◉Liber iste dividitur in quinque partes. Pars prima proemium libri; secunda in declaratione quod lucis accidit reflexio ex politis corporibus; tertia in modo reflexionis forme; pars quarta in ostensione quod comprehensio forme ex corporibus politis non est nisi ex reflexione; pars quinta in modo comprehensionis formarum per reflexionem.

◉This book is divided into five parts. The first part [constitutes] the book’s prologue; the second [is devoted] to showing that the reflection of light occurs from polished bodies; the third [focuses] on how the [visible] form is reflected; the fourth part [is concerned] with showing that the perception of a form in polished bodies is due solely to reflection; the fifth part [deals] with how forms are perceived through reflection.

◉[Capitulum 1]

◉[Chapter 1]

1.1 ◉ Iam explanavimus in libris tribus modum comprehensionis formarum in visu cum fuerit directus, et enumeravimus singula que in rebus visis comprehendit visus. Sed diversificatur adquisitio visus tripliciter: aut enim directe, sicut diximus; aut per reflexionem in politis corporibus; aut per penetrationem, ut in raris, quorum non est raritas sicut raritas aeris. Nec potest diversificari visus nisi hiis modis tribus. [1.2] Et hiis duobus modis posterioribus comprehendit visus in rebus visis ea que supra exposuimus et quorum adquisitionem in visu directo patefecimus. Et forsitan visus in hiis incurret errorem aut consequitur veritatem. Et nos assignabimus in hoc libro quomodo per reflexionem fiat formarum adquisitio, et quomodo erit reflexio, et quis linearum reflexarum situs. Et preponemus quedam antecedentia preponenda.

1.1 ◉ We have already explained in the [previous] three books how [visible] forms are perceived by sight when it is direct [and uninterrupted], and we have listed the particular characteristics of visible objects that the visual faculty perceives.⁑ But visual perception occurs in three different ways: i.e., directly, as we [just] mentioned; or by means of [rays] reflecting from polished bodies; or by means of [rays] passing through transparent media, there being none more transparent than [pure] air. These three are the only ways in which vision can occur.⁑

◉[Capitulum 2]

◉[Chapter 2]

2.1 ◉ Planum ex libro primo quoniam lux a corpore lucido luce ei propria vel accidentali dirigitur in omne corpus ei oppositum, et eodem modo color cum in eo lux fuerit mittitur. Igitur corpore polito opposito corpori lucido, mittitur ad ipsum lux lucidi mixtim cum colore, et reflectitur lux cum colore, sive fuerit fortis sive debilis, sive prima sive secondaria.

2.1 ◉ From the first book it is clear that light [emanating] from an illuminated body, whether that body is intrinsically or extrinsically illuminated, shines upon every object that faces it, and likewise color is propagated [toward all facing objects] as long as [that color] is illuminated.⁑ Thus, when a polished body faces an illuminated body, the light from the illuminated body, which is mingled with its color, shines upon that polished body, and the resulting light is reflected along with the color, no matter whether it is bright or faint or whether it is primary or secondary.⁑

2.2 ◉ Et quod fiat in luce forti reflexio patere potest opposito luci forti speculo ferreo, et etiam oppositus sit paries speculo; et descendat super ipsum lux declinata non recta. Videbitur in pariete lux fortis reflexa, que quidem non videbitur super eundem locum si speculum auferatur, nec videbitur super eundem locum si speculum moveatur; immo secundum motum speculi mutabitur locus lucis reflexe in pariete. Quare palam reflexionem fieri in luce forti.

2.2 ◉ That reflection may occur in the case of bright light can be demonstrated by directing an iron mirror⁑ toward an intense light-source, having a wall face the mirror, and letting the light shine upon the mirror at a slant, not orthogonally. The bright light will [then] appear reflected upon the wall, but it will definitely not appear at the same spot if the mirror is removed, nor will it appear at the same spot if the mirror is moved; on the contrary, the spot at which the light is reflected upon the wall will shift with the motion of the mirror. So it is clear that reflection occurs in the case of bright light.

2.3 ◉ In luce debili patere potest defacili. Intra domum aliquam per foramen unicam a terra elongatum, sed non multum, descendat lux diei, non solis, super aliquod corpus. Et circa corpus statuatur speculum ferreum, et circa speculum corpus aliquod album. Apparebit in secundo corpore albo lux maior quam sine speculo, et augmentum illius lucis non est nisi ex speculi reflexione, quoniam ablato speculo, sola lux secundaria debilis apparebit in corpore albo.

2.3 ◉ This can be demonstrated as easily in the case of faint light. Within a room that has a single window that is not too high above the ground, let daylight, but not [direct] sunlight, shine through that window upon some object.⁑ Place an iron mirror near that object, and put a white body near the mirror. On that second white body the light will appear brighter than [it did] when the mirror was absent, and the increase in that light is due solely to reflection from the mirror, for when the mirror is removed, only faint, secondary light will appear on the white body.

2.4 ◉ Amplius, si diligens figatur intuitus in lineis per quas a corpore primo lux in speculum mittitur, perpenditur quidem linearum illarum declinatio super speculum et super idem linearum reflexionis declinatio eadem. Et est proprium reflexioni ut eadem sit declinatio et idem angulus linearum venientium et reflexarum. Quod si moveatur corpus album a loco reflexionis in alium locum, tamen circa speculum, non videbitur in eo lucis augmentum, nec videri poterit nisi in illo situ tantum. Quare planum proprium esse reflexioni hunc situm.

2.4 ◉ Furthermore, if one carefully directs his sight along the lines according to which the light shines from the first object to the mirror, he cannot help but realize that the inclination of those lines upon the mirror is the same as the inclination of the lines of reflection to the [white body] itself. So it is characteristic of reflection that the inclination is the same and that the angle of the incident and reflected rays is the same. But if the white body is moved from the location of reflection to somewhere else, even [if that new location is] near the mirror, the increase in light will not appear in it, nor can it be detected except in that particular location [of reflection]. So it is clear that this location is peculiar to reflection.⁑

2.5 ◉ Hoc idem poteris videre secundaria luce si predictum speculum sit argenteum et corpus tertium album sit ex alia parte speculi. Apparebit quidem super corpus tertium lux secundaria, et super corpus secundum lux maior illa, et palam huius maioritatis causam solam esse reflexionem. Patebit autem lucis reflexio in omni loco ubi super corpus descendat per foramen aliquod lux fortis, adhibito luci speculo et ei corpore albo opposito modo supraposito.

2.5 ◉ You can observe this same thing in the case of secondary light, if the aforementioned mirror is silver and a third white body lies on the other side of the mirror. Secondary light will in fact appear on the third body, whereas on the second [white] body the light will be brighter than the secondary light [on the third body], and it is obvious that the only reason for this increased brightness is reflection.⁑ Moreover, the reflection of light will be obvious everywhere strong light shines upon a body through any window when a mirror is disposed to face the light and a white body faces the mirror in the way that has just been described.

2.6 ◉ Verum locum reflexionis proprium et linearum situm explanabimus. Iam patuit in libro primo quod lux reflexa sequitur rectitudinem linearum, quare ex corporis politis fit reflexio secundum processum rectitudinis in situ proprio.

2.6 ◉ But we shall describe the appropriate location of reflection and the disposition of the rays. It has already been demonstrated in the first book that reflected light follows straight lines, so reflection occurs from polished bodies to a definite spot according to a rectilinear course.⁑

2.7 ◉ Amplius, planum ex superioribus quod lux secunda a corpore illuminato accidentali luce procedens secum fert colorem corporis. Ab omni igitur corpore illuminato seu lucido color mixtus cum luce ad corpora opposita polita mittitur, et mixtim in partem debitam reflectitur. [2.8] Et huius rei fides poterit fieri si intra domum unius foraminis tantum descendit lux solis super corpus forti et specioso colore. Et statuatur circa ipsum speculum ferreum, et circa speculum corpus concavum ad ciphi modum intra quod sit corpus album, et aptetur hoc vas in loco reflexionis ut lux reflexa incidat in corpus album. Apparebit quidem super faciem albi corporis color illius in quo fit descensus lucis, quod quidem non accideret si extra proprium reflexionis situm statuatur corpus album. Et secundum diversas colorum species hec probatum invenies, velut colori celesti, rubore, viriditate, et huiusmodi. Quare planum colorem mixtum cum luce remitti, et certior est coloris reflexi apparentia si speculum fuerit argenteum.

2.7 ◉ Now, it is evident from earlier discussions that secondary light emanating from a body that is extrinsically illuminated carries the body’s color with it.⁑ Hence, from every illuminated body or light-source, color mingled with light shines on facing polished bodies and, so mingled, is reflected in a determinate direction.

2.9 ◉ Quare autem non appareat hec probatio—scilicet, quod comprehendatur color reflexus cuicumque corpori opponatur speculum et ei adhibeatur album—hec est ratio. Sicut supradictum est, colores debiles, licet simul cum luce mittantur, non sentiuntur, quia forme que reflectuntur debiliores sunt formis a quibus reflexio oritur. Et hoc in luce potest patere, quoniam luce forti in speculum cadente et reflexa in pariete, debilior videbitur lux parietis quam speculi, et notabilis est inter eas proportio.

2.9 ◉ Here, however, is why this demonstration may not be evident—i.e., [the demonstration] that reflected color may be perceived on every body that faces the mirror and presents a white surface to it. As has been asserted earlier, weak colors, even though they shine along with light, are not perceived because the forms that are reflected [in such cases] are weaker than the forms from which the reflection originates.⁑ And this can be shown in the case of light, for when bright light strikes a mirror and is reflected to a wall, the light on the wall will appear weaker than it does in the mirror, and there is a noticeable difference [in intensity] between them.

