forma pervenit ad visum manifestum est, et causa eius declarata est in predictis tractatibus, et est quoniam visus nichil comprehendit nisi in rectitudine linearum radialium, non enim patitur nisi in verticationibus istarum linearum.

5.20 ◉ Why the visual faculty perceives a visible object at the image-location and why the image lies at the intersection of the line along which the form reaches the eye and the normal dropped from the visible point to the surface of the transparent body will be explained as follows. Now it is manifest that the visual faculty perceives the form of a visible point that it perceives by means of refraction on the straight line along which the form reaches the eye, and the reason for this has been discussed in the preceding chapters, and it is because the visual faculty perceives nothing unless [it does so] along straight radial lines, for it is only affected along these lines.⁑

5.21 ◉ Quare autem comprehendit formam per perpendiculares exeuntes a re visa super superficiem corporis diaffoni est quia, ut in secundo declaravimus, quando lux extenditur in corpore diaffono, extenditur per motum velocissimum. Et in quarto capitulo huius tractatus declaravimus quod motus lucis in corpore diaffono super lineam declinem super superficiem illius corporis est compositus ex motu super perpendicularem exeuntem a puncto in quo extenditur lux super superficiem illius corporis diaffoni et ex motu super lineam que est perpendicularis super hanc perpendicularem. Forma autem que extenditur a puncto viso reflexive ad locum reflexionis, que est forma lucis existens in puncto viso mixta cum forma coloris, semper extenditur super lineam declinem super superficiem corporis diaffoni. Hec igitur forma extenditur ad locum reflexionis motu composito ex motu super perpendicularem que exit a puncto viso super superficiem corporis diaffoni et ex motu super lineam que est perpendicularis super hanc perpendicularem.

5.21 ◉ But the reason it perceives the form on the normals dropped from the visible object to the surface of the transparent body is that, as we showed in the second [Chapter], when light extends through a transparent body, it does so with an extraordinarily swift motion. And in the fourth chapter of this book [actually chapter 2, paragraphs 2.81-87] we showed that the motion of light through a transparent body along a line slanted to the surface of that body is composed of motion along the normal dropped from the point to which the light extends on the surface of that transparent body and motion along a line orthogonal to this perpendicular line. Consisting of the form of the light in a visible point mingled with the form of color, the form that extends according to refraction from the visible point to the point of refraction always extends along a line that is oblique to the surface of the transparent body. Hence, this form extends to the point of refraction with a motion composed of the motion along the normal dropped from the visible point to the surface of the transparent body and the motion along a line orthogonal to this normal.

5.22 ◉ Est ergo motus forme que movetur super perpendicularem que est super superficiem corporis diaffoni, et deinde translata est ab hac perpendiculari alio motu, aut super perpendicularem que existit super primam perpendicularem, et translata est post motum ipsius super primam perpendicularem motu composito ex predictis duobus motibus. Hoc autem punctum comprehenditur a visu in rectitudine linee per quam forma pervenit ad visum. Forma ergo existens in loco reflexionis pervenit ad ipsum per motum forme que movetur super lineam perpendicularem super superficiem corporis diaffoni, deinde translata est ab hac perpendiculari per motum in rectitudine linee per quam forma pervenit ad visum.

5.22 ◉ The motion of a form moving along the normal to the surface of the transparent body and then diverted from that normal by another motion or along the perpendicular to the first perpendicular and diverted after its motion on the first perpendicular therefore occurs with a motion composed of the two aforementioned motions. But this point is perceived by the visual faculty on the straight line along which the form reaches the center of sight. Hence, the form at the point of refraction reaches it according to the motion of a form moving along a line normal to the surface of the transparent body and then diverted from that normal by the motion on the straight line along which the form reaches the center of sight.

5.23 ◉ Forma autem que est super perpendicularem existentem super superficiem corporis diaffoni deinde movetur in rectitudine linee per quam forma extenditur ad visum est forma que extenditur a puncto viso in rectitudine perpendicularis exeuntis ex ipso super superficiem corporis diaffoni donec perveniat ad punctum sectionis inter hanc perpendicularem et lineam per quam forma extenditur ad visum. Forma igitur puncti quam visus comprehendit reflexive ultra corpus diaffonum est per motum forme que pervenit ad visum a loco ymaginis. Visus autem comprehendit hanc formam ex loco ymaginis, quia est per motum forme quam visus comprehendit recte et sine reflexione, et est locus qui distat a visu quantum punctus ymaginis, cuius situs in respectu visus est situs forme que est in loco ymaginis, unde visus comprehendit illud punctum secundum reflexionem in loco ymaginis.

5.23 ◉ Moreover, a form that lies on the normal to the surface of the transparent body and then moves on the straight line along which the form extends to the center of sight is a form that extends from the visible point in a straight line with the normal dropped from it to the surface of the transparent body until it reaches the point of intersection between this normal and the line along which the form extends to the center of sight. Thus, the form of a point that the visual faculty perceives by means of refraction through a transparent body results from the motion of the form that reaches the center of sight from the image-location. And the visual faculty perceives this form at the image-location because it results from the motion of the form that the visual faculty perceives straight on without refraction, and this location lies as far from the center of sight as the image-point whose location with respect to the center of sight is the location of the form at the image-location, so the visual faculty perceives that form at the image-location according to refraction.

5.24 ◉ Hec ergo est causa propter quam visus comprehendit rem visam ultra corpus diaffonum in loco ymaginis et propter quam ymago cuiuslibet puncti rei vise comprehense secundum reflexionem est in loco in quo linea per quam forma pervenit ad visum secat perpendicularem exeuntem ab illo puncto super superficiem corporis diaffoni.

5.24 ◉ This, then, is why the visual faculty perceives a visible object through a transparent body at the image-location and why the image of any point on a visible object that is perceived by means of refraction lies where the line along which the form reaches the eye intersects the normal dropped from that point to the surface of the transparent body.⁑

5.25 ◉ Hoc autem declarato, dicamus quod nullum visum comprehensum a visu ultra aliquod corpus diaffonum quod differt in diaffonitate a corpore quod est in parte visus, si corpus fuerit ex corporibus communibus, nichil habet nisi unam solam ymaginem. Corpora autem diaffona assueta sunt celum, et aer, et aqua, et vitrum, et lapides diaffoni, et superficies celi que est ex parte visus est sperica concava, unde omnis superficies equalis plana que secat eam facit in ea lineam circularem cuius concavitas est ex parte visus. Superficies autem aeris que tangit illam est sperica convexa, unde, si secetur a superficie equali, fiet in ipsa linea circularis cuius convexum est ex parte celi. Superficies vero aque que est ex parte visus est sperica convexa, et si secetur a superficie equali, fiet in ipsa linea circularis cuius convexum est ex parte visus.

5.25 ◉ Now that this has been shown, we should point out that no visible object perceived by the visual faculty through any transparent body that differs in transparency from the body on the side of the eye has more than one single image if the [transparent] body is among the common [transparent] bodies.⁑ The heavens, air, water, glass, and transparent stones are customary transparent bodies, and the surface of the heavens that faces the eye is concave spherical, so every plane that cuts it forms a circular line on it with its concavity facing the eye. The surface of the air in contact with it is convex spherical, so, if it is cut by a plane, [that cut] will form a circular line on it with its convexity facing the heavens. The surface of water facing the eye, on the other hand, is spherical convex, and if it is cut by a plane surface, [that cut] will form circular line on it with its convexity facing the eye.⁑

5.26 ◉ Vitrorum autem et lapidum diaffonorum figure assuete sunt rotunde aut plane, unde, si secentur a planis superficiebus, fient in illis aut circuli aut linee recte. Et universaliter dicimus quod omne punctum comprehensum a visu ultra quodcumque corpus diaffonum cuius superficies que opponitur visui est una superficies et quod, si secetur a superficie equali, fiet in superficie eius linea recta aut circularis non habet hoc punctum nisi unam ymaginem, nec comprehenditur a visu nisi unum punctum tantum.

5.26 ◉ The customary shapes of glass [objects] and transparent stones are round or flat, so if they are cut by planes, [those cuts] will form either circles or straight lines on them. And [so] we say that, generally speaking, every point perceived by the visual faculty through any transparent body whose surface facing the eye is a single [simple] surface, has only one image and is perceived as only one point, if, when [that surface] is cut by a plane surface, [that cut] will form a straight or circular line on its surface.

5.27 ◉ [PROPOSITIO 2] Sit ergo visus A [FIGURE 7.5.2, p. 441] et punctum visibile B. Et corpus diaffonum ultra quod est B sit illud in cuius superficie est G, et sit diaffonitas huius corporis grossior diaffonitate corporis quod est ex parte visus. Et sit superficies eius que est ex parte visus equalis, et extrahamus a puncto A perpendicularem AGC. Punctum ergo B aut erit super lineam AGC, aut erit extra ipsam.

5.27 ◉ [PROPOSITION 2] So let A [in figure 7.5.49, p. 184] be the center of sight and B the visible point. Let the transparent body in which B lies have [point] G on its surface, and let the transparency of this body be denser than the transparency of the body on the side of the eye [at A]. Let the surface of the body on the side of the eye be plane, and let us erect perpendicular AGC from point A. Hence, point B will either lie on line AGC, or it will lie outside it.

5.28 ◉ Si ergo punctum B fuerit in linea GC, tunc visus A comprehendet B recte et sine reflexione, nam forma B, quando extenditur per BG, exit ad corpus quod est in parte A in rectitudine BG, nam BG est perpendicularis super superficiem corporis diaffoni quod est ex parte visus. Visus ergo A comprehendit B in suo loco et in rectitudine AGB.

5.28 ◉ Accordingly, if point B lies on line GC, the center of sight at A will perceive B straight on without refraction, for when the form of B extends along BG, it passes out to the [transparent] body on the side of A straight along BG because BG is perpendicular to the surface of the [transparent] body on the side of the eye. Therefore, the center of sight at A perceives B in its [actual] location along straight line AGB.

5.29 ◉ Dicimus ergo quod punctum B extra hanc lineam numquam reflectitur ad A, quod si sit possibile, reflectitur forma B ad A ex T. Et extrahamus superficiem in qua est perpendicularis AGB et punctum T. Faciet ergo in superficie corporis diaffoni lineam rectam. Sit ergo GTD, et extrahamus a puncto T perpendicularem super lineam GD, et sit KTL, Erit ergo KTL perpendicularis super superficiem corporis diaffoni. Et continuemus BT, et extrahamus illam ad H.

5.29 ◉ We say, then, that [the form of] point B is never refracted to A outside this line, but if it is possible [for it to do so, let us suppose that] the form of B is refracted to A from T. Let us produce the plane containing perpendicular AGB and point T. It will therefore cut a straight line on the surface of the transparent body. Let it be GTD, then, and let us extend a perpendicular to line GD from point T, and let it be KTL. KTL will thus be perpendicular to the transparent body’s surface. Let us then connect BT and extend it to H.

5.30 ◉ Erit ergo angulus KTH ille quem continet linea per quam extenditur forma et perpendicularis exiens a loco reflexionis super superficiem corporis diaffoni. Quia ergo corpus quod est ex parte A est subtilius illo quod est ex parte B, cum B pervenit ad T, reflectetur ad partem contrariam illi in qua est perpendicularis TK. Non ergo pervenit forma reflexa ad lineam AB, sed est ex parte reflexa ad punctum A, quod est impossibile. Non ergo reflectetur forma B ad A ex T, nec ex alio puncto. A ergo non comprehendet B nisi ex rectitudine AGB; non ergo comprehendit ipsum nisi ex puncto uno tantum, et hoc voluimus declarare.

5.30 ◉ Angle KTH will thus be [equal to] the angle [BTL] that the line along which the form extending [to the surface of the transparent body] makes with the normal dropped from the point of refraction to the surface of the transparent body. Consequently, since the [transparent] body on the side of A is rarer than the one on the side of B, then when [the form of] B reaches T, it will be refracted away from normal TK. Hence, the refracted form does not reach line AB, yet it is [supposedly] refracted to A, which is impossible. The form of B will therefore not be refracted to A from T, nor from any other point [outside line AGB]. Hence, [the center of sight at] A will only perceive B along straight line AGB, so it perceives it at only one point, and this [is what] we wanted to prove.

5.31 ◉ [PROPOSITIO 3] Si ergo B fuerit extra AGC [FIGURE 7.5.3, p. 441], extrahamus superficiem in qua est AGC linea et punctum B. Ergo erit perpendicularis super superficiem corporis diaffoni, et fiat in superficie huius corporis linea GD. Erit ergo GD recta. Non ergo reflectetur forma B ad A nisi in superficie in qua est GD, non enim transit per duo puncta A, B superficies perpendicularis super superficiem corporis diaffoni aut superficies transiens per perpendicularem AC, et non transit per perpendicularem AC et per punctum B superficies equalis nisi una sola tantum. Forma ergo B non reflectitur ad A nisi ex linea GD.

5.31 ◉ [PROPOSITION 3] If B lies outside AGC [as in figures 7.5.50 and 7.5.50a, p. 185], let us form the plane containing line AGC and point B. It will thus be perpendicular to the surface of the transparent body, and let it form line GD on the surface of this body. GD will thus be straight. Hence, the form of B will be refracted to A only in the plane containing GD, for only one plane perpendicular to the surface of the transparent body passes through the two points A and B, or passes through perpendicular AC, and [also] passes through perpendicular AC and point B. Thus, the form of B is refracted to A only from line GD.

5.32 ◉ Reflectatur ergo forma B ad A a puncto E, et continuemus duas lineas BE, EA, et extrahamus ex E perpendicularem super lineam GED. Sit ergo HEZ. Erit ergo HEZ perpendicularis super duas superficies duorum corporum diaffonorum. Et extrahamus BE recte ad T. Erit ergo ET inter duas lineas EH, EA, nam corpus diaffonum quod est ex parte A est subtilius illo quod est ex parte B. Forma ergo B, que extenditur per lineam BE, cum pervenerit ad E, reflectetur ad partem contrariam parti perpendicularis ZEH; ideo erit linea ET inter duas lineas EH, EA.

5.32 ◉ So let the form of B be refracted to A from point E, and let us connect the two lines BE and EA and erect a perpendicular from E on line GED. Let it be HEZ, then. HEZ will thus be perpendicular to the two surfaces of the two [contiguous] transparent bodies [on the sides of A and B]. Let us extend BE straight to T. ET will therefore lie between the two lines EH and EA, for the transparent body on the side of A is rarer than that on the side of B. Consequently, when the form of B, which extends along line BE, reaches E, it will be refracted away from normal ZEH, so line ET will lie between the two lines EH and EA.

5.33 ◉ Et extrahamus ex B perpendicularem super lineam GD, scilicet BK. Erit ergo BK perpendicularis super superficiem corporis diaffoni quod est ex parte B. Et extrahamus AE recte ut secet angulum BEK, et secet lineam BK in M. M ergo erit ymago puncti B, et angulus TEA erit angulus reflexionis. Dico ergo quod B non habebit aliam ymaginem preter M, nec forma eius reflectetur ad A ex alio puncto quam ex E.

5.33 ◉ From B let us drop a perpendicular, i.e., BK, to line GD. BK will thus be perpendicular to the surface of the transparent body on the side of B. Let us also extend AE straight so that it cuts angle BEK and intersects line BK at M. M will thus be the image of point B, and angle TEA will be the angle of refraction. I say, then, that B will have no image other than M, and its form will not be refracted to A from any point other than E.

5.34 ◉ Huius demonstratio est quoniam demonstratum est quod B non comprehenditur a visu nisi per perpendicularem BK. Si ergo B aliam habuerit ymaginem, erit in linea BK et inter duo puncta B, K, corpus enim quod est ex parte B est grossius illo quod est ex parte A. Sit ergo illa alia ymago, si possibile est, punctum N. Erit ergo N aut inter duo puncta M, K aut inter duo puncta M, B.

5.34 ◉ The proof of this claim is based on the proof that [the form of] B is perceived by the center of sight only on normal BK [i.e., the cathetus of incidence]. Thus, if B has another image, it will lie on line BK and between the two points B and K, for the [transparent] body on the side of B is denser than that on the side of A. So let that other image be point N, if such is possible. N will therefore lie either between the two points M and K or between the two points M and B.

5.35 ◉ Et continuemus AN. Secabit ergo lineam GD in puncto O. Et continuemus BO, et transeat usque ad L. Erit ergo O punctum reflexionis, quia linea AON est illa per quam extenditur forma, que est apud N, ad A, et erit angulus LOA angulus reflexionis. Et extrahamus ex O perpendicularem super lineam GD, et sit FOQ. Erit ergo linea FOQ perpendicularis super superficiem corporis diaffoni, et erit angulus LOF angulus quem continet perpendicularis et linea per quam extenditur forma ad locum reflexionis.

5.35 ◉ [Let it lie between M and K, as in figure 7.5.50, p. 185] and let us connect AN. Hence, it will intersect line GD at point O. Let us connect BO, and let it continue to L. O will thus be the point of refraction because line AON is the one along which the form [of the point] at N extends to A, and [so] angle LOA will be the angle of refraction. From O let us drop perpendicular FOQ to line GD. Line FOQ will thus be perpendicular to the surface of the transparent body, and angle LOF will be [equal to] the angle [of incidence BOQ] that the normal makes with the line along which the form extends to the point of refraction.

5.36 ◉ Si autem N fuerit inter duo puncta M, K, tunc O erit inter duo puncta E, K; angulus ergo EBK erit maior angulo OBK. Angulus ergo TEH est maior angulo LOF. Et angulus TEA est angulus reflexionis ex angulo TEH, et angulus LOA est angulus reflexionis ex angulo LOF. Angulus ergo TEA est maior angulo LOA, ut declaratum est in tertio capitulo huius tractatus; angulus ergo AEH est maior angulo AOF quod est impossibile.

5.36 ◉ Now if N lies between the two points M and K, O will lie between the two points E and K, so angle EBK > angle OBK. Consequently, angle TEH [which = corresponding angle EBK] > angle LOF [which = corresponding angle OBK]. But angle TEA is the angle of refraction for angle [of incidence BEZ, which = angle] TEH, whereas angle LOA is the angle of refraction for angle [of incidence BOQ, which = angle] LOF. Therefore, angle TEA > angle LOA, as was demonstrated in the third chapter of this book [where it was shown that a larger angle of incidence yields a larger angle of refraction], so angle AEH > angle AOF, which is impossible.⁑

5.37 ◉ Si autem N fuerit inter duo puncta M, B [FIGURE 7.5.3a, p. 441], tunc punctus E erit inter duo puncta O, K, et erit angulus EBK minor angulo OBK. Erit ergo angulus TEH minor angulo LOF; erit ergo angulus TEA, qui est angulus reflexionis, minor angulo LOA, qui est angulus reflexionis. Angulus ergo AEH est minor angulo AOF, quod est impossibile. Ergo impossibile est quod punctum N sit ymago puncti B, nec aliud punctum ab M; ergo punctum B respectu visus A nullam habet ymaginem preter quam punctum M, et hoc declarare debuimus.

5.37 ◉ If, however, N lies between the two points M and B [in figure 7.5.50a, p. 185], then point E will lie between the two points O and K, and angle EBK < angle OBK. Therefore, angle TEH < angle LOF, so angle TEA, which is the angle of refraction [for angle of incidence BEZ], is less than angle LOA, which is the angle of refraction [for angle of incidence BOQ]. Hence, angle AEH < angle AOF, which is impossible. It is therefore impossible for point N or any point other than M to be the image of point B, so with respect to center of sight A, point B has no image other than point M, and this [is what] we had to demonstrate.

5.38 ◉ [PROPOSITIO 4] Et iterum sit corpus grossius ex parte visus et subtilius ex parte rei vise, et sit differentia communis inter hanc superficiem et superficiem corporis diaffoni linea GD [FIGURE 7.5.4, p. 441]. Extrahamus ex B perpendicularem super lineam GD, et sit BK. Erit ergo BK perpendicularis super superficiem corporis diaffoni. Et reflectatur forma B ad A ex E, et continuemus BE, EA. Et extrahamus perpendicularem HE, et extrahamus BE recte ad T.

5.38 ◉ [PROPOSITION 4] To continue, let the denser [transparent] body lie on the side of the eye [at A in figures 7.5.51 and 7.5.51a, p. 186] and the rarer one on the side of the visible object [B], and let the common section of this plane [containing A and B] and the surface of the transparent body be line GD. Let us drop a perpendicular from B to line GD, and let it be BK. BK will thus be perpendicular to the surface of the transparent body. And let the form of B be refracted to A from E, and let us connect [lines] BE and EA. Let us then erect perpendicular HE and extend BE straight to T.

5.39 ◉ Erit ergo AE linea media inter duas lineas ET, EH, nam prima linea per quam extenditur forma ad locum reflexionis est linea BET. Reflexio autem est ad partem perpendicularis EH, nam corpus quod est ex parte A est grossius illo quod est ex parte B; linea ergo AE est media inter duas lineas ET, EH. Et extrahamus directe AE ad partem E quousque occurrat linee BK, secabit enim HEZ. Occurret ergo illi in puncto M. M ergo erit ymago puncti B, nam corpus quod est ex parte B est subtilius illo quod est ex parte A. Dico ergo quod B non habet ymaginem nisi M.

5.39 ◉ Line AE will therefore lie between the two lines ET and EH, for the first line along which the form extends to the point of refraction is line BET. However, the refraction is toward normal EH because the [transparent] body on the side of A is denser than the one on the side of B, so line AE lies between the two lines ET and EH. Let us extend AE straight on the side of E until it meets line BK, for it will cut HEZ [parallel to BK]. So let it meet it at point M. M will therefore be the image of point B because the [transparent] body on the side of B is rarer than the one on the side of A. I say, then, that B has no image except for M.

5.40 ◉ Habeat ergo N, si possibile est. N ergo erit in perpendiculari BK et infra punctum B, quia corpus quod est ex parte B est subtilius illo quod est ex parte A. Est ergo aut inter duo puncta M, B aut infra M. Et continuemus AN. Secabit ergo lineam GD in O; O ergo est punctum reflexionis. Et continuemus BO; et transeat usque ad L et extrahamus ex O perpendicularem FOQ. Linea ergo BO est illa per quam extenditur forma ad locum reflexionis; ergo linea AO erit inter duas lineas OL, OF, reflexio enim est ad partem perpendicularis.

5.40 ◉ So if it is possible, let it have N [as an image]. N will therefore lie on normal BK and below point B, since the [transparent] body on the side of B is rarer than that on the side of A. Hence, it lies between the two points M and B or below M. Let us connect AN. It will therefore intersect line GD at O, so O is the point of refraction. Let us connect BO; let it continue to L, and let us erect perpendicular FOQ at O. Hence, line BO is the one along which the form extends to the point of refraction; so line AO will lie between the two lines OL and OF because the refraction occurs toward the normal [FO].

5.41 ◉ Si ergo N fuerit inter duo puncta M, B, tunc punctum O erit inter duo puncta E, K. Angulus ergo OBK est minor angulo EBK; ergo angulus LOF est minor angulo TEH. Ergo angulus LOA, qui est angulus reflexionis, est minor angulo TEA, qui est angulus reflexionis. Et angulus AOF, qui remanet post angulum reflexionis, est minor angulo AEH, qui remanet post angulum reflexionis, sicut declaravimus in tertio capitulo huius tractatus. Sed angulus AOF est equalis angulo ANK, et angulus AEH est equalis angulo AMK; ergo angulus ANK est minor angulo AMK, quod est impossibile.

5.41 ◉ If, therefore, N lies between the two points M and B [in figure 7.5.51], point O will lie between the two points E and K. Consequently angle OBK < angle EBK, so angle LOF < angle TEH. Angle LOA, which is the angle of refraction [for angle of incidence BOQ, which = angle LOF], is therefore less than angle TEA, which is the angle of refraction [for angle of incidence BEZ, which = angle TEH]. As we demonstrated in the third chapter of this book, moreover, angle AOF, which remains after angle of refraction [LOA is subtracted from LOF], is less than angle AEH, which remains after angle of refraction [TEA is subtracted from TEH]. But angle AOF = [corresponding] angle ANK, and angle AEH = [corresponding] angle AMK, so angle ANK < angle AMK, which is impossible.

5.42 ◉ Si autem N fuerit infra M [FIGURE 7.5.4a, p. 442], tunc E erit inter duo puncta O, K, et erit angulus OBK maior angulo EBK; angulus ergo LOF erit maior angulo TEH. Ergo angulus LOA est maior angulo TEA. Et angulus AOF est maior angulo AEH; ergo angulus ANK est maior angulo AMK, quod est impossibile. N ergo non est ymago B, nec aliud punctum preter quam M; B ergo non habet ymaginem nisi M, et hoc est quod voluimus.

5.42 ◉ On the other hand, if N lies below M [in figure 7.5.51a], E will lie between the two points O and K, and angle OBK > angle EBK, so angle LOF > angle TEH. Therefore, angle LOA > angle TEA. But [that requires that] angle AOF > angle AEH, from which it follows that angle ANK > angle AMK, which is impossible. N is therefore not an image for B, nor is any point other than M, so B has no image other than M, and this is what we wanted [to demonstrate].

5.43 ◉ [PROPOSITIO 5] Ad duas autem lineas circulares convexam et concavam premittemus hoc: cum due corde sese secuerint in circulo, angulus sectionis erit equalis angulo qui est apud circumferentiam quem cordant duo arcus quos distinguunt ille due corde, et si due linee secuerint circulum et secuerint se extra circulum, angulus sectionis erit equalis angulo qui est apud circumferentiam quem cordat excessus maioris illorum duorum arcuum quos distinguunt ille due linee super reliquum.

5.43 ◉ [PROPOSITION 5, LEMMA 1] We will [now] make the following preliminary point about two convex or concave circular lines: if two chords intersect each other within a circle, the angle of intersection will be equal to the angle at the circumference that the two arcs those two chords mark off subtend, and if the two lines intersect the circle but intersect each other outside the circle, the angle of intersection will be equal to the angle at the circumference subtended by the difference between the larger and smaller of those two arcs that those two lines mark off [on the circumference].

5.44 ◉ Verbi gratia, in circulo ABG [FIGURE 7.5.5, p. 442] secent se due corde AG, BD in E. Dico ergo quod angulus AEB est equalis angulo qui est in circumferentia quem respiciunt duo arcus AB, GD, et quod angulus BEG est equalis angulo in circumferentia quem respiciunt duo arcus AD, GB.

5.44 ◉ For instance, [in figure 7.5.52, p. 187] let the two chords AG and BD in circle ABG intersect one another at E. I say, then, that angle AEB is equal to the angle on the circumference that the two arcs AB and GD subtend, and that angle BEG is equal to the angle on the circumference that the two arcs AD and GB subtend.

5.45 ◉ Probatio huius: extrahemus ex B lineam HBZ equidistantem linee AG. Arcus ergo GZ est equalis arcui AB, et arcus GD est communis; ergo arcus DZ est equalis duobus arcubus AB, GD. Sed arcus DZ respicit angulum DBZ; ergo DZ respicit arcus equales duobus arcubus AB, GD. Et angulus DBZ est equalis angulo AEB; ergo angulus AEB equalis angulo qui est in circumferentia quem respiciunt duo arcus AB, GD, et hoc est quod voluimus.

5.45 ◉ The proof of this [claim is as follows]. We will draw line HBZ from [point] B parallel to line AG. Thus, arc GZ = arc AB, and arc GD is common, so arc DZ is equal to the two arcs AB + GD. But arc DZ subtends angle DBZ, so DZ spans an arc equal to the two arcs AB + GD. But angle DBZ = [alternate] angle AEB, so angle AEB = the angle on the circumference that the two arcs AB + GD subtend, and this is what we wanted [to demonstrate].

5.46 ◉ Item continuemus DZ. Erit ergo angulus HBE equalis duobus angulis BDZ, BZD, et duo anguli BZD, BDZ respiciuntur a duobus arcubus DB, BZ; angulus ergo HBE est equalis angulo quem respicit arcus DB, BZ. Et arcus AB est equalis arcui ZG, et arcus DABZ est equalis duobus arcubus DA, BG; ergo angulus HBE est equalis angulo quem respiciunt duo arcus DA, BG. Et angulus HBE est equalis angulo BEG; ergo angulus BEG est equalis angulo qui est in circumferentia quem respiciunt duo arcus DA, BG, et hoc est quod voluimus declarare.

5.46 ◉ To continue, let us connect DZ. Therefore, [exterior] angle HBE [of triangle BDZ] is equal to the two [interior] angles BDZ + BZD, and the two angles BZD and BDZ are subtended by the two arcs DB and BZ, so angle HBE is equal to the angle that arcs DB + BZ subtend. Arc AB = arc ZG [by construction], and arc DABZ is equal to the two arcs DA + BG, so angle HBE is equal to the angle that the two arcs DA + BG subtend. But angle HBE = [alternate] angle BEG, so angle BEG is equal to the angle on the circumference that the two arcs DA + BG subtend, and this is what we wanted to demonstrate.

5.47 ◉ Et si linea HBZ fuerit contingens circulum [FIGURE 7.5.5a, p. 442], tunc angulus EBZ erit equalis angulo cadenti in portione BAD, et sic arcus BGD respicit angulum apud circumferentiam equalem angulo EBZ. Et angulus EBZ est equalis angulo BEA. Ergo angulus BEA est equalis angulo qui est apud circumferentiam quem respicit arcus BGD, et arcus BG est equalis arcui BA, quia diameter qui exit ex B est perpendicularis super lineam AG, quare dividit ipsum in duo equalia. Ergo arcus AB erit equalis arcui BG; arcus ergo BGD erit equalis duobus arcubus BA, GD. Ergo angulus BEA est equalis angulo qui est apud circumferentiam quem respiciunt duo arcus AB, GD. Et similiter declarabitur quod angulus BEG est equalis angulo qui est apud circumferentiam quem respiciunt duo arcus BG, AD, et hoc est quod voluimus.

5.47 ◉ If, moreover, line HBZ is tangent to the circle [in figure 7.5.52a, p. 187], then angle EBZ will be equal to an angle [with its vertex] falling on segment BAD [of the circle, by Euclid, III.32], and so arc BGD subtends an angle at the circumference equal to angle EBZ. But angle EBZ = [alternate] angle BEA. Therefore, angle BEA is equal to an angle at the circumference that arc BGD subtends, and arc BG = arc BA because the diameter that passes from B is perpendicular to line AG, since it bisects it.⁑ Therefore, arc AB = arc BG, so arc BGD will be equal to the two arcs BA + GD. Angle BEA is thus equal to the angle at the circumference that the two arcs AB and GD subtend. And it will be demonstrated likewise that angle BEG is equal to the angle at the circumference that the two arcs BG and AD subtend, and this is what we wanted [to demonstrate].⁑

5.48 ◉ Item sit E extra circulum ABGD [FIGURE 7.5.5b, p. 443], et extrahamus ex E duas lineas secantes circulum ABGD, et sint EAD, EBG. Dico ergo quod angulus GED est equalis angulo qui est apud circumferentiam quem respicit excessus arcus DG super arcum AB.

5.48 ◉ Now let E lie outside circle ABGD [in figure 7.5.52b, p. 188], let us extend two lines from E to intersect circle ABGD, and let them be EAD and EBG. I say, then, that angle GED is equal to the angle at the circumference that the difference between arc DG and arc AB subtends.

5.49 ◉ Huius demonstratio: extrahamus lineam equidistantem linee BG. Erit ergo arcus ZG equalis arcui AB; erit ergo arcus DZ excessus arcus DG super arcum AB. Sed arcus DZ respicit angulum DAZ, et angulus DAZ est equalis angulo GED. Ergo angulus GED est equalis angulo qui est apud circumferentiam DAZ, et hoc est quod voluimus.

5.49 ◉ The demonstration of this [claim is as follows]. Let us draw line [AZ] parallel to line BG. Arc ZG will therefore be equal to arc AB, so arc DZ is the difference between arc DG and arc AB. But arc DZ subtends angle DAZ, and angle DAZ = [corresponding] angle GED. Therefore, angle GED = angle DAZ on the circumference, and this is what we wanted [to demonstrate].

5.50 ◉ [PROPOSITIO 6] Hiis declaratis, sit visus punctum A [FIGURE 7.5.6, p. 443], et sit punctum B in aliquo viso, et sit ultra corpus diaffonum grossius corpore quod est in parte visus. Et sit superficies corporis diaffoni quod est ex parte B superficies circularis convexa ex parte visus. Ergo per duo puncta A, B transit superficies perpendicularis super superficiem corporis diaffoni, et non transit per illa superficies perpendicularis super superficiem corporis diaffoni in qua reflectitur forma B ad A nisi una tantum. Hanc ergo superficiem corporis diaffoni signet circulus GED, cuius centrum quidem sit Z, et continuemus AGD. Linea ergo GZD erit perpendicularis super superficiem corporis diaffoni; punctum autem B aut erit extra lineam GD aut in ipsa.

5.50 ◉ [PROPOSITION 6] Now that these [points] have been established, let point A [in figure 7.5.53, p. 188] be the center of sight, let point B lie on some visible object, and let it lie in a transparent body denser than the body that lies on the side of the center of sight. Let the surface of the transparent body on the side of B be a circular surface with its convexity facing the center of sight. Therefore, [there is] a plane [that] passes through both points A and B and is perpendicular to the surface of the transparent body, but only one plane passes through the two points A and B orthogonal to the surface of the transparent body [and] in that plane the form of B is refracted to A. Let circle GED, whose center lies at Z, represent this plane on the transparent body, and let us draw AGD. Line GZD will thus be perpendicular to the surface of the transparent body, and point B will lie either outside line GD or on it.

5.51 ◉ Si ergo B fuerit in linea GD, tunc visus A comprehendet B recte et sine reflexione, nam forma que extenditur per lineam GD extenditur recte in corpore diaffono quod est ex parte visus A, quia linea GD est perpendicularis super superficiem corporis diaffoni quod est ex parte visus. Visus ergo A comprehendit B in suo loco et recte. Dico ergo quod forma B quod est in linea GD numquam reflectitur ad A.

5.51 ◉ Accordingly, if B lies on line GD, then center of sight A will perceive B straight on without refraction, for the form extending along line GD extends straight through the transparent body on the side of center of sight A, since line GD is perpendicular to the surface of the transparent body on the side of the center of sight. Consequently, center of sight A perceives B in its [actual] location and [it does so] directly. I say, then, that the form of B on line GD is never refracted to A.

5.52 ◉ Huius demonstratio, quoniam punctum B aut erit in centro aut extra centrum. Si ergo fuerit in centro, tunc omnis linea per quam extenditur forma B ad circumferentiam GED in rectitudine eius extenditur in corpore diaffono quod est ex parte eius, nam omnis linea exiens a centro circuli GED est perpendicularis super superficiem corporis diaffoni, et non exit a centro circuli GED ad visum A linea recta nisi linea ZA. Ergo forma B que est in centro non reflectitur ad A ex circumferentia GED; ergo forma B numquam reflectitur ad A, si B fuerit in centro.

5.52 ◉ The proof of this [point depends on the fact] that point B will lie either at the center or outside the center [of the circle]. If, then, it lies at the center, every line along which the form of B extends straight to circumference GED extends through the transparent body on its side, for every line extending from the center of circle GED is perpendicular to the surface of the transparent body, and no straight line other than line ZA extends from the center of circle GED to center of sight A. Thus, the form of B at the center is not refracted to A at circumference GED, so the form of B is never refracted to A if B lies at the center.

5.53 ◉ Si vero fuerit extra centrum, aut erit in linea ZG aut in ZD. Sit ergo primo in linea ZG. Dico quod forma B non reflectitur ad A, quod si fuerit possibile, reflectatur ex puncto E. Et continuemus BE, et extrahamus illud ad H, et continuemus ZE, et extrahamus ipsum ad T. Erit ergo linea ZET perpendicularis super superficiem corporis diaffoni quod est ex parte visus. Forma ergo B, quando extenditur ad lineam BE et reflectitur in puncto E, transit a perpendiculari TE ad partem H ad partem contrariam illi in qua est perpendicularis. Forma ergo B non perveniet ad A secundum reflexionem, si B fuerit in linea ZG.

5.53 ◉ If, on the other hand, it lies outside the center, it will lie either on line ZG or on [line] ZD. For a start, then, let it lie on line ZG. I say that the form of B is not refracted to A, but if it is possible, let it be refracted at E. Let us connect BE and continue [that] line to H, and let us connect ZE and continue [that] line to T. Line ZET will therefore be normal to the surface of the transparent body on the side of the center of sight [at A]. Hence, when it extends along line BE and is refracted at point E, the form of B passes through normal TE[Z] toward H away from the normal. Thus, the form of B will not reach A by refraction if B lies on line ZG.

5.54 ◉ Item sit B in linea DZ [FIGURE 7.5.6a. p. 443]. Dico ergo quod forma B non reflectitur ad A, quod si est possibile, reflectatur ex E. Et continuemus BE, et extrahamus lineam ad R, et continuemus ZE, et extrahamus lineam usque ad T. Et reflectatur forma B ad A per lineam EA. Sic ergo angulus REA erit angulus reflexionis; angulus autem RET erit angulus quem continet linea per quam extenditur forma et perpendicularis exiens a loco reflexionis. Angulus ergo REA est minor angulo RET, et linea BZ aut minor linea ZE aut equalis ei, nam B aut est inter duo puncta D, Z aut in puncto D. Ergo angulus EBZ aut est maior angulo BEZ aut equalis ei. Sed angulus AER est maior angulo EBZ; ergo angulus AER est maior angulo BEZ. Ergo angulus AER est maior angulo RET, quo prius erat minor, quod est impossibile.

5.54 ◉ Now let B lie on line DZ [at point B’]. I say, then, that the form of B’ is not refracted to A, but if it is possible, let it be refracted at E. Let us connect [line] B’E and extend the line to R, and let us connect ZE and extend the line to T. Let the form of B’ be refracted to A along line EA. Thus, angle REA will be the angle of refraction, and angle RET will be [equal to] the angle [of incidence B’EZ] that line [B’E] along which the form extends makes with normal [ZET] dropped from the point of refraction. Angle REA is therefore smaller than angle RET [by rule 5, p. 260 above], and line B’Z is either less than or equal to line ZE, for B’ lies either between points D and Z or at point D. Consequently, angle EB’Z is either greater than or equal to angle B’EZ. But angle AER > angle EB’Z, so angle AER > [angle] B’EZ. Therefore, angle AER > angle RET, than which it was previously [assumed to be] smaller, which is impossible.

5.55 ◉ Ergo forma B non reflectetur ad A ex E, nec ex alio puncto circumferentie GED, neque ex alia circumferentia circulorum qui fuerit in superficie corporis diaffoni in quo est B. B ergo existente in linea GD non comprehenditur a visu reflexive, quare non comprehenditur nisi unum punctum solum.

5.55 ◉ [No matter where B is on line GD], then, the form of B will not be refracted to A from E, nor from any other point on circumference GED, nor from any other circumference on the [great] circles that lie on the surface of the transparent body in which B lies. Hence, if B lies on line GD, it is not perceived by the visual faculty according to refraction, so it is perceived as only one point.

5.56 ◉ Item sit B extra lineam GD [FIGURE 7.5.6b, p. 444], et extrahemus superficiem in qua est perpendicularis AD et punctum B. Hec ergo superficies erit perpendicularis super superficiem corporis diaffoni, et punctum B non reflectitur ad A nisi in hac superficie, non enim transit per duo puncta A, B superficies perpendicularis super superficiem corporis diaffoni nisi illa que transit per lineam AD, et non exit ex linea AD superficies que transit per B nisi una tantum. Hec ergo superficies signet in superficie corporis diaffoni circulum GED. Forma ergo B non reflectetur ad A nisi ex circumferentia GED; reflectatur ergo ex E. Dico ergo quod non reflectitur ex alio puncto quam E.

5.56 ◉ To continue, let B lie outside line GD [in figure 7.5.53a, p. 189], and we will produce the plane containing normal AD and point B. This plane will therefore be perpendicular to the surface of the transparent body, and [the form of] point B is refracted to A only in this plane, for no plane perpendicular to the transparent body’s surface passes through the two points A and B except the one passing through line AD, and only one plane through line AD passes through B. So let this plane cut circle GED on the surface of the transparent body. The form of B will therefore be refracted to A only from the circumference of GED, so let it be refracted at E. I say, then, that it is refracted from no point other than E.

5.57 ◉ Reflectatur ergo, si possibile est, ex alio puncto, qui, ut dictum est, erit in circumferentia GED. Sit ergo M. Et continuemus lineas BE, EA, BM, MA, ZE, ZM, et secent se linee BM, ZE in C. Et extrahamus BE usque ad H, et BM ad N, et EZ ad T, et ZM ad L. Erit ergo angulus HET ille quem continet linea per quam extenditur forma et perpendicularis exiens a loco reflexionis, et angulus HEA erit angulus reflexionis, et NML angulus ille quem continet linea per quam extenditur forma et perpendicularis exiens a loco reflexionis, et angulus NMA erit angulus reflexionis.

5.57 ◉ So, if it is possible, let it be refracted from another point, which, as was [just] said, will lie on circumference GED. Let [that point] be M. Let us connect lines BE, EA, BM, MA, ZE, and ZM, and let lines BM and ZE intersect one another at C. Let us extend BE to H, BM to N, EZ to T, and ZM to L. Therefore, angle HET will be [equal to] the angle [of incidence BEZ] that the line along which the form extends makes with the normal dropped from the point of refraction, and angle HEA will be the angle of refraction; NML will also be [equal to] the angle [of incidence BMZ] that the line along which the form extends makes with the normal dropped from the point of refraction, and angle NMA will be the angle of refraction.

5.58 ◉ Angulus HET aut erit equalis angulo NML, aut erit minor, aut maior. Si equalis, angulus HEA, qui est angulus reflexionis, erit equalis angulo NMA, qui est angulus reflexionis. Angulus ergo AMB erit equalis angulo AEB, quod est impossibile. Si minor, erit angulus HEA minor angulo NMA; angulus ergo AMB erit minor angulo AEB, quod est impossibile.

5.58 ◉ Angle HET will be equal to, less than, or greater than angle NML. If it is equal, then angle HEA, which is the angle of refraction [tied to HET], will be equal to angle NMA, which is the angle of refraction [tied to NML]. Therefore, angle AMB [supplementary to NMA] = angle AEB [supplementary to HEA], which is impossible. If it [i.e. HET] is less [than NML], then angle HEA < angle NMA, so angle AMB < angle AEB, which is impossible.

5.59 ◉ Si maior, extrahemus lineam EB in partem B ad F, et extrahemus MB usque ad O. Angulus ergo EBM erit equalis angulo qui est apud circumferentiam quem respiciunt duo arcus EM, FO, et cum angulus HET fuerit maior angulo NML, erit angulus ZEB maior angulo NML. Et cum angulus ZEB fuerit maior angulo NML, angulus MZT erit maior angulo MBE, et excessus anguli MZE super angulum MBE erit equalis excessui anguli ZEB super angulum ZMB, nam duo anguli apud C sunt equales. Arcus ergo qui respicit angulum MZE, cum fuerit apud circumferentiam, erit duplus ad arcum ME.

5.59 ◉ If it [i.e. HET] is greater [than NML, which is the case in figure 7.5.53a], we will extend line EB to F on the side of B, and we will extend MB to O. Thus, angle EBM will be equal to the angle at the circumference that the two arcs EM + FO subtend [by proposition 5, lemma 1], and since angle HET > angle NML [by supposition], angle ZEB > angle NML. Since, moreover, angle ZEB > angle NML, angle MZT > angle MBE,⁑ and angle MZE – angle MBE = angle ZEB – angle ZMB, for the two angles at C are equal.⁑ Therefore, the arc subtending angle MZE will be twice arc ME, when [the angle subtended by it] lies at the circumference.

5.60 ◉ Si ergo angulus MZE fuerit maior angulo MBE, tunc arcus ME duplicatus erit maior duobus arcubus ME, FO. Et erit excessus arcus ME duplicati super duos arcus ME, FO equalis excessui arcus ME super arcum FO. Excessus ergo anguli MZE super angulum MBE est ille quem respicit apud circumferentiam excessus arcus ME super arcum FO. Sed excessus arcus ME super arcum FO est minor duobus arcubus ME, FO; ergo excessus anguli MZE super angulum MBE est minor angulo MBE. Ergo excessus anguli ZEB super angulum ZMB est minor angulo MBE. Ergo excessus anguli HET super angulum NML est minor angulo MBE. Ergo excessus anguli HEA, qui est angulus reflexionis, super angulum NMA, qui est angulus reflexionis, est multo minor angulo MBE.

5.60 ◉ If, therefore, angle MZE > angle MBE, twice arc ME [subtending an angle equal to MZE at the circumference] will be greater than the two arcs ME + FO [subtending an angle at the circumference equal to angle MBE]. And 2 arc ME – (arc ME + arc FO) = arc ME – arc FO. Therefore, angle MZE – angle MBE is equal to the angle at the circumference subtended by arc ME – arc FO. But arc ME – arc FO < arc ME + arc FO, so angle MZE – angle MBE < angle MBE. Consequently, angle ZEB – angle ZMB [which = MZE – MBE by previous conclusions] < angle MBE. Therefore, angle HET – angle NML < angle MBE. As a result, the difference between angle HEA, which is the angle of refraction [tied to HET], and angle NMA, which is the angle of refraction [tied to NML] is a fortiori less than angle MBE.

5.61 ◉ Sed excessus anguli HEA super angulum NMA est excessus anguli AMB super angulum AEB; ergo excessus anguli AMB super angulum AEB est minor angulo MBE. Sed excessus anguli AMB super angulum AEB est duo anguli MAE, MBE. Ergo duo anguli MAE, MBE sunt minores angulo MBE, quod est impossibile. Forma ergo B non reflectetur ad A ex alio puncto preter quam ex E, et hoc est quod voluimus.

5.61 ◉ But angle HEA – angle NMA = angle AMB – angle AEB, so angle AMB – angle AEB < angle MBE. However, angle AMB – angle AEB = the two angles MAE + MBE. Therefore, the two angles MAE + MBE < angle MBE, which is impossible. The form of B will thus not be refracted to A from any point other than E, and this is what we wanted [to demonstrate].⁑

5.62 ◉ [PROPOSITIO 7] Cum ergo forma B non reflectitur ad A nisi ex uno puncto, non habebit nisi unam ymaginem. Sed locus ymaginis diversatur secundum diversitatem loci in quo est B. Continuemus ergo BZ [FIGURE 7.5.6b]. Linea ergo BZ aut concurret cum linea EA, aut erit ei equidistans, et concursus aut erit in parte EB, ut in K, aut in parte A, ut in R. Et cum BZ fuerit equidistans linee EA, erit ut linea BZ, que est media inter duas lineas KBZ, BZR.

5.62 ◉ [PROPOSITION 7] Consequently, since the form of B is refracted to A from only one point, it will have only one image. The location of the image, however, varies according to the variation in where B lies. So let us draw [lines] BZ [, B’Z, and B«Z in figure 7.5.53b, p. 189, according to three locations for B]. Line BZ [or B’Z, or B«Z] will intersect line EA, or it will be parallel to it, and the intersection will be either on the side of EB, as in K, or on the side of A, as in R. And when B’Z is parallel to line EA, it will be like line B’Z[X], which lies between the two lines KBZ and B«ZR.

5.63 ◉ Si ergo concursus harum duarum linearum fuerit in K, erit ymago ante visum, et erit forma manifesta et comprehensa a visu in K. Si vero concursus fuerit in R, erit ymago punctum R, et tunc forma comprehendetur a visu in eius oppositione, sed non manifeste, tamen quia comprehenditur a visu extra suum locum. Hoc autem declaratum est in loco in quo locuti sumus de reflexione. Et si linea BZ fuerit equidistans linee EA, tunc ymago erit indeterminata, et forma comprehendetur in loco reflexionis. Huius autem causa similis est illi quam diximus in loco reflexionis, cum fuerit reflexio per lineam equidistantem perpendiculari.

5.63 ◉ Hence, if the intersection of these two lines is at K, the image will lie in front of the center of sight, and the form will be clear and [will be] perceived by the visual faculty at K. If, however, the intersection is at R, point R will be the image, and in that case the form will be perceived by the visual faculty straight ahead, but not clearly because it is still perceived outside of its [true] location by the visual faculty. This was shown in the place where we discussed reflection. If line BZ is parallel to line EA the image will be indefinite, and the form will be perceived at the point of refraction. The reason for this is similar to the one we discussed in regard to reflection when reflection occurs along a line parallel to the normal.⁑

5.64 ◉ Ex predictis ergo patet quod res que comprehenditur a visu ultra corpus diaffonum grossius corpore quod est ex parte visus non habet nisi unam ymaginem, neque comprehendetur nisi unum tantum. Hec vero reflexio est a concavitate corporis diaffoni quod est ex parte visus contingentis convexum corporis diaffoni quod est ex parte rei vise, et hoc est quod voluimus.

5.64 ◉ From the foregoing it is therefore evident that an object perceived by the visual faculty through a transparent body that is denser than the body on the side of the eye has only one image, and it will only be perceived as single. This refraction, however, occurs at the concave surface of the transparent body on the side of the eye [and containing it], which is in contact with the convex [surface] of the transparent body on the side of the visible object [and containing it], and this is what we wanted [to demonstrate].⁑

5.65 ◉ [PROPOSITIO 8] Et si corpus diaffonum grossius fuerit ex parte visus et subtilius ex parte rei vise, tunc visus non habebit nisi unam solam ymaginem, nam tunc visus erit ut B [FIGURE 7.5.6b] et res visa ut A, et cum forma A reflectetur ad B, reflexio erit in superficie perpendiculari super superficiem corporis diaffoni, et erit differentia communis inter illam superficiem et superficiem corporis diaffoni circulus ut circulus GED. Et erit punctus reflexionis ut E, et erit linea reflexa ut EH.

5.65 ◉ [PROPOSITION 8] Furthermore, if the denser transparent body lies on the side of the eye and the rarer one on the side of the visible object, then the visual faculty will grasp only one image, for in that case the center of sight will be at B and the visible object at A [in figure 7.5.53a, p. 189], and since the form of A will be refracted to B, the refraction will occur in a plane perpendicular to the surface of the transparent body, and the common section of that plane and the surface of the transparent body will be a circle, such as circle GED. Also the point of refraction will be E, and the line of refraction will be [B]EH.⁑

5.66 ◉ Sequitur ergo ut forma que extenditur per lineam AE et reflectetur per BE, quando extenditur ex B per lineam BE quod reflectitur per lineam AE. Si ergo forma A reflectatur ad B ex alio puncto quam ex E, sequitur quod forma B reflectatur ad A ex illo puncto. Sed iam declaratum est quod, cum forma extensa fuerit per lineam BE et reflexa per lineam AE, numquam reflectetur ex B alia forma ad A, quare A non reflectetur ad B nisi ex uno puncto, nec habebit nisi unam ymaginem. Et si A fuerit in perpendiculari exeunti ex B ad centrum spere, tunc B comprehendet A in rectitudine perpendicularis, et patet quod forma A non reflectetur ad B, ex quo patuit quod forma B, cum fuerit in perpendiculari, non reflectetur ad A. Cum ergo grossius corpus fuerit ex parte visus et subtilius ex parte rei vise, tunc res visa non habebit nisi unam ymaginem et unam formam tantum, et hoc voluimus.

5.66 ◉ It follows, then, that the form that extends along AE and that will be refracted along BE [according to the analysis in proposition 4] is refracted along line AE when it extends from B along line BE. Therefore, if the form of A is refracted to B from some point other than E, it follows that the form of B should be refracted to A from that [same] point. But it was just demonstrated that, when the form extends along line BE and is refracted along AE, another form from B will never be refracted to A, since [the form] of A will be refracted to B from only one point and will only have one image. Moreover, if A lies on the normal dropped from B to the center of the sphere, B will perceive A straight along the normal, and it is clear that the form of A will not be refracted to B, from which it was evident that, when it lies on the normal, the form of B will not be refracted to A. Hence, when the denser body lies on the side of the eye and the rarer one on the side of the visible object, the visible object will have only one image and only one form, and this [is what] we wanted [to demonstrate].⁑

5.67 ◉ [PROPOSITIO 9] Item, iteremus figuram, ponentes in circumferentia GED punctum ex parte G [FIGURE 7.5.7, p. 444]. Et sit E, ex quo extrahemus lineam equidistantem linee AD, et sit linea ET. Et continuemus ZE, et extrahamus illam usque ad H. Et sit proportio anguli ZEK ad angulum KET duplicatum maxima proportio quam angulus quem continet linea per quam extenditur forma cum perpendiculari possit habere ad angulum reflexionis quem exigit ille angulus, quoad sensum. Anguli enim reflexionis qui fuerint inter duo corpora diaffona diversa in diaffonitate a luce transeunte per illa diversantur, quorum diversitas quoad sensum habet finem quem, si excesserint, sensus non comprehendet quantitatem reflexionis, comprehendet enim centrum lucis transeuntis per duo corpora in rectitudine linee per quam lux extenditur, cum videlicet expertum fuerit hoc per instrumentum.

5.67 ◉ [PROPOSITION 9] To continue, let us copy the diagram [in figure 7.5.54, p. 190], locating the point [of refraction] on circumference GED of the circle on the side of G. Let it be E, and from it we will draw line ET parallel to line AD. Let us connect ZE and extend it to H. And let the ratio of angle ZEK to twice angle KET be the size of the largest possible ratio between the angle that the line along which the form extends makes with the normal and the angle of refraction mandated by that angle, as far as can be empirically determined.⁑ For the angles of refraction that are [produced] between two transparent bodies differing in transparency through which light passes vary in that regard, and, as far as the sense [of sight] is concerned, that difference has a limit, and if it is exceeded, the sense [of sight] will not perceive the amount of refraction, for it will perceive the center of the light passing through the two bodies on the straight line along which the light extends, i.e., as this was observed in the [experimental] apparatus [constructed in chapter 3].⁑

5.68 ◉ Et ponamus angulum DZT equalem angulo KET. Erit ergo angulus ZKE duplus ad angulum KET, et sic proportio anguli ZEK ad angulum ZKE erit maxima proportio inter angulum quem continet prima linea et perpendicularis et inter angulum reflexionis. Sed linea EK concurret cum linea AD; concurrant ergo in B. Et extrahamus ex E lineam equidistantem TZ. Concurret ergo cum ZG extra circulum ex parte G. Sit concursus in A. Extrahamus BE usque ad L. Erit ergo angulus LEA equalis angulo ZKE, et angulus LEH equalis angulo ZEK. Erit ergo angulus LEA angulus reflexionis quem exigit angulus LEH. Si ergo B fuerit in aliquo viso, et corpus diaffonum cuius convexum est ex parte A fuerit continuatum ex E usque ad B, et non fuerit distinctum apud circumferentiam GED ex parte B, tunc forma B extendetur per lineam BE, et reflectetur per lineam EA, et comprehenditur a visu A per verticationem AE.

5.68 ◉ Let us suppose that angle DZT = angle KET. Angle ZKE will thus be twice angle KET, and so angle ZEK to angle ZKE will be the largest possible ratio between the angle that the first line makes with the normal and the angle of refraction.⁑ But line EK will intersect line AD, so let them intersect at B. Let us draw a line from E parallel to TZ. It will thus intersect ZG outside the circle on the side of G. Let the intersection be at A. Let us extend BE to L. Consequently, angle LEA = angle ZKE, and angle LEH = angle ZEK.⁑ Angle LEA will therefore be the angle of refraction mandated by angle LEH [which = angle of incidence BEZ]. So if B lies on some visible object, and if the transparent body with its convex surface facing A is continuous from E to B and has no [refractive] interface at circumference GED on the side of B, the form of B will extend [straight] along line BE and will be refracted along line EA, and it is perceived by the center of sight at A along line AE [so its image will be located at center of sight A, where reflected ray AE and normal BZA intersect].

5.69 ◉ Et angulus AEH potest dividi pluribus proportionibus earum que fuerint inter angulos reflexionis et angulos quos continent perpendiculares cum primis lineis qui fuerint inter duo corpora diaffona. Sic ergo in linea DB erunt plura puncta quorum forme extendentur ad arcum GE et reflectentur ad A, et forma totius linee in qua est ille punctus reflectetur ad A ex arcu GE.

5.69 ◉ But angle AEH can be subdivided into several ratios of those obtaining between the angles of refraction and the angles that the normals make with the first lines [of incidence] between the two transparent bodies. There will thus be several points on line DB whose forms will extend to arc GE and will be refracted to A, and [so] the form of the entire line containing [each] such point will be refracted to A from arc GE.⁑

5.70 ◉ Cum ergo visus fuerit in corpore diaffono et res visa fuerit in alio diaffono grossiori, et fuerit superficies corporis diaffoni grossioris que est ex parte visus sperica convexa, et visus fuerit extra circulum cuius convexum est ex parte visus, et fuerit remotior a visu quam punctum remotius ex duobus punctis sectionis facte inter perpendicularem et circumferentiam, et corpus diaffonum grossum quod est ex parte visus fuerit continuum usque ad locum in quo est res visa, et non fuerit decisum apud circulationem que est ex parte rei vise, tunc visus poterit comprehendere illam rem visam et reflexe et recte, et ymago huius rei vise erit centrum visus.

5.70 ◉ Therefore, when the eye lies in a transparent body and the visible object lies in another, denser transparent body, and when the surface of the denser transparent body that faces the eye is convex spherical, and when the visible object lies outside the circle whose convexity faces the eye, and when it is farther from the center of sight than the farther of the two points of intersection of the normal and the circumference [i.e., beyond point D in figure 7.5.54], and when the dense, transparent body facing the eye is continuous up to where the visible object lies and there is no refractive interface at the segment on the circle facing the visible object, then the visual faculty will be able to perceive that visible object both directly and according to refraction, and the image of this visible object will lie at the center of sight.

5.71 ◉ Item, si fixerimus lineam AGB et revolverimus figuram AEB in circuitu AB, et pars superficiei corporis diaffoni quod est ex parte rei vise fuerit sperica, tunc punctum E signabit circumferentiam in superficie circulari convexa que est ex parte visus, ex qua circumferentia reflectetur B ad A. Sed ymago in tota circumferentia reflexionis erit una, scilicet centrum visus. Ymago ergo huiusmodi rei vise etiam est una. Sed ex hac positione accidit quod visus comprehendat formam rei vise apud locum reflexionis, ea causa quam diximus in conversione ex speculis, cum fuerit conversio a circumferentia in aliqua spera, et fuerit ymago centrum visus.

5.71 ◉ Furthermore, if we hold line AGB fixed and revolve the figure [containing triangle] AEB around AB [as an axis], and if the portion of the surface of the transparent body facing the visible object is spherical, then point E will describe a circumference in a convex circular plane facing the center of sight, and [the form of] B will be refracted to A from that circumference. But the image on the entire circumference of refraction will be single, i.e., [coincident with] the center of sight. Hence, the image of this kind of visible object is also single. But in this situation it happens that the visual faculty perceives the form of the visible object at the point of refraction for the reason that we discussed in [the case of] reflection from mirrors, when the reflection occurs from a circumference in some sphere and when the image is [at] the center of sight.⁑

5.72 ◉ Ergo huius rei vise forma comprehenditur a visu circularis apud circulum reflexionis et in rectitudine perpendicularis transeuntis per visum et rem visam simul, et hoc est quod voluimus.

5.72 ◉ Consequently, the form of this visible object is perceived by the visual faculty [as] circular on the circle of refraction and [is perceived] at the same time in a straight line along the normal passing through the center of sight and the visible object, and this is what we wanted [to demonstrate].⁑

5.73 ◉ [PROPOSITIO 10] Item sit A visus [FIGURE 7.5.8, p. 444], et sit B in aliquo visu et ultra corpus diaffonum grossius illo in quo est visus. Et sit superficies corporis quod est ex parte visus circularis concava, cuius concavitas sit ex parte visus. Dico ergo quod B unam solam habebit ymaginem et unam tantum formam apud A.