2.10 ◉ Idem patebit in luce debili pari modo. In domo prima dispositione prima, si corpus tertium album ponatur loco speculi ferrei vel circa ipsum, maior apparebit lux super hoc corpus quam super secundum, quod non accideret nisi reflexio lucem debilitaret.

2.10 ◉ This will become apparent in the same way for weak light. In the case of the original room disposed as it initially was, if the third white body is placed at or near the [original] location of the iron mirror, the light will appear brighter upon that body than it does upon the second one, which would not happen unless reflection weakened the light.⁑

2.11 ◉ Sed dicet aliquis causam huius rei esse nigredinem speculi ferrei, que admixta luci in speculum cadenti ipsam obumbrat, et inde, reflexa in corpus secundum, debilis et fusca apparet. Sed in corpus tertium loco speculi vel circa positum, non descendit lux nisi a corpore primo nulli admixta nigredini. Verum quod hoc non sit in causa palam eo quod, loco speculi ferrei argenteo posito, eadem accidet probatio.

2.11 ◉ But someone will argue that the reason for this phenomenon is the iron mirror’s black coloring, which, having mingled with the light shining upon the mirror, darkens that light, so, when [the resulting light] is reflected to the second body, it appears faint and dark. On the other hand, in the case of the third [white] body placed at or near the [original] location of the mirror, the light shines [upon it] only from the first body with no mingling whatever of dark coloring. That this is actually not the case, however, is evident from the fact that, when a silver mirror replaces the iron one, the same thing will be demonstrated.

2.12 ◉ Pari modo reflexus color debilior erit colore a quo fit reflexio, quod in domo reflexionis coloris patere poterit, si corpus album loco speculi ponatur vel circa. Fortior apparebit in ipso color quam in corpore albo intra vas posito. Et idem patebit si in loco ferrei argenteum ponatur speculum. Igitur reflexio debilitat et luces et colores, sed colores amplius quam luces secundum utrumque speculum. Et est quoniam colores accedunt debiliores quam luces, unde facile efficiuntur in reflexione debiliores.

2.12 ◉ In precisely the same way [as before], the reflected color will be fainter than the color from which the reflection originates. This can be shown for the reflection of color if the white body in the room is placed at or near the location of the mirror. The color will appear stronger on that body than [it does] on the white body placed inside the vessel.⁑ The same will be evident if a silver mirror replaces the iron one. Therefore, reflection weakens both light and colors, but colors more than light, according to whichever mirror [is used]. And this happens because colors shine more weakly than light, so they are easily weakened in reflection.

2.13 ◉ Amplius, color debilis, cum ad speculum pervenit, miscetur colori eius, quare reflexus apparet debilis et tenebrosus; et forme debiliores sunt reflexe quam in loco reflexionis, et reflexio causa est debilitatis. [2.14] Poterit aliquis dicere non esse debilitatem formarum in reflexione nisi ex elongatione earum a sua origine. Sed explanabitur quod, licet ab ortu equaliter elongentur lux directa et lux reflexa, tamen debilior erit reflexa.

2.13 ◉ Now, when a weak color reaches a mirror, it does mingle with the mirror’s color, so when it is reflected it appears faint and dark; and reflected forms are weaker than [the forms] at the point of reflection, and reflection is the cause of weakening.

2.15 ◉ Intret radius solis domum aliquam per foramen, et opponatur foramini in aere speculum ferreum minus foramine. Et lux foraminis residua cadat in terram super corpus album, et lux a speculo reflexa cadat in corpus album elevatum. Hoc observato ut eadem sit elevati et iacentis a foramine longitudo, videbitur quidem super elevatum lux minor quam super iacens. Et huius minoritatis non potest assignari causa nisi reflexio sola. Idem accidet si speculum fuerit argenteum.

2.15 ◉ Let a shaft of sunlight enter a room through a window, and place an iron mirror that is smaller than the window in the air facing it. Let the rest of the light coming through the window shine upon a white body on the ground, and let the light that is reflected from the mirror shine upon a white body that is raised [above the ground]. When this is set up so that the elevated body and the body lying upon the ground lie the same distance [from the window], the [reflected] light shining upon the elevated body will definitely appear fainter than the light shining [directly] upon the body lying on the ground.⁑ And this weakening [of light] can be imputed to no other cause than reflection. The same thing will happen if the mirror is silver.

2.16 ◉ Idem in colore potest patere, luce solis in domum aliquam per foramen descendente super corpus coloris fortis cui circa adhibeatur speculum, et aliud corpus concavum intra quod sit corpus album in quod cadit reflexio. Et statuatur in domo aliud corpus album eiusdem modi cum eo quod est in concavo, et sit elongatio huius albi a corpore colorato in quod cadit lux foraminis eadem cum elongatione albi quod est in concavo ab eodem, et elongatione speculi ab eodem. Perpendi quidem poterit color debilior in albo quod est intra concavum quam in eo quod est extra, licet equidistant ab actu suo—id est a corpore colorato. Et in causa est reflexio colorem debilitans.

2.16 ◉ This very point can be demonstrated in the case of color, when sunlight passes through a window into the room [and shines] upon a brightly colored object facing a nearby mirror, and when another hollow object⁑ with a white body inside it is positioned to catch the reflection. Within the room place another white body of the same kind as the body in the vessel, and let the distance of this white body from the colored body struck by the light shining through the opening be the same as the distance of the white body in the hollow vessel from that same [colored body] combined with the distance of the mirror from that same [colored body].⁑ It can be determined [on this basis] that the color [appears] weaker on the white body in the vessel than [it does] on the body that lies outside it, even though they are [both] equidistant from their source—i.e., from the colored body. And the reason for this is the reflection that weakens color.

2.17 ◉ Amplius, lux reflexa fortior est luce secundaria, licet eiusdem sit elongationis ab origine sua. Luce etenim reflexa cadente in corpus aliquod, si aliud eiusmodi corpus ponatur extra locum reflexionis, et sit cum eo eiusdem elongationis a speculo, videbitur super ipsum lux minor quam in illo.

2.17 ◉ Moreover, reflected light is brighter than secondary light, even when the [two kinds of light] lie the same distance from their source. For in fact, when reflected light shines on some object, if another object of the same kind is placed outside the location of reflection, and if it lies the same distance from the mirror [as the first object], the light on the second object will appear fainter than on the first one.

2.18 ◉ Idem etiam planum erit in domo, si deprimatur in terra in directo foraminis speculum quod accipit totam foraminis lucem. Erit lux fortior super corpus in loco reflexionis positum quam super aliud eiusdem modi extra hunc locum tantumdem a speculo elongatum.

2.18 ◉ The same thing will also be evident in the room [used in the previous experiments] if the mirror is laid upon the ground in a direct line with the opening so that it receives all the light [coming] from the window. The light will be brighter on an object lying at the location of reflection than on another object of the same kind lying outside that location at the same distance from the mirror.

2.19 ◉ Eodem modo, si excedat lux foraminis quantitatem speculi, et cadat circa speculum lux in terram aut corpus album a quo aliud corpus tantum elongatur quantum corpus reflexionis a speculo, debilior apparebit in eo lux quam super reflexionis corpus.

2.19 ◉ In the same way, if [the shaft of] light shining through the window is wider than the mirror, if the [excess] light around the mirror falls upon the ground or upon a white body, and if another body lies as far from the mirror as the object upon which the light is reflected, the light upon that body will appear fainter than the light upon the body [at the location] of reflection.⁑

2.20 ◉ Similiter accidit in colore. Si corpus aliquod tantum distet a speculo extra situm reflexionis quantum aliud ei simile quod est in situ, apparebit quidem super corpus quod est in situ reflexionis color reflexus; super aliud forsitan nullus. Si enim ferreum fuerit speculum aut fere nullus videbitur, aut omnino nullus; si vero argenteum fuerit speculum, apparebit super ipsum color aliquis, sed valde debilis, et longe debilior quam in corpore quod est in situ reflexionis.

2.20 ◉ The same happens in the case of color. If some object lies as far from the mirror outside the location of reflection as another identical object lies from the mirror at the location [of reflection], the reflected color will definitely appear on the object lying at the location of reflection; [whereas] on the other object no color at all may appear. In fact, if the mirror is iron, almost no color, or none at all, will appear, but if the mirror is silver, some color, albeit very faint, will appear upon the object, but it will be far fainter than [the color] on the object that lies at the location of reflection.

2.21 ◉ Et iam igitur planum quod forme lucium et colorum reflectuntur ex corporibus politis et in reflexione debilitantur. Et erit forma directa fortior reflexa cum eadem earum origo et equalis ab ea elongatio. Et reflexa fortior secundaria cum idem vel equalis earum ortus et par elongatio.

2.21 ◉ It is therefore now evident that the forms of lights and colors are reflected from polished bodies and that they are weakened in reflection. And a form [of light or color] that shines directly [upon an object] will be brighter than one that is reflected [to it] when they share the same source and lie the same distance from it. But a reflected form is brighter than a secondary one when they share the same source or [originate from] sources of equal [intensity] and lie the same distance [from their source].

◉[Capitulum 3]

◉[Chapter 3]

Pars tertia: in modum reflexionis formarum in corporibus politis

3.1 ◉ Politum est lene multum in superficie, et lenitas est quod sint partes superficiei continue sine pororum multitudine. Lenitas intensa est ubi multa partium superficiei continuitas et pororum parvitas et paucitas, et finis lenitatis est privatio pororum et privatio divisionis partium. Igitur politas est politiva continuitas partium superficiei cum poris raris et exiguis, et finis politive est vera continuitas partium et privatio pororum.