5.73 ◉ [PROPOSITION 10] Now let A [in figure 7.5.55, p. 192] be the center of sight, and let B lie on some visible object in a transparent body denser than the one in which the center of sight lies. Let the [great circle on the] surface of the body facing the eye be concave circular, with its concavity facing the eye. I say, then, that B will have only one image and only one form from [the perspective of point] A.

5.74 ◉ Et sit centrum concavitatis G, et continuemus AG, et extrahemus ipsam recte usque ad Z. Erit ergo AZ perpendicularis super superficiem concavam, et B aut erit in AZ, aut extra. Sit ergo primo in linea AZ. A ergo comprehendet B in rectitudine AB, cum AB sit perpendicularis super superficiem concavam, et numquam ipsam reflexe; quod si est possibile, reflectatur forma B ad A ex E, et continuemus BE, GE. Extrahamus BE usque ad T.

5.74 ◉ Let G be the center of concave curvature, and let us connect AG; and we will extend it straight to Z. AZ will thus be perpendicular to the concave surface, and B will lie either on AZ or outside it. First, let it lie on line AZ. A will therefore perceive B straight along AB, since AB is perpendicular to the concave surface, and it is never refracted. But if that is possible, let the form of B be refracted to A from E, and let us draw BE and GE [and] extend BE to T.

5.75 ◉ Angulus ergo TEG est ille quem continet linea per quam extenditur forma et perpendicularis exiens a loco reflexionis, et quia corpus quod est ex parte A est subtilius illo quod est ex parte B, erit reflexio ad partem contrariam illi in qua est EG. Linea ergo ET, quando reflectetur, removetur a linea EG, et linea ET non concurrit cum linea BA aliquo modo. Forma ergo B non reflectetur ad A; non ergo comprehendetur reflexe, sed comprehendetur recte. Ergo non habebit apud visum nisi unam formam tantum, et hoc est quod voluimus.

5.75 ◉ Angle TEG is thus [equal to] the [angle of incidence] that the line along which the form extends makes with the normal dropped from the point of refraction, and since the [transparent] body on the side of A is rarer than the one on the side of B, the refraction will occur away from [normal] EG. Consequently, when it will be refracted, line [B]ET is diverted away from line EG, and line ET does not intersect line BA in any way. The form of B will therefore not be refracted to A, so it will not be perceived according to refraction but will be perceived directly. Hence, from [the perspective of] the center of sight [at A] it will have only one form, and this is what we wanted [to demonstrate].⁑

5.76 ◉ [PROPOSITIO 11] Item iteremus figuram, et sit B extra lineam AZ [FIGURE 7.5.9, p. 445], et extrahemus superficiem in qua est AZ et B. Hec ergo superficies erit perpendicularis super superficiem concavam, et non reflectetur forma B ad A nisi in hac superficie, non enim erigitur perpendicularis super superficiem concavam aliqua superficies equalis que transit per A nisi illa que transit per AZ. Sed per AZ et per B non transit nisi una sola tantum. Forma ergo B non reflectetur ad A nisi in superficie transeunte per lineam AZ et per B. Et sit differentia communis inter hanc superficiem et superficiem concavam arcus HDE, et reflectatur forma B ad A ex H.

5.76 ◉ [PROPOSITION 11] To continue, let us redraw the diagram [figure 7.5.56, p. 193]; let B lie outside line AZ, and we will produce the plane containing AZ and B. This plane will thus be perpendicular to the concave surface, and the form of B will be refracted to A only in this plane, for no [plane] is erected perpendicular to the concave surface to pass through A unless it passes through AZ. But only one plane passes through AZ and through B. Thus, the form of B will be refracted to A only in the plane passing through [both] line AZ and [point] B. Let the common section of this plane and the concave surface be arc HDE, and let the form of B be refracted to A from H.

5.77 ◉ Dico ergo quod non reflectetur ex alio puncto; quod si fuerit possibile, reflectatur ex M. Et continuemus lineas AH, BH, GH, AM, BM, GM, et extrahamus HB recte usque ad T, et BM recte usque ad N, et GH recte usque ad L, et GM recte usque ad O. Et perficiamus circumferentiam HDE, et secet lineam AG in K. A ergo aut erit in linea KD aut extra in partem K. Si ergo A fuerit in linea KD, aut erit in G aut in altera duarum linearum GD, GK.

5.77 ◉ I say, then, that it will not be refracted from any other point, but if it is possible, let it be refracted from M. Let us then connect lines AH, BH, GH, AM, BM, and GM, and let us extend HB straight to T, BM straight to N, GH straight to L, and GM straight to O. Let us complete circumference HDE, and let it cut line AG at K. A will therefore lie on line KD or beyond it on the side of K. So if A lies on line KD, it will either lie at G, or [it will lie] on one of the two line-segments GD or GK.

5.78 ◉ Si ergo fuerit A in G, tunc forma B non reflectetur ad A, linee enim que continuant corpus circulare cum G sunt perpendiculares super superficiem corporis quod est ex parte A. Reflexio autem non erit per ipsam perpendicularem sed ab ipsa; forma B ergo non reflectitur ad A, si A fuerit in G.

5.78 ◉ Accordingly, if A lies at G, the form of B will not be refracted to A, for the lines that connect the circular [section on the surface of the transparent] body and G are perpendicular to the surface of the body on the side of A. Refraction, however, will not occur along the normal itself but away from it, so the form of B is not refracted to A, if A lies at G.

5.79 ◉ Et si A fuerit in GD, tunc linea HT erit inter duas lineas HA, HG, et ideo linea MN erit inter duas lineas MA, MG, nam reflexio est ad partem contrariam parti perpendicularis, nam corpus diaffonum quod est ex parte visus est subtilius illo quod est ex parte rei vise. Et si linea HT fuerit inter duas lineas HA, HG, et A fuerit in linea GD, tunc angulus BHA erit ex parte D, et similiter angulus BMA erit ex parte D, et erit B ultra lineam GHL, videlicet ex parte K a linea GHL. Et erit angulus THG ille quem continet linea per quam extenditur forma cum perpendiculari, et similiter angulus NMG, et erit angulus THA angulus reflexionis, et similiter angulus NMA.

5.79 ◉ Furthermore, if A lies on GD, then line HT will lie between the two lines HA and HG, and so line MN will lie between the two lines MA and MG, for the refraction occurs away from the normal because the transparent body on the side of the eye is rarer than the one on the side of the visible object. And if line HT lies between the two lines HA and HG, and if A lies on line GD, angle BHA will lie on the side of D, and likewise angle BMA will lie on the side of D, and B will lie beyond line GHL, i.e., on the side of K away from line GHL. Angle THG will be [equal to angle of incidence BHL] that the line along which the form extends makes with the normal, and so will angle NMG [be equal to angle of incidence BMO], and angle THA will be the angle of refraction [for angle of incidence BHL], and so will angle NMA [be the angle of refraction for angle of incidence BMO].

5.80 ◉ Angulus autem NMG aut erit equalis angulo THG, aut maior, aut minor. Si equalis, AMN erit equalis angulo AHT; ergo angulus BHA erit equalis angulo BMA, quod est impossibile. Si maior, tunc angulus AMN erit maior angulo AHT, et sic angulus BMA erit minor angulo BHA, quod est impossibile.

5.80 ◉ However, angle NMG will be equal to, greater than, or less than angle THG. If it [i.e., NMG] is equal [to THG], then [angle] AMN = angle AHT, so angle BHA = angle BMA, which is impossible. If it [i.e., NMG] is greater [than THG], then angle AMN > angle AHT, and so angle BMA < angle BHA, which is impossible.

5.81 ◉ Si minor, tunc angulus AMN erit minor angulo AHT, et sic totus angulus AMG erit minor toto angulo AHG. Et erit diminutio anguli AMN ab angulo AHT minor quam diminutio anguli AMG ab angulo AHG. Sed diminutio anguli AMG ab angulo AHG est equalis diminutioni HGM ab angulo HAM, duo enim anguli qui sunt in sectione linearum AH, MG sunt equales. Ergo diminutio anguli AMN a diminutione de angulo AHT est minor quam diminutio anguli HGM ab angulo HAM.

5.81 ◉ If it [i.e., NMG] is less [than THG], then angle AMN < angle AHT, and so the entire angle AMG < the entire angle AHG. And [so] angle AHT – angle AMN < angle AHG – angle AMG. But angle AHG – angle AMG = [angle] HAM – angle HGM, for the two angles at the intersection of lines AH and MG are equal.⁑ Therefore, angle AHT – angle AMN < angle HAM – angle HGM.

5.82 ◉ Et extrahamus duas lineas HA, MA ad duo puncta E, C. Erit ergo angulus HAM ille quem respiciunt in circumferentia duo arcus HM, EC, et angulum HGM respicit in circumferentia arcus HM duplicatus, et cum angulus HGM est minor angulo HAM, erit arcus HM duplicatus minor duobus arcubus HM, EC. Et erit diminutio arcus HM duplicati a duobus arcubus HM, EC sicut diminutio arcus HM ab arcu EC. Ergo diminutio anguli AMN ab angulo AHT erit minor angulo quem respicit apud circumferentiam diminutio arcus HM ab arcu EC; est ergo minor angulo HAM. Excessus ergo anguli BMA super angulum BHA est minor quam angulus HAM. Sed excessus anguli BMA super angulum BHA sunt duo anguli HAM, HBM. Ergo duo anguli isti simul sunt minores angulo HAM, quod est impossibile.

5.82 ◉ Let us extend the two lines HA and MA to the two points E and C. Angle HAM will therefore be [equal to] the angle on the circumference that the two arcs HM + EC subtend [by proposition 5, lemma 1], and angle HGM [will be equal to the angle] on the circumference that twice arc HM subtends [by Euclid, III.20, since the vertex of angle HGM is at the center of the circle], and since angle HGM < angle HAM, 2 arc HM < arcs MH + EC. But arcs HM + EC ­– 2 arc HM = arc EC – arc HM. Therefore angle AHT – angle AMN is less than the angle at the circumference that arc EC – arc HM subtends, so it is less than angle HAM. Angle BMA [supplementary to AMN] – angle BHA [supplementary to AHT] is therefore less than angle HAM. But angle BMA – angle BHA = angle HAM + angle HBM.⁑ Therefore those two angles [HAM + HBM] together are less than angle HAM, which is impossible.

5.83 ◉ Et si A fuerit in linea GK, tunc linea HT erit inter duas lineas HG, HA, et similiter linea MN erit inter duas lineas MG, MA. Erit ergo angulus BHA ex parte K, et similiter angulus BMA erit ex parte K, et erit B infra lineam GMO, scilicet ex parte D a linea GMO. Et uterque angulus THG, NMG est ille quem continent linea per quam extenditur forma et perpendicularis, et uterque angulus THA, NMA erit angulus reflexionis.

5.83 ◉ If A lies on line GK [in figure 7.5.56a, p. 193], then line HT will lie between the two lines HG and HA, and likewise line MN will lie between the two lines MG and MA. Consequently, angle BHA will face toward K, and likewise angle BMA will face K, and B will lie below line GMO, i.e., on the side of D from line GMO. Both angles THG and NMG are [equal to] the ones that the line along which the form extends makes with the normal [i.e., angles of incidence BHL and BMO], and both angles THA and NMA will be the [respective] angles of refraction.

5.84 ◉ Si ergo angulus THG fuerit equalis angulo NMG, tunc angulus THA erit equalis angulo NMA, et sic angulus BHA erit equalis angulo BMA, quod est impossibile. Si vero fuerit maior, tunc angulus THA erit maior angulo NMA, et sic angulus BHA erit minor angulo BMA, quod est impossibile.

5.84 ◉ Hence, if angle THG = angle NMG, then angle THA = angle NMA, and so angle BHA = angle BMA, which is impossible. If, however, it [i.e., THG] is greater [than NMG], then angle THA > angle NMA, and so angle BHA < angle BMA, which is impossible.

5.85 ◉ Et si fuerit minor, tunc angulus THA erit minor angulo NMA, et sic totus angulus GHA erit minor angulo GMA. Ergo angulus HGM erit minor angulo HAM, et erit diminutio anguli HGM ab angulo HAM minor quam angulus GMA, ut prius declaravimus, et diminutio anguli THA ab angulo NMA est minor quam diminutio anguli GHA ab angulo GMA; est ergo minor quam diminutio anguli HGM ab angulo HAM. Ergo diminutio anguli THA ab angulo NMA est minor quam angulus GMA. Sed diminutio anguli THA ab angulo NMA est excessus anguli BHA super angulum BMA. Sed excessus anguli BHA super angulum BMA sunt duo anguli HAM, HBM; ergo isti duo anguli simul sunt minores angulo HAM, quod est impossibile.

5.85 ◉ If it [i.e., THG] is less [than NMG], then angle THA < angle NMA, and so the entire angle GHA < [the entire] angle GMA. Therefore, angle HGM < angle HAM, and angle HAM – angle HGM < angle GMA, as we demonstrated earlier [in paragraph 5.81], and angle NMA – angle THA < angle GMA – angle GHA, so it is less than angle HAM – angle HGM. Hence, angle NMA – angle THA < angle GMA. But angle NMA – angle THA = angle BHA – angle BMA. Angle BHA – angle BMA, however, is equal to the two angles HAM + HBM, so those two angles together are smaller than angle HAM, which is impossible.

5.86 ◉ Si vero A fuerit extra lineam KZ et ad partem K [FIGURE 7.5.9b, p. 445], et corpus in quo est A fuerit continuum usque ad A, continuabimus duas lineas AH, AM, et secabunt circumferentiam in R et in Q. Et si angulus THG fuerit equalis angulo NMG, tunc angulus BHA erit equalis angulo BMA, quod est impossibile. Et si fuerit maior, tunc angulus THA erit maior angulo NMA, et sic angulus BHA erit minor angulo BMA, quod est impossibile.

5.86 ◉ But if A lies beyond line KZ on the side of K [in figure 7.5.56b, p. 193], and if the body in which A lies is continuous up to A, we will draw the two lines AH and AM, and they will intersect the circumference at R and Q. If angle THG = angle NMG, then angle BHA = angle BMA, which is impossible. If it [i.e., THG] is greater [than NMG], then angle THA > angle NMA, and so angle BHA < angle BMA, which is impossible.

5.87 ◉ Si vero fuerit minor, tunc angulus THA erit minor angulo NMA, et totus angulus GHA erit minor toto angulo GMA; ergo angulus HGM erit minor angulo HAM. Sed angulus MGH est ille quem respicit apud circumferentiam arcus HM duplicatus, et angulus HAM est ille quem respicit in circumferentia excessus arcus HM super arcum RQ. Ergo arcus HM duplicatus est minor excessu arcus HM super arcum RQ, quod est impossibile.

5.87 ◉ If it [i.e., THG] is smaller [than NMG], though, then angle THA < angle NMA, and the entire angle GHA < the entire angle GMA, so angle HGM < angle HAM. But angle MGH is [equal to the angle] at the circumference that twice arc HM subtends, and angle HAM is [equal to the angle] at the circumference that arc HM – arc RQ subtends [by proposition 5, lemma 1]. Thus, 2 arc HM < arc HM – arc RQ, which is impossible.

5.88 ◉ Ergo si punctum B fuerit extra lineam AKG, tunc forma eius non reflectetur ad A nisi ex uno puncto tantum, quapropter non habebit nisi unam solam ymaginem, que ymago aut erit ante visum, aut retro, aut in loco reflexionis, ut in precedentibus declaravimus, et hoc est quod voluimus.

5.88 ◉ Therefore, if point B lies outside line AKG, its form will be refracted to A from only one point, so it will have only one image, and that image will be in front of the center of sight, behind it, or at the point of reflection, as we showed earlier, and this is what we wanted [to demonstrate].⁑

5.89 ◉ Si vero corpus diaffonum grossius fuerit ex parte visus et subtilius ex parte rei vise, eisdem permanentibus figuris, tunc etiam res visa non habebit nisi unam solam ymaginem, et hoc demonstrabitur, ut in conversa septime figure. Et omnia que declaravimus in reflexionibus a convexo et concavo circuli sequuntur in superficiebus spericis et columpnalibus, preter reflexionem circularem a circumferentia circuli, que non fit nisi in superficiebus spericis tantum. Hec autem que diximus sunt ymagines visibilium que comprehenduntur a visu ultra corpora diaffona simplicia, que sunt unius substantie et quarum figura que est ex parte visus est una figura.

5.89 ◉ If, however, the denser transparent body lies on the side of the eye and the rarer one on the side of the visible object, all things remaining the same in the figures, then the visible object will still have only one image, and this will be proven as [it was] in the converse of the seventh proposition [of this chapter, i.e., proposition 8, pp. 290-291 above].⁑ And everything we have demonstrated about refraction from convex and concave circular sections applies to spherical and cylindrical surfaces, except the circular refraction from the circumference of a circle, which only occurs on spherical surfaces. Moreover, what we have discussed involves images of visible objects that are perceived by the visual faculty through simple transparent bodies, which consist of one substance [with a surface] whose shape facing the eye is unitary [i.e., that yields circles or straight lines when cut by a plane].

5.90 ◉ Si vero corpus diaffonum fuerit diversum aut non consimilis diaffonitatis, tunc ymagines rei vise diversantur, et si superficies corporis diaffoni que est ex parte visus fuerit diversa, tunc loca ymaginum rei vise diversantur, cum forme reflexionum ex superficie corporis diversantur etiam. Et si aliquis respexerit ad parvam speram, aut aliquod corpus rotundum parvum, aut columpnale vitri, aut cristalli, ultra quod corpus fuerit aliquod visibile, inveniet ymaginem illius alio modo quam sit res visa in se, et forte inveniet rei vise ymaginem aliam, et sic ambigetur super hoc. Sed huiusmodi reflexio non est una sed due reflexiones, forma enim rei vise extenditur a re ad speram aut ad aliud corpus rotundum columpnale, et reflectitur a convexo spere aut columpne ad interius corporis, et extenditur intra corpus quousque pervenit ad superficiem eius, et deinde reflectitur a spera aut columpna apud concavitatem aeris continentis speram aut columpnam. Et sic comprehensio huiusmodi rerum erit duabus diversis reflexionibus, quapropter ymago eius erit diversa ab ymagine eius quod comprehenditur una reflexione. Nos autem loquemur de hoc parum, quando tractabimus de deceptionibus visus que fiunt per reflexionem.

5.90 ◉ But if the transparent body consists of different [surface-shapes] or is not uniform in its transparency, the images of a visible object [seen through them] vary, and if the surface of the transparent body facing the eye consists of different [surface-shapes], the image-locations of the visible object vary, since the modes of refraction from the body’s surface also vary. And if one looks through a small sphere, a small round body, or a cylindrical piece of glass or crystal, behind which a visible object lies, he will find that the image of that object is different from the visible object itself, and he may find another image of the visible object and may thus be confused about it. But this sort of refraction is not singular but double, for the form of the visible object extends from the object to the sphere or other round, cylindrical body and is refracted into the body at the convexity of the sphere or cylinder, and it extends through the body until it reaches its [inner] surface and is then refracted from the sphere or cylinder at the concavity of the air enclosing the sphere or cylinder. Consequently, the perception of this sort of object will be due to two different refractions, so its image will be different from the image that is perceived according to one refraction.⁑ But we will discuss this a bit, when we deal with the visual misperceptions that arise from refraction.

◉Capitulum sextum

◉Chapter Six

Qualiter visus comprehendit visibilia secundum reflexionem

6.1 ◉ In precedentibus iam declaravimus quod, cum forma reflectitur ab aliquo corpore diaffono ad aliud corporis alterius diaffonitatis, extenditur per lineam rectam donec perveniat ad superficiem corporis diaffoni in quo est; deinde reflectitur in illo alio corpore diaffono per lineam aliam rectam que continet cum prima linea angulum. Et cum forma extenditur per hanc lineam aliam super quam reflectitur forma in secundo corpore, alia quecumque sit forma in secundo corpore usque ad punctum sectionis inter duas lineas rectas reflectetur per primam lineam rectam.

6.1 ◉ We just showed in the previous analysis that, when a form is refracted from one transparent body to another body of different transparency, it extends along a straight line until it reaches the surface of the transparent body in which it lies; then it is refracted in that other transparent body along another straight line that forms an angle with the first line. And as the form extends along this other line according to which the form is refracted in the second body, some other form in the second body [which] is [extending] to the point of intersection between the two straight lines will be refracted [reciprocally] along the first straight line.

6.2 ◉ Et est manifestum per experientiam quod, si aliquis inspexerit aliquod corpus diaffonum quod differt in sua diaffonitate a diaffonitate aeris, comprehendet omnia que sunt ultra de illis que opponuntur visui, et si cooperuerit alterum visum et aspexerit reliquo, comprehendet etiam quecumque sunt ultra, sive illud sit corpus sive aer, aut aqua, aut vitrum. Et similiter, si homo posuerit visum intra aut in aliquo corpore grossiori aere, aut vitro, aut cristallo, videbit omnia que sunt ultra de illis que sunt in aere. Et si aspiciens moverit visum suum dextrum, aut sinistrum, et in omnem partem, et non removerit ipsum multum a suo primo loco, tunc comprehendet etiam omnia que prius comprehendebat, sive motus visus fuerit in aere aut in vitro.

6.2 ◉ And it is obvious from experience that, if someone looks through some transparent body that differs in transparency from the transparency of the air, he will perceive everything beyond [the two transparent bodies] that faces the eye, and if he closes either eye and looks with the other, he will still perceive what lies beyond, whether the body [through which he looks] is air, water, or glass. Likewise, if a person places his eye in or on another body denser than air, be it glass or crystal, he will see everything in the air behind it. And if the viewer moves his eye to the right, or left, or in any direction whatever, and if he does not move it too far from its original location, he will still perceive everything he perceived before, whether the eye’s motion occurs in air or in glass.

6.3 ◉ Sed iam declaravimus experientia et demonstratione quod nichil comprehendit visus de illis que sunt ultra corpora diaffona que differunt a diaffonitate ab aere nisi secundum reflexionem, preter quam unum punctum quod est in perpendiculari exeunti a centro visus super superficiem corporis diaffoni. Ergo omne punctum comprehensum a visu ultra corpus diaffonum, preter illud punctum predictum, comprehenditur ex forma que extenditur ex illo puncto ad superficiem corporis diaffoni ultra quod est, et reflectetur a superficie illius corporis ad visum. Et cum unus visus comprehendit omnia que sunt ultra corpus diaffonum, omne punctum existens ultra illud corpus diaffonum extenditur forma eius per lineam rectam ad superficiem illius corporis diaffoni, et reflectetur ad illum visum unum, preter quam illud punctum predictum.

6.3 ◉ But it has already been shown by experiment and [rational] demonstration that the visual faculty only perceives things through transparent bodies that differ in transparency from the air according to refraction, except for the one point that lies on the normal dropped from the center of sight to the surface of the transparent body. Thus, every point that is perceived by the visual faculty through a transparent body, except for that aforementioned point, is perceived by means of a form that extends from that point to the surface of the transparent body behind which it lies, and it will be refracted to the center of sight from the surface of that body. And since one eye perceives everything that lies behind the transparent body, the form of every point lying behind that transparent body extends along a straight line to the surface of that transparent body, and it will be refracted to that one eye, except for that aforementioned point.

6.4 ◉ Et cum forme omnium punctorum que sunt in omnibus visibilibus existentibus ultra corpus diaffonum reflectuntur in eodem tempore ad centrum visus unius, forma puncti quod existit apud centrum illius visus, cum fuerit in aliquo visibili, reflectetur ad omnia puncta que sunt in omnibus visibilibus existentibus ultra corpus diaffonum oppositum visui in eodem tempore et eodem modo. Et similiter est de omni puncto propinquo puncto quod est apud centrum visus, nam si visus motus fuerit ad omnem partem et non fuerit remotus a suo situ, comprehendet visibilia. Ergo forma cuiuslibet puncti cuiuslibet visi, cum fuerit ultra aliquod corpus diaffonum, extenditur ad superficiem corporis diaffoni ultra quod est, et reflectitur ad universum eius quod opponitur ei ex corpore aeris. Et non est aliquod tempus magis appropriatum huic quam aliud, sed hoc est proprium nature lucis et coloris que sunt in visibilibus, scilicet quod semper extendantur a quolibet puncto cuiuslibet corporis lucidi per lineam rectam que extenditur ab illo puncto, et reflectuntur in omni corpore diaffono diverso, preter quam punctum quod est in perpendiculari.

6.4 ◉ And since the forms of all the points on all the visible objects lying behind the transparent body are refracted to the center of the one eye at the same time, the form of the point that lies at the center of that eye, if it lies on some visible object, will be refracted at the same time and in the same way to all the points on all the visible objects that lie behind the transparent body and face the eye. The same holds for every point near the point at the center of the eye, for if the eye is moved in every direction but does not lie far from its [original] place, it will perceive [all those] visible objects. When it lies behind some transparent body, then, the form of any point on any visible object extends to the surface of the transparent body behind which it lies, and it is refracted throughout the body of the air that faces it. And there is no time more appropriate than another for this [to happen], but it is a natural characteristic of the light and color in visible objects that they always extend [continually] from any point on any luminous or illuminated body along a straight line extending from that point, and they are refracted in every body of differing transparency, except for the point on the normal.

6.5 ◉ Et omnis forma puncti cuiuslibet visibilis existentis in aliquo corpore diverso ab aere extenditur in illo corpore in quo existit et reflectitur in universo corpore aeris sibi opposito. Et illa forma exit apud quodlibet punctum aeris, quapropter forma totius rei vise coniungitur apud quodlibet punctum aeris, et forma totius cuiuslibet visi existentis in aliquo corpore diverso ab aere existit apud unumquodque punctum aeris oppositi illius rei vise. Et illa forma extenditur a quolibet puncto rei vise in corpore in quo est, et reflectitur apud superficiem illius corporis, et pervenit ad illud punctum aeris. Et ideo, si visus aspexerit aliquod corpus diaffonum diversum ab aere ultra quod fuerit aliqua res visibilis, visus comprehendit illam rem, nam forma illius existit apud punctum apud quod existit centrum visus, propter hoc quod etiam, si visus comprehenderit aliquam rem visibilem ultra aliquod corpus diaffonum diversum ab aere, deinde motus fuerit a suo loco dextro aut sinistro, dum in suo motu fuerit oppositus corpori diaffono et rei que est ultra, semper comprehendet illam rem. Unde etiam plures aspicientes comprehendent unam rem in celo et in aqua, et in uno eodem tempore, et hoc etiam est in eodem corpore diaffono, scilicet quod forma visi congregatur apud quodlibet punctum corporis in quo est, nam forma puncti cuiuslibet eius extenditur per lineam rectam, et inter quodlibet punctum corporis in quo est visus et quodlibet punctum rei vise est linea recta.

6.5 ◉ Moreover, every form of a point on any visible object lying in any [transparent] body different [in transparency] from the air extends through that body in which it lies and is refracted throughout the body of the air facing it. And that form leaves at any given point in the air, so the form of the entire visible object converges at any given point in the air, and the form of any entire visible object lying in any [transparent] body different from the air lies at any given point in the air facing that visible object. And that form extends from any given point on the visible object in the [transparent] body within which it lies, and it is refracted at the surface of that body to reach that point in the air. And so, if the visual faculty looks through any given transparent body different from the air behind which some visible object lies, the visual faculty perceives that object, for its form lies at the point at which the center of sight lies, and so also, if the visual faculty perceives some visible object through some transparent body different from the air, and if it then moves to the right or left of its [original] place while it remains facing the object and the visible object behind it during its motion, it will always perceive that object.⁑ So, too, several viewers will perceive a single object in the heavens or in water, and [they will do so] at the same time, and this still occurs in the same transparent medium, i.e., [it remains the case] that the form of the visible object converges at any given point in the [transparent] body in which [the viewer] lies, for the form of any point on it extends along a straight line, and there is a straight line between any given point in the [transparent] body in which the center of sight lies and any given point on the visible object.

6.6 ◉ Forma ergo cuiuslibet puncti rei vise extenditur ad quodlibet punctum corporis diaffoni in quo est res visa, et forma cuiuslibet rei vise lucide congregatur apud quodlibet punctum corporis in quo existit, et congregatur apud quodlibet punctum cuiuslibet corporis diaffoni diversi a corpore in quo existit, quando inter rem visam et illud corpus diaffonum diversum non interfuerit aliquod impedimentum. Et forma rei vise que est apud quodlibet punctum corporis diaffoni in quo extenditur ad illud punctum recte et forma illius apud quodlibet punctum corporis diaffoni diversi extenditur ad illud punctum reflexive, quando inter quodlibet punctum aeris et quamlibet rem visibilem existentem in aliquo corpore diaffono diverso ab aere, fit piramis reflexa, cuius capud est in aere punctus, et cuius basis est illa res visa, et erit reflexio eius ad superficiem corporis diaffoni ab aere diversi. Omnis ergo res visa in corpore diaffono diverso ab aere, quando comprehenditur a visu, comprehenditur a forma extensa in piramide reflexa adunata apud punctum existens in centro visus. Hoc igitur modo comprehendit visus ea que comprehendit reflexive.

6.6 ◉ Hence, the form of any point on a visible object extends to any point in the transparent body in which the visible object lies, and the form of any luminous or illuminated visible object converges at any given point in the [transparent] body in which it lies, and it converges at any given point in any transparent body different from the body in which it lies, as long as there is no obstacle between the visible object and that different transparent body. And the form of the visible object that lies at any given point in the transparent body through which it extends straight to that point and the form of that object at any given point in the different transparent body extends to that point by means of refraction, according to which a cone of refraction is formed between any given point in the air and some visible object lying in another body of different transparency from the air, the vertex of this cone lying at the point in the air, and its base lying on that visible object, and its refraction will occur at the surface of the transparent body different from the air. Consequently, when it is perceived by the visual faculty, every visible object in a transparent body different from the air is perceived according to a form extending through a cone of refraction and united at the point lying at the center of sight. This, then, is how the visual faculty perceives what it perceives according to refraction.

6.7 ◉ In capitulo autem ymaginis declaravimus quod omne visum comprehenditur a visu ultra ymaginem, et locus ymaginis est punctum in quo secant se linea radialis per quam forma extenditur ad visum et perpendicularis exiens a puncto viso. Si ergo ymaginati fuerimus quod ab unoquoque puncto rei vise exit perpendicularis ad superficiem corporis diaffoni in quo est res visa, tunc habebimus quoddam corpus exiens a viso ad superficiem corporis diaffoni, unde sequitur quod istud corpus secet piramidem reflexam, et illa superficies in qua se secant est ymago illius rei vise.

6.7 ◉ Now in chapter [five] on image[s] we showed that every visible object [seen according to refraction] is perceived by the visual faculty through an image, and the image-location is the point at which the radial line along which the form extends to the center of sight and the normal dropped from the visible point intersect. Hence, if we imagine a normal dropped from every given point on a visible object to the surface of the transparent body in which the visible object lies, we will have a body of sorts extending from the visible object to the surface of the transparent body, so it follows that this body should cut the refracted cone, and the plane in which they intersect forms the image of that visible object.

6.8 ◉ Si ergo superficies corporis diaffoni in quo est res visa fuerit equalis, tunc corpus ymaginatum continens omnes perpendiculares erit equalis superficiei, quare ymago addet parum super rem visam. Et si corpus fuerit spericum, et convexum eius fuerit ex parte visus, et centrum eius fuerit super illam rem visam, tunc corpus ymaginatum erit piramidale, cuius capud erit centrum spere, et quanto magis extenditur ad superficiem corporis sperici, tanto magis amplificabitur. Et si sectio fuerit inter rem visam et superficiem spericam, tunc ymago erit amplior ipsa re visa. Si autem sectio fuerit ultra rem visam, tunc ymago erit strictior re visa. Si vero res visa fuerit ultra superficiem spericam, tunc corpus ymaginatum erit due piramides opposite, quarum capud centrum spere. Quare loca sectionis non cadent inter corpus ymaginatum et piramidem, forte locus sectionis in quo est ymago erit maior viso, forte minor, forte equalis.

6.8 ◉ Accordingly, if the surface of the transparent body in which the visible object lies is plane, the body imagined to comprise all the normals will have a plane surface, so the image will be somewhat larger than the visible object. If the [transparent] body is [convex] spherical, and if its convexity faces the center of sight, while its center lies on that visible object, then the imaginary body will be conical, with its vertex at the center of the sphere, and the farther [that cone] extends toward the surface of the [transparent] spherical body, the more it will spread out. On the one hand, if the intersection [of this cone and the cone of refraction] lies between the visible object and the spherical surface, the image will be larger than the visible object itself. On the other hand, if the intersection lies beyond the visible object, the image will be smaller than the visible object. But if the visible object lies beyond the spherical surface, the imagined body will form two opposite cones [each] with its vertex at the center of the sphere. Because the intersection-points will not fall between the imagined body and the cone, the area of intersection at which the image lies may be larger than, smaller than, or the same size as the visible object.⁑

6.9 ◉ Si vero corpus diaffonum fuerit spericum, et concavitas eius fuerit ex parte visus, tunc corpus ymaginatum erit piramis cuius capud est centrum spere; quanto ergo magis extenditur hoc corpus in partem superficiei spere, tanto magis adunatur et constringitur, et quanto magis extenditur in aliam partem, tanto magis amplificatur, superficies enim continua parva erit media inter centrum eius et speram. Si vero locus sectionis huius corporis cum piramide reflexa fuerit propinquior centro concavitatis quam res visa, erit ymago minor ipsa re visa. Si autem fuerit remotior a centro concavitatis quam res visa, et ymago maior est quam res visa.

6.9 ◉ If the transparent body is [concave] spherical and its concavity faces the eye, however, the imagined body [comprising the normals] will be a cone with its vertex at the center of the sphere, so the farther this body extends toward the surface of the sphere, the narrower and more constricted it gets, and the farther it extends in the opposite direction, the more it spreads out, for the surface between the sphere’s center and the sphere itself is [relatively] small. On the one hand, if the area of intersection between this [imagined] body and the cone of refraction lies nearer the center of concave curvature than the visible object, the image will be smaller than the visible object itself. On the other hand, if it lies farther from the center of concave curvature than the visible object, the image is larger than the visible object itself.⁑

6.10 ◉ Et cum una res visa comprehenditur a pluribus visibus in uno momento, omnes ymagines quas comprehendunt illi visus erunt illo tempore in uno ymaginato quod est perpendiculare super superficiem corporis diaffoni. Et una res visibilis comprehenditur ab uno homine in uno tempore ultra corpus diaffonum diversum a diaffonitate corporis in quo est visus utroque visu, et tamen comprehendit illam unam. Si enim homo comprehenderit aliquod de eis que sunt in celo, aut in aqua, aut ultra vitrum, et cooperuerit alterum visum, nichilominus comprehendet illud reliquo, ex quo patet quod una res existens ultra corpus diaffonum diversum ab aere comprehendetur utroque visu et altero visu.

6.10 ◉ In addition, when a single visible object is perceived by several eyes at once, all the images that those eyes perceive at that time will [each] lie in one imagined [cone] that is perpendicular to the surface of the transparent body. And a single visible object is perceived by a single person at one time with both eyes through a transparent body differing in transparency from the body in which his eye lies, and yet he perceives it as single. For if a person perceives a given body among those in the heavens, or in water, or through glass, and if he closes either eye, he will nonetheless perceive it with the remaining eye, from which it is clear that a single object lying behind a transparent body different from air will be perceived by both eyes and by either eye.

6.11 ◉ Causa autem huius est, ut in tertio huius libri diximus, quoniam in omni puncto cuiuslibet visi comprehensibilis recte et utroque visu in quo coniuncti fuerint duo radii utriusque visus consimilis positionis quantum ad duos axes visuum comprehendetur unum, et si in ipso congregati fuerint radii diverse positionis quantum ad duos axes visuum, comprehendentur duo. Sed in maiori parte earum que comprehenduntur positio est consimilis. Hec autem que sunt diverse positionis respectu utriusque visus sunt valde rara, ut in tertio diximus.

6.11 ◉ As we explained in the third book, moreover, the reason for this is that, when any point on any visible object is perceived directly by both eyes, and when two rays from both eyes that have a corresponding situation with respect to the two visual axes intersect on that point, it will be perceived as single, but if the rays converging on it have a divergent [non-corresponding] situation with respect to the two axes of the eyes, it will be perceived double. For the most part, however, the situation [of two such rays] corresponds when things are perceived. And the cases in which they have a divergent situation with respect to both eyes are quite rare, as we said in the third [book].⁑

6.12 ◉ Et illud quod comprehenditur reflexe comprehenditur in loco ymaginis. Forma autem que est in loco ymaginis comprehenditur a visu recte; est ergo quasi esset in aere et comprehenderetur a visu recte. Positio autem huius forme, que est ymago respectu visus, est sicut positio alicuius rei vise earum que videntur recte, unde positio harum ymaginum respectu visus est in maiori parte consimilis. Et in omni puncto ymaginis congregantur duo radii duorum visuum consimilis positionis, quare una res videbitur una utroque visu.

6.12 ◉ What is perceived by means of refraction is perceived at an image-location. Moreover, the form at the image-location is perceived by the visual faculty [as if it were perceived] directly, so it is as if it lay in air and were perceived directly by the visual faculty. In addition, the situation of this form, which is the image with respect to the center of sight, is like the situation of any of those visible objects that are perceived directly, so for the most part the situation of these images with respect to the center of sight is corresponding. And from both eyes two rays having a corresponding situation converge at every point on an image, so a single object will be seen as single by both eyes.

6.13 ◉ Et ut hoc evidentius declaretur, dicamus quod iam diximus quod omne punctum eius quod comprehenditur reflexive comprehenditur in loco ymaginis, que est inter punctum sectionis ex perpendiculari exeunti ex illo puncto super superficiem corporis diaffoni in quo est res visa et inter lineam radialem per quam extenditur forma ad visum. Cum ergo aspiciens comprehenderit punctum alicuius rei utroque visu, ymago illius puncti respectu utriusque visus est in perpendiculari exeunti ex illo puncto, que est eadem linea. Et cum forma illius puncti pervenerit ad duo puncta superficierum visuum, quorum situs respectu axis visuum est consimilis, tunc due linee per quas forme extenduntur ad utrumque visum perveniunt ad duo centra duorum visuum. Sunt ergo axes aut habentes ex axibus positionem consimilem.

6.13 ◉ In order for this to be shown more clearly, we should point out that we have just claimed that every point on an object that is perceived according to refraction is perceived at the image-location, which is the point of intersection between the normal dropped from that point to the surface of the transparent body in which the visible object lies and the radial line along which the form extends to the center of sight. Hence, when the viewer perceives a point on any object with both eyes, the image of that point with respect to both eyes lies on the normal dropped from that point, which is the same line. And when the form of that point reaches two points on the surfaces of the eyes that have a corresponding situation with respect to the visual axes, the two lines along which the forms extend to both eyes reach the two centers of both eyes. [These lines] are therefore either the [visual] axes [themselves] or [lines] that have a corresponding situation with respect to the [visual] axes.

6.14 ◉ Et duo axes visuum semper sunt in eadem superficie, et omnes linee exeuntes a centro duorum visuum habentes positionem consimilem ab axe communi erunt in eadem superficie, axis enim communis semper est in eadem superficie, nam, si aliquod comprehenditur utroque visu in eodem tempore vera comprehensione, tunc axes concurrunt in uno puncto illius rei, quare sunt in eadem superficie. Item positio visuum naturalis est consimilis, et non exit a naturali nisi per accidens aut violentiam, quare axes eorum sunt in eadem superficie, principium enim axium est unum punctum quod est in medio concavitatis communis nervi, a qua exit communis axis.

6.14 ◉ But the two visual axes always lie in the same plane, and all of the lines extending from the center of the two eyes that have a corresponding situation with respect to the common axis will lie in the same plane, for the common axis always lies in the same plane because, if something is perceived correctly by both eyes at the same time, the axes intersect at one point on that thing, so they lie in the same plane. Furthermore, the eyes are naturally disposed to correspond in orientation, and they do not deviate from [their] natural [disposition] unless by accident or force, so their axes lie in the same plane, for the origin of the axes is a single point at the center of the hollow in the common nerve, from which the common axis extends.

6.15 ◉ Existentibus ergo duobus visibus in sui naturali positione, semper axes erunt in eadem superficie, sive sint moti, sive quiescentes. Si autem positio alterius visuum mutata fuerit respectu reliqui propter aliquod impedimentum, tunc res una videbitur due, ut in primo declaravimus. Duo ergo axes in maiori parte sunt in eadem superficie, quare omnes duo radii habentes positionem similem ex duobus axibus erunt in eadem superficie. Due ergo linee per quas extenduntur forme unius puncti ad duo loca consimilis positionis sunt in eadem superficie. Sed ymagines illius puncti respectu duorum visuum sunt in illis duabus lineis; ergo sunt in eadem superficie, sed ymagines illius puncti sunt in perpendiculari exeunti ex illo puncto; ergo sunt in loco sectionis inter superficiem in qua sunt linee radiales, que est una superficies, et inter perpendicularem, que est una linea.

6.15 ◉ Therefore, when the two eyes are in their natural situation, the axes will always lie in the same plane, whether they are moved or at rest. If, however, the situation of either of the eyes is altered with respect to the other on account of some obstacle, a single object will appear double, as we showed in the first [book].⁑ Hence, for the most part the two axes lie in the same plane, so every pair of rays having a corresponding situation with respect to the two axes will lie in the same plane. The two lines along which the forms of one point extend to two places [on the eye] that have a corresponding situation thus lie in the same plane. But the images of that point with respect to [each of] the two eyes lie on those two lines, so they lie in the same plane, and the images of that point lie on the normal dropped from that point, so they lie at the intersection of the plane in which the radial lines lie, which is a single plane, and the normal, which is a single line.

6.16 ◉ Sectio autem unius superficiei cum una linea est unum punctum; ergo ymagines unius puncti respectu duorum visuum, quando perveniunt ad duo loca consimilis positionis, sunt unum punctum, ex quo patet quod ymago totius rei vise respectu duorum visuum erit una, si positio ymaginis fuerit consimilis, quare res comprehenditur una utroque visu. Si vero positio fuerit parum diversa et videbitur res una, sed non vere sed cavillose. Si autem diversitas positionis fuerit multa, tunc forma rei videbitur due, sed hoc fit rarissime. Hec est ergo qualitas comprehensionis visus de visibilibus secundum reflexionem.

6.16 ◉ The intersection of a plane with a line, however, is a point, so, when they reach two places with a corresponding situation, the images of one point with respect to the two eyes form a single point, from which it is evident that the image of the entire visible object will be single with respect to both eyes if the situation of the image is corresponding, so the object is perceived as single by both eyes. If the situation is slightly divergent, though, the object will still be seen as single, but in a confused [or blurred] rather than a correct [or distinct] way. But if the divergence in situation is considerable, the form of the object will be seen double, although this happens extremely rarely. This, then, is how the visual faculty perceives visible objects according to refraction.

6.17 ◉ Hoc autem declarato, dicamus universaliter quod omnia que comprehenduntur a visu comprehenduntur reflexive, sive visus et visum sunt in eodem diaffono aut in diversis, sive visum sit in oppositione visus, sive comprehendatur ab ipso reflexive. Nichil enim comprehenditur sine reflexione facta apud superficiem visus, nam tunice visus, que sunt cornea, albuginea, et glacialis, sunt etiam diaffone et spissiores aere. Et iam declaratum est quod forme eorum que sunt in aere et in aliis corporibus diaffonis extenduntur in illis corporibus, et si occurrerint corpori diverse diaffonitatis ei in quo sunt, reflectuntur in illo corpore diaffono. Forma ergo eius quod est in aere semper extenditur in aere; cum ergo aer tetigerit superficiem alicuius visus, tunc illa forma que est in aere reflectitur in superficie visus, et tunc reflectetur omnimodo in corpore cornee et albuginee, reflexio enim proprie est de numero formarum. Recipere autem formas et reflexiones est proprium corporibus diaffonis; forme ergo eorum que visui opponuntur semper reflectuntur in tunicis visus.

6.17 ◉ Now that this has been confirmed, we may make the general claim that everything perceived by sight is perceived according to refraction, whether the eye and the visible object lie in the same or different transparent [bodies], or whether the visible object faces the eye directly, or whether it is perceived by the eye according to refraction. In fact, nothing is perceived without refraction occurring at the surface of the eye, for the tunics of the eye, which are the cornea, the albugineous [humor], and the glacial [humor], are also transparent and denser than the air. And it has just been shown that the forms of objects in the air and in other transparent bodies extend through those bodies, and if they encounter a body whose transparency differs from that of the one in which they lie, [those forms] are refracted in that transparent body. Accordingly, the form of what is in the air always extends through the air, [and] so, if the air is in contact with the surface of any eye, the form in the air is refracted at the surface of the eye, and it will then be refracted throughout in the body of the cornea and the albugineous humor, for refraction is characteristic of all forms. Moreover, the reception and refraction of forms is characteristic of transparent bodies, so the forms of objects that face the eye are always refracted in the tunics of the eye.⁑

6.18 ◉ Et iam patuit quod, cum forme extenduntur super lineas perpendiculares super secundum corpus, pertranseunt recte in secundo corpore; forme ergo eorum que opponuntur superficiei visus reflectuntur omnes in tunicis visus, et que fuerint ex eis in extremitatibus linearum radialium perpendicularium super superficiem visus pertransibunt recte cum reflexione formarum earum in tunicis visus. Parti enim superficiei visus que opponitur foramini uvee multa opponuntur visibilia, quorum alia sunt apud extremitates linearum radialium, et alia extra.

6.18 ◉ It was just shown as well that, when forms extend along perpendicular lines to a second [transparent] body, they pass straight through the second body, so the forms of objects that face the surface of the eye are all refracted in the tunics of the eye [when they reach its surface at a slant], and the forms of those [points on the object] that lie at the endpoints of the radial lines that are perpendicular to the surface of the eye will pass straight through while at the same time their forms are [also] refracted in the tunics of the eyes. For many visible objects face the portion of the eye’s surface in line with the aperture in the uvea [i.e., the pupil], and some of them lie at the endpoints of radial lines, and others lie outside [such lines].⁑

6.19 ◉ Omnes enim linee radiales que sunt perpendiculares super superficies tunicarum visus continentur in piramide, cuius capud est centrum visus et cuius basis est circumferentia foraminis uvee, et quanto magis extenditur hec piramis et removetur a visu, tanto magis amplificatur. Et omnes forme eorum que sunt intra piramidem extenduntur in rectitudine linearum radialium, et pertranseunt in tunicis visus recte, et hec piramis dicitur piramis radialis. Linee autem que extenduntur in hac piramide quarum extremitates sunt apud centrum visus dicuntur linee radiales.

6.19 ◉ Indeed, all the radial lines that are perpendicular to the surfaces of the eye’s tunics are contained within a cone, whose vertex lies at the center of sight and whose base is [formed by] the circumference of the aperture in the uvea, and the farther this cone extends outward from the center of sight, the more it spreads out. All the forms of those objects lying within the cone extend straight along radial lines and pass straight through the tunics of the eye, and this cone is called the »cone of radiation.« Furthermore, the lines that extend within this cone and have their endpoints at the center of sight [i.e., converge at that point] are called »radial lines.«

6.20 ◉ Forme vero eorum que sunt extra hanc piramidem numquam extenduntur per aliquam linearum radialium, tamen extendentur per lineas rectas que sunt inter ipsam superficiem visus que opponitur foramini uvee. Et forme que extenduntur per has lineas reflectuntur a diaffonitate tunicarum visus, et forma cuiuslibet puncti eorum que sunt intra piramidem radialem extenditur ad superficiem visus que opponitur foramini uvee in piramide cuius capud est illud punctum et cuius basis est superficies que opponitur foramine uvee. Et una linea earum que ymaginatur in hac piramide est linea radialis; cetere autem omnes que sunt in hac piramide non sunt radiales, et nulla earum est perpendicularis super superficies tunicarum visus.

6.20 ◉ But the forms of objects that are outside this cone never extend along any of the radial lines, yet they will extend along straight lines [that pass] between [the object and] the surface of the eye facing the aperture in the uvea. The forms that extend along these lines are refracted by the transparency of the eye’s tunics, and the form of any point on objects that lie within the cone of radiation extends to the surface of the eye facing the aperture of the uvea within a cone whose vertex lies at that point and whose base is the surface facing the aperture of the uvea.⁑ Of [all] the lines imagined [to exist] in this cone, [only] one is a radial line; all the rest that are within this cone are not radial [lines], and none of them is perpendicular to the surfaces of the eye’s tunics.

6.21 ◉ Et forma cuiuslibet puncti eorum que sunt intra piramidem radialem extenditur super lineam omnem que potest cadere in illa piramide cuius capud est illud punctum et cuius basis est superficies rei vise que opponitur foramini uvee. Et per unam istarum linearum transit forma que extenditur per illam in tunicis visus secundum rectitudinem, et omnes forme alie extense in residuo piramidis reflectuntur in tunicis visus et non pertranseunt recte. Omnia ergo que opponuntur parti superficiei visus que opponitur foramini uvee ex illis que sunt in aere, aut in celo, aut in aqua, aut consimilibus, et ex illis que convertuntur a tersis corporibus, que perveniunt ad hanc partem superficiei visus omnes reflectuntur in tunicis visus. Et forme eorum que sunt intra piramidem pertranseunt recte in tunicis visus cum reflexione formarum earum que extenduntur super piramidem que remanent ex universo huius partis superficiei visus. Restat ergo declarare quod forme que reflectuntur in tunicis visus comprehenduntur a visu et sentiuntur a virtute sensibili.

6.21 ◉ Moreover, the form of any point among those within the cone of radiation extends along every line that can fall within that cone, whose vertex lies at that point and whose base is the surface of the visible object that faces the aperture in the uvea.⁑ And the form that extends straight through the tunics of the eye follows one of those lines, whereas all the other forms extending within the rest of the cone are refracted in the eye’s tunics and do not pass straight through. Consequently, [the form of] everything in the air, the heavens, water, or the like, as well as what is reflected from polished bodies, is refracted in the tunics of the eye [when that form extends from an object] in front of the portion of the eye’s surface facing the aperture in the uvea and [when it] reaches that portion of the eye’s surface. And the forms of those objects that lie within the cone pass straight through the tunics of the eyes while the remaining forms of those objects, which extend within the cone [along oblique lines] are refracted throughout this portion of the eye’s surface. So it remains to demonstrate that forms refracted in the eye’s tunics are perceived by the visual faculty and sensed by the sensitive power.⁑

6.22 ◉ In primo autem declaravimus quod, si membrum sensibile sentiret ex quolibet puncto sue superficiei omnem formam ad ipsam pervenientem, tunc sentiret formas rerum mixtas, unde membrum sensibile non sentit formas nisi ex rectitudine linearum perpendicularium super superficiem ipsius tantum, quare transeunt forme visibilium nec admiscentur apud ipsum forme visibilium. In hoc vero tractatu monstravimus quod forme reflexe numquam comprehenduntur nisi in perpendicularibus exeuntibus a visibilibus super superficies corporum diaffonorum. Ergo forme reflexe in tunicis visus non comprehenduntur a visu nisi in perpendicularibus exeuntibus a visibilibus super superficies tunicarum visus, et hee perpendiculares linee sunt exeuntes a centro visus.

6.22 ◉ Now in the first [book] we showed that, if the sensitive organ were to sense every form reaching it at every point on its surface, it would sense the forms of things all mingled together, which is why the sensitive organ senses forms exclusively along straight lines perpendicular to its surface so [that] the forms of visible objects pass [straight] through, and the forms of visible objects do not intermingle at it. In this book, in fact, we have demonstrated that refracted forms are perceived only on the normals extending from visible objects to the surfaces of transparent bodies. Consequently, the forms that are refracted in the eye’s tunics are perceived by the visual faculty only along the normals extending from visible objects to the surfaces of the eye’s tunics, and these normals are lines extending from the center of sight.

6.23 ◉ Forme igitur omnes reflexe in tunicis visus comprehenduntur a visu in rectitudine linearum exeuntium a centro visus; forme ergo omnium visibilium que opponuntur parti superficie visus que opponitur foramini uvee existunt in hac parte superficiei visus, et reflectuntur in diaffonitate tunicarum visus, et perveniunt ad membrum sensibile, quod est humor glacialis, et comprehenduntur a virtute sensibili per lineas rectas que continuant centrum visus cum illis visibilibus. Sed quod forma cuiuslibet puncti visi oppositi superficiei visus que opponitur foramini uvee existit in universo superficiei huius partis, et reflectitur a tota hac parte, et pervenit ad humorem glacialem, et tunc ille humor sentit formam ad se venientem. Et virtus sensibilis comprehendit omnia que perveniunt ad glacialem ex forma visus puncti super unam lineam continuantem centrum visus cum illo puncto. Hoc igitur modo comprehendit visus omnia visibilia.

6.23 ◉ Hence, all forms that are refracted in the eye’s tunics are perceived by the visual faculty along straight lines extending from the center of sight, so the forms of all visible objects that are in front of the portion of the eye’s surface facing the aperture in the uvea lie on this portion of the eye’s surface, and they are refracted according to the transparency of the eye’s tunics, then reach the sensitive organ, which is the glacial humor, and are perceived by the sensitive power along straight lines connecting the center of sight with those visible objects. But the form of any point on the object in front of the surface of the eye facing the aperture in the uvea lies on the entire surface of this portion, and it is refracted at this entire portion and reaches the glacial humor, at which time the [glacial] humor senses the form reaching it. And the sensitive power perceives all the forms of the visible point that reach the glacial humor along a single line connecting the center of sight with that point. So this is how the visual faculty perceives all visible objects.

6.24 ◉ In hoc capitulo diximus quod eorum que opponuntur superficiei visus alia sunt intra piramidem radialem et alia extra, et cum dixero superficiem visus, intellige nunc et amodo partem oppositam superficiei uvee. Visibilia ergo que sunt intra piramidem radialem comprehenduntur a visu ex rectitudine linearum radialium recte ex formis eorum que extenduntur ad visum in rectitudine harum linearum etiam, hee linee enim sunt perpendiculares que exeunt a punctis visibilibus que sunt intra piramidem super superficiem tunicarum visus. Illa autem que sunt extra piramidem radialem comprehenduntur a visu ex formis reflexis et in rectitudine linearum exeuntium a centro visus existentium extra piramidem radialem, et hee linee que sunt extra piramidem possunt etiam dici linee radiales transumptive, assimulantur enim lineis radialibus in eo quod exeunt a centro visus. Restat ergo declarare per experientiam quod visus comprehendit ea que sunt extra piramidem radialem.

6.24 ◉ In this chapter we claimed that some objects that face the surface of the eye lie within the cone of radiation and some lie outside it, and when I say »the surface of the eye« you must henceforth understand [that I mean] the portion facing the surface of the uvea. Thus, visible objects that lie within the cone of radiation are perceived by the visual faculty along straight radial lines according to the forms [of those objects] that extend to the center of sight straight along these same lines, for these lines are the normals within the cone that extend to the surface of the eye’s tunics from visible points. On the other hand, those [points] outside the cone of radiation are perceived by the visual faculty according to refracted forms but along straight lines extending from the center of sight [to points] lying outside the cone of radiation, and these lines that lie outside the cone can also be called radial lines by analogy, for they are like [actual] radial lines in that they extend from the center of sight. So it remains [for us] to show by experiment that the visual faculty perceives objects outside the cone of radiation.

6.25 ◉ Dicimus ergo quod manifestum est quod lacrimalia et ea que continent oculum sunt extra piramidem cuius capud est centrum visus et cuius basis est circumferentia foraminis uvee, quod est parvum foramen in medio nigredinis oculi. Et si aliquis sumpserit acum subtilem gracilem et posuerit extremitatem eius in postremo oculi et inter palpebras, et quieverit visus, tunc videbit extremitatem acus. Et similiter, si posuerit extremitatem acus in lacrimali, et si miserit illam in oculo, et applicaverit extremitatem in latere nigredinis oculi aut prope, videbit extremitatem acus. Item omnia que equidistant superficiei rei vise ex locis continentibus visum sunt extra piramidem radialem, et cum dico loca continentia visum intelligo illa a quibus linee exeuntes ad medium superficiei visus secant axem piramidis radialis. Et si homo erexerit indicem suum in parte sue faciei et prope palpebram, videbit indicem, et similiter, si applicaverit indicem cum inferiori palpebra ita quod superior superficies eius indicis sit equidistans superficiei visus quantum ad sensum, videbit superficiem indicis.

6.25 ◉ Accordingly, we maintain that it is obvious that the tear ducts and what surrounds the eye lie outside the cone whose vertex lies at the center of sight and whose base is [formed by] the circumference of the aperture in the uvea, which is a small opening in the middle of the black portion of the eye.⁑ And if one takes a small, thin needle and places its endpoint at the lower extremity of the eye between the eyelids, and if he holds his eye steady, he will see the needle’s endpoint. By the same token, if he places the needle’s endpoint at the tear ducts and brings it up to the eye, or if he puts the endpoint at or near the side of the eye’s black portion, he will see the needle’s endpoint. Furthermore, everything parallel to the surface of the visible object in the areas surrounding the eye lies outside the cone of radiation, and when I say »areas surrounding the eye« I mean those [areas] from which lines extending to the middle of the eye’s surface intersect the axis of the cone of radiation.⁑ Also, if a person holds his forefinger up to the side of his face and near the eyelid, he will see his forefinger, and likewise, if he puts his finger at the lower eyelid so that the top surface of his forefinger is parallel to the eye’s surface as far as can be empirically determined, he will see the forefinger’s surface.

6.26 ◉ Sed omnia ista loca sunt extra piramidem radialem, et hoc patet, nam piramis radialis quam continet foramen uvee est valde subtilis, et extenditur recte, et piramidalitas eius non est ampla, unde nichil ex ipso pervenit ad loca que circumdant oculum, et appropinquant corpori oculi, et equidistant superficiei oculi. Et inter omnia loca continentia oculum et equidistantia superficiei visus et inter superficiem visus sunt linee recte propter reflexionem earum a corporibus densis; cum aer qui est inter ipsam et superficiem visus fuerit continuus, tunc forma horum visibilium pervenit ad superficiem visus super has lineas que sunt extra piramidem. Et cum hec forma pervenit ad visum non per lineas radiales, et tamen comprehendetur a visu, patet quod visus comprehendit illam reflexive. Ex hac igitur experientia patet quod visus comprehendit multa eorum que sunt extra piramidem radialem reflexive.

6.26 ◉ But all these areas lie outside the cone of radiation, and this is obvious because the cone of radiation that the aperture in the uvea circumscribes is quite small and continues straight out [through the aperture], and its amplitude is slight, so none of it reaches the areas that surround the eye or approach the eye’s body while remaining parallel to the eye’s surface. But there are straight lines between the eye’s surface and all the areas surrounding the eye and parallel to the eye’s surface according to their refraction in the dense bodies [comprising the eye’s tunics, and] since the air between those areas and the eye’s surface is continuous, the form of these visible objects reaches the eye’s surface along these lines that lie outside the cone. And since this form does not reach the eye along radial lines yet will be perceived by the visual faculty, it is clear that the visual faculty perceives it according to refraction. From this experiment, therefore, it is evident that the visual faculty perceives many objects that lie outside the cone of radiation by means of refraction.⁑

6.27 ◉ Inductione etiam possumus ostendere quod visus comprehendit illa que sunt intra piramidem reflexive cum hoc quod comprehendit illa recte hoc modo. Accipias acum subtilem, et sedeas in loco opposito albo parieti, et cooperias alterum oculorum, et ponas acum in oppositione alterius oculi, et facias acum appropinquare ita quod applicetur palpebre. Et ponas acum in oppositione medii visus, et aspicias parietem oppositum, tunc enim videbis acum quasi corpus diaffonum in quo est aliquantula densitas, et videbis quicquid est ultra acum ex pariete et apud acum quasi corpus latum, cuius latitudo est multiplex ad latitudinem acus.