3.1 ◉ A polished body has an extremely smooth surface, and smoothness consists in the parts of the surface being continuous without many pores. Extreme smoothness exists when there is considerable continuity of the parts of the surface and the pores are few and small; and perfect smoothness consists in the absence of pores and the absence of gaps between the segments [of the surface]. Hence, the polish in a polished [body] consists in the continuity of the parts of the surface with very few and very small pores, and perfect polish consists in absolute continuity of the parts along with the [complete] absence of pores.⁑

3.2 ◉ In omnibus politis superficiebus, licet diversis subiaceant figuris, accidet reflexio, et idem reflexionis modus et eadem proprietas: et est quod in omni polita superficie ab omni puncto fit reflexio; et sumpto quocumque puncto in superficie a quo fiat reflexio, linea accessus forme alicuius ad illum punctum et linea reflexionis in eadem superficie erunt cum linea perpendiculari super illud punctum erecta; et tenebunt hee linee eundem situm respectu perpendicularis et equalitatem angulorum. Et volo dicere perpendicularem que sit perpendicularis super superficiem tangentem corpus politum in illo puncto, et due linee cum perpendiculari sunt in eadem superficie ortogonaliter cadente super superficiem corpus politum in puncto a quo fit reflexio tangentem.

3.2 ◉ Reflection will occur in the case of all polished bodies, even though they may be subdivided into different shapes, and [they are all subject to] the same mode of reflection and share the same specific characteristic[s]: i.e., [that] in every polished surface reflection occurs from every point; [that] whatever point on the surface from which reflection occurs is taken, the line of incidence for any form to that point and the line of reflection [extending from that point] will lie in the same plane as the normal dropped to that point; and [that] these lines will maintain an equivalent situation with respect to [that] normal and will form equal angles [with it]. Now, by »normal« I mean the perpendicular to the plane tangent to the polished body at that point [of reflection], and the two lines [of incidence and reflection] lie along with the perpendicular in the same plane, which falls orthogonally to the plane that is tangent to the polished body at the point from which reflection occurs.⁑

3.3 ◉ Si autem linea per quam accidit ad speculum forma cadat perpendiculariter super illud, fiet reflexio forme per ipsam, non per aliam, et hoc est proprium in omni reflexione in omni polito corpore. Si ergo corpus politum fuerit planum, superficies tangens punctum reflexionis erit una et eadem cum superficie corporis. Si vero fuerit columpnare speculum interius aut extra politum, erit contactus superficiei speculi et superficiei contingentis linea tantum secundum longitudinem speculi intellecta. Idem in speculo piramidali intus vel extra polito. In sperico, sive concavo interius sive exterius polito, contingens superficies tangit in solo puncto.

3.3 ◉ If, however, the line along which the form reaches the mirror falls perpendicular to it, the form will reflect [back] along that same line and along no other, and this is characteristic of every reflection from every polished body. Thus, if the polished body is flat, the plane tangent [to it] at the point of reflection will be one and the same as the surface of the body. On the other hand, if the mirror is cylindrical and is polished on either the inside or outside [surface], the contact between the mirror’s surface and the plane tangent to it will consist only of a line imagined along the length of the mirror.⁑ The same holds for a conical mirror, [whether it is] polished on the inner or on the outer [surface]. In a spherical [mirror], be it polished on the concave inner [surface] or on the [convex] outer [surface], the plane of tangency touches at only one point.

3.4 ◉ Quomodo autem ad oculum pateat hic modus reflexionis in speculis omnibus explanabimus. Accipe tabulam eneam spissam ut firmior sit, eius longitudo non minus quam 12 digitorum, et sit latitudo sex digitorum. Et fiat linea equidistans extremitati longitudinis et circa illam extremitatem. Et super punctum huius linee medium ponatur pes circini, et fiat semicirculus cuius semidyameter sit latitudo tabule. [3.5] Et extrahatur a puncto quod est centrum linea ortogonaliter super dyametrum iam factum. Et erit linea illa semidyameter dividens semicirculum per equalia. Et in hoc semidyametro sumatur mensura unius digiti et, posito pede circini super centrum, fiat semicirculus secundum quantitatem partis residue semidyametri, residue scilicet secundum semidyametrum quinque digitorum.

3.4 ◉ We will explain, moreover, how this account of reflection can be empirically demonstrated for all mirrors. Take a bronze plaque not less than 12 digits long that is thick [enough] to be quite rigid, and let it be 6 digits wide.⁑ Draw a line right along the lengthwise edge [of the plaque] and parallel to it. Place the point of a compass on the midpoint of this line and draw a semicircle whose radius is the width of the plaque.

3.6 ◉ Et dividantur semicirculi primi medietates in quot libuerit partes ita quod sibi respondeant in qualitate prima—scilicet prima prime, secunda secunde, et sic de aliis—et protrahantur linee a centro ad puncta divisionum.

3.6 ◉ Divide the intermediate portions of the first semicircle into as many parts as you please so that they correspond in kind with the first—i.e., [so that] the first [division corresponds to] the first, the second to the second, and so forth—and draw [straight] lines from the centerpoint [of the semicircle] to the points of division.

3.7 ◉ Deinceps in semidyametro mensura digiti signetur, et ex parte centri, et super punctum signatum protrahatur linea equidistans dyametro semicirculi, sive tabule extremitati, quod idem est. Et secetur ex tabula quod interiacet hanc lineam et semidyametrum usque ad centrum et lineas primas ad divisiones semicirculi protractas—id est ad lineas tales semidyametro propinquiores.

3.7 ◉ Then, mark off 1 digit on the [perpendicular] radius on the side of the centerpoint, and draw a line through the point [just] marked off parallel to the diameter of the semicircle, or to the [lower] edge of the plaque, which is the same thing.⁑ Cut off from the plaque the portion bounded by this line and the radius [of the larger semicircle along the lower edge] to the centerpoint as well as by the first lines dividing the semicircle—i.e., to such lines as lie nearest the [semicircle’s lower] radius.

3.8 ◉ Post secetur tabula circa semicirculum maiorem ut solum remaneat semicirculus. Et secetur tabula sub centro ubi centri locus acuatur quasi punctus hoc tamen modo ut in eadem superficie plana remaneat cum semicirculo et aliis lineis.

3.8 ◉ Afterwards, cut the plaque around the [circumference of] the larger semicircle so that all that remains is the semicircle. Then cut the plaque below the center, sharpening that spot at the center to a point in such a way that it lies on the same plane as the [larger] semicircle and all the other lines.⁑

3.9 ◉ Post sumatur tabula lignea plana excedens eneam in longitudinem duobus digitis, et sit quadrata, et eius altitudo, sive spissitudo, septem digitorum. Signetur ergo in hac tabula punctum medium, et super ipsum fiat circulus excedens maiorem circulum tabule enee super quantitatem digiti magni. Et fiat super idem centrum circulus equalis circulo minori tabule enee.

3.9 ◉ Next, take a flat, square block of wood that is broader than the bronze plaque by 2 digits [i.e., 14 by 14], and let its height, or thickness, be 7 digits. Then mark the midpoint [on the top surface] of this block, and from it draw a circle that is a full digit larger than the larger circular segment of the bronze plaque. From the same centerpoint draw a circle equal to the smaller circle on the bronze plaque.

3.10 ◉ Et dividatur circulus maior in partes in equalitate respondentes partibus semicirculi tabule enee, ut scilicet prima respondeat prime, secunda secunde, et sic de aliis. Et secetur circumquaque tabula lignea ut solum remaneat maior circulus, et erit hec sectio usitato secandi modo. Secetur etiam pars tabule circulo minori contenta, et modus sectionis erit ut huic tabule associetur alia tabula ita ut linea a centro huius ad centrum illius transiens sit perpendicularis super illam. Et adhibito tornatili instrumento centris earum, fiat sectio partis circularis iam dicte. Est autem alterius tabule associatio, ut fixa stet in sectione.

3.10 ◉ Then divide the larger circle into corresponding sections equal to the sections [marked off] on the semicircle of the bronze plaque—i.e., so that the first [section marked off on the bronze plaque’s circle] corresponds to the first [section marked off on the circle drawn on the wooden block], the second to the second, and so forth.⁑ Cut all around the [circumference of the larger circle] on the wooden block so that only the [part bounded by the] larger circle is left; this section will now serve as a template for cutting. Cut out the portion on the block bounded by the smaller circle as well, and the way to do this is to fit another block to this one so that the line passing through the centerpoint of the former and the centerpoint of the latter is perpendicular to [the top surface] of the latter. Then, fitting a lathe to their centerpoints, form the aforementioned [hollow] circular section. Moreover, the other block should fit firmly so that it remains rigidly in place during the cutting.

3.11 ◉ Igitur restabit tabula quasi anulus circularis cuius latitudo duorum digitorum, longtitudo 14, altitudo septem, et sit hec altitudo optime circulata ad modum columpne. Remanent autem in latitudine huius anuli linee dividentes circulum eius secundum divisionem semicirculi tabule enee.

3.11 ◉ What will remain of the wooden block, then, is a circular ring that is 2 digits thick, 14 digits across, and 7 digits high, and it should be rounded along its height to form a cylinder. But the lines that divide the circumference of this ring according to the divisions [marked off] on the bronze plaque’s semicircular circumference are left [on the portion of the top surface that remains].⁑

3.12 ◉ A capitibus harum linearum producantur linee in superficie altitudinis exterioris perpendicularis super superficiem latitudinis, et poterit hoc modo fieri. Queratur regula bene acuta cuius capiti linee adhibeantur, et regula moveatur donec tangat superficiem altitudinis in qualibet parte acuminis. Signa eius capita, et fac lineam, quoniam illa erit perpendicularis quam queris. Et eadem sit operatio secundum quamlibet dividentem lineam.

3.12 ◉ From the endpoints of these lines draw lines along the length of the outer surface [of the ring, and draw them] perpendicular to the [top] surface [of the ring], and this can be done as follows. Find a sharply pointed ruler to whose endpoint the lines [on the ring’s top surface] are applied, move the ruler around until it touches the outer surface of the ring somewhere on the [ruler’s] point. Mark [the points where] the endpoints [of the measuring line touch the surface of the ring], and draw the line [through those points], for that will be the perpendicular you seek.⁑ The same procedure can be followed for each of the lines of division.