6.27 ◉ We can also show empirically that the visual faculty perceives objects within the cone by means of refraction, notwithstanding that it perceives them directly [and we can do so] as follows. Take a thin needle, sit down in a place that faces a white wall, close one eye, put the needle in front of the other eye, and bring the needle near [the eye] so that it touches the eyelid. Then put the needle directly in front of the midpoint of [the front surface of the] eye, and look at the facing wall, for in that case you will see the needle as a transparent object with a bit of opacity in it, and you will see that portion of the wall beyond the needle and in line with the needle as a wide object that is many times wider than the needle.

6.28 ◉ Causa autem huius in secundo tractatu declarata est, scilicet quod, si res visibilis multum fuerit propinqua visui, videbitur maior quam sit, et quanto magis fuerit illa propinqua, tanto magis videbitur maior. Diaffonitas autem eius est quia visus comprehendit quicquid est ultra. Acus autem est corpus densum cooperiens quod est ultra, et quia acus est valde propinqua visui, ideo cooperuit de pariete multiplex ad sui latitudinem, piramis enim, cuius capud est centrum visus et basis est latitudo acus, basis eius erit multiplex ad latitudinem acus. Et cum hoc visus comprehendit quicquid est ultra acum, nec cooperuit a visu aliquid de pariete, sed comprehendit quod est ultra quasi ultra corpus diaffonum.

6.28 ◉ The reason for this has been shown in the second book, i.e., that, if a visible object lies quite close to the eye, it will appear larger than it is, and the closer it gets, the larger it will appear.⁑ Its [apparent] transparency, moreover, is due to the visual faculty’s perceiving what lies beyond it. But the needle is an opaque body blocking what lies beyond, and since the needle is quite close to the eye, it blocks a portion of the wall that is many times its width, for [when it is extended to the wall] the base of the cone whose vertex lies at the center of sight and whose base is the width of the needle will be many times as wide as the width of the needle. And yet the visual faculty perceives what lies behind the needle, and it blocks none of the wall [behind] from the center of sight, but [the visual faculty] perceives what lies behind as if [it lay] behind a transparent body.

6.29 ◉ Et cum acus fuerit opposita medio visui, tunc non cooperiet totam superficiem visus propter subtilitatem eius sed aliquam partem quanta est latitudo eius, et remanet ex superficie visus aliquid a lateribus acus, et exit forma eius ad illud quod est a lateribus acus de superficie visus. Forma autem exiens ad acum numquam perveniet ad visum, nec comprehendetur ab ipso; forma autem que pervenit ad latera superficiei visus reflectitur ad visum, cum non recte perveniat ad centrum visus. Si ergo visus non comprehenderet illud quod opponitur acui ex pariete nisi recte, tunc illud quod opponitur acui ex pariete esset coopertum a visu. Cum igitur comprehendatur et non recte, patet ipsum comprehendi reflexive per formam que reflectitur a lateribus acus ex superficie visus. Et hoc manifestatur etiam, si experimentator posuerit loco acus aliquod corpus latum cuius latitudo sit maior latitudine foraminis uvee, tunc enim nichil videbit omnino de pariete, nec videbit illud corpus diaffonum sed densum.

6.29 ◉ But since the needle faces the middle of the eye directly, it will not block the entire surface of the eye but only a part commensurate with its width because of its narrowness, and [so] some of the surface of the eye remains [uncovered] to the sides of the needle, and the form of the object extends to that area of the eye’s surface to the sides of the needle. But the form extending to the needle [itself] will never reach the center of sight or be perceived by it, whereas the form that reaches the sides of the eye’s surface is refracted at the eye, since it cannot reach the center of sight directly. Consequently, if the eye were only to perceive the part of the wall that lies directly in line with the needle by means of direct vision [i.e., only along the orthogonals], then the part of the wall directly in line with the needle would be blocked from the visual faculty. Since, then, it must be perceived, but not directly, it is clear that is is perceived by means of refraction according to a form that is refracted at the surface of the eye to the sides of the needle. And this is also made evident if the experimenter replaces the needle with a wide object that is wider than the aperture in the uvea, for in that case he will seen none of the wall whatever, nor will he see that body [as if it were] transparent but [as completely] opaque.

6.30 ◉ Ex hoc ergo quod paries comprehenditur ultra acum ex gracilitate eius et non comprehenditur ultra corpus latum scimus quod illa comprehensio est ex forma que pervenit ad acum ex superficie visus et reflectitur in tunicis visus. Et quia quicquid a visu comprehenditur reflexive comprehenditur in rectitudine perpendicularium, ideo illud quod comprehendit comprehendit reflexive ex forma eius quod opponitur acui per rectitudinem linearum exeuntium a centro visus que continuant centrum visus cum eo quod opponitur acui ex pariete, et hee linee secantur acu. Et visus comprehendit illud quod est ultra acum etiam in rectitudine harum linearum, et comprehendit acum etiam in rectitudine illarum, quare totam quasi formam comprehendit ultra corpus diaffonum in quo est aliquantula densitas.

6.30 ◉ From the fact that the wall is perceived behind the needle because of its narrowness but is not perceived behind a wide object, then, we know that the perception is due to a form that reaches the eye’s surface at [the sides of] the needle and is refracted in the eye’s tunics. And because whatever is perceived by the visual faculty according to refraction is perceived straight along the perpendiculars, it follows that it perceives what it perceives according to refraction by means of a form on the [part of the object] directly in line with the needle along straight lines extending from the center of sight and connecting the center of sight with the portion of the wall facing the needle, and these lines are cut by the needle. And [so] the visual faculty perceives what lies behind the needle straight along these lines, and it perceives the needle as well along those straight lines, which is why it perceives the entire form as if through a transparent body in which there is a modicum of opacity.

6.31 ◉ Et si experimentator scripserit in bombace subtiliter et applicaverit ipsum parieti, et remotus fuerit a pariete in quantum possit legere scripturam, et posuerit acum in oppositione medii visus, ut primo fecit, et aspexerit bombacem, tunc poterit legere scripturam, sed tamen videbit eam quasi ultra vitrum aut ultra corpus diaffonum in quo est aliqua densitudo. Si ergo visus non comprehenderet illud quod opponitur acui de bombace secundum reflexionem, tunc aliquid lateret de scriptura, acus enim debet cooperire de scriptura multo magis se in quantitate latitudinis diaffonitatis quam tunc comprehendit propter remotionem bombacis a visu. Sed quia visui non latet aliquid de scriptura, patet ipsum comprehendere illud quod opponitur acui, sed hoc non potest fieri recte. Restat ergo quod fiat reflexive.

6.31 ◉ Moreover, if the experimenter writes something small on [a piece of] paper and attaches it to the wall, and if he stands far enough from the wall that he can read the writing and puts the needle directly in front of the middle of the eye, as he did before, then when he looks at the paper he can read the writing, and yet he will still see it as if through glass or through a transparent body in which there is some opacity. Hence, if the visual faculty did not perceive that portion of the paper behind the needle according to refraction, the writing would lack something in the way of visibility, for the needle ought to block the writing all the more in relation to the width of its [apparent] transparency than [the portion of the paper behind the needle] that he perceives in that case because of the distance of the paper from the center of sight. But since none of the writing is hidden from the visual faculty, it is clear that it perceives what lies behind the needle, yet it cannot do so directly. It follows, therefore, that it must be done according to refraction.

6.32 ◉ Et si experimentator abstulerit acum, non destruetur reflexio que prius erat, non enim propter acum erat reflexio, sed crescit reflexio eo quod reflectitur ex loco acus. Et cum experimentator abstulerit acum, comprehendet illud quod opponitur visui manifestius, nam comprehendet illud recte quod cooperiebatur acu, cum hoc quod comprehendit illud reflexive, sicut comprehendebat cum cooperiebatur, et propter hanc additionem comprehendit illud manifestius quam antequam auferabat acum, ex qua experientia patet quod illud quod opponitur visui de illis que sunt intra piramidem radialem comprehenditur reflexive et recte.

6.32 ◉ Furthermore, if the experimenter removes the needle, he will not eliminate the refraction that occurred previously, for the refraction was not due to the needle; instead, the refraction increases according to what is refracted from the area [uncovered] from the needle. And when the experimenter removes the needle, he will perceive what faces the eye more clearly, for he will perceive directly what was blocked by the needle, notwithstanding that he [also] perceives it by means of refraction, just as he perceived it when it was blocked, and according to this reinforcement [by the added orthogonal and oblique impingments] he perceives it more clearly than [he did] before he removed the needle, and from this experiment is it evident that what faces the eye within the cone of radiation is perceived both directly and by means of refraction.⁑

6.33 ◉ Ex hiis ergo omnibus declaratur quod omnia que comprehenduntur a visu quorum forme perveniunt ad visum recte, aut conversive, aut reflexe omnia comprehenduntur secundum reflexionem factam apud superficiem visus, et quod illa que comprehenduntur secundum reflexionem factam a superficie visus quedam comprehenduntur reflexe et recte simul. Et ideo illud quod opponitur medio visus est manifestius illo quod est in circuitu medii, et cum visus comprehenderit aliquod latum, comprehendet illud quod est in medio manifestius illo quod est in lateribus. Hoc autem declaratum est in secundo tractatu, in quo declaravimus qualiter hoc posset experiri et diximus quod causa huius est propter lineas radiales, et hoc est in illis que sunt intra piramidem radialem. In illis autem que sunt extra causa est reflexio. Causa ergo universalis in hoc quod illud quod opponitur medio visus est manifestius quam illud quod est in circuitu est quoniam illud quod opponitur medio visui comprehenditur recte et reflexe simul. Hoc autem quod quicquid comprehenditur a visu comprehenditur reflexe a nullo antiquorum dictum est.

6.33 ◉ From all these [observations], then, it is manifest that everything whose form reaches the center of sight directly, by reflection, or by refraction is perceived by the visual faculty according to refraction occurring at the eye’s surface, and [it is also manifest] that some of the things perceived according to refraction at the surface of the eye are perceived [both] directly and by means of refraction at the same time. And so whatever faces the middle of the eye is clearer than what surrounds the middle, and when the visual faculty perceives something broad, it will perceive what is at the middle more clearly than what lies at the sides. This, moreover, was demonstrated in the second book, where we showed how this could be empirically tested and explained why this is so on the basis of radial lines, and this applies to those things that lie within the cone of radiation.⁑ In the case of things outside [the cone], however, refraction is the cause. Thus, the overarching reason that what faces the middle of the eye is clearer than what surrounds [that point] is that what faces the middle of the eye is perceived both directly and by means of refraction at the same time. Furthermore, that whatever is perceived by the visual faculty is perceived by means of refraction was mentioned by none of the ancients.

◉Capitulum septimum

◉Chapter Seven

In deceptionibus visus que fiunt secundum reflexionem

7.1 ◉ Fallacie que accidunt secundum reflexionem similes sunt eis que accidunt per conversionem, quod enim comprehenditur reflexive comprehenditur non in suo loco, cum comprehendatur in loco ymaginis, quapropter positio forme comprehense erit alia a positione rei vise, et similiter remotio in eis. Item reflexio debilitat formam reflexam, scilicet formam lucis et coloris que sunt in re visa. Et hoc potest intelligi quoniam, si aspexeris aliquid existens in aqua, et tu sis obliquus a perpendicularibus exeuntibus in re visa super superficiem aque multa obliquatione, et intuearis illud vere, deinde movearis et moveas visum donec ponas ipsum in aliqua perpendiculari exeunte a re visa super superficiem aque, et aspexeris, tunc videbis illud manifestius quam cum eras obliquus. Et nulla est differentia inter duos situs nisi quia in primo forma que exit ad visum est reflexa et multum obliqua; in secundo autem forma exit recte, aut quedam pars ipsius exit recte, et quedam modicum oblique, aut fere recte. Ex hac igitur experimentatione declaratur quod reflexio debilitat formas reflexas.

7.1 ◉ The misperceptions⁑ that arise according to refraction are similar to those that arise in reflection, for what is perceived according to refraction is not perceived in its [actual] place, since it is perceived at the image-location, so the location of the form that is perceived will be different from the location of the visible object, and the same [holds for] their [relative] distance [from the center of sight]. Moreover, refraction weakens the refracted form, i.e., the form of the light and color in the visible object. This can be understood from the fact that, if you look at something in water and are far to the side of the normals dropped from the visible object to the water’s surface, and if you inspect it properly, then change position, move your eye until you line it up on any normal dropped from the visible object to the surface of the water, and look [at the same object under water], you will see it more clearly than [you did] when you were off to the side. But there is no difference in the two situations except that in the first one the form that extends to the center of sight is refracted and sharply slanted [with respect to the refracting surface], whereas in the second [situation] the form, or some part of it, extends straight [through the refracting surface], while some [neighboring parts of the form extend] at a gentle slant or nearly straight [through the refracting surface]. From this experiment, then, it becomes clear that refraction weakens refracted forms.

7.2 ◉ Item ea que sunt in aqua, et ultra vitrum, et consimilia, quando reflectuntur ad visum, deferunt secum colorem corporis in quo existunt. In illis ergo que comprehenduntur reflexe ultra corpora diaffona accidunt propter reflexionem fallacie que non accidunt in eis que videntur recte, scilicet diversitas positionis et distantie et debilitas lucis et coloris. Preterea accidunt eis ea que accidunt illis que recte videntur, forme enim eorum que comprehenduntur reflexive comprehenduntur in oppositione visus et in rectitudine linearum radialium. Quicquid ergo accidit eis que videntur in rectitudine linearum radialium accidit istis. Et in tertio declaravimus omnes illas fallacias et causas earum, et que sunt etiam cause istorum. Sed in hiis accidit magis et citius propter debilitatem huiusmodi formarum.

7.2 ◉ Furthermore, when they are refracted to the eye, [the forms of] things in water, or behind glass, or [behind] similar [media] convey the color of the body in which they lie.⁑ Hence, in the case of things that are perceived according to refraction through transparent bodies, misperceptions arise on account of the refraction that do not arise in the case of things that are seen directly, i.e., a disparity in location and distance, as well as a weakening of light and color. In addition, [the misperceptions that] arise in the case of things that are seen directly also arise in those [things that are seen by means of refraction], for the forms of things perceived by means of refraction are perceived directly in front of the eye along straight radial lines. Consequently, what happens in the case of things that are seen along straight radial lines [also] happens in the case of things [seen according to refraction]. In the third [book], moreover, we explained all those misperceptions and their causes, and these are also the causes for [misperceptions] in the case of things [that are seen according to refraction].⁑ In the latter case, however, [misperception] arises more frequently and quickly on account of the weakening of such [refracted] forms.

7.3 ◉ Particulares autem deceptiones que accidunt propter figuras superficierum corporum diaffonorum sunt multimode, sed accidunt raro visui, ea enim que comprehenduntur ultra corpora diaffona diversa ab aere sunt stelle et ea que sunt in aqua; illa autem que sunt ultra vitrum et lapides diaffonos diversarum figurarum raro comprehenduntur a visu. Et non est ita de istis corporibus diaffonis ut de speculis, specula enim sepius aspiciuntur ab hominibus, ut videant in eis suas formas, et habentur in domibus. Et similiter, quando homo aspicit in quolibet corpore terso, etiam videbit formam eorum que sunt in oppositione, et similiter, si aspexerit aquam, videbit formam sui in ea, et videbit ea que sunt in oppositione, et non est ita illud quod videtur ultra vitrum et lapides diaffonos, quia homines raro aspiciunt ad illud quod est ultra vitrum et lapides diaffonos. Et quia ita est, dicamus de deceptionibus reflexionis particularibus que semper accidunt et sine difficultate, scilicet que accidunt in eis que videntur in celo et aqua, et dicemus parum de hiis que videntur ultra vitrum et lapides.

7.3 ◉ In addition, specific misperceptions that arise on account of the shapes of the surfaces of transparent bodies are manifold, but they rarely occur in vision, for the things that are [normally] perceived through transparent bodies different from the air include the stars and things in water, whereas things that lie behind glass and transparent stones of various shapes are rarely perceived by the visual faculty. And what applies in the case of these transparent bodies is not what applies to mirrors, for mirrors are often regarded by people so that they can see their images in them, and they are kept in homes. Likewise, when a person looks into any polished body, he will also see the forms of things [surrounding him] that face [that body], and by the same token, if he looks into water, he will see his own form in it, and he will see what faces [the water], but such is not the case for what is seen through glass and transparent stones, for people rarely look at what lies behind glass and transparent stones. And since this is the case, let us talk about specific misperceptions that arise from refraction all the time and without complication, i.e., the ones that arise in the case of things that are seen in the heavens and in water, and we will discuss [only] briefly those things that are seen through glass and stones.

7.4 ◉ Dicimus ergo quod semper visus fallitur in eis que comprehenduntur ultra corpus diaffonum diversum ab aere preter quam in positione, et remotione, et coloribus, et lucibus eorum, ut in magnitudine eorum, et figuris quorumdam, ea enim que videntur in aqua et ultra vitrum et lapides diaffonos videntur maiora. Stelle autem et distantie inter stellas quandoque videntur maiores, quandoque minores.

7.4 ◉ Accordingly, we maintain that the visual faculty is always deceived about things that are perceived through a transparent body different from air, and aside from [misperceptions] of their location, distance, colors, and light, [it is deceived about] their size and about the shapes of some of them, for things that are seen in water and through glass or transparent stones appear magnified. In addition, the stars and the intervals between stars sometimes appear magnified and sometimes diminished.

7.5 ◉ [PROPOSITIO 12] Sit ergo visus A [FIGURE 7.7.10, p. 446], et sit BG ultra corpus diaffonum grossius aere. Dico quod BG videtur maior quam sit.

7.5 ◉ [PROPOSITION 12] Accordingly, let A [in figure 7.7.61, p. 199] be a center of sight, and let BG lie behind a transparent body denser than air. I say that BG appears larger than it [actually] is.

7.6 ◉ Sit ergo in primo superficies corporis diaffoni plana. A autem aut est in perpendiculari exeunti a medio BG super superficiem corporis, aut extra. Sit ergo in primis in ipsa, et sit illa perpendicularis AMZ. Et extrahamus superficiem in qua sunt linee AZ, BG, et faciat in superficie corporis diaffoni lineam DME. Linea ergo AM est perpendicularis super lineam DME, et superficies in qua sunt due linee AZ, BG erit perpendicularis super superficiem corporis diaffoni.

7.6 ◉ To start with, then, let the surface of the transparent body be plane. [Center of sight] A either lies on the normal dropped from the middle of BG to the surface of the [transparent] body, or [it lies] outside [that normal]. First, let it lie on it, and let that normal be AMZ [and let Z be the midpoint of BG]. Let us produce the plane containing lines AZ and BG, and let it form line DME on the surface of the transparent body. Thus, line AM is perpendicular to line DME, and the plane within which the two lines AZ and BG lie will be perpendicular to the surface of the transparent body.

7.7 ◉ Et non transit per A et per aliquod punctum linee BG superfices que sit perpendicularis super superficiem corporis diaffoni nisi illa in qua sunt linee AZ, BG, non enim transit per A superficies perpendicularis super superficiem corporis diaffoni nisi illa que transit per AZ, que linea est perpendicularis super superficiem corporis diaffoni, nec exit ex A perpendicularis super superficiem corporis diaffoni nisi linea AZ. Non ergo transit per A superficies que sit perpendicularis super superficiem corporis diaffoni nisi illa que transit per lineam AZ, et non transit per aliquod punctum linee BG et per lineam AZ nisi illa superficies in qua sunt due linee AZ, BG. Non ergo transit per A et per aliquod punctum linee BG superficies perpendicularis super superficiem corporis diaffoni nisi illa in qua sunt linee AZ, BG; non ergo reflectitur forma alicuius puncti eorum que sunt in BG nisi ex linea DE.

7.7 ◉ Moreover, the only plane perpendicular to the surface of the transparent body that passes through A and through any point [in addition to Z] on line BG is the one in which lines AZ and BG lie, for no plane passes through A perpendicular to the surface of the transparent body unless it passes along AZ, which is a line normal to the surface of the transparent body, and AZ is the only line extending from A that is normal to the surface of the transparent body. Hence, no plane passes through A perpendicular to the surface of the transparent body unless it passes along AZ, and only the plane containing the two lines AZ and BG passes through any point on line BG as well as along line AZ. Thus, only the plane containing the two lines AZ and BG and perpendicular to the surface of the transparent body passes through A as well as through any point [in addition to Z] on line BG, so the form of any point on BG is refracted only at line DE [within that plane].

7.8 ◉ Et extrahamus ex B et G duas perpendiculares. Cadant ergo in lineam DE in duobus punctis D, E, scilicet BD, GE. Et sit BG in primis equidistans linee DE, et reflectatur forma B ad A ex T et forma G ad A ex H. Et continuemus lineas BT, TA, GH, HA, et extrahamus AT ad L et AH ad K. Quia ergo Z positum fuit in medio linee BG, positio B ex A erit equalis positioni G ex A, et sic distantia T ex A erit sicut distantia H ex A, et sic angulus DTL erit equalis angulo EHK. Sed duo anguli D, E sunt recti, et linea DT est equalis linee EH, quia TM est equalis linee MH. Ergo DL est equalis EK.

7.8 ◉ Let us then drop two perpendiculars from B and G. Accordingly, let them fall at the two points D and E on line DE, [and let them be] BD and GE. Let [line] BG be parallel to line DE at the outset, and let the form of B be refracted to A from T, and [let] the form of G [be refracted] to A from H. Let us then connect lines BT, TA, GH, and HA, and let us extend AT to L and AH to K. Therefore, since Z was assumed to lie at the middle of line BG, the location of B with respect to A will be equivalent to the location of G with respect to A, so the distance of T from A will be equal to the distance of H from A, and so angle DTL = angle EHK. But the two angles [BDE and GED] at D and E are right, and line DT = line EH, since [line] TM = line MH. Thus, DL = EK.

7.9 ◉ Et continuemus LK. Erit ergo equalis linee BG. Et continuemus AB, AG. Angulus ergo GAB erit minor angulo KAL, et linea LK est diametrum ymaginis linee BG, nam omne punctum linee BG reflectitur ab aliquo puncto linee TH, nam si forma B reflectitur ex T, punctum quod est inter B et Z reflectitur ab aliquo puncto inter T et M. Et ponamus super lineam BZ punctum N. Si ergo forma eius reflecteretur ab aliquo puncto extra lineam MT ex parte D, tunc linea per quam extenditur forma N secaret lineam BT, et sic forma puncti sectionis reflectetur ad A ex duobus punctis, quod est impossibile, ut diximus in capitulo de ymagine. N ergo non reflectitur ad A nisi ex aliquo puncto inter T, M, et similiter omne punctum in ZG non reflectetur ad A nisi ex linea MH. Linea ergo LK est diameter ymaginis linee BG; forma ergo BG videbitur in LK.

7.9 ◉ Let us connect LK. It will therefore be equal to line BG. And let us connect AB and AG. Hence, angle GAB < angle KAL, and line LK is the cross-section of the image of line BG, for [the form of] every point on line BG is refracted from some point on line TH because, if the form of B is refracted from T, the [form of a] point that lies between B and Z is refracted from some point between T and M. Let us [for instance] take [some] point N on line BZ. If, therefore, its form were refracted from some point [R] outside line MT on the side of D, the line along which the form of N extends would intersect line BT, and so the form of the point of intersection [at X] will be refracted to A from two points [i.e., T and R], which is impossible, as we said in the chapter on images [i.e., chapter 5, proposition 3 above]. Thus, [the form of] N is refracted to A only from some point between T and M, and likewise [the form of] every point on ZG will be refracted to A only from line MH. Line LK is therefore the cross-section of the image of line BG, so the form of BG will be seen on LK.

7.10 ◉ Item iam declaravimus quod forma reflexa est debilior recta. Ergo forma BG, que comprehenditur reflexe, est debilior forma eius que comprehenditur recte, et propter debilitatem forme rei assimilat eam visus forme rei que videtur a maiori remotione, maior enim distantia debilitat formam. Et iam declaravimus in secundo quod visus comprehendit ymaginem rei vise secundum quantitatem anguli respectu remotionis et positionis rei vise apud visum. Et angulus KAL est maior angulo GAB, et positio LK est sicut positio BG, et BG videtur in LK, et LK comprehenditur quasi in maiori distantia BG propter debilitatem forme. Visus ergo comprehendit BG reflexive ex comparatione anguli maioris angulo GAB ad distantiam maiorem distantia BG et ad positionem equalem positioni BG, quapropter BG comprehenditur reflexive maior, et hoc duabus de causis, scilicet magnitudine anguli et debilitate forme. Causa autem magnitudinis anguli est propinquitas anguli ex visu, et causa propinquitatis est reflexio. Causa ergo qua BG comprehenditur maior est reflexio.

7.10 ◉ Now we have already shown that a refracted form is weaker than [one seen in] direct [vision]. Hence, the form of BG, which is perceived by means of refraction, is weaker than the form of BG that is perceived directly, and because of the weakening of the object’s form, the visual faculty matches it to the form of the object seen from a greater distance, for an increase in distance weakens the form. We have also shown in the second [book] that the visual faculty perceives the image of a visible object according to the size of the angle [it subtends at the center of sight] in relation to the distance and orientation of the visible object vis-à-vis the center of sight.⁑ But angle KAL > angle GAB, the orientation of LK is equivalent to the orientation of BG, BG appears at LK, and LK is perceived as if [it lay] at a greater distance than BG because of the weakening of the form. Therefore, the visual faculty perceives BG according to refraction by correlating an angle [KAL] greater than angle GAB to a distance greater than the distance of BG [i.e., according to the magnification of the image’s apparent distance by the weakening of the form] as well as to an orientation equivalent to the orientation of BG, so BG is perceived according to refraction as magnified, and this for two reasons: i.e., the size of the [visual] angle and the weakness of the form. And the reason for the [increased] size of the angle is the proximity of the [vertex of the] angle to the center of sight, and the reason for its proximity is refraction.⁑ Hence, refraction is the reason BG is perceived as magnified.

7.11 ◉ Item iteremus formam, et sit BG [FIGURE 7.7.10a, p. 446] non equidistans linee DE. Et extrahamus a remotiore extremitatum BG ex linea DE lineam equidistantem linee DE, et sit GQ. Et extrahamus AZ ad O. Erit ergo O in medio GQ, quare Z est in medio BG, quia BQ est equidistans ZO. Ergo proportio QO ad OG est sicut BZ ad ZG, Et reflectatur forma Q ad A ex T, et forma G ad A ex H. Et continuemus AT, et pertranseat usque ad L, et continuemus AH, et pertranseat usque ad K, et continuemus LK. Erit ergo LK diameter ymaginis QG. Et continuemus AQ, AG. Erit ergo angulus KAL maior angulo GAQ; A ergo comprehendit ymaginem QG maiorem quam QG. ut prius diximus.

7.11 ◉ To continue, let us redraw the diagram [in figure 7.7.61a, p. 199], but let BG not be parallel to line DE. Let us draw a line from endpoint [G] of BG away from DE parallel to line DE, and let it be GQ. Let us then extend AZ to O. O will therefore lie in the middle of GQ, since Z lies in the middle of BG because BQ is parallel to ZO. Hence, QO:OG = BZ:ZG. Let the form of Q be refracted to A from T, and [let] the form of G [be refracted] to A from H. Let us connect AT and let it pass to L, and let us connect AH and let it pass to K, and let us connect LK. LK will therefore be the cross-section of the image of QG. Let us connect AQ and AG. Angle KAL will therefore be larger than angle GAQ, so [the center of sight at] A perceives the image of QG as larger than QG, as we said earlier.

7.12 ◉ Linea autem QT secabit lineam BG in R. R ergo reflectetur ad A ex T; ergo B reflectetur ad A ex puncto inter duo puncta T, D, nam si reflecteretur ex puncto inter T, M, accideret predictum impossibile. Reflectatur ergo B ad A ex F. Et continuemus AF, et pertranseat ad I. Et continuemus IK. Ergo IK erit diameter ymaginis BG, et positio IK in respectu A est similis positioni BG, quia IK aut erit equidistans ad BG, aut non erit inter illum et equidistantem diversitas que mutet positionem, non enim est inter distantiam IK et distantiam BG a visu grandis diversitas, quare declinatio IK a linea equidistanti BG que exit ex K erit valde parva. Angulus ergo IAK est maior angulo BAG, et positio IK est similis positioni BG, et IK comprehenditur quasi remotior propter debilitatem forme eius. Linea ergo IK videtur maior quam BG, ut in precedenti figura declaravimus. Sed IK est ymago BG; ergo BG videbitur maior quam sit, et hoc est quod voluimus.

7.12 ◉ Moreover, line QT will intersect line BG at R. [The form of] R will therefore be refracted to A from T, so [the form of] B will be refracted to A from a point between the two points T and D, for if it were refracted from a point between T and M, the previously-mentioned impossibility would result [i.e., the form at the point of intersection would be refracted from two points]. Therefore, let [the form of] B be refracted to A from F. Let us connect AF, and let it pass to I. Then let us connect IK. IK will thus be the cross-section of the image of BG, and IK’s orientation with respect to A is equivalent to the orientation of BG [with respect to A] because IK will either be parallel to BG, or there will not be [enough] difference between them to affect the orientation, for there is no significant difference between the distance of IK and the distance of BG from the center of sight, so the divergence of IK from a line extending from K parallel to BG will be quite small.⁑ Therefore, angle IAK > angle BAG, the orientation of IK is equivalent to the orientation of BG, and IK is perceived as if [it were] more distant because of the weakening of its form. Thus, line IK appears larger than line BG, as we demonstrated in the previous [part of this] theorem. But IK is the image of BG, so BG will appear larger than it [actually] is, and this is what we wanted [to demonstrate].

7.13 ◉ [PROPOSITIO 13] Item sit visus A [FIGURE 7.7.11, p. 447], et res visa BG, et extrahamus perpendiculares BD, GE, et continuemus DE. Et sit BG equidistans DE, et sit A extra superficiem BDGE, cum eo quod continuatur cum ipsa. Et dividamus BG in duo equalia in Z, et extrahamus perpendicularem AH, et continuemus AZ, et sit AZ posita perpendicularis super BZG. Positio ergo B respectu A est similis positioni G respectu A, et distantia B ex A est equalis distantie G ex A. Et reflectatur B ad A ex T, et G ad A ex K. Positio ergo T respectu A est similis positioni K respectu A, et distantia T ex A est sicut distantia K ex A.

7.13 ◉ [PROPOSITION 13] Now let A [in figure 7.7.62, p. 200] be the center of sight and BG the visible object, and let us draw perpendiculars BD and GE and connect DE. Let BG be parallel to DE, and let A lie outside plane BDGE in a plane [AHZ] that joins it. Let us bisect BG at Z, let us draw perpendicular AH and connect AZ, and let AZ be presumed to be perpendicular to BZG. Thus, the location of B with respect to A is equivalent to the location of G with respect to A, and the distance of B from A is equal to the distance of G from A. Let [the form of] B be refracted to A from T [in plane of refraction AHBD], and [let the form of] G [be refracted] to A from K [in plane of refraction AHGE]. The location of T with respect to A is thus equivalent to the location of K with respect to A, and the distance of T from A is equal to the distance of K from A.

7.14 ◉ Et continuemus lineas BT, TA, GK, KA. Est ergo superficies in qua sunt due linee AT, BT perpendicularis super superficiem corporis diaffoni, quia est superficies reflexionis; perpendicularis ergo BD erit in hac superficie, et perpendicularis que exit ex T. Linea ergo AT secat BD. Extrahatur ergo AT, et secet BD in L, et extrahatur AK, et secet GE in O. Erit ergo AL sicut AO, et erit BL sicut GO. Et continuemus LO, que est diameter ymaginis BG, et erit LO equalis BG. Et continuemus AB, AG. Utraque ergo superficies ALB, AOG est perpendicularis super superficiem corporis diaffoni, et tres superficies perpendiculares super superficiem corporis diaffoni que transeunt per puncta B, Z, G secant se in perpendiculari exeunti ex A super superficiem corporis diaffoni.

7.14 ◉ Let us then connect lines BT, TA, GK, and KA. The plane in which the two lines AT and BT lie [i.e., ATBH] is therefore perpendicular to the surface of the transparent body, since it is the plane of refraction; so normal BD will lie in this plane, and [so will] the normal extending from T. Line AT therefore intersects [line] BD. Let AT be extended, then, let it intersect BD at L, and let AK be extended to intersect GE at O. Hence, AL = AO, and BL = GO. Let us connect LO, which is the cross-section of the image of BG, and [so] LO = BG. Let us connect AB and BG. Hence, both planes ALB and AOG are perpendicular to the surface of the transparent body, and the three planes [ADB, AZH, and AGE] that pass through points B, Z, and G perpendicular to the transparent body’s surface intersect one another on the normal [AH] dropped from A to the surface of the transparent body.

7.15 ◉ Et erit angulus BTL angulus reflexionis, et linea BLD est perpendicularis super superficiem corporis. Ergo linea AL est obliqua super ipsum. Linea ergo AT continet cum perpendiculari exeunti ex T super superficiem corporis angulum acutum ex parte L. Et extrahamus perpendicularem, et sit TC. TC ergo erit equidistans LD; angulus ergo TLD est acutus. Ergo angulus ALB est obtusus. Linea ergo AL est minor quam linea AB, et similiter declaratur quod AO erit minor AG. Sed linee AL, AO sunt equales, et AB, AG sunt equales, et linea LO est equalis linee BG. Ergo angulus OAL est maior angulo GAB.

7.15 ◉ Angle BTL will be [equal to] the angle of refraction, and line BLD is perpendicular to the surface of the [transparent] body. Thus, line AL is oblique to it. Line AT therefore makes an acute angle with the normal dropped from T to the surface of the [transparent] body on the side of L. Let us then draw the normal, and let it be TC. TC will thus be parallel to LD, so angle TLD is acute. Hence, angle ALB is obtuse. Consequently, line AL < line AB, and it is demonstrated the same way that AO < AG. But lines AL and AO are equal, [lines] AB and AG are equal, and line LO = line BG. Therefore, angle OAL > angle GAB.

7.16 ◉ Et positio LO est similis positioni BG, quia linea que exit ex A ad medium LO est perpendicularis super lineam LO, quia LO est equidistans BG, et BG est perpendicularis in qua sunt AZ, DB. Linea ergo LO est perpendicularis super lineam AZ. Linea ergo LO est perpendicularis super superficiem que continuat A cum medio LO; positio ergo LO respectu A est sicut positio BG respectu A. Sed LO comprehenditur remotius propter debilitatem forme; ergo LO videbitur maior quam BG. Sed LO est ymago BG; ergo BG videbitur maior quam sit.

7.16 ◉ Furthermore, the orientation of LO is equivalent to the orientation of BG, since the line extending from A to the middle of LO is perpendicular to line LO because LO is parallel to BG, and BG is perpendicular [to the planes] in which AZ and DB lie [i.e., BG is perpendicular to both AZ and DB]. Hence, line LO is perpendicular to line AZ.⁑ Line LO is therefore perpendicular to the plane [AZH] that connects A with the middle of LO, so the orientation of LO with respect to A is equivalent to the orientation of BG with respect to A. But LO is perceived as farther [away than it would normally be judged to be] because of the weakening of its form, so LO will appear larger than BG. But LO is the image of BG, so BG will appear larger than it [actually] is.

7.17 ◉ Item iteremus formam [FIGURE 7.7.11a, p. 447], et sit BG non equidistans DE, et extrahamus GF equidistantem ad DE. Et continuemus AF, et sit T punctum ex quo reflectitur F ad A; B autem reflectitur ad A ex Q. Et continuemus AQ, et protrahamus illam ad C. Sic ergo erit C altius quam L, nam B est ultra lineam AF, unde linea AC est ultra lineam AL. Ergo C est altius quam L.

7.17 ◉ Let us now recapitulate the diagram [in figure 7.7.62a, p. 200], let BG not be parallel to DE, and let us draw GF parallel to DE. Let us connect AF, and let T be the point from which [the form of] F is refracted to A, while [the form of] B is refracted to A from Q. Let us connect AQ and extend it to C. C will therefore be higher than L, for B lies beyond line AF, so line AC lies beyond line AL. Hence, C is higher than L.

7.18 ◉ Et continuemus CO. Erit ergo CO diameter ymaginis BG, et erit CO maior LO, et AC minor AL. Et due linee AC, AO sunt in duabus superficiebus secantibus se, scilicet ACB, AOG, et differentia communis inter has duas superficies transit per A. Et due linee que exeunt ex A perpendiculariter super hanc differentiam communem inter has duas superficies sunt altiores duabus lineis AC, AO. Ergo angulus CAO est maior angulo BAG, et remotiones CO, BG ex A non differunt multum, et linea CO aut erit equidistans BG, aut non erit ibi differentia sensibilis in positione. Positio ergo CO respectu A est sicut positio BG respectu A, et inter distantias CO, BG respectu A non est diversitas sensibilis, quapropter CO videbitur maior quam BG. Sed CO est ymago BG; ergo BG videtur maior quam sit, et hoc voluimus.

7.18 ◉ Let us connect CO. Thus, CO will be the cross-section of the image of BG, and CO > LO, and AC < AL. The two lines AC and AO lie in two intersecting planes, i.e., ACB and AOG, and the common section [AH] of these two planes passes through A. In addition, the two lines dropped orthogonally from A to this common section within these two planes [i.e., AX in plane ACB and AY in plane AOF] are higher than the two lines AC and AO. Therefore, angle CAO > angle BAG, and the distances of CO and BG from A are not different by much, and line CO will either be parallel to BG, or there will be no perceptible difference in orientation between the two.⁑ Thus, the orientation of CO with respect to A is equivalent to the orientation of BG with respect to A, and there is no perceptible difference between the distances of CO and BG with respect to A, so CO will appear larger than BG. But CO is the image of BG, so BG appears larger than it [actually] is, and this [is what] we wanted [to demonstrate].

7.19 ◉ [PROPOSITIO 14] Item iteremus formam primam huius capituli [FIGURE 7.7.12, p. 448], et sit perpendicularis secans LK AMOZ. Erit ergo LO medietas LK. Sed punctus Z videbitur in O, quia videtur in perpendiculari ZM. Ergo BG videbitur in linea LK. Et BZ est medietas BG, et LO est medietas LK, et LK videtur maior quam BG; ergo LO videtur maior quam BZ.

7.19 ◉ [PROPOSITION 14] To continue, let us copy the first diagram for this chapter [i.e., figure 7.7.61, redrawn as figure 7.7.63, p. 201], and let AMOZ be a normal intersecting LK. LO will thus be half LK. But point Z will be seen at O because it is seen on normal ZM. Thus, BG will be seen on line LK. But BZ is half BG, LO is half LK, and LK appears larger than BG, so LO appears larger than BZ.

7.20 ◉ Causa autem magnitudinis BG est reflexio; ergo causa magnitudinis BZ est reflexio. A autem est in perpendiculari AZ, que exit ab extremitate BZ super superficiem corporis diaffoni. Et hoc idem sequitur in tribus figuris sequentibus primam, scilicet in secunda, et tertia, et quarta huius capituli, scilicet quod visus comprehendit medietates visibilium maiores quam sint. Et visus est in perpendiculari exeunte ab extremitate medietatis aut super superficiem transeuntem per extremitatem medietatis perpendicularis super superficiem corporis diaffoni, nam punctus quod est medium ymaginis est in perpendiculari exeunti a medio rei vise, sive res visa sit equidistans superficiei corporis diaffoni, sive non.

7.20 ◉ The reason for BG’s magnification is refraction, so the reason for BZ’s magnification is [also] refraction. Moreover, A lies on normal AZ, which extends from the end of BZ to the surface of the transparent body. This same point obtains in the [demonstrations based on] the three figures following the first one, i.e., in the second, third, and fourth [figures] of this chapter [figures 7.7.61a-7.7.62a, pp. 199-200], that is, the visual faculty perceives the halves of the visible objects [in those three cases] as larger than they [actually] are. Moreover, the center of sight lies on the normal dropped from the end of the half-segment, or [it lies] in a plane that is perpendicular to the surface of the transparent body and that passes through the end of the half-segment, for the point that constitutes the midpoint of the image lies on the normal dropped from the midpoint of the visible object, whether the visible object is or is not parallel to the surface of the transparent body.

7.21 ◉ Item BN est quedam pars linee BZ. Et extrahamus perpendicularem NC. Ymago ergo N erit in linea NC. Sit ergo C ymago N. C ergo aut erit in linea LC aut prope illam, quapropter LC aut erit equalis linee BN aut fere. Sed in prima figura huius capituli declaravimus quod BG comprehenditur maior quam sit, et causa huius est reflexio. Et reflexiones formarum que remotiores sunt a perpendiculari cadenti a centro visus super superficiem corporis diaffoni sunt maiores reflexionibus formarum que sunt propinquiores perpendiculari. Reflexio ergo forme BN ad A est maior quam reflexio forme partis linee ZN ad A. Causa ergo que facit formam BZ videri maiorem facit ut BN habeat maiorem proportionem ad ipsum quam illam quam habet BZ ad BN. Ergo LC, que est ymago BN, comprehenditur maior quam BN.

7.21 ◉ Now BN is a segment of line BZ. Let us draw normal NC. The image of N will thus lie on line NC. Accordingly, let C be the image of N. C will therefore lie either on line LC or near it, so [line] LC will either be equal to line BN or nearly so.⁑ In the first theorem of this chapter [i.e., proposition 12], however, we demonstrated that BG is perceived as larger than it [actually] is, and the reason for this is refraction. Furthermore, the refractions of forms that lie farther from the normal dropped from the center of sight to the surface of the transparent body are more pronounced than the refractions of forms that lie nearer the normal. Hence, the refraction of the form of BN to A is more pronounced than the refraction of the form of line-segment ZN to A. Thus, what causes the form of BZ to appear magnified causes BN to appear proportionately larger in relation to that very [form] than BZ does in relation to BN.⁑ Consequently, LC, which is the image of BN, is perceived as larger than BN.

7.22 ◉ Item si A non comprehenderit ymaginem BN maiorem quam BN, non comprehendet ymagines ceterarum partium linee BN que sunt propinquiores ad Z maiores ipsis partibus, nam forme ceterarum partium sunt minoris reflexionis quam forma BZ. Sed reflexio est causa ymaginis; ergo A non comprehenderet LO maiorem quam BZ, nam comprehendit LO maiorem quam BZ. Ergo comprehendet BN maiorem quam sit. Et A est extra perpendiculares exeuntes ex BZ super superficiem corporis diaffoni, et linea que exit ex A ad medium BZ non est perpendicularis super BZ, et hoc idem sequitur in tribus figuris, in secunda scilicet, et tertia, et quarta huius capituli.

7.22 ◉ Furthermore, if [the center of sight at] A does not perceive the image of BN as larger than BN [itself], it will not perceive the images of the remaining segments of line BN that are nearer to Z as larger than the segments themselves, for the forms of the remaining segments are refracted less than the form of BZ. But refraction causes the image [to be magnified], so [the visual faculty at] A would not perceive LO as larger than BZ, but in fact it does perceive LO as larger than BZ. Hence, it will perceive BN as larger than it [actually] is. Furthermore, A lies outside the normals dropped from BZ to the surface of the transparent body, and the line extending from A to the midpoint of BZ is not perpendicular to BZ, and this same situation follows in the [demonstrations based on the] three [previous] figures, i.e., in the second, third, and fourth of this chapter [figures 7.7.61a-7.7.62a, pp. 199-200].

7.23 ◉ Omne ergo quod comprehenditur a visu ultra aliquod corpus diaffonum grossius aere, cuius superficies fuerit plana, comprehenditur maius quam sit, sive sit visus in aliqua perpendiculari exeunti ex illo visu super superficiem corporis, sive sit extra, et indifferenter si diameter rei vise fuerit equidistans superficiei corporis, aut non equidistans.

7.23 ◉ Accordingly, everything perceived by the visual faculty through any transparent body [that is] denser than air [and] whose surface is plane is perceived as larger than it [actually] is, whether the center of sight lies on any normal dropped from the object seen to the surface of the [transparent] body, or whether it lies outside [all such normals], and it is irrelevant whether the cross-section of the visible object is parallel or not parallel to the surface of the [transparent] body.

7.24 ◉ [PROPOSITIO 15] Item sit superficies corporis sperica, cuius convexum sit ex parte visus, et sit grossius aere. Et sit visus A [FIGURE 7.7.13, p. 448] et res visa BG, et sit centrum spere ultra BG in respectu visus. Et sit centrum D, Z medium BG, et continuemus DB, DZ, DG, et extrahamus has lineas quousque concurrant cum superficie spere ad E, et M, et N. Et extrahamus ZM in parte M.

7.24 ◉ [PROPOSITION 15] Now let the surface of the [transparent] body be spherical [figure 7.7.64, p. 201], let its convexity face the eye, and let it be denser than the air [in which the eye is located]. Let A be the center of sight and BG the visible object, and let the center of the sphere lie beyond BG with respect to the center of sight. Let D be the center and Z the midpoint of BG, and let us connect DB, DZ, and DG and extend these lines until they intersect the surface of the sphere at E, M, and N. Then let us extend ZM in the direction of M.

7.25 ◉ In primo sit visus in linea ZM. Erit ergo AMZ linea recta. Et in primo sit BD equalis GD. Sic ergo erit AZ perpendicularis super BG; positio ergo B respectu A erit similis positioni G respectu A. Et extrahamus superficiem in qua sunt DE, DN. Faciet ergo in superficie sperica arcum circuli magni. Sit ergo arcus EMN, et hec superficies est perpendicularis super superficiem spericam, nec fit reflexio extra hanc superficiem, nam AZ est perpendicularis super superficiem corporis. Non ergo reflectitur forma alicuius partis BG ad A nisi ex circumferentia EMN.

7.25 ◉ First, let the center of sight lie on ZM. AMZ will thus be a straight line. Let BD = GD for a start. Therefore, AZ will be perpendicular to BG, so the location of B with respect to A will be equivalent to the location of G with respect to A. Let us produce the plane in which DE and DN lie. It will therefore form an arc of a great circle on the spherical surface. Let EMN be that arc, then, and this [lies within a] plane [that] is perpendicular to the spherical surface, and no refraction occurs outside this plane, for AZ is perpendicular to the surface of the [transparent] body. Hence, the form of any portion of BG is refracted to A only from the circumference [of the great circle containing] EMN.

7.26 ◉ Reflectatur ergo B ad A ex H, et G ad A ex T. Positio ergo H respectu A et distantia eius est equalis positioni et distantie T. Et continuemus BH, HA, GT, TA, et extrahamus AH ad K et AT ad L, et continuemus KL. Erit ergo AK equalis AL, et erit LK ymago BG, et erit equidistans BG; erit ergo maior quam BG. Et continuemus AB, AG. Erit ergo angulus KAL maior angulo BAG, et erit positio KL similis positioni BG. Et inter KL et GB non est differentia in distantia, ut in precedentibus diximus; ergo KL videbitur maior quam BG. Sed KL est ymago BG; ergo BG videbitur maior quam sit, quia ymago eius est maior se. Et hoc est quia forma eius est debilior quam vera forma, et hoc est quod voluimus.

7.26 ◉ Accordingly, let [the form of] B be refracted to A from H, and [let the form of] G [be refracted] to A from T. Consequently, the location of H with respect to A, as well as its distance [from A], is equivalent to the location and distance of T [with respect to A]. Let us connect BH, HA, GT, and TA, let us extend AH to K and AT to L, and let us connect KL. Therefore, AK = AL, LK will be the image of BG, and it will be parallel to BG, so it will be larger than BG. Let us connect AB and AG. Hence, angle KAL > angle BAG, and the orientation of KL will be equivalent to the orientation of BG. Between KL and GB, moreover, there is no [perceptible] difference in distance [with respect to center of sight A], as we claimed earlier, so KL will appear larger than BG. But KL is the image of BG, so BG will appear larger than it [actually] is because its image is larger than it. And this [is so] because its form is weaker than its true form [would be if seen from the same distance], and this is what we wanted [to demonstrate].⁑

7.27 ◉ [PROPOSITIO 16] Si ergo BD, GD fuerint inequales, tunc AK, AL erunt inequales, et sic BG, KL erunt oblique super lineam AD. Erit ergo KL, ut in secunda figura huius capituli diximus, maior quam BG in visu.

7.27 ◉ [PROPOSITION 16] Accordingly, if BD and GD are unequal, then AK and AL will be unequal, so BG and KL will be oblique to line AD. Consequently, as we claimed [on the basis of the demonstration tied to] the second figure of this chapter [i.e., figure 7.7.61a, p. 199], KL will be [perceived as] larger than BG in the visual faculty.

7.28 ◉ Item si A fuerit extra superficiem BZG, et BD, GD fuerint equales aut inequales, declarabitur, ut in tertia et quarta figura huius capituli, quod KL videbitur maior quam BG. Sed secet ante DM lineam KL in O. Erit ergo KO ymago BZ. Et erit angulus KAO maior angulo BAZ, et positio KO est similis positioni BZ, et distantie KO, BZ respectu A non differunt multum, quapropter KO videbitur maior quam BZ.

7.28 ◉ Moreover, if A [in figure 7.7.65, p. 202] lies [in a plane] outside [of and oblique to] the [vertical] plane [containing] BZG [and DZM perpendicular to BG], it will be demonstrated, as [it was] in the proofs [based on] the third and fourth figures of this chapter [i.e., figures 7.7.62 and 7.7.62a, p. 200], that, whether BD and GD are equal or unequal, KL [seen under angle KAL] will appear larger than BG [seen under angle BAG]. But let [AO] intersect KL at O in front of DM.⁑ KO will therefore be the image of BZ. Furthermore, angle KAO > angle BAZ, the orientation of KO is equivalent to the orientation of BZ, and the distances of KO and BZ with respect to A are not much different, so KO will appear larger than BZ.

7.29 ◉ Et A est in perpendiculari ZM, que exit ab extremitate BZ super superficiem corporis; sit autem BC pars BZ, et sit KR ymago BC. Ergo, ut in quinta figura huius capituli diximus, patet quod KR videbitur maior quam BC. A autem est extra omnes perpendiculares exeuntes ex BC super superficiem corporis, nam linea que exit ex A ad medium BC non est perpendicularis super BC. Et quia BG, KL sunt oblique super AZD aut super superficiem que transit per lineam MD, et KO est ymago BZ, et LO est ymago ZG, et angulus quem respicit KO apud centrum visus est maior angulo quem respicit BZ apud centrum visus, et similiter angulus quem respicit OL est maior angulo quem respicit ZG, ergo KO videbitur maior quam BZ, et similiter KR videbitur maior quam BC. Et omnia hec declarantur in quinta figura huius capituli. Sed in hac positione est quedam additio, scilicet quod KL, que est ymago BG, est maior in veritate quam BG, et KO est maior quam BZ.

7.29 ◉ And [if] A lies on normal ZM [in figure 7.7.65a, p. 203], which extends to the surface of the [transparent] body from the endpoint [Z] of BZ, let BC be a segment of BZ, and let KR be the image of BC. Hence, as we claimed in the [demonstration based on the] fifth figure of this chapter [i.e., figure 7.7.63, p. 201], it is evident that KR will appear larger than BC. Now A lies outside all the normals extending from BC to the surface of the [transparent] body, for the line that extends from A to the midpoint of BC is not perpendicular to BC. And since BG and KL are oblique to AZD or to the plane that passes through line MD, and since KO is the image of BZ and LO the image of ZG, and since the angle that KO subtends at the center of sight is greater than the angle that BZ subtends at the center of sight, and likewise, since the angle that OL subtends is greater than the angle that ZG subtends, KO will appear larger than BZ, and by the same token KR will appear larger than BC. And all these [conclusions] are established in the [demonstration based on the] fifth figure of this chapter [i.e., figure 7.7.63, p. 201]. But in this case there is something additional: namely, that KL, which is the image of BG, is actually larger than BG, and KO is [also actually] larger than BZ.

7.30 ◉ In prima autem positione, scilicet in plana superficie, due ymagines sunt equales duobus visis; ymago ergo KL et ymago KO sunt in visu maiores ipsis rebus, et sic sunt in veritate. Et patet quod angulus quem respicit KL apud centrum visus est maior angulo quem respicit BG apud centrum visus, et angulus quem respicit KO apud centrum visus est maior illo quem respicit BZ, cum visus fuerit extra superficiem in qua sunt DE, DZ, ut in quarta huius capituli diximus. Ergo si visus comprehenderit aliquid ultra corpus grossius aere, cuius superficies fuerit sperica, et cuius convexum fuerit ex parte visus, et cuius centrum fuerit ultra rem visam quantum ad visum, comprehendet illud maius quam sit, sive fuerit visus in perpendiculari exeunti a re visa super superficiem spericam, sive extra, sive linea que exit a centro visus ad medium rei vise fuerit perpendicularis super rem visam aut obliqua, et hoc est quod voluimus declarare.

7.30 ◉ In the first case, i.e., in [that of] a plane [refracting] surface, the two images are the same size as the two objects that are seen, so [in the case of the spherical surface] the image of KL and the image of KO are larger to the visual faculty than the objects themselves, and so they are in actuality. But it is clear that, when the center of sight lies outside the plane in which DE and DZ lie, as we claimed in the [demonstration based on the] fourth [figure] of this chapter [i.e., figure 7.7.62, p. 200], the angle that KL subtends at the center of sight is greater than the angle that BG subtends at the center of sight, and the angle that KO subtends at the center of sight is greater than the one BZ [subtends at the center of sight]. Therefore, if the visual faculty perceives something through a [transparent] body denser than air, if that body’s surface is spherical and its convexity faces the center of sight, and if its center lies beyond the visible object as far as the center of sight is concerned, it will perceive that object as larger than it [actually] is, whether the center of sight lies on or outside the normal dropped from the visible object to the spherical surface, or whether the line extending from the center of sight to the middle of the visible object is perpendicular or oblique to the visible object, and this is what we wanted to demonstrate.

7.31 ◉ Et hoc accidit in eis qui videntur in aqua, nam convexum superficiei aque sperice est ex parte visus, et centrum superficiei aque est ultra illa que comprehenduntur in aqua, et aqua est grossior aere. Sed illud quod videtur in aqua, si aqua fuerit clara et pauca, forte non comprehendet visus ipsum esse maius in aqua quam si esset in aere, non enim differt quantitas eius tunc quantum ad sensum, scilicet quantitas eius in aqua et aere, tunc enim illa additio in aqua erit parva, et ideo sensus non distinguet tunc illam additionem.

7.31 ◉ This happens in [the case of] things seen in water, for the convex spherical surface of the water faces the eye, and the center [of curvature] of the water’s surface lies beyond the things perceived in the water, and water is denser than air.⁑ However, if the water is clear and shallow, the visual faculty may not perceive what is seen in the water as larger in the water than [it would appear to be] if it were in air, for in that case its size does not vary according to sense-perception, i.e., its size in water and in air, because the magnification in the water will be slight, so the sense [of sight] will not discern that magnification.

7.32 ◉ Tamen experientia potest comprehendi hoc modo: accipe corpus columpnale, cuius longitudo non sit minor uno cubito, et sit aliquante grossitiei album, nam albedo in aqua manifestius distinguitur. Et sit superficies basis eius plana, ita quod per se stet equaliter super faciem terre. Hoc preservato, accipe vas amplum, et sit superficies eius plana, et infunde in vas aquam claram in altitudine minori longitudine corporis columpnalis. Deinde mitte illud corpus columpnale in aquam, et pone ipsum super suam basim in medio vasis. Erit ergo aliqua pars huius corporis extra aquam, nam altitudo aque est minor longitudine huius corporis. Tunc enim, cum quieverit aqua, videbis partem corporis que est intra aquam grossiorem illa que est extra aquam. Patet ex hac experientia quod omne visum comprehensum in aqua comprehenditur maius quam sit in veritate.

7.32 ◉ Nonetheless, it can be understood experimentally as follows. Take a cylindrical object not less than a cubit in length, and let it be somewhat thick and white, for white is quite clearly discerned in water. Let the surface at its base be plane, so that it can stand upright by itself on the ground. When this condition is met, take a capacious vessel, let its [bottom] surface be plane, and pour clear water into the vessel to a height less than the length of the cylindrical object. Then put that cylindrical object in the water, and place it in the middle of the vessel on its base. Consequently, some portion of this body will lie outside the water, for the height of the water is less than the length of this object. In that case, when the water calms down, you will see the portion of the object under water [as] thicker than the portion above the water. From this experiment it is evident that every visible object perceived in water is perceived as larger than it is in actuality.

7.33 ◉ Item sit corpus spericum, cuius convexum sit ex parte visus, et res visa sit ultra centrum superficiei sperice, et sit illud corpus grossius aere. Sed in assuetis visibilibus non est tale aliquid quod videatur ultra corpus diaffonum spericum grossius aere ultra centrum spere et res visa, cum hoc erit intra corpus spericum, hoc enim non fit nisi corpus spericum fuerit vitreum aut lapideum, et fuerit totum corpus spericum solidum, et res visa fuerit intra ipsum, aut ut corpus spericum sit portio spere maior semispera, et res visa sit applicata cum basi eius. Sed hii duo situs raro accidunt. Huiusmodi ergo res non sunt de assuetis visibilibus; non ergo debemus negotiari circa ea que accidunt huiusmodi visibilibus.

7.33 ◉ To continue, let there be a [transparent] spherical body with its convexity facing the eye, let the visible object lie beyond the center [of curvature] of the spherical surface, and let that [transparent] body be denser than air. Now in [the case] of customary visible objects there is no such thing to be seen beyond a transparent spherical body denser than air and lying beyond the sphere’s center while the visible object will nonetheless lie inside the sphere, for this happens only if the spherical body is made of glass or stone and if the entire spherical body is solid and the visible object lies inside it, or if the spherical body constitutes a portion of the sphere greater than a hemisphere, while the visible object is attached to its base. But these two situations rarely occur. Hence, things of this sort are not among customary visible objects, so we need not deal with what happens with these sorts of visible objects.

7.34 ◉ Sed sunt quedam assueta que videntur ultra corpus diaffonum spericum grossius aere, cuius convexum erit ex parte visus, cum res visa fuerit ultra speram cristallinam aut vitream, et res visa fuerit in aere non intra speram; positiones autem huiusmodi visibilium sunt multimode. Sed hec raro comprehenduntur, et si comprehendantur, raro videntur. Non ergo est conveniens distinguere omnes illas positiones; sumus ergo contenti una sola positione, scilicet quod visus et res visa sunt in eadem perpendiculari super superficiem corporis sperici.

7.34 ◉ There are, however, some customary visible objects that are seen through a transparent spherical body that is denser than air and that has its convex [surface] facing the eye, [such as] when the visible object lies behind a crystalline or glass sphere, and the object lies in the air instead of inside the sphere; but the conditions under which such visible objects [are seen] are manifold. Such situations are rarely perceived, however, and if they are perceived, they are rarely noticed. Consequently, it is not appropriate to differentiate all these situations, so we are satisfied [to deal with] one single situation, i.e., the one in which the center of sight and the visible object lie on the same normal [passing] through the surface of the [transparent] spherical body.⁑

7.35 ◉ [PROPOSITIO 17] Sit ergo visus A [FIGURE 7.7.14, p. 449] et corpus spericum BGDZ, et centrum eius sit E. Et continuemus AE, et extrahamus eam recte, et secet superficiem spere in duobus punctis B, D. Et extrahamus ipsam in parte D usque ad H. Et extrahamus ex linea HAB superficiem equalem secantem speram. Faciet ergo in superficie spere circulum BGDZ.

7.35 ◉ [PROPOSITION 17] Accordingly, let A [in figure 7.7.66, p. 204] be the center of sight and BGZD the [transparent] spherical body, and let E be its center. Let us connect AE and extend it straight, and let it intersect the surface of the sphere at the two points B and D. Then let us extend it to H on the side of D. Let us also produce a plane surface along line HBA to cut the sphere.⁑ It will thus form [great] circle BGZD on the sphere’s surface.⁑

7.36 ◉ In nona autem figura de capitulo ymaginis, diximus quod in linea BD sunt plura puncta quorum forme reflectuntur ad A ex circumferentia BGDZ, et quod forma totius illius linee reflectitur ad A, si BGDZ fuerit continuum et non fractum in parte D. Reflectatur HL ad A ex circumferentia BGDZ, et reflectatur H ad A ex G et L ad A ex T. Forma ergo HL reflectetur ad A ex arcu GT. Et continuemus lineas GMH, GA, LZT, TA. H ergo extenditur per GH et reflectitur per GA, et L extenditur per LT et reflectitur per TA. Et continuemus lineas EG, ET, EM, EZ, et extrahamus EM ad C et EZ ad F.