3.13 ◉ Aliter poterit hoc idem fieri. Ponatur pes circini super terminum linee dividentis, et fiat semicirculus secundum altitudinem anuli, qui dividatur per equa. Et protrahatur a puncto ad punctum linea, et ita in singulis. Pari modo, a terminis illarum dividentium protrahantur perpendiculares ex parte interioris altitudinis.

3.13 ◉ Another way of doing this is as follows. Place the point of a compass at the endpoint of [a given] line of division, draw on the outer surface of the ring a semicircle whose radius is the height of the cylinder, and divide it in half. Draw a line from [one] point to [the other] point, and so on for each [line of division]. In the same way, draw the perpendiculars to the endpoints of the lines of division on the inner surface [of the ring].⁑

3.14 ◉ Amplius, sumatur in altitudine interiori ex parte faciei non divisa altitudo duorum digitorum, et in perpendicularibus fiat signum. Et in signis illis fiat circulus equidistans faciei anuli hoc modo. Tabula aliqua plana fiat circularis equalis circulo minori tabule enee, et secetur ex ea pars aliqua usque ad centrum quasi triangulus ex duobus semidyametris et arcu circuli secundum quod libuerit ut possis tabulam cum manu imponere et locis assignatis aptare. Apta ergo locis illis ut sit equidistans faciei anuli, et fac circulum secundum ipsam.

3.14 ◉ To continue, on the inner surface [of the ring] mark off points on the [aforementioned] perpendiculars at a height of 2 digits above the [bottom] face [of the ring] that is not subdivided. Through these points draw a circle parallel to the [bottom] face [of the ring] in the following way. Form a flat circular template the same size as the smaller circle on the bronze plaque [i.e., 5 digits in radius], and, up to its centerpoint, cut out from it a triangular section of whatever size you please [whose sides are formed] from two radii and an arc on the circle, which allows you to insert the template manually [into the ring], and fit it up to the points that have been marked off. Place it in this way to those points so that it is parallel to the [bottom] face of the ring, and draw a circle [on the inner surface of the ring] according to [the circumference of] the inserted tablet.

3.15 ◉ Sumatur etiam in hunc circulum altitudo medietatis grani ordei, et fiant signa, et in punctis signatis fiat circulus per aptationem tabule. Et in hoc circulo postremo fiat circularis concavitas, et sit unius digiti eius profunditas et altitudo tamquam altitudo tabule enee. Et sit altitudo hec intra altitudinem duorum digitorum ut eadem sit postremi circuli et concavitatis superficies.

3.15 ◉ Then, at the height of this circle, mark off points at a level of half a grain of barley below it,⁑ and through the points [just] marked off draw a circle using the [aforementioned] template as a guide. Along this latter circle scoop out a circular cavity that is 1 digit deep and of the same thickness as the bronze plaque.⁑ And let this cavity lie within the [previously measured-off] altitude of 2 digits so that the latter circle [i.e., the upper one defining the top of the cavity] and the [top edge of the] cavity fall in the same plane.

3.16 ◉ Aptetur autem huic concavitati tabula enea, que quidem intrabit concavitatem usque ad circulum minorem, cum distantia minoris a maiori sit unius digiti, et concavitas similiter. Igitur circulo postremo et tabule enee communis erit superficies, et linee perpendiculares in altitudine anuli tangunt lineas divisionis tabule enee, et cadent perpendiculariter super tabulam eneam. Sit autem facies tabule enee divisa ex parte faciei anuli divise.

3.16 ◉ Now, insert the bronze plaque into this cavity, and it will fit into the cavity all the way to the smaller circle [drawn on its surface], since the [difference in] length between the [radius of] the smaller circle and [that of] the larger one is 1 digit, which is also the depth of the cavity.⁑ Hence, the latter circle and the bronze plaque will lie in a common plane, and the perpendicular lines drawn along the height of the ring [along its inner surface] intersect the lines of division [drawn] on the bronze plaque, and they will fall orthogonally to the [surface of] the bronze plaque. Make sure, however, that the surface of the bronze plaque that is subdivided faces the [upper] face of the ring that is [equivalently] subdivided.

3.17 ◉ Amplius, in exteriori altitudine anuli signetur punctus a longitudine duorum digitorum, et posito pede circini super punctum signatum, fiat circulus secundum quantitatem unius grani ordei. Et instrumento ferreo cuius similiter latitudo sit quantitas unius grani ordei perforetur foramine columpnari. Et baculus ligneus foramini aptetur, qui quidem, cum transierit ad interiorem concavitatem, tanget tabule enee superficiem. Pari modo, super singulas exterioris altitudinis perpendiculares similia et equalia efficiantur foramina in quantitate et altitudine.

3.17 ◉ Next, on the outer surface of the ring mark a point at a height of 2 digits [from the bottom], and, with the point of a compass placed on the point [just] marked, draw a circle with a diameter of a single grain of barley. With an iron drill whose diameter is likewise a single grain of barley, punch a cylindrical hole [through the ring’s wall]. Insert a wooden peg into the hole so that it penetrates to the inner hollow [of the ring] and will [therefore] touch the surface of the bronze plaque. In the same way, drill holes of the same kind and size, at the same height, through each of the perpendicular lines on the outer surface [of the ring].⁑

3.18 ◉ Deinde sumatur tabula lignea quadrata cuius latus est equale dyametro anuli, et protrahatur in eius superficie linea dividens quadratum per medium equidistans lateribus. Et ab una parte sumatur longitudo duorum digitorum, et fiat signum. Post sumatur longitudo semidyametri minoris circuli tabule enee, et posito pede circini, fiat circulus transiens per signum, qui circulus erit equalis minori circulo tabule enee et concavitati anuli.

3.18 ◉ Now, take a square wooden block [each] of whose sides is equal to the diameter of the ring [i.e., 14 digits], and through the midpoint of its surface draw a line parallel to its sides and bisecting it. From one end [of that line] measure off a length of 2 digits and mark it off. Then [from that point] mark off a distance equal to the radius of the smaller circle on the bronze plaque [i.e., 5 digits], and placing the point of the compass [at that distance], draw a circle passing through the point [just marked off], that circle being the same size as both the smaller circle on the bronze plaque and the hollow of the [cylindrical] ring.

3.19 ◉ Deinde supra centrum huius circuli sumatur longitudo duorum digitorum, et infra centrum similiter, et signentur puncta. Ab utroque in utramque partem protrahatur linea equidistans lateribus quadrati, et in utraque harum linearum signetur longitudo duorum digitorum ex utraque parte puncti signati. Et a punctis unius linee signatis protrahantur linee equidistantes ad puncta alterius linee signata, et fiet quadratum quatuor digitorum. Fodiatur hoc quadratum secundum altitudinem unius digiti, et concavationis latera efficiantur plana et ortogonalia, et fundus similiter planus.

3.19 ◉ Above the centerpoint of this circle measure off a distance of 2 digits, and do the same below that centerpoint, and mark off the points. From each point draw a line to both sides [of the block] that is parallel to the sides of the square, and on each of these lines mark off a distance of 2 digits on each side from each of the points [just] marked. Then, from the points marked on one of the lines draw parallel lines to the points marked on the other line, and [thus] a square 4 digits on a side will be produced.⁑ Hollow out this square to a depth of 1 digit, and smooth the sides of the cavity to square them off while likewise making its bottom flat.⁑

3.20 ◉ Deinde aptetur hec tabula faciei anuli ita ut circulus minor applicetur foramini anuli, et extremitas eius extremitati. Et firmetur hec applicatio cum clavis ut immota maneat tabula. Nota quod in omnibus predictis duorum digitorum mensura certa debet esse et determinata, et ob hoc in linea aliqua fiat immutabili ne ex mutatione mensure error accidat.

3.20 ◉ Next, attach this block to the [bottom] face of the ring so that the smaller circle [just drawn on it] coincides with the hollow of the ring and so that its edges meet the [outer] circumference [of the ring]. Make this attachment snug with nails so that the block remains perfectly fixed. Bear in mind that the measure of 2 digits applied in all the previous situations must be accurate and definite, so make sure to maintain that measure for each [relevant] edge so that no error arises from a change in measurements.⁑

3.21 ◉ Amplius, fiat columpna ferrea concava plana aliquantulum spissa ut nec statim intret nec immutari queat, et sit quantitas dyametri circuli eius unius grani ordei. Et ponatur columpna in foraminibus, que quidem cum ad interiora anuli pervenerit, continget lineas in tabula enea factas. Et erit operis eius complementum si linea tabule enee sit contingens circulo columpne in puncto linee altitudinis anuli perpendicularis super tabulam eneam et transeuntis per centrum circuli columpne.

3.21 ◉ Next, make a hollow iron tube that is even [in size throughout] and [whose walls are] fairly thick so that it does not readily penetrate [the small holes drilled into the cylindrical ring’s wall] and so that it cannot be squeezed out of shape, and let the diameter of its [outer] circumference be 1 grain of barley. Insert the tube into the holes so that when it reaches the inside of the [cylindrical] ring, it will touch the lines drawn on the bronze plaque. And this procedure will be carried out perfectly if the line on the bronze plaque touches the circumference of the tube at the point where the line [that is drawn] perpendicular to the bronze plaque along the height of the ring [on its inner surface] passes through the center of the circular section of the tube.

3.22 ◉ Fiat autem in capite columpne anulus aut repagulum quod non permittat columpnam intrare nisi ad locum determinatum. Fit autem huiusmodi longitudinis columpna ut procedens supra tabulam eneam attingat lineam equidistantem dyametro tabule intra quas facta est sectio. Et est linea illa equidistans basi trianguli tabule enee.

3.22 ◉ Attach a ring or a bolt to the end of the tube so that the tube is allowed to penetrate [into the hollow of the ring] only to a determinate point. But make the tube long enough that, as it passes over the bronze plaque, it reaches the line that is parallel to the diameter of the plaque, these two lines defining the [lower] section [of the plaque]. This line is [thus] parallel to the base of the triangle [inscribed] on the bronze plaque.⁑

3.23 ◉ Amplius, fabricentur septem specula ferrea quorum unum planum; duo sperica, unum concavum intra politum, aliud extra; duo columpnaria, unum concavum, aliud in superficie politum; duo piramidalia, unum politum in facie, aliud in concavitate. Speculum autem planum fit circulare, et sit eius dyameter longitudinis trium digitorum.

3.23 ◉ Then, make seven iron mirrors: one of them plane; two spherical, of which one is concave and polished on its inner surface, the other [convex and polished] on its outer surface; two cylindrical, of which one is concave, the other polished on its [outer, convex] surface; and two conical, of which one is polished on its [outer, convex] surface, the other on its [inner] concave surface. Let the plane mirror be circular, and let it be 3 digits in diameter.

3.24 ◉ Speculum columpnare politum in superficie sit lucidum et perfecte politum, et sit dyameter circuli longitudinis sex digitorum, qui circulus est basis eius. Longitudo autem columpne sit trium digitorum. In base columpne sumatur corda longitudinis trium digitorum. Similiter in base eiusdem columpne opposita sumatur equalis huic corda et ei opposita ut linee a capitibus unius corde ad capita alterius producte sunt recte. Et secetur hec columpna secundum harum linearum processum ut restet nobis pars columpne cuius capita portiones cordarum earum, aut altitudo axis portionis remanentis minus quam dimidii digiti. Axem autem dico lineam a medio puncto arcus ad medium corde punctum productum.

3.24 ◉ Let the cylindrical mirror that is polished on its [outer] surface be clear and highly polished, and let the diameter of the circle at its base be 6 digits. Furthermore, let the cylinder be 3 digits high. At the base of the cylinder measure off a chord 3 digits in length. Likewise, at the opposite base of this same cylinder measure off a chord of equal length directly facing the other chord so that the lines drawn from the endpoints of one chord to the [corresponding] endpoints of the other are perpendicular [to both bases]. Cut the cylinder along these lines so that what we have left is a portion of the cylinder whose bases consist of the sections of those chords, or [to put it another way, so that] the height of the remaining portion’s axis is less than half a digit. By »axis« I mean the line [extending] from the midpoint of the arc [at the base] to the midpoint produced on the chord.

3.25 ◉ Columpne concave longitudo sit trium digitorum, et dyameter basis eius sex digitorum, et in ea sumatur corda trium digitorum, et fiat sectio sicut in prima. Et erit altitudo axis partis remanentis minus quam dimidii digiti. Sit autem in hiis omnibus politura exquisita et equalitas omnimoda.

3.25 ◉ Let the height of the concave cylinder be 3 digits, let the diameter of its base be 6 digits, measure off a chord of 3 digits on that base, and cut off a section as [you did] in the first [cylinder]. So the height of the remaining portion’s axis will be less than half a digit. Make sure that all these mirrors are exquisitely polished and even throughout.

3.26 ◉ Speculum piramidale queratur dyameter basis cuius quantitas sit sex digitorum, et corda trium, et longitudo piramidalis quatuor digitorum et dimidii. Et fiat sectio secundum lineas rectas, et axis portionis altitudo minor quam dimidii digiti, et hoc in unoquoque piramidali intellige.

3.26 ◉ Find a conical mirror whose base circle is 6 digits in diameter, the chord at its base being 3 digits, and let it be 45 digits high along the longitude. Cut out the [appropriate] portion along the straight lines [connecting the endpoints of the chord to the vertex of the cone] so that the axis of the [base] of the remaining portion is less than half a digit long, and see that this [is done] in each cone [i.e., concave and convex].

3.27 ◉ Speculum spericum sit portio sperica cuius dyameter sit sex digitorum, et dyameter basis huius speculi trium digitorum, et erit axis minus quam dimidii digiti. Item operare in speculo sperico concavo.

3.27 ◉ Let the [convex] spherical mirror be a portion of a sphere with a diameter of 6 digits, and let the base [of the cut-off portion] of the mirror be 3 digits in diameter, so the axis [of this portion] will be less than half a digit. The same should be done in the case of the concave spherical mirror.⁑

3.28 ◉ Deinde facias septem regulas ligneas planas quarum latera equidistantia et ortogonalia super capita equidistantia in fine possibilitatis, et sit longitudo regularum sex digitorum, latitudo quatuor. Postea quadrato concavo adaptetur alia regularum ita ut ortogonaliter cadat super inferiorem concavi quadrati superficiem, et vide ut facile intret quadratum ne comprimens immutetur.

3.28 ◉ Then you should produce seven flat wooden panels whose sides are parallel and orthogonal [at the corners] so that the edges are as parallel as possible, and the panels should be 6 digits long and 4 digits wide. Next, fit [one or] another of the panels into the hollow square [in the block attached at the base of the cylindrical ring] so that it stands perfectly erect on the bottom of the hollow square, and see to it that it fits easily into the [hollow] square without being squeezed out of shape.

3.29 ◉ Cadat igitur super faciem lateris regule acumen tabule enee, et ubi continuabitur ei fiat signum, et a puncto assignato producatur in extremitates regule linea equidistans lateribus regule ut sit linea illa linea longitudinis regule. Deinceps in longiori parte illius linee circa punctum sumptum sumatur altitudo medii grani ordei, et fiat punctum. Dico quod ille est punctus medius regule, qui etiam centro foraminum opponitur recte.

3.29 ◉ Accordingly, let the point on the bronze plaque touch the face [of the panel], mark off the point where it will reach to it, and from the point [just] marked to the [upper and lower] edges of the panel draw a line parallel to the sides of the panel so that this line forms a line of longitude on the panel.⁑ Then, on the longer segment of that line, from the point just marked off, measure a distance of half a grain of barley, and mark the point. I say that this point lies at the midpoint of the panel and that it also lies directly in line with [each] centerpoint of the holes [in the wall of the ring].

3.30 ◉ Probatio: quoniam centra foraminum elongantur super superficiem tabule enee in medii grani quantitate, et distant a superficie anuli per duos digitos, igitur punctus ille distat ab eadem per duos digitos. Et regula in quadrato concavo, per digitum unum. Quare ab extremitatibus regule ad punctum sunt tres digiti, quare punctus ille est medius. Super hunc medium punctum producatur in utramque partem linea secundum latitudinem equidistans extremitatibus. Et medietates linee longitudinis super quam hec est perpendicularis dividantur per equalia per lineas latitudinis perpendiculares extremitatibus equidistantes. Et ita divisa erit regula in quatuor equales partes. Similis fiat in aliis regulis operatio.

3.30 ◉ [Here is] the proof. Since the centerpoints of the holes lie half a grain of barley above the surface of the bronze plaque, and since those centerpoints lie 2 digits above the [bottom] surface of the ring, then that point lies 2 digits above that same surface. But the panel is sunk to a depth of 1 digit in the hollow square [in the block attached to the bottom of the ring]. Since the distance between the [top and bottom] edges of the panel and the point is 3 digits, then that point constitutes the midpoint. At this midpoint draw a line from side to side across the panel and parallel to the [top and bottom] edges. Then bisect the [two remaining sections of the] line of longitude to which this line is perpendicular with orthogonal lines that are parallel to the [top and bottom] edges. Hence the panel will be subdivided into four equal portions. Do the same thing with the [six] other panels.⁑

3.31 ◉ Hiis completis, adaptetur speculum planum uni regularum. Et est ut sit regula cavata secundum altitudinem speculi ita ut superficies speculi sit in eadem superficie cum superficie regule, et ita ut medium superficiei speculi punctum directe supponatur medio superficiei regule puncto, et ita ut linea dividens superficiem regule in duo equalia dividat etiam superficiem speculi per equalia, et ut continuentur partes speculi cum linea dividente. Et observetur in possibilitatis fine.

3.31 ◉ When all this is finished, fit the plane mirror into one of the panels. In order to do this, scoop out a hollow [in the panel] as deep as the thickness of the mirror so that the surface of the mirror lies in the same plane as the surface of the panel, the midpoint of the mirror’s surface lies directly upon the midpoint of the panel’s surface, the line bisecting the surface of the panel also bisects the surface of the mirror, and the [top and bottom] points of the mirror coincide with that dividing-line. Be as careful as possible to follow this procedure accurately.⁑

3.32 ◉ Deinde speculum columpnare politum in facie applicetur alicui regule ita ut medius eius punctus cadat super medium regule punctum, et ita ut linea in longitudine speculi sumpta dividens ipsum per equalia continuetur cum partibus linee longitudinis superficiei regule eque dividenti, et ut media longitudinis speculi linea sit in superficie regule. Et hoc sic fieri poterit utriusque basis speculi arcus per equalia dividantur et a puncto divisionis signato ad oppositum signatum linea producatur, et linee medie longitudinis regule aptetur et continuetur. [3.33] Speculum columpnare concavum aptetur regule ut media longitudinis eius linea secundum equalem arcuum basium divisionem sumpta equidistans sit medie linee longitudinis regule, et etiam ut utriusque arcus corda cum lineis longitudinis extremis sint in superficie regule.