7.36 ◉ Now in the ninth proposition of chapter [five] on images [i.e., proposition 9, pp. 291-292 above], we said that there are several points on line BD whose forms are refracted to A from circumference BGZD and that the form of that entire line is refracted to A, if BGZD is continuous and not interrupted [by a refractive interface] on the side of D.⁑ [Let us therefore start by assuming that the entire area below arc ZD is of the same density as the area within the sphere. Accordingly] let [the form of line-segment] HL [on the extension of BD] be refracted to A from [arc BT on] circumference BGZD, and let [the form of] H be refracted to A from G and [the form of] L [be refracted] to A from T. Hence, the [entire] form of HL will be refracted to A from arc GT. Let us connect lines GMH, GA, LZT, and TA. [The form of] H therefore extends along GH and is refracted along GA, and [the form of] L extends along LT and is refracted along TA. Let us then connect lines EG, ET, EM, and EZ, and let us extend EM to C and EZ to F.

7.37 ◉ Forma ergo que extenditur per AG reflectitur per GH, et pervenit ad H, et forma que extenditur per AT reflectitur per TL et pervenit ad L. Hoc est si corpus diaffonum fuerit continuum usque ad G. Si ergo corpus spericum fuerit signatum apud superficiem spericam, tunc forma que extenditur per AG reflectitur per GM in partem perpendicularis que est EG, et cum forma pervenerit ad M, reflectetur secundo in contrariam partem perpendicularis que est EMC. Reflectatur ergo ad K. Et ideo forma que extenditur per AT reflectitur per TZ, et cum fuerit reflexa ad Z, reflectetur secundo ad contrariam partem perpendicularis que est EZF. Sit ergo reflexio forme que pervenit ad Z per lineam ZO.

7.37 ◉ Accordingly [if we switch the direction of radiation], the form [of point A] extending along AG is refracted along GH and reaches H, whereas the form [of point A] extending along AT is refracted along TL and reaches L. This [is the case] if the transparent body is continuous all the way up to G. If, therefore, the spherical body is demarcated [by a refractive interface] at the spherical surface [containing arc ZD], the form that extends along AG is refracted along GM toward normal EG, and when the form reaches M, it will be refracted again away from normal EMC. So let it be refracted to K. Hence, the form that extends along AT is refracted [toward normal ET] along TZ, and when it is refracted at Z, it will be refracted again away from normal EZF. Let the refraction of the form reaching Z be along line ZO, then.

7.38 ◉ Forma ergo K extenditur per KM, et reflectetur per MG; deinde reflectitur secundo per GA. Et similiter forma O extenditur per OZ, et reflectitur per ZT; deinde secundo reflectitur per TA. Forma ergo totius KO reflectitur ad A ex arcu GT. Et si linea AK fuerit fixa, et ymaginati fuerimus figuram AGMK circumvolvi circa AK, tunc arcus GT faciet figuram circularem, ut armillam, a cuius universo reflectetur forma KO ad A, et erit ymago KO centrum visus, quod est A. Forma ergo KO videbitur in tota superficie circulari, que est locus reflexionis, que est in rectitudine linearum radialium, que est figura armille. Forma ergo KO erit maior se, et erit figura forme diversa a figura KO.

7.38 ◉ Consequently [if we again switch the direction of radiation], the form of K extends along KM, and it will be refracted along MG; then it is refracted again along GA. Likewise, the form of O extends along OZ, and it is refracted along ZT; then it is refracted again along TA. Hence, the form of the entire [line] KO is refracted to A from arc GT. And if line AK is [held] stationary, and if we imagine the figure [delineated by] AGMK to be rotated around AK, arc GT will form a circular figure, like a ring, from whose entire [surface] the form of KO will be refracted to A, and the image of KO will be [located at] the center of sight, which is A. Thus, the form of KO will appear on the entire circular [ringlike] surface that constitutes the area of refraction, which lies on [the endpoints of] the straight, radial lines [extending from A] and is shaped like a ring.⁑ Hence, the form [TG] of KO will be larger than [KO] itself, and the shape of the form [which is convex with respect to A] will be different from the shape of KO [which is straight].⁑

7.39 ◉ Hoc autem potest experimentari sic: accipe speram cristallinam aut vitream rotundissimam, et accipe corpus parvum vel ceram parvam, ut granum ciceris, nam experientia per corpus parvum erit manifestior. Et tingat ipsam colore nigro, et sit figura cere sperica. Deinde pones ipsam in capite acus, et pone speram cristallinam in oppositione alterius oculorum, et claude alterum oculum. Et eleva acum ultra speram, et aspice ad medium spere, et pone ceram in oppositione medii forme, ita quod sit opposita medio spere in una linea recta quoad sensum. Et respice ad superficiem spere, tunc enim videbis in ipsa superficie nigredinem rotundam in figura armille. Si vero non videbis ipsam, move ceram ante et post donec videas nigredinem rotundam. Tunc aufer ceram, et abscindetur nigredo; deinde redeat cera ad suum locum, et videbis illam nigredinem rotundam.

7.39 ◉ This, moreover, can be determined empirically as follows. Take a perfectly round crystalline or glass sphere, and take a small object or a small piece of wax the size of a chick-pea, for the experiment will be clearer [when conducted] with a small object. It should be colored black, and the shape of the wax should be spherical. You will then put it on the point of a needle, place the crystalline sphere in front of either eye, and close the other eye. Then raise the needle behind the sphere, look toward the center of the sphere, and pose the wax [sphere] directly opposite the middle of its image, so that, as far as can be sensibly determined, it is opposite the center of the sphere in a straight line. Then look at the surface of the sphere, for in that case you will see a round black [area] in the shape of a ring on that surface. In case you will not see this, however, move the wax to and fro until you see the round black [image].⁑ Remove the wax at that time, and the blackness will disappear; then the wax should be returned to its [original] place, and you will [again] see that round black [image].

7.40 ◉ Ex hac ergo experientia patebit quod, si res visa fuerit ultra corpus diaffonum spericum grossius aere, et visus, et res visa, et centrum corporis sperici fuerint in eadem linea recta, tunc visus comprehendet illam rem visam in figura armille.

7.40 ◉ It will thus be manifest from this experiment that, if the visible object lies behind a transparent spherical body that is denser than air, and if the center of sight, the visible object, and the center of the spherical body lie on the same straight line, the visual faculty will perceive that visible object in the form of a ring.

7.41 ◉ [PROPOSITIO 18] Si vero BGDZ fuerit in corpore columpnali, et corpus fuerit grossius aere, tunc forma KO videbitur apud arcum GT et apud arcum sibi equalem et sibi respondentem ex arcu BD, sed hec forma non erit circularis, quia figura AHMG, cum fuerit circumvoluta circa AK, non transibit per illam lineam arcus GT per totam superficiem columpnalem. Sed forte reflectetur forma ex aliquibus portionibus columpne, sed erit continua recte, nam superficies ex LK que transit per axem columpne facit in superficie columpne que est ex parte A lineam rectam que transit per B et extenditur in longitudine columpne. Et non reflectitur forma KO ex illa linea recta, nam KZ erit perpendicularis super illam lineam rectam. Non ergo erit forma rotunda, si fuerit corpus columpnale, sed erunt due forme, quarum altera reflectitur super alteram. Videbitur ergo KO esse duo, quorum utrumque erit maior KO, et forma utriusque erit diversa a forma KO, et tamen ille due forme erunt idem punctum, scilicet centrum visus.

7.41 ◉ [PROPOSITION 18] On the other hand, if BGZD [in figure 7.7.66, p. 204] lies on a [transparent] cylindrical body, and if the body is denser than air, the form of KO will be seen on arc GT as well as on an equal arc corresponding to it on arc BD, but this form will not be circular because, when it is revolved about AK, the figure [delineated by] AHMG will not carry arc GT through the entire surface of the cylinder along that line. Instead, the form may be refracted from some portions of the [surface of the] cylinder, but it will be straight and continuous, for the plane through LK that passes along the cylinder’s axis forms a straight line [of longitude] on the surface of the cylinder facing A [and] that [plane] passes through B and [forms a line through B that] extends along the length of the cylinder. But the form of KO is not refracted from that straight line, for KB will be perpendicular to that straight line.⁑ Therefore, if the [transparent] body is cylindrical, the form will not be round [and annular], but there will be two images, each one of them refracted to [one or] the other [side of B].⁑ Hence, KO will appear double, both of [its images] larger than KO, and the shape of both will be different from the shape of KO, yet those two forms will be [located at] the same point, i.e., the center of sight.

7.42 ◉ In visibilibus autem assuetis nichil est quod comprehenditur a visu ultra corpus diaffonum spericum grossius aere, cuius concavum sit ex parte visus, nam si fuerit ex vitro aut aliquo lapide, oportet quod sit portio spere concava et quod res visa sit intra illam speram, aut quod superficies eius que est ultra concavitatem sit plana et res visa adhereat illi. Et illi duo situs non inveniuntur, aut raro; non ergo solicitemur circa huiusmodi.

7.42 ◉ Now among customary visible objects there are none perceived by the visual faculty behind a spherical transparent body denser than air with its concave surface facing the eye, for if [that transparent body] were made of glass or some [transparent] stone, it follows that a portion of the sphere must be concave and that the visible object must lie inside [the transparent body containing the concave portion of] that sphere, or else the [back] surface [of the transparent body] beyond its concavity must be plane, and the visible object must be attached to it. But these two situations are not encountered, or only rarely, so we need not concern ourselves with such a case.

7.43 ◉ Item non invenitur aliquod corpus subtilius aere, cuius superficies que est ex parte visus sit plana aut convexa, et non invenitur aliquod corpus subtilius aere ultra quod comprehenditur aliquid nisi corpus celi et ignis. Et non dividetur a corpore aeris superficies que distinguat unam partem ab alia, sed quanto magis appropinquat aer celo, tanto magis purificatur donec fiat ignis. Subtilitas ergo eius sit ordinate secundum succesionem, non in differentia determinata. Forme ergo eorum que sunt in celo, quando extenduntur ad visum, non reflectuntur apud concavitatem spere ignis, cum non sit ibi superficies concava determinata. Nullum ergo invenitur corpus subtilius aere in quo extenduntur forme visibilium et reflectuntur apud superficiem eius ad visum nisi corpus celeste, et corpus celeste est spericum concavum ex parte visus. Ergo omnes stelle que sunt in celo extenduntur in corpore celi, et reflectuntur apud concavitatem celi, et extenduntur in corpore ignis et in corpore aeris recte donec perveniant ad visum, et centrum concavitatis celi est centrum terre.

7.43 ◉ Furthermore, no [transparent] body rarer than air with a plane or convex surface facing the eye is encountered, and the only body rarer than air through which anything is perceived is the body of the heavens and [the body of] fire. The body of air will not be subdivided by [any] surface that demarcates one part from another, but [instead], the closer the air approaches the heavens, the more it is purified until it becomes fire. Its rarity is therefore continuously graduated, not [defined] according to a determinate differentiation. Consequently, when they extend to the eye, the forms of objects in the heavens are not refracted at the concave surface of the sphere of fire [demarcating that sphere from the sphere of air below it], since there is no determinate concave surface there. Thus, there is found no body rarer than air, other than the celestial body, through which the forms of visible objects extend and at whose surface they are refracted to the eye, and the celestial body is spherical concave with respect to the eye. Hence, [the forms of] all the stars in the heavens extend through the body of the heavens, are refracted at the concave surface of the heavens, and extend straight through the body of fire and through the body of air until they reach the eye, and the center of the heavens’s concave curvature is the center of the earth. ⁑

7.44 ◉ Dico ergo quod stelle in maiori parte comprehenduntur non in suis locis et quod semper comprehenduntur non in suis magnitudinibus, et cum hoc diversatur magnitudo uniuscuiusque earum secundum locorum diversitatem. Diversitas autem locorum est propter positionem radiorum reflexorum, ut prius diximus. Diversitas autem quantitatum est propter remotionem, propter remotionem enim comprehendentur minores quam sint in veritate, ut diximus in tertio tractatu, scilicet quod illa que in maxima remotione sunt comprehenduntur minora. Diversitas autem quantitatum secundum diversitatem locorum accidit propter reflexionem, cuius causam hic declaravimus. Et in quarto capitulo declaravimus quod forme stellarum que comprehenduntur a visu sunt reflexe.

7.44 ◉ I say, then, that for the most part the stars are not⁑ perceived in their [actual] places and that they are never perceived according to their [proper] sizes, yet nonetheless the size of each one of them varies according how their location varies. A variation in location, on the one hand, is due to the orientation of the refracted rays, as we said earlier. A variation in size, on the other hand, is due to the distance [the stars lie from the center of sight], for according to [such] distance they will be perceived as smaller than they are in reality, as we said in the third book, i.e., those that lie extremely far away are perceived as smaller [than they actually are].⁑ But a variation in size due to a variation in location occurs according to refraction, and we have [already] shown why here [in this chapter].⁑ In the fourth chapter, moreover, we showed that the forms of the stars that are perceived by the visual faculty are refracted.⁑

7.45 ◉ Dico ergo quod omnis stella comprehenditur ex omnibus locis celi per quos movetur in minori quantitate quam sit in veritate, secundum quod exigit remotio eius, scilicet minor, si visa fuerit recte, cum non fuerit inter illam et visum aliqua nubes aut vapor grossus. Et omnis stella in vertice capitis aspicientis existens videtur minor quam in alio loco celi, et quanto magis removetur a vertice capitis, tanto magis apparet maior, ita quod in orizonte apparet maior quam in alio loco. Et hoc est commune omnibus stellis remotis et propinquis.

7.45 ◉ I say, therefore, that every star is perceived at every location in the sky through which it moves as smaller than it actually is according to [the size] dictated by its distance, i.e., smaller [than] if it were seen directly, as long as there are no clouds or thick vapor between it and the center of sight. Moreover, every star lying at the viewer’s zenith appears smaller than [it does] in any other location in the sky, and the farther it lies from the zenith, the larger it appears, so that at horizon it appears larger [than it does] at any other location. And this applies universally to all heavenly bodies [whether they lie] at a great distance or [relatively] near [as do the sun and moon in respect to Saturn and the fixed stars].

7.46 ◉ Item si in aere fuerit vapor grossus ultra quam fuerit aliqua stella, tunc comprehendetur maior quam si esset sine illo vapore, et multum accidit quod vapor grossus sit in orizontibus, unde stelle in maiori parte videntur in orizonte maiores quam in medio celi. Et hoc apparet in distantiis que sunt inter stellas magis quam in magnitudinibus ipsarum stellarum, nam quantitas stelle quoad visum est parva, et excessus in diversitate distantie inter stellas, cum fuerit in orizonte, est grandis manifestius sensui, et maxime in distantiis spatiosis, et maxime si in orizonte fuerit vapor grossus.

7.46 ◉ Furthermore, if there is thick vapor in the air behind which a star lies, it will be perceived as larger than [it would be] if it were [perceived] without that [intervening] vapor, and it frequently happens that thick vapor lies at the horizon, so for the most part stars appear larger at the horizon than [they do] in the middle of the sky. In [the case] of the intervals between stars this [effect] is [even] more apparent than in [the case of] the sizes of the stars themselves, for the size of a star is small as far as sight is concerned, whereas at the horizon the magnification of the intervals between stars is quite obvious to the sense [of sight], especially in [the case of] vast intervals, and especially if there is thick vapor at the horizon.

7.47 ◉ [PROPOSTIO 19] Sit ergo circulus meridiei in aliquo orizonte BK [FIGURE 7.7.15, p. 450], et differentia communis inter hunc circulum et concavitatem celi circulus MEZ. Et sit centrum mundi G et centrum visus T, et extrahamus GT in partem T. Et occurrat circulo meridiei in B, et secet circulum qui est in concavitate orbis in E. Erit ergo B vertex capitis quoad visum T. Sit KL diameter alicuius stelle aut distantia inter aliquas duas stellas, et linea TB transeat per medium KL, et secabit illam in C. Ergo erit arcus KB equalis arcui BL. Et continuemus duas lineas TK, TL. Erit ergo angulus KTL ille a quo T comprehendit KL si recte comprehenderetur.

7.47 ◉ [PROPOSITION 19] Accordingly, let BK [figure 7.7.67, p. 207] be the circle of the meridian in some horizon-plane [XT], and let the common section of this circle and the concave surface of the heavens be circle MEZ. Let G be the center of the world and T the center of sight [GT thus being the radius of the earth], and let us extend GT on the side of T. Let it intersect the circle of the meridian at B, and let it intersect the circle on the concave surface of the heavens at E. B will thus be the zenith for center of sight T. Let [straight line] KL be the cross-section of some star or the [rectilinear] interval between two stars, and let line TB pass through the middle of KL; it will intersect it at C. Hence, arc KB = arc BL. Then let us connect the two lines TK and TL. Angle KTL will thus be the one under which [the center of sight at] T perceives KL, if it were perceived directly.

7.48 ◉ Et reflectatur K ad T ex M, et L ad T ex Z. Et continuemus GM, GZ, et pertranseant ad F, O. Et continuemus lineas KM, MT, LZ, ZT. Forma autem que extenditur ex K per KM reflectitur per MT, et GM est perpendicularis exiens ex M, quod est punctum reflexionis, super superficiem corporis quod est in parte T, et quia corpus ZM est subtilius corpore GT, erit reflexio MT ad partem perpendicularis MG. M ergo erit inter duas lineas TB, TK, nam si M esset ultra TK, tunc perpendicularis que exit ex G esset ultra TK, et forma K, cum extenderetur ad illud punctum, reflecteretur ad partem perpendicularis, et non perveniret ad perpendicularem, et non perveniret ad T. M ergo est inter duas lineas TK, TB, et similiter declarabitur quod Z est inter duas lineas TB, TL.

7.48 ◉ Let [the form of] K be refracted to T from M, and [let the form of] L [be refracted] to T from Z.⁑ Let us connect GM and GZ, and let them continue to F and O. Let us then connect lines KM, MT, LZ, and ZT. The form that extends from K along KM is refracted along MT, and GM is the normal dropped from M, which is the point of refraction, to the surface of the [transparent] body on the side of T, and since the [transparent] body [beyond] ZM is rarer than the [transparent] body [containing] GT, MT’s refraction will occur toward normal MG. M will therefore lie between the two lines TB and TK, for if M lay beyond TK, the perpendicular dropped from G would lie beyond TK, and when it extended to that point, the form of K would be refracted toward the normal but would not reach the normal [itself], and [so] it would not reach T. Hence, M lies between the two lines TK and TB, and it will be demonstrated the same way that Z lies between the two lines TB and TL.

7.49 ◉ Et extrahamus TM ad Q et TZ ad R. Erit ergo arcus QK equalis arcui LR, et angulus QTR erit minor angulo KTL. Et angulus QTR est ille per quem T comprehendit KL reflexive, et angulus KTL est ille per quem T comprehenderet KL si recte comprehenderetur. Sed remotio KL a visu est maxima, quapropter quantitas eius non certificatur, quare T estimat remotionem KL, sicut in secundo huius libri diximus. Sed estimatio eius, quando comprehendit reflexe, non differt ab estimatione eius, quando comprehendit recte, nisi quod putat se recte comprehendere, cum reflexe comprehendat. T ergo comprehendit KL reflexive ex angulo minori illo ex quo comprehendit illam recte et secundum comparationem ad illam eandem remotionem ad quam comparet illam, si recte comprehenderet. Sed visus comprehendit magnitudinem ex quantitate anguli respectu remotionis; T ergo comprehendit quantitatem KL reflexe minorem quam si comprehenderet illam recte.

7.49 ◉ Let us extend TM to Q and TZ to R. Therefore, arc QK = arc LR, and angle QTR < angle KTL. But angle QTR is the one under which [the center of sight at] T perceives KL according to refraction, and angle KTL is the one under which [the center of sight at] T would perceive KL if it were perceived directly. KL’s distance from the center of sight is vast, however, so its size is not accurately determined, [and] so [the center of sight at] T estimates the distance of KL, as we explained in the second [book] of this treatise.⁑ But when it perceives [things] according to refraction, the way it estimates [distance] does not differ from the way it estimates [distance] when it perceives [things] directly, except that [the visual faculty] judges that it perceives [it] directly when [in fact] it perceives it by means of refraction. According to refraction, then, [the center of sight at] T perceives KL under a smaller angle than the one under which it perceives it directly and [it perceives it] by correlating it to the same distance to which it would correlate it if it perceived it directly. But the visual faculty perceives size on the basis of the size of the angle with respect to distance, so according to refraction [the center of sight at] T perceives KL to be smaller than if it perceived it directly.

7.50 ◉ Et si circumvolvamus figuram KTL circa TB immobili, faciet circulum, et erunt anguli qui sunt apud T quos continent due linee KT, TL et suos compares equales. T ergo comprehendit KL reflexive in omni situ in respectu circuli meridiei, cum fuerit in vertice capitis, minorem quam si comprehenderet eam recte. Et si TB secuerit KL in duo equalia, tunc duo puncta Q, R erunt etiam inter duo puncta K, L, et erit angulus QTR minor angulo KTL, et erit omnis angulus eius exiens a puncto secans stellam, et linea que exit ex T in superficie illius circuli secabit circulum, et comprehendetur minor quam sit. Et sic tota stella videbitur minor quam sit.

7.50 ◉ If, moreover, we rotate figure KTL about TB [while holding TB] stationary, it will form a circle, and the angles that the two lines KT and TL, as well as their counterparts [during the revolution], form at T will be equal. Thus, according to refraction [the visual faculty at] T perceives KL, when it lies at the zenith, to be smaller everywhere with respect to the circle of the meridian than if it perceived it directly. And if TB bisects KL, the two points Q and R will also lie between the two points K and L, and angle QTR < angle KTL, and every [counterpart of] angle [QTR within the circle of revolution and] issuing from point [T] to intersect the star [will be smaller than KTL], and a line passing from T on the surface of that circle will intersect the circle, and [so] it will be perceived as smaller than it is. And so the entire star will appear smaller than it is.

7.51 ◉ Stella ergo in vertice capitis comprehenditur minor quam si comprehenderetur recte, et similiter distantia inter duas stellas, cum vertex capitis fuerit inter duas extremitates distantie, comprehendetur in omnibus positionibus minor quam si recte comprehenderetur, et hoc est quod voluimus.

7.51 ◉ Therefore, a star at the zenith is perceived as smaller than if it were perceived directly, and likewise the interval between two stars will be perceived in every situation as smaller than if it were perceived directly, when the zenith-point lies between the two endpoints of the interval, and this is what we wanted [to demonstrate].

7.52 ◉ [PROPOSITIO 20] Item si stella sive distantia fuerit infra verticem capitis et orizonta, aut in orizonte, aut inter orizonta et verticem capitis.

7.52 ◉ [PROPOSITION 20] Likewise, if the star or the interval lies below the zenith and the horizon, on the horizon [itself], or between the horizon and the zenith [it will appear smaller according to refraction than if seen directly].⁑

7.53 ◉ Et sit visus A [FIGURE 7.7.16, p. 450] et vertex capitis B, et continuemus AB. Et sit diameter stelle aut distantia DE equidistans orizonti, et sit circulus verticalis qui transit per alteram extremitatem diametri vel distantie circulus BD, et ille qui transit per aliam extremitatem circulus BE. Et sint due differentie communes inter duos circulos et inter concavitatem orbis duo circuli HG, GZ. Forma ergo D reflectitur ad A in superficie circuli BD. Et continuemus AD, AE. Arcus ergo BD erit equalis arcui BE, quia DE est equidistans orizonti, et reflectitur D ad A ex H, et E ad A ex Z.

7.53 ◉ Let A [in figure 7.7.68, p. 208] be the center of sight and B the zenith, and let us connect AB. Let DE, the diameter of the star or the interval [between stars], be parallel to the horizon [XY], and let circle BD be the vertical circle passing through one endpoint of the diameter or interval and BE the [vertical] circle passing through the other endpoint. Let the two common sections of [these] circles and the concave surface of the [inner] sphere [defined by the heavens] be the two circles HG and GZ. Hence, the form of D is refracted to A in the plane of circle BD. Let us connect AD and AE. Accordingly, arc BD = arc BE because DE is parallel to the horizon [by construction], and [the form of] D is refracted to A from H, whereas [the form of] E [is refracted] to A from Z.

7.54 ◉ Et continuemus lineas AH, HD, AZ, ZE. Et sit centrum mundi M, et continuemus MH, MZ, et pertranseant ad F, N. Erit ergo MH perpendicularis exiens ex H super superficiem corporis diaffoni, et erit HA reflexa ad partem HM; erit ergo reflexa ad partem contrariam illi in qua est HF. H ergo est altius quam AD, et similiter declarabitur quod Z est altius quam AE. Duo ergo puncta F, N sunt inter duo puncta D, E, et angulus reflexionis qui est apud H est equalis angulo reflexionis qui est apud Z, positio enim duorum punctorum D, E respectu A est consimilis; tantum ergo distat F ex D quantum N ex E.

7.54 ◉ Let us connect lines AH, HD, AZ, and ZE. Let M be the center of the world, and let us connect MH and MZ, and let them continue to F and N. MH will thus be the normal dropped from H to the surface of the transparent body, and [ray] HA will be refracted toward HM, so it will be refracted away from HF [i.e., toward A in the opposite direction from F]. H is therefore higher than AD, and it will be demonstrated in the same way that Z is higher than AE. Thus, the two points F and N lie between the two points D and E,⁑ and the angle of refraction at H is equal to the angle of refraction at Z, for the two points D and E are equivalently situated with respect to A, so F lies as far from D as N [lies] from E.

7.55 ◉ Et extrahamus AH ad T et AZ ad K. Distabit ergo T ex D tantum quantum K ex E. Et continuemus TK. Erit ergo equidistans DE; est ergo minor. Et linee AT, AK, AD, AE sunt equales, quia A est quasi centrum duobus circulis BD, BE. Due ergo linee AT, AK sunt equales duabus lineis AD, AE. Et basis TK est minor quam basis DE; ergo angulus TAK est minor angulo DAE, et angulus TAK est ille quo DE comprehenditur reflexe, et angulus DAE est ille quo DE comprehenditur recte.

7.55 ◉ Let us extend AH to T and AZ to K. Accordingly, T will lie as far from D as K [lies] from E. Let us connect TK. It will thus be parallel to DE, so it is smaller. But lines AT, AK, AD, and AE are equal because A is virtually the center of the two circles BD and BE.⁑ The two lines AT and AK are thus [virtually] equal to the two lines AD and AE. But base TK [subtending angle TAK] is smaller than base DE [subtending angle DAE], so angle TAK < angle DAE, and angle TAK is the one under which DE is perceived according to refraction, and angle DAE is the one under which DE is perceived directly.

7.56 ◉ Si ergo stella fuerit in orizonte aut inter orizonta et circulum meridiei, et fuerit diameter eius equidistans orizonti, videbitur minor quam si videretur recte, et hoc idem de distantia inter duas stellas, si distantia fuerit equidistans orizonti, et hoc est quod voluimus.

7.56 ◉ Therefore, if the star lies on the horizon [itself] or between the horizon and the meridian circle, and if its diameter is parallel to the horizon, it will appear smaller than if it were seen directly, and the same applies to the interval between two stars, if the interval is parallel to the horizon, and this is what we wanted [to demonstrate].

7.57 ◉ [PROPOSITIO 21] Item iteremus formam [FIGURE 7.7.17, p. 451], et sit diameter aut distantia erecta, scilicet in eodem circulo verticali. Et sit ille diameter aut distantia linea DE in circulo verticali BDE, et sit differentia communis inter hunc circulum et inter concavitatem orbis circulus GHZ. Et contineumus AD, AE, et reflectatur D ad A ex H, et E ad A ex Z. Patet ergo, ut in precedenti figura, quod H est altius quam AD et quod Z est altius quam AE. Et continuemus lineas AH, HD, AZ, ZE, MH, MZ, et extrahamus MH ad T et MZ ad K. Erit ergo angulus AZM valde parvus, et angulus reflexionis eius erit pars illius; erit ergo angulus EZK acutus, et similiter DHT acutus, et uterque angulus AHD, AZE est obtusus.

7.57 ◉ [PROPOSITION 21] To continue, let us recapitulate [the previous] diagram [in figure 7.7.69, p. 209], but let the diameter [of the star] or the interval [between stars] be upright, i.e., on the same vertical circle. Let that diameter or interval be line DE on vertical circle BDE, and let the common section of this circle and the concave surface of the [inner] sphere [defined by the heavens] be circle GHZ. Let us connect AD and AE, and let [the form of] D be refracted to A from H, and [let the form of] E [be refracted] to A from Z. As in the preceding figure, it is clear that H is higher than AD and that Z is higher than AE. Let us connect lines AH, HD, AZ, ZE, MH, and MZ, and let us extend MH to T and MZ to K. Angle AZM will thus be tiny, and angle of refraction [AZY] will [only] be a part of it, so [vertical] angle EZK [which = AZM + AZY] will be acute, and likewise [vertical angle] DHT [which = AHM + AHX, will be] acute, and [so] both angles AHD and AZE are obtuse.⁑

7.58 ◉ Z autem aut erit in orizonte aut altius; erit ergo in extremitate perpendicularis exeuntis ex A super AB, aut altius illa, et H est altius quam Z. Ergo angulus AHM est minor angulo AZM; ergo angulus DHT est minor angulo EZK. Ergo angulus AHD est maior angulo AZE. Et due linee MT, MK sunt diametri circuli BDE et diametri circuli GHZ; ergo MT est equalis MK, et MH est equalis MZ. Ergo HT est equalis ZK, et angulus DHT est minor angulo EZK; ergo linea HD est minor quam EZ.