3.32 ◉ Then fit the cylindrical mirror that is polished on its [outer convex] surface into one of the panels so that its midpoint coincides with the midpoint of the panel, the line taken along the length of the mirror that bisects it coincides with the [remaining] segments of the line along the length of the panel that bisects its surface, and the midline along the length of the mirror lies in [the plane] of the panel’s surface. And this can be accomplished if the arcs at both bases of the mirror are bisected, and a line is drawn from one point of bisection to the other. Match this line to the midline along the length of the panel so as to coincide with it.⁑

3.34 ◉ Piramidale speculum extra politum applicetur regule ut acumen eius sit in termino medie longitudinis regule linee, et linea dividens portionem piramidis per equa—que scilicet a cono ad medium arcus basis punctum producitur—sit in superficie continuata cum parte restante medie linee longitudinis regule.

3.34 ◉ Fit the conical mirror that is polished on its outer [convex] surface into a panel so that its vertex lies at the end of the midline along the length of the panel, and the line bisecting the section of the cone [from which the mirror is formed]—i.e., the line that extends from the vertex to the midpoint of the arc at its base—lies in [the plane of] the panel’s surface and coincides with the remaining segment of the midline along the length of the panel.⁑

3.35 ◉ Speculum piramidale concavum applicetur regule ita ut acumen eius sit in directo medie linee longitudinis regule; corda vero arcus basis sit in superficie, scilicet regule. Linea a cono ad medium arcus basis punctum ducta sit equidistans medio linee longitudinis regule. Cum autem longitudo piramidum sit quatuor digitorum et dimidius, restabunt ex longitudine regule digitus et medius.

3.35 ◉ Fit the concave conical mirror into a panel so that its vertex lies right at the midline along the length of the panel, and let the chord of the arc at its base lie in [the plane of] the panel’s surface. The line drawn from the vertex to the midpoint of the arc at the base [of the mirror] should be parallel to the midline along the length of the panel. Furthermore, since the cone is 4.5 digits long, 1.5 digits will be left along the longitudinal midline of the panel [on its bottom half].⁑

3.36 ◉ Adaptandum regule speculum spericum extra politum, fiat in regula circulus secundum quantitatem trium digitorum. Eius sit centrum medium regule punctum. Et cava et apta speculum ut medium superficiei eius punctum sit in superficie regule et in medio puncti medie linee longitudinis regule, quod quidem sciri poterit per applicationem alterius regule acute equalis huic in longitudine et divise per equalitatem et applicate medie linee longitudinis regule ita ut medius huius regule acute punctus tangat medium speculi sperici punctum.

3.36 ◉ In order to fit the spherical mirror that is polished on its outside [convex] surface into a panel, draw a circle 3 digits in diameter on the panel. Let its centerpoint be the centerpoint of the panel. Then scoop out [a hollow according to that circle], and fit the mirror into it so that the midpoint on its surface lies in the plane of the panel[‘s surface] and coincides with the midpoint of the midline along the panel’s length. One can make sure that this is done correctly by applying another sharp-edged ruler of the same length [as the panel] and identically subdivided upon the midline along the length of the panel so that the midpoint of this sharp-edged ruler touches the midpoint of the spherical mirror.⁑

3.37 ◉ Spericum concavum, facto in regula circulo secundum quantitatem trium digitorum cuius centrum medius regule punctus, cavato circulo, imponatur ita ut circulus basis speculi sit in superficie regule, et punctum medium concavitatis speculi directe oppositum medio regule puncto. Et dyameter basis speculi continuetur medie linee regule, quod ita perpendetur. In regula acuta punctus signetur, et ab illo puncto longitudo semidyametri basis speculi notetur ex utraque parte. Et ita hec acuta regula medie linee regule applicetur ut punctum signatum in ea directe opponatur medio concavitatis speculi puncto et dyameter in ea factus similis sit cum basis dyametro.

3.37 ◉ Having drawn a circle 3 digits in diameter on a panel, the centerpoint of that circle being the panel’s centerpoint, and having scooped out this circular section, fit the concave spherical mirror into [the resulting cavity] so that the circle at the base of the mirror lies in the plane of the panel[‘s surface], and so that the midpoint of the concave surface of the mirror is directly opposite the panel’s midpoint. To ensure that the diameter of the mirror’s base coincides with the midline [along the length] of the plank, do the following. Mark a point on [the edge] of the sharp-edged ruler, and on each side of that point measure off a distance equal to the radius of the [circle forming the] mirror’s base. Accordingly, apply this sharp-edged ruler to the midline of the panel so that the point marked on it lies directly opposite the midpoint of the concave surface of the mirror and so that the diameter marked off on it is the same as the diameter of the [mirror’s] base.⁑

3.38 ◉ Hiis peractis, in semidyametro tabule enee triangulum per equalia dividente signetur ab acumine eius longitudo equalis axi huius speculi concavi, et fiat punctum. Axis autem sic dinoscitur. Regula acuta superficiei applicetur ut acuitas directe sit super mediam longitudinis lineam, puncto eius super medium concavi punctum directe statuto. Deinde acus recta et subtilis secundum illud regule acute punctum perpendiculariter cadat in speculum. Descendet quidem super medium concavi punctum. Signetur autem in acu punctum quod post eius descensum tangit acuitas regule sive punctum signatum, et sit modicum declinata regula ut certius possit fieri in acu signum. Postea secundum longitudinem acus a puncto signato in ea metire ab acumine tabule enee in linea triangulum dividente, et fac punctum.

3.38 ◉ Once all this is accomplished, measure off from the endpoint of the radius that bisects the triangle on the bronze plaque at its sharpened point a distance equal to the axis of this concave mirror [i.e., ca. 4 digits], and mark this point. The axis, moreover, can be determined as follows. Place a sharp-pointed ruler on the surface [of the panel bearing the mirror] so that its sharp point lies directly on the midline along the panel’s length and directly above the midpoint on the concave surface [of the mirror]. Then, from that point on the ruler, lower a thin, sharp needle perpendicular to the mirror. It will, of course, reach to the centerpoint of the concave [surface of that mirror]. On the needle mark the point where, after it has reached [the mirror’s surface], it touches the point of the ruler or a point [previously] marked [on the ruler], and slant the ruler slightly so that the mark can be made accurately on the needle. Then, from the vertex of the triangle on the bronze plaque and along the line that bisects that triangle, measure off the distance from the needle’s point to the point [just] marked on it, and mark that point [on the triangle’s line of bisection].⁑

3.39 ◉ Deinceps hanc regulam facias intrare quadratum concavum ita ut acumen tabule enee descendat supra speculum; et adhibeatur regula acuta ut signetur punctum in linea dividente triangulum, quem tetigerit ex ea regula acuta, cum acumen trianguli descenderit usque ad superficiem speculi concavi. Signa igitur punctum.

3.39 ◉ Next, you should insert the panel [containing the concave spherical mirror] into the hollow square [at the bottom of the cylindrical ring] so that the point of the bronze plaque touches the mirror; [against the face of the panel] apply the sharply pointed ruler [orthogonally] to the line that bisects the triangle [on the bronze plaque] so that a point may be marked on that line, which is touched by that sharply pointed ruler, since the vertex of the triangle reaches all the way to the surface of the concave mirror. Accordingly, mark that point.

3.40 ◉ Erit autem hoc secundum punctum minus distans ab acumine quam primum, superficies enim tabule enee distat a superficie anuli sive tabule in qua est quadratum concavum per duos digitos minus medietate grani ordei. Punctus autem medius regule directe est oppositus medio speculi sperici concavi puncto, qui quidem distat ab eadem superficie tabule per duos digitos. Cum ergo acumen tabule enee ortogonaliter descendat, non cadet super medium concavi, qui est terminus axis, sed in puncto altiori, quare propositum.

3.40 ◉ This second point, however, will lie a smaller distance from the vertex [of the triangle] than the first point, because the surface of the bronze plaque lies 2 digits minus half a grain of barley from the [bottom] surface of the [cylindrical] ring, or from the [top] surface of the block containing the square cavity. On the other hand, the midpoint of the panel [containing the mirror] is directly opposite the midpoint of the concave spherical mirror, that point of course lying 2 digits above the same surface [i.e., the top surface] of the block [at the base of the ring]. Consequently, since the vertex of the bronze plaque extends orthogonally [to the panel containing the mirror], it will not reach the midpoint of the concave [surface], which is the end of the [mirror’s] axis, but to a point that is higher [with respect to the axial height of the mirror], so [we have established] what we set out [to show].⁑

3.41 ◉ Signetur vero in speculo concavo punctum in quod accidit acumen tabule enee, et extracto in puncto illo foramine ortogonaliter descendente et modico ad hanc quidem mensuram ut in eo descendat acutum donec acuitas regule adhibite contingat punctum linee dividentis triangulum primo signatum. Quod cum fuerit, erit quidem acumen tabule enee in eadem superficie cum termino axis speculi, que superficies sit equidistans superficiei regule. Et erit linea a termino axis ad acumen ducta perpendicularis super superficiem tabule enee. Axis autem speculi in eadem superficie cum centris foraminum, quoniam distantia eorum a superficie anuli duorum est digitorum, et medius terminus axis similiter.

3.41 ◉ Mark the spot on the concave mirror where the vertex of the bronze plaque reaches, and, having bored a hole at that point, move the vertex orthogonally [into the hole] just far enough that the sharp edge of the ruler that is applied [to the face of the panel] touches the point first marked on the line bisecting the triangle. That being so, the point of the bronze plaque will lie in the same plane as the endpoint of the mirror’s axis [on the convex surface of the mirror], and that plane is parallel to the surface of the panel [containing the mirror]. And the line drawn from the endpoint of the axis to the point [of the triangle] will be perpendicular to the surface of the bronze plaque.⁑ Moreover, the axis of the mirror lies in the same plane as the centers of the holes, because they lie 2 digits from the [bottom] surface of the [cylindrical] ring, and so does the endpoint of the axis [at the] midpoint [of the mirror].

3.42 ◉ Hiis cum diligentia preparatis, poterit videri quod promisimus. Immitatur anulo regula super quam est speculum planum donec acumen tabule enee cadat super speculum, et infigatur regula quadrato concavo, et in eo subtus regulam aliquid opponatur quod ei firmitatem conserat ne vacillet. Deinde opponatur pargamenum foraminibus, et cum digito fiat impressio ut obturentur et impressionem percipere possis. Et signum foraminis fiat in pargameno cum incausto, vel aliquo alio. Unum autem foraminum relinquatur apertum declinatum non super regulam mediam, et adhibeatur radio solis foramen apertum. Certioratum autem erit huius rei comprehensio si adhibeatur radio solis per foramen domus intranti.

3.42 ◉ When all of this is carefully done, what we predicted can be empirically verified. Insert the panel with the plane mirror upon it into the ring until the point of the bronze tablet touches the mirror, fix the panel into the hollow square [at the bottom], and apply something [adhesive] to the bottom of the panel in order to keep it firmly nested so it does not wobble. Then press [a piece] of parchment up to the holes [in the wall of the cylindrical ring], and make an impression [of each hole] with the finger in order to fill in the holes so that you can make out the impression. Then mark the impression of [each] hole on the parchment with red ink or something else [of that kind]. Leave one of the holes open, however, but not the one directly facing the middle of the tablet, and point the open hole at an [incoming] beam of sunlight. The result of this operation will be clearer if the apparatus is held up to a ray of sunlight entering a room through a window.

3.43 ◉ Cum igitur radius foramen intrans ad speculum pervenerit, videbis ipsum reflecti ad foramen illud respiciens super lineam tabule enee equalem angulum continentem cum linea triangulum per equa dividente et angulo quam tenet linea a foramine discooperto cum illo tabule semidyametro. Si vero foramen in quod fit reflexio discoopertum opponas radio priore cooperto, videbis reflecti radium in coopertum.

3.43 ◉ Accordingly, when the beam passing into the hole reaches the mirror, you will see it reflected to the corresponding hole [on the wall of the cylindrical ring] along the line on the bronze plaque that forms with the line bisecting the triangle an angle equal to the angle formed by the line from the open hole with the same radius [that bisects the triangle]. On the other hand, if you uncover the hole to which the ray was previously reflected and shine the light through it, you will see the ray reflected to the [previously] open hole.⁑

3.44 ◉ Si vero foramini imponatur columpna ferrea concava, quam ad quantitatem foraminum fieri precipimus (ut firmius stet modicum cere circa eam apponatur), descendet lux per columpne concavitatem sicut descendit per foramen. Et reflectetur in foramen respiciens, et super lineas tabule enee erit descensus et reflexio pari modo, ut prius. Et si ad secundum foramen columpnam transtulerimus, in primum lucem reflexam videbimus. Erit autem debilior lux per columpnam descendens quam sine columpna per foramen. Erit autem videre eundem reflectendi modum in debiliori luce.

3.44 ◉ Furthermore, if into the hole you insert the hollow iron tube that we made earlier according to the size of the hole (apply a bit of wax around it to make it nest snugly), the light will pass through the hollow of the tube just as it passes through the hole. And it will be reflected to the corresponding hole, and the incidence and reflection will follow the [corresponding] lines on the bronze plaque as before. Also, if we transfer the tube to the second hole, we will see the light reflected to the first one. However, the light passing through the tube will be weaker than it was when it passed through the hole without the tube in it. Still, the same way of reflecting will be observed in the case of the weaker light [as in the case of the more intense light].

3.45 ◉ Obturetur foramen cum cera ut modicum circa centrum eius restet vacuum, et videbitur lucis reflexio in foramine, sive circa centrum. Pari modo, si concavitatem columpne cum cera obturaveris ut remaneat quasi terminus solius axis, descendet lux super axem columpne et reflectetur ad centrum foraminis similis. Eodem modo, alterata columpna imposita, cum descenderit lux super axem unius foraminis, reflectetur super axem similis, centrum enim foraminis directe axi opponitur. Et cum lucis reflexio cadat in centrum nec moveatur nisi per lineam rectam, oportet ut procedat secundum axem. [3.46] Obturatis autem foraminibus singulis preter medium quod directe super tabulam eneam incidit, fiat baculus columpnaris ad quantitatem foraminis, et extremitas eius acuatur ut remaneat solus terminus axis eius. Et descendat per foramen, et signa punctum speculi in quod ceciderit. Deinde descendat radius solis per foramen illud. Cadet quidem super punctum signatum, et circa ipsum efficiet circulum. [3.47] Signetur autem in fine huius lucis circularis punctum, et secundum quantitatem linee interiacentis puncta signata fiat circulus. Erit quidem circulus iste maior circulo foraminis, quoniam processus lucis per foramen ingredientis est in modum piramidis. Verum in nullo foraminum videbitur lucis reflexio, unde palam quod lux descendens per axem reflectitur super eundem. Verumptamen apparebit lux circularis circa basem interioris foraminis maioris quidem capacitatis radio, maioris etiam lucis interioris circulo.

3.45 ◉ Block the tube with wax in such a way that a tiny hole remains at its center, and the light will be seen to reflect to [the corresponding] hole, or, rather, to its centerpoint. Likewise, if you fill the hollow of the tube with wax in such a way as to leave [a tiny channel] virtually the size of the [tube’s] axis, the light will pass along the axis of the tube and will be reflected to the centerpoint of the corresponding hole. By the same token, if the tube is inserted into the other hole, then when the light passes along the axis of the one hole, it will be reflected along the axis of the corresponding hole, for the center of the hole lies directly in line with the axis.⁑ And since the reflected light falls at the center and only radiates along a straight line, it follows that it must proceed along the axial line.

3.48 ◉ Et palam hanc lucem apparentem esse per reflexionem, verum non per reflexionem lucis super axem descendentis, quod ex hoc poterit patere. Obturata utraque foraminis base ut quasi sola remaneat axis via, et radio solis per viam axis descendente, non apparebit lux illa circularis circa inferiorem basem foraminis, quare non procedebat ex reflexa luce axis.

3.48 ◉ So it is obvious that the way this light appears is due to reflection, but not to the reflection of light passing along the axial line, a fact that can be demonstrated as follows. If both bases of the hole are blocked so that all that is left is a narrow opening along the axis, and if a ray of sunlight shines along the axial line, that light will not appear to form a circle around the interior base of the hole because it was not formed by light reflected along the axis.⁑

3.49 ◉ Amplius, supra quamdam regulam supposuimus ut ortogonaliter caderet in quadratum concavum. Si aliquantulum ex eis auferatur ut regula declinetur ita ut extremitas a quadrato remotior sit dimissior radio descendente super foramen medium, non cadet perpendiculariter supra speculum, et apparebit lux reflexa a foramine medio remota. Et quanto maior erit declinatio maior erit lucis reflexe a foramine remotio. Si vero ad rectitudinem regula reducatur, lux reflexa circa inferiorem foraminis basem, ut prius, videbitur.

3.49 ◉ Earlier, moreover, we set up [this] particular panel so that it stood orthogonally on the [bottom of the] hollow square. If the conditions [under which we set it up] are slightly changed so that the panel may be slanted in such a way that the end farther from the square is inclined down toward the ray passing through the middle hole, the ray will not fall orthogonally upon the mirror, so the light will appear reflected away from the middle hole. And the greater the slant, the farther away from the hole the reflected light will be. But if the panel is restored to an upright position, the reflected light will appear around the inner base of the hole as it did before.

3.50 ◉ Palam igitur quod luce super speculum perpendiculariter cadente, regreditur ad foramen per quod ingressa est. Cum vero lux axis declinata ceciderit, reflectitur non ad foramen, sed apparebit super lineam superficiei anuli perpendicularem super tabulam eneam et descendentem per centrum foraminis medii. [3.51] Quecumque autem dicta sunt in duobus foraminibus primis declinatis intellige in singulis. Et quod dictum est in speculo plano, luce per foramen declinatum seu medium descendente, regula recta seu declinata, in aliis speculis intellige.

3.50 ◉ It is therefore evident that, when light falls orthogonally upon the mirror, it returns back to the hole through which it entered. When, however, the light falls along a line slanted to the axis, it does not reflect to the hole but will appear upon the line on the [inner] surface of the ring that is perpendicular to the bronze plaque and that passes through the centerpoint of the middle hole.⁑

3.52 ◉ Si autem regula in qua fuerit speculum columpnare extra politum declinetur in quadrato ita ut non ortogonaliter cadat super quadratum sed declinetur super partem dextram vel sinistram, apparebit tamen lux reflecti super foramen simile eius descensui, et medium lucis super medium foraminis, sicut visum est regula non declinata.

3.52 ◉ If, however, the panel containing the cylindrical mirror that is polished on its outer [convex] surface is held at a slant in the [hollow] square so that it does not stand upright in the square but is inclined to the right or left, the light will still appear to be reflected upon the hole that corresponds to the hole through which it enters, and the light [passing through] the middle [hole will be seen reflected back] to the middle hole, just as was seen when the panel was not slanted.⁑

3.53 ◉ Regulam in quam situm est columpnare concavum impones, et descendat acumen tabule enee donec tangat superficiem speculi, et declinabis hoc speculum secundum latus suum sicut declinasti extra politum.

3.53 ◉ You should [next] insert the panel with the concave cylindrical mirror in it, and let the point on the bronze plaque approach it until it touches the surface of the mirror. You will slant this mirror to the side just as you slanted the one polished on its outer [convex] surface.

3.54 ◉ Idem in speculis piramidalibus concavis operaberis.

3.54 ◉ You will follow the same procedure for concave conical mirrors.