7.58 ◉ Now Z will lie either on the horizon or above it, so it will lie at the endpoint of the normal [CA] dropped to AB from A [within the plane of the horizon], or [it will lie] above it, and H is higher than Z. Therefore, angle AHM < angle AZM, so angle DHT < angle EZK. Consequently, angle AHD > angle AZE. But the two lines MT and MK are diameters of circle BDE and [coincide with] diameters [MH and MZ] of circle GHZ, so MT = MK, and MH = MZ. Hence, HT = ZK, and angle DHT < angle EZK, so line HD < [line] EZ.

7.59 ◉ Et due linee AD, AE sunt equales, et A est quasi centrum circuli BDE; ergo circulus qui continet triangulum AHD est maior circulo qui continet triangulum AZE, quia angulus AHD est maior angulo AZE. Et linea HD est minor, ut declaratum est, quam ZE; ergo HD distinguit de circulo continenti triangulum AHD arcum minorem arcu simili arcui quem dividit ZE ex circulo continenti triangulum AEZ. Angulus ergo HAD est minor angulo ZAE.

7.59 ◉ Also, the two lines AD and AE are [virtually] equal, and A is virtually the center of circle BDE, so the circle that circumscribes triangle AHD is larger than the circle that circumscribes triangle AZE because angle AHD > angle AZE. However, as has been demonstrated [earlier], line HD < [line] ZE, so HD cuts off a smaller arc on the circle circumscribing triangle AHD than the equivalent arc that ZE cuts off on the circle circumscribing triangle AEZ. Therefore, angle HAD < angle ZAE.⁑

7.60 ◉ Sit ergo angulus ZAD communis. Ergo angulus HAZ est minor angulo DAE, et angulus HAZ est ille quo A comprehendit DE reflexive, et angulus DAE est ille quo comprehendit DE, si illud comprehenderet recte. A ergo comprehendit DE reflexe minorem quam recte, et hec demonstratio sequetur si circulus BDE fuerit circulus meridiei.

7.60 ◉ So let angle ZAD be common. Therefore, angle HAZ [which = ZAD + HAD] < angle DAE [which = ZAD + ZAE], and angle HAZ is the one under which [the center of sight at] A perceives DE according to refraction, whereas angle DAE is the one under which it perceives DE, if it were to perceive it directly. Consequently, according to refraction [the center of sight at] A perceives DE as smaller [that it would if it perceived it] directly, and this demonstration will follow if circle BDE is the circle of the meridian [as in proposition 19 above].

7.61 ◉ Diameter ergo stelle, cum fuerit directus et rectus, et distantia inter duas stellas recta comprehenditur reflexive minor quam recte, et hoc est quod voluimus.

7.61 ◉ When it is upright and faces the eye directly, then, the diameter of a star or the [rectilinear] interval between two stars is perceived as smaller according to refraction than [it would be if seen] directly, and this is what we wanted [to demonstrate].

7.62 ◉ Et omnis stella in celo comprehenditur rotunda; ergo diametri eius comprehenduntur equales. Et cum sit manifestum quod uterque diameter eius rectus et transversus secundum latitudinem comprehenditur minor quam si comprehenderetur recte, ergo uterque diameter eius declinis comprehenditur minor quam si comprehenderetur recte. Et similiter distantie inter stellas comprehenduntur in omnibus locis et in omnibus sitibus minores quam si comprehenderentur recte. Item diximus quod omnis stella existens in vertice capitis comprehenditur minor quam in omnibus partibus celi, et quanto magis fuerit remotior a vertice capitis, tanto magis comprehendetur maior, et quod maxima comprehenditur quando comprehenditur in orizonte. Et hoc quidem testatur esse, scilicet quod stella videtur in medio celi minor quam si fuerit intra medium celi, et similiter de distantiis, et stella maxima in orizonte et distantie similiter. Restat ergo declarare causam quare hoc sit.

7.62 ◉ Every star in the sky is perceived as round, so its diameters are perceived [to be] equal.⁑ And since it is evident that both its upright and transverse diameters across the width are perceived as smaller than if they were perceived directly, it follows that each one of its inclined diameters is perceived as smaller than if it were perceived directly. By the same token, the intervals between stars are perceived everywhere and in every case as smaller than if they were perceived directly. Furthermore, we have said that every star lying at the zenith is perceived as smaller than anywhere else in the sky, and [we have said that] the farther it lies from the zenith, the larger it will be perceived and that it is perceived as largest when it is perceived at the horizon. This fact, i.e., that the star appears smaller in the middle of the sky than when it lies at an intervening part of the sky, is certainly attested to be [the case], and the same [holds] for intervals [between stars], both the star and the intervals being likewise largest at horizon. So it remains to explain why this is so.

7.63 ◉ Dico quod in secundo huius libri declaravimus, cum tractavimus de magnitudine, quod si visus comprehenderit magnitudines visibilium, comprehendet illas ex quantitatibus angulorum quos respiciunt visibilia apud centrum visus, et ex quantitatibus remotionum, et ex comparatione angulorum ad remotiones. Et declaravimus quod numquam visus comprehendit visibilium quantitates nisi remotiones eorum sint in rectitudine corporum propinquorum continuorum et quod, si visus non certificaverit de remotionibus visibilium, non certificabit quantitates visibilium. Et declaravimus illic etiam quod, si visus non certificaverit distantiam visi, potest perpendere distantiam eius et assimulare eam distantiis visibilium assuetorum quibus tale visibile comprehenditur in tali forma et in tali figura; deinde comprehendit magnitudinem illius ex quantitate anguli quem respicit illud visibile apud centrum visus respectu remotionis quam perpendit.

7.63 ◉ I maintain that, when we dealt with [the perception of] size in the second [book] of this treatise, we showed that, if the visual faculty perceives the sizes of visible objects, it will perceive them according to the sizes of the angles that the visible objects subtend at the center of sight as well as [according to their] distances [from the center of sight], and [the final judgment of size will be based] on correlating the angles to the distances. We also showed that the visual faculty never perceives the sizes of visible objects [properly] unless the distances they span lie in line with [a] continuous [succession of] neighboring objects and that, if the visual faculty does not determine the distances of visible objects accurately, it will not determine the sizes of [those] visible objects accurately. At that same point we also showed that, if the visual faculty does not determine the distance of a visible object accurately, it can assess its distance [by estimation] and match it to the distances of customary visible objects according to which such a visible object is perceived in such a form and shape, and then it perceives its size according to the size of the angle that the visible object subtends at the center of sight with respect to the distance it assesses [by estimation].⁑

7.64 ◉ Et remotiones stellarum non sunt in rectitudine corporum propinquorum, quare visus non comprehendit quantitates eorum, nec visus certificat distantias stellarum; visus ergo perpendit distantias stellarum et assimulat illas distantiis eorum que sunt terrestria que comprehenduntur ex distantia maxima, et perpendit quantitates eorum. Corpus autem celi non videtur sensui quod sit spericum et quod concavum eius sit ex parte visus, nec visus sentit corporeitatem celi, nec visus sentit de celo nisi solum colorem glaucum solummodo. Corporeitas vero, et extensio secundum tres dimensiones, et rotunditas, et concavitas nullo modo possunt comprehendi. Et cum visus non certificaverit aliquid, tunc assimilabit ipsum illis quibus assimilatur de rebus assuetis, unde comprehendit solem et lunam planas, et corpora convexa et concava a distantia maxima plana, et arcus quorum convexum aut concavum est ex parte visus comprehendet rectas, nam si non comprehenderit propinquitatem medii et remotionem extremitatum, et remotionem in mediis concavis et propinquitatem extremitatum, tunc assimilabit superficies convexas et concavas superficiebus planis, et assimilabit arcus lineis rectis, assueta enim visibilia in maiori parte sunt plana et recta.

7.64 ◉ But the distances of the stars do not lie in line with [a continuous succession of] neighboring bodies, so the visual faculty does not perceive their sizes [properly], nor does the visual faculty determine the distances of the stars accurately, so the visual faculty assesses the distance of the stars [by estimation] and matches their [estimated] distances to vast terrestrial distances that are perceived, and it assesses their sizes [accordingly]. Moreover, that the body of the heavens is spherical and that its concave [outer] surface faces the eye is not discernible to the sense [of sight], nor does the visual faculty sense the bodily mass of the heavens,⁑ nor does the visual faculty sense anything about the heavens except its bluish color.⁑ Indeed, its bodily mass, its extension in three dimensions, its roundness, and its concavity can in no way be perceived. And when the visual faculty does not perceive something accurately, it will match it to those things corresponding to it among customary objects, so it perceives the sun and moon as flat, and at a vast distance [it perceives] convex and concave objects as flat, and it will perceive arcs whose convexity or concavity faces the eye as straight, for if it does not perceive the nearness of the middle and the remoteness of the edges [of a convex surface] or the remoteness of the middle and the nearness of the edges of a concave [surface], it will match [remote] convex and concave surfaces to flat surfaces, and it will match [remote] arcs to straight lines because customary visible objects are generally flat and straight.⁑

7.65 ◉ Nec visus, cum forma stelle pervenit ad ipsum, sentit quod illa forma sit reflexa aut quod reflectetur ex superficie concava, et quod corpus in quo est stella sit subtilius corpore in quo est visus. Sed forma stelle comprehenditur sicut forme aliarum rerum qui comprehenduntur in aere recte, et forme visibilium non reflectuntur quando occurrunt corpori diverso ab aere propter visum, nec visus sentit reflexionem eorum, nec superficiem a qua reflectuntur forme in corporibus diversis in diaffonitate in proprietate naturali forme lucis et coloris que extenduntur in corporibus diaffonis. Forme ergo stellarum reflexarum perveniunt ad visum sicut perveniunt forme eorum que sunt in aere ad visum, et comprehenduntur sicuti comprehenduntur in aere.

7.65 ◉ Also, when the form of a star reaches it, the visual faculty does not sense that the form is refracted or that it will be refracted at the concave [inner] surface [of the heavens], and [it does not sense] that the [transparent] body [of aither] in which the star lies is rarer than the [transparent] body [of air] in which the eye lies. On the contrary, the form of the star is perceived in the same way as the forms of other things that are perceived directly in the air, and as far as the visual faculty is concerned, the forms of visible objects are not refracted when they encounter a [transparent] body different from air, nor does the visual faculty sense their refraction, and [it does not sense] the surface at which the forms are refracted in bodies whose transparency is different on account of the natural tendency of the form of light and color to extend through transparent bodies [along straight lines]. Consequently, [when their light is] refracted, the forms of the stars reach the eye in the same way as the forms of objects in the air reach the eye, and they are perceived in the same way they are perceived in air.

7.66 ◉ Visus autem comprehendit colorem celi, nec tamen certificat formam eius nudo sensu. Et cum visus comprehenderit aliquem colorem existens in longitudine et latitudine super hoc quod comprehendet figuram et formam, comprehendet ipsum planum, assimilabit enim ipsum aliquibus superficiebus assuetis, ut parieti et aliis, et hoc modo comprehendet superficies convexas et concavas a maxima remotione. Visus etiam comprehendit planitiem terre planam omnino, nec sentit convexitatem eius nisi fuerint ibi montes et valles. Visus ergo comprehendit superficiem celi planam, et comprehendet stellas sicut comprehendit visibilia assueta separata que sunt in locis spatiosis. Et cum visus comprehenderit aliqua visibilia separata in aliquo loco spatioso, et comprehenderit illa angulis equalibus, et comprehenderit quantitates distantiarum illorum visibilium, tunc illud quod est remotius comprehendetur maius, nam quantitates magnitudinis remotionis comprehendentur ex comparatione anguli quem respicit illa remotio apud centrum visus ad distantiam remotam, et comprehendet quantitatem magnitudinis propinque ex comparatione anguli quem respicit illud propinquum qui est equalis angulo quem respicit distantia ad distantiam propinquam.

7.66 ◉ Furthermore, the visual faculty perceives the color of the sky, but yet it does not accurately determine its shape by brute sensation.⁑ And when the visual faculty perceives some color extending the length and breadth of something whose shape and form it will thereby perceive, it will perceive it as flat, for it will match it to various customary surfaces, such as walls and other [flat surfaces], and this is how it will perceive convex and concave surfaces at a vast distance. The visual faculty also perceives the face of the earth as entirely flat, and it does not sense its convexity unless there are mountains and valleys there. Hence, the visual faculty perceives the surface of the heavens as flat, and it will perceive the stars in the same way it perceives customary visible objects scattered over vast areas [on a flat surface]. And when the visual faculty perceives any visible objects scattered over a vast area, and when it perceives them under the same angles and perceives the distances of those visible objects, it will perceive what lies farther away as larger, for the extents of the distances will be perceived according to a correlation of the angle that the distance subtends at the center of sight to a remote distance, and it will perceive how far away a near object lies according to a correlation of the angle that the near object subtends, which corresponds to the angle that the distance subtends to a near distance.⁑

7.67 ◉ Et hoc patet et esse testatur ei, scilicet quod duorum visibilium que a visu comprehenduntur duobus angulis equalibus, quorum distantie sunt diverse sensibiliter, remotior videbitur maior, et remotius videbitur maius. Nam si homo opposuerit se spatioso parieti, deinde elevaverit manum donec opponat illam visui, et cooperuerit alterum visum, et aspexerit reliquo, et posuerit manum mediam inter visum suum et illum parietem, tunc manus eius cooperiet portionem et latitudinem illius parietis, et comprehendet parietem et manum suam simul. Comprehendet ergo manum suam angulo acuto, et in hoc statu comprehendet latitudinem parietis maiorem quam latitudinem manus multiplicem. Deinde, si moverit manum ita quod detegatur illud quod manus cooperuerat de pariete, et aspexerit illud quod detectum fuerit de pariete, et aspexerit ad manum, videbit illud quod est detectum de pariete maius quam sit sua manus multipliciter. Et ipse comprehendet manum suam, et comprehendet parietem duobus angulis equalibus, ex quo patet quod visus comprehendit magnitudinem ex comparatione anguli ad remotionem.

7.67 ◉ This is clear and is attested to by the fact that, when two visible objects are perceived by the visual faculty under two equal angles, and when their distances are perceptibly different, the farther [distance] will appear longer, and the farther [object] will appear larger. For if a person faces a broad wall, raises his hand until it is in front of the eye, closes the other eye, looks with the remaining one [facing the hand], and poses his hand in between his eye and that wall, his hand will block a portion or the [entire] width of that wall, and he will perceive the wall and his hand at the same time. Hence, he will perceive his hand under an acute angle, and in this case he will perceive that the wall is many times wider than his hand. Then, if he moves his hand so that the portion of the wall that his hand blocked is uncovered, and if he looks at what has been uncovered of the wall and [also] looks at his hand, he will see that the uncovered portion of the wall is many times larger than his hand. But he will perceive his hand, and he will perceive the wall under two equal angles, from which it is evident that the visual faculty perceives size by correlating the angle to the distance.

7.68 ◉ Visus ergo comprehendet superficiem celi planam, nec sentiet concavitatem eius, et comprehendet stellas separatas in ipso. Comprehendet ergo stellas equales separatas inequales, nam comparat angulum quem respicit stella extrema propinqua orizonti apud centrum visus ad distantiam remotam, et comparat angulum quem respicit stella que est in medio celi et propinqua medio remotioni propinque. Et similiter comprehendit stellam que est in orizonte aut prope maiorem ea que est in medio celi aut prope. Comprehendit ergo eandem stellam et distantiam in diversis locis celi diverse quantitatis; sic ergo comprehendit eandem stellam aut distantiam in orizonte aut prope maiorem quam in medio celi aut prope, nam comparat angulum quem respicit illa stella apud centrum visus, stella existente in orizonte, distantie remote, et comparat angulum quem respicit illa stella apud centrum visus, stella existente in medio celi, distantie propinque. Sed inter angulum quem respicit stella apud centrum visus, stella existente in medio celi, et inter angulum quem respicit apud centrum visus, stella existente in orizonte, non est maxima diversitas, sed duo anguli sunt propinqui, quamvis diversi, et similiter distantie inter stellas. Et cum sensus comparaverit duos angulos propinquos in magnitudine ad duas diversas distantias in magnitudine, tunc remotior comprehenditur maior.

7.68 ◉ Accordingly, the visual faculty will perceive the surface of the heavens as flat and will not sense its concavity, and it will perceive the stars scattered on it. It will therefore perceive the stars scattered on it, [which are all] the same size, as unequal [in size], for it correlates the angle that a star nearest the horizon subtends at the center of sight to a far distance, and it correlates the angle that a star at or near the middle of the sky subtends to a near distance. By the same token, it perceives a star at or near the horizon as larger than one at or near the middle of the sky. Hence, it perceives the same star or interval [between stars] to be different sizes at different places in the sky, so it perceives the same star or interval at or near the horizon as larger than [when it lies] at or near the middle of the sky, for when the star is at the horizon, [the visual faculty] correlates the angle that the star subtends at the center of sight to a far distance, and when the star is in the middle of the sky, it correlates the angle the star subtends at the center of sight to a near distance. But there is hardly any difference between the angle the star subtends at the center of sight when the star lies in the middle of the sky and the angle it subtends at the center of sight when the star lies at the horizon; on the contrary, although different, the two angles are close [to equal], and the same holds for the intervals between stars.⁑ And when the sense [of sight] correlates two angles nearly equal in size to two [noticeably] different distances, it perceives the farther one as larger.

7.69 ◉ Et quod certificat hanc causam est quod illi anguli quos eadem stella respicit apud centrum visus ex omnibus partibus celi, cum linee que continent ipsos fuerint reflexe, quoniam locus visus est centrum celi, et reflexiones formarum stellarum non diminuunt ex istis angulis diminutione maxima, et cum iste diminutiones non sint maxime, tunc diversitas inter angulos reflexos quibus stella comprehenditur et inter remotionem inter stellas a locis diversis celi non erit maxima diversitas. Et cum diversitas istorum angulorum non est maxima, tunc magnitudo stelle non comprehendetur diversa maxima diversitate. Et quod demonstrat diminutiones angulorum reflexionis ab angulis quos continent linee recte non sunt maxime magnitudinis et quod sunt valde parve est quod dictum est in predicta experientia de capitulo reflexionis, in quo declaravimus quod visus comprehendit stellam reflexive, et videt stellam fixam ex polo mundi, et remotio est eius ab ipso in una revolutione, nam hec diversitas invenitur parva, ex quo patet quod anguli reflexionis sunt parvi. Unde per illam diversitatem que est inter ipsos non diversantur anguli quibus stella comprehenditur in locis diversis celi maxima diversitate.

7.69 ◉ What substantiates this reasoning is that, since the lines that form the angles subtended by the same star at the center of sight anywhere in the heavens are refracted because the center of sight is at the center of the heavens, and since the refracted forms of the stars are not much reduced in size according to those angles, and since those reductions in size are not significant, the difference between the refracted angles under which the star is perceived or the difference among intervals between stars at various places in the sky will not be significant. And since the difference between those angles is not significant, the size of the star will not be perceived as significantly different. The demonstration that the angles after refraction are not significantly smaller than the angles formed by the straight lines and that [the reductions in size of those refracted angles] are quite small follows from what was claimed in the earlier experiment in chapter [4] on refraction, where we showed that the visual faculty perceives a star according to refraction and that it observes a fixed star [according to its angular distance] from the celestial pole, and its distance is [measured] from that [pole] during one revolution, for the difference [between its distance from the pole at one point in the sky and its distance from the pole at another point in the sky] turns out to be slight, from which it is evident that the angles of refraction are slight.⁑ Hence, according to the difference between them the angles under which the star is perceived at various places in the sky do not differ very greatly.

7.70 ◉ Sed magnitudo stelle et distantia stellarum differunt multum cum sint in orizonte et in medio celi; ergo causa diversitatis stelle et distantie in magnitudine in locis diversis celi non est diversitas angulorum reflexionis. Et iam declaravimus quod visus comprehendit magnitudinem comparando angulos ad remotiones; ergo si diversitas inter angulos fuerit modica et inter distantias et remotiones multa, tunc res videbitur ex maiori distantia maior. Causa ergo per quam videntur distantie stellarum in orizonte maiores quam in medio celi aut prope est illud quod sensus estimat illas distare in orizonte magis quam in medio celi, et hoc quod visus comprehendit stellas in locis diversis celi diversas in magnitudine est error perpetualis, quia causa eius perpetua, et est quoniam visus comprehendit superficiem celi planam, nec sentit concavitatem eius et equalitatem distantiarum eius a visu. Et constat in anima quod in superficie plana, que extenditur ad omnem partem, differunt distantie eius in visu et quod illud quod est propinquius est illud quod est proximius capiti. Comprehendit ergo illud quod est in orizonte remotius quam illud quod est in medio celi, et quod anguli quos respicit eadem stella apud centrum visus ex omnibus partibus celi non maxime diversantur. Et quod visus comprehendit magnitudinem rei ex comparatione anguli quem res respicit ad remotionem illius rei a visu, comprehendit ergo quantitatem stelle et quantitatem distantie que est inter stellas, cum fuerit in orizonte aut prope, comparatione anguli ad distantiam remotam, et cum fuerint in medio celi aut prope, ex comparatione anguli equalis primo aut fere ad distantiam propinquam, et inter ipsam et inter distantiam orizontis videtur diversitas maxima.

7.70 ◉ Nevertheless, the [apparent] size of a star and the [apparent size of an] interval between stars differ considerably when they are at horizon and when they are in the middle of the sky, so the difference in size of the star or the interval at various places in the sky is not due to a difference in the angles of refraction. Moreover, we have already shown that the visual faculty perceives size by correlating angles to distances, so if the difference between the angles is slight but [the difference] between distances is great, an object will appear larger at a longer distance. Hence, the reason the distances of the stars at horizon appear longer than [they do] at or near the middle of the sky is because the sense [of sight] estimates that at the horizon they lie farther away than [when they lie] in the middle of the sky, and the visual faculty’s perceiving the stars as different sizes at various places in the sky is a constant error because its cause is invariant, and it is due to the visual faculty’s perceiving the surface of the heavens as flat, and it [therefore] does not sense its concavity or the equality of the distances [of all points] on it from the center of sight. But it is established in the soul that on a flat surface that extends in every direction the distances on it differ in [relation to] the center of sight and that what lies closer is what lies nearer the head. Accordingly, it perceives what lies at the horizon as farther away than what lies in the middle of the sky, and [it also perceives] that the angles the same star subtends at the center of sight anywhere in the sky do not differ by much. Moreover, because the visual faculty perceives the size of an object by correlating the angle the object subtends to the distance of that object from the center of sight, it perceives the size of the star or the size of the interval between stars by correlating the angle [subtended by the star or interval] to a remote distance, when the star is at or near the horizon, or by correlating the angle, [which is] equal or nearly equal to the first [angle], to a near distance, when the [star and interval] are at or near the middle of the sky, and between that distance and the distance to the horizon there appears to be a vast difference.

7.71 ◉ Hec igitur est causa propter quam errat visus in diversitate magnitudinis stellarum et distantiarum, et hec causa est fixa, perpetua, immutabilis. Et visus comprehendit stellas parvas propter remotionem earum, respiciunt enim apud centrum visus angulos parvos. Sed sensus non certificat quantitatem remotionis stelle, sed estimat et comparat remotiones stellarum cum remotionibus visibilium assuetorum que sunt in terra ita quod opinatur quod remotio stelle est sicut remotio alicuius maxime remoti in terra. Comparat ergo angulum quem facit stella apud visum, qui est parvus, ad remotionem sicut remotio est eorum que sunt super terram, et sic comprehendit stellam propter hanc comparationem parvam. Et si visus esset certus de quantitate remotionis stelle, tunc comprehenderet eam magnam, et similiter est de omnibus que sunt super terram maxime remotis, si comprehendantur parva est quia non certificatur remotio eorum, et iam declaravimus hoc perfecte in tertio huius libri. Et sicut visus errat in quantitate remotionis stelle, quia non est certus de ipsa, et assimulat eam remotionibus que sunt super terram, sic errat in hoc quod distantie earum in locis diversis celi sunt diverse, cum sint equales, quia assimulat eas etiam distantiis diversis que sunt super terram dextram et sinistram et opposite de quibus non est dubium ipsas esse diversas. Et sicut error in remotione et magnitudine stelle est perpetuus, sic error in diversitate distantiarum stellarum in locis diversis celi et in diversitate magnitudinis est perpetuus, nam forme harum distantiarum non diversantur apud visum in diversis temporibus sed semper sunt eodem modo, et visus assimulat eas distantiis assuetarum rerum que maxime distant a visu super faciem terre.

7.71 ◉ This, then, is why the visual faculty errs in [perceiving] a difference in the size of stars or the intervals [between stars], and the cause for this [error] is fixed, constant, and immutable. But the visual faculty perceives the stars as small because of their remoteness, for they subtend tiny angles at the center of sight. Yet the sense [of sight] does not determine the distance of the star accurately; rather, it estimates [the distance] and correlates the [estimated] distances of the stars to the distances of customary visible objects on earth so that it assumes that the distance of the star is like the distance of something extremely far away on earth. Hence, it correlates the angle the star subtends at the center of sight, which is small, to a distance that is like the distance of things that are on earth, and so it perceives the star as small according to this correlation. But if the visual faculty were certain about the size of the star’s distance, it would perceive it as large, and in the same vein, if they are perceived as small, all objects that lie at an extremely remote distance on earth are [so perceived] because their distance is not accurately determined, and this has already been demonstrated completely in the third book of this [treatise].⁑ And just as the visual faculty errs in [judging] the size of the star’s distance because it is not certain about it, and [just as it] matches it to distances on earth, so it errs in [judging] that the distances of things at various places in the sky are different, when they are [in fact] equal, because it also matches them to different distances to the right or left or in the facing direction on the earth’s surface, there being no doubt that they are different. And just as an error about the distance and size of the star is constant, an error in [perceiving] a difference in the distances and size of the stars at various places in the sky is constant, for the forms of these distances do not differ according to the visual faculty at various times but are always of the same sort, and the visual faculty matches them to the distances of customary objects that lie extremely far from the center of sight on the face of the earth.⁑

7.72 ◉ Accidit etiam eis que sunt in celo alia causa ad hoc quod videantur maiora in orizonte in maiori parte, scilicet vapores grossi qui sunt oppositi inter visum et stellas. Et cum vapor fuerit in orizonte aut prope, et non fuerit continuus usque ad medium celi, erit portio spere, cuius centrum erit centrum mundi, quod continet terram, et sic abscindetur ex parte medii celi, et erit superficies eius que est ex parte visus plana, quare forma aut distantie que sunt ultra illum vaporem videbuntur maiores quam sine illo vapore, in illo enim loco concavitatis celi ex quo loco reflectitur forma stelle ad visum existit forma stelle, et ex ipso extenditur recte ad visum, si in orizonte non fuerit vapor grossus.

7.72 ◉ There also happens [to be] another factor that causes things in the heavens to appear for the most part larger at the horizon, namely, thick vapors that lie between the eye and the stars. When the vapor lies at or near the horizon but does not rise uninterrupted to the middle of the sky, it will form a segment of a sphere whose center will be the center of the world, which includes the earth, and so it will be truncated on the side of the sky’s middle, and the surface on it that faces the eye will be plane, so the form [of a star] or the intervals [between stars] that lie beyond that [wall of] vapor will appear larger [than they would] without that vapor, for the form of the star lies at the spot on the [inner] concave surface of the heavens from which the star’s form is refracted to the eye, and it extends straight to the center of sight from that point if there is no thick vapor at the horizon.

7.73 ◉ Si vero affuerit vapor grossus, tunc hec forma extendetur ad superficiem vaporis qui est ex parte celi et existet in illa superficie, et sic visus comprehendet illam sicut comprehendet ea que sunt in vapore, scilicet quod illa forma extenditur in vapore grosso recte, deinde reflectitur apud superficiem vaporis que est ex parte visus ad contrariam partem perpendicularis existentis super superficiem vaporis (que est plana), nam aer qui est ex parte visus est subtilior illo vapore; ex quo sequitur quod forma videatur maior quam si videretur recte, ut in prima figura huius capituli diximus. Et est cum corpus subtilius fuerit ex parte visus et grossius ex parte rei vise, erit superficies corporis grossioris plana. Forma ergo que pervenit ad superficiem vaporis que est ex parte celi est res visa, et corpus in quo extenditur hec forma est vapor grossus, et aer in quo est visus est subtilior illo.

7.73 ◉ On the other hand, if thick vapor is present, this form will extend to the surface of the vapor facing the heavens and will lie on that surface, and so the visual faculty will perceive it as it will perceive objects that lie in vapor, that is, the form extends straight through the thick vapor and is refracted away from the normal dropped to the vapor’s surface at the surface of the vapor on the side of the eye (that surface being plane), for the air on the side of the eye is rarer than the vapor; so it follows from this that the form should appear larger than [it would] if it were seen directly, just as we claimed in the first proposition of this chapter [i.e., proposition 12, pp. 309-311]. And since the rarer body lies on the side of the eye and the denser one on the side of the visible object, the surface of the denser body will be flat. Hence, the form that reaches the surface of the vapor on the side of the heavens constitutes the visible object, and the body through which this form extends is the thick vapor, whereas the air in which the eye lies is rarer than it.⁑

7.74 ◉ Causa ergo principalis quare stelle et distantie stellarum videntur in orizontibus maiores quam in medio celi est illa predicta, et est fixa perpetua. Si vero acciderit quod sit vapor grossus, crescit magnitudo earum. Sed hec causa est in quibusdam locis semper et in quibusdam quandoque. Omnia ergo que diximus in hoc capitulo de illis que accidunt visui propter reflexionem deceptiones sunt ille que semper accidunt, aut in maiori parte, et sufficiunt in hoc quo indigemus de deceptionibus quarum causa est reflexio.

7.74 ◉ So the main reason that the stars and the intervals between stars appear larger at the horizon than in the middle of the sky is the one described before, and it is fixed and invariant. If it happens that there is thick vapor, however, their [apparent] size increases. But this factor applies invariably in some places and [only] occasionally in others.⁑ Thus, all the misperceptions befalling the visual faculty according to refraction that we have discussed in this chapter are ones that always occur, or [at least] for the most part, and they suffice for what we need [to show] concerning the misperceptions caused by refraction.

7.75 ◉ Nunc autem terminemus hunc tractatum, qui est finis libri, etc.

7.75 ◉ Let us now conclude this book, which is the end of the treatise, etc.