3.55 ◉ Spericum concavum aptetur donec descendat acumen tabule enee in foramen speculi factum secundum acuminis descensum. [3.56] Spericum extra politum sic imponatur ut acumen tabule enee sit in superficie regule et in eadem superficie cum medio speculi puncto, quod sic fieri poterit. Adhibeatur regula acuta regule et puncto speculi medio, et descendat acumen tabule enee quousque sit in directo acuitatis regule. Et tunc cogatur sistere.

3.55 ◉ Set up the concave spherical mirror so that the point of the bronze plaque enters the hole in the mirror that was made according to the insertion of that point.⁑

3.57 ◉ In speculis columpnaribus videbis reflexionem hoc modo. Aptetur speculum, sicut dictum est, et per foramen medium descendat baculus columpnaris, sicut factum est in speculis planis. Cadet quidem baculus super mediam longitudinis speculi lineam, et erit eius terminus in superficie regule. Super mediam lineam signetur punctum in quod cadit, et ab hoc puncto in superficie regule sumatur longitudo semidyametri circuli facti in regula ad discernendum circularem lucis casum. Et ex alia parte puncti sumatur longitudo eadem, et habebitur linea equalis dyametro predicti circuli. Videbitur autem lux cadens extendi supra predictam lineam tantum, et reflectitur ad foramen medium. Et circa eius basem inferiorem videbitur lux circularis maior circulo inferiori, sicut in speculis planis visum est.

3.57 ◉ In the case of cylindrical mirrors, you will observe [what happens in] reflection as follows. Set up the mirror as described before, and pass the cylindrical pointer through the middle hole, just as was done in the case of plane mirrors. Of course the pointer will fall upon the midline of the mirror along its length, and its point will lie in the plane of the panel. On this midline mark the point where it falls, and from this point measure off on the surface of [the mirror inserted in] the panel the distance of the radius of the circle drawn [earlier] on the panel for the purpose of observing how the impinging light forms a circle [on the mirror’s surface]. Measure off the same distance on the other side of the point, and a line equal to the diameter of the aforementioned circle will be determined. Moreover, the impinging light will be observed to extend only upon the aforementioned line, and it is reflected to the middle hole. And the circle of light at its inner base will appear larger than the circle of light [on the mirror] inside [the ring], just as was observed in the case of plane mirrors.

3.58 ◉ Idem in speculis piramidalibus videre poteris.

3.58 ◉ You can observe the same thing in the case of conical mirrors.

3.59 ◉ Pari modo in speculis spericis, luce per foramen medium descendente, fiat circulus in superficie regule ad quantitatem circuli iam dicti. Et videbitur lux extendi super hunc circulum et reflecti ad foramen medium modo iam dicto. Et apparebit in hiis omnibus rectis reflexionibus linea perpendicularis in interiori superficie anuli secare lucem circularem reflexam et dividere circulum eius per medium.

3.59 ◉ Likewise, in the case of spherical mirrors, when light passes [to them] through the middle hole, draw a circle on the surface of the [the mirror inserted in the] panel the same size as the aforementioned circle. The light will be observed to fill this circle and then reflect to the middle hole in the way already described. In all these straight-on reflections the perpendicular line [of longitude] drawn along the inner surface of the [cylindrical] ring will appear to transect the circle of reflected light and divide it in half.

3.60 ◉ Quod dictum est de luce naturali videri poterit in luce accidentali. Domus unici foraminis opponatur parieti in quam descendit solis radius, et applicetur instrumentum foramini cum intraverit lux accidentalis per foramen non medium. Videbitur reflecti per eius oppositum, et si aptetur instrumentum ut intret per duo foramina, reflectetur per duo similia.

3.60 ◉ What has been described for natural [or primary] light can be observed for accidental [or secondary] light. In a room with one window place a screen facing [the window] so that a beam of sunlight shines upon it, and set up the apparatus with respect to the window so that the accidental light [radiating from the screen] passes through one of the holes, but not the middle one. The light will be seen to reflect to the [corresponding] hole opposite it, and if the instrument is set up so that the light enters through two holes, it will be reflected to the two corresponding holes.

3.61 ◉ Verum ut possis perpendere lucem cum intraverit directe et ad ipsam transierit, appone superius pargamenum album, et inclina instrumentum donec videas lucem cadentem super pargamenum. In speculis etenim non plene comprehenditur lucis accidentalis casus propter debilitatem eius. Idem autem in hac luce patebit quod in naturali patuit, non enim est diversitas in earum natura nisi quod una fortis et alia debilis.

3.61 ◉ Moreover, in order to be able to determine that the light enters directly and passes straight through, place the aforementioned piece of parchment [inside the cylindrical ring], and turn the apparatus until you see the light shining on the parchment. In fact, the shining of accidental light on mirrors is not clearly perceived because of its faintness. However, in this kind of light the same thing will be evident as was evident in natural [i.e., primary] light, for there is no difference in their nature except that the one is bright, the other faint.

3.62 ◉ Palam ergo quod luces propter diversas lineas ad specula accidentes per diversas reflectuntur lineas. Et si eadem parte ad speculum venerit, in eandem gradiuntur partem, et declinatio linearum reflexionis equalis declinationi erit linearum accessus. Et planum quod linee lucis reflexe et advenientis sunt in eadem superficie ortogonaliter super superficiem politi et contingenti punctum a quo fit reflexio. Et si super perpendicularem venerit, reflectetur super perpendicularem, et in quemcumque punctum cadit reflectitur in superficie perpendiculari super superficiem tangentem illud punctum.

3.62 ◉ Thus, it is clear that lights reaching mirrors along various lines are reflected along various [corresponding] lines. And if light reaches the mirror from the same direction, it continues [from it] in an equivalent direction, and the inclination of the reflected rays will be equal to the inclination of the incident rays. And it is evident that the incident and reflected rays of light lie in the same plane, which is perpendicular to the polished surface or [the surface] tangent to the point from which reflection occurs. Moreover, if the light arrives along the perpendicular, it will reflect along the perpendicular, and no matter what point it strikes, it is reflected in a plane that is perpendicular to the plane tangent to that point.

3.63 ◉ Et semper linea reflexa cum perpendiculari super illud punctum equalem tenet angulum angulo quem includit linea veniens cum eadem perpendiculari. Et huius rei probatio est quia palam quod, si descendat lux quecumque per foramen aliquod, reflectitur per ipsum respiciens. Et si constringatur foramen ut restet quasi solus axis, reflectitur per axem respicientis foraminis. Et si fiat alteratio descensus lucis, reflectitur lux per lineas per quas prius descenderat. Et palam quod foramina se respicientia eundem habent situm respectu medii, et cum non procedat lux nisi per lineas rectas, planum quod reflectitur per lineas eiusdem situs respectu medii cum lineis descensus.

3.63 ◉ In addition, the reflected ray invariably forms with the normal to that point an angle equal to the angle formed by the incident ray with that same normal. And the proof of this point is evident from the fact that, if any light shines through any hole [in the ring], it is reflected to the corresponding hole. And if the hole is narrowed so that all that is left is virtually equivalent to the axial line, the light is reflected along the axis of the corresponding hole. And if the light shines through the other hole, it is reflected along the lines according to which it was incident before. And it is obvious that the corresponding holes are equivalently disposed with respect to the middle hole, and since light only propagates along straight lines, it is evident that it is reflected along lines that are equivalently disposed with respect to the middle hole as the lines of incidence.

3.64 ◉ Unde cum accidit per ortogonale, per eam reflectitur solam, quare semper linee reflexionis eundem servant situm cum lineis descensus respectu superficiei contingentis punctum reflexionis. Et hoc substantiale sive in substantiali sive in accidentali luce, sive forti sive debili, et generaliter in omni.

3.64 ◉ Hence, when it arrives along the orthogonal, it is reflected along that line alone, because the lines of reflection invariably maintain the same disposition as the lines of incidence with respect to the plane that is tangent to the point of reflection. And this is [an] essential characteristic [of light], whether the light be essential or accidental, or whether it be intense or faint, and it applies universally in every case.⁑

3.65 ◉ Et nos ostendemus ydemptitatem situs. Iam scimus quod superficies regule cadit super tabulam in qua quadratum fecimus ortogonaliter. Igitur linea media tabule ortogonaliter est super lineam communem ei et regule, et est super lineam latitudinis regule. Et tabula equidistans enee tabule, et linea eius media equidistans linee medie tabule enee, et est linee a centro tabule enee producte et dividentis arcum per equalia.

3.65 ◉ Now we shall demonstrate the equivalent disposition [of the lines of incidence and reflection]. We already know that the surface of the panel stands orthogonal to the [bottom of the] block [at the base of the ring] in which we dug out the [hollow] square. Hence, the midline of the base-block is perpendicular to the common section formed by the [top surface of the] base-block and the panel, and it is [therefore perpendicular] to the line across the width of the panel [formed by this common section]. Moreover, the [top surface of] the base-block is parallel to the [top surface of the] bronze plaque, and its midline is parallel to the midline of the bronze plaque, that is, to the line drawn from the center of the bronze plaque [at the point of its triangle] that bisects its arc.

3.66 ◉ Linea autem communis tabule enee et regule, que est linea latitudinis regule, est equidistans linee communi tabule et regule, quare linea media tabule enee cadit perpendiculariter super lineam communem regule et tabule enee. Et regula perpendicularis est super superficiem quadrati, et superficies quadrati equidistans superficiei tabule, quare superficies tabule ortogonaliter super superficiem regule. [3.67] Et similiter superficies tabule enee ortogonaliter super eandem, et linea media longitudinis regule est perpendicularis super latitudinem eius, quare linea media tabule erit perpendicularis super mediam longitudinis regule lineam, cum cadit super eam; et similiter linea media tabule enee erit perpendicularis super eandem. Et ita media linea tabule enee est perpendicularis super superficiem regule et super mediam longitudinis eius lineam, et ita est perpendicularis super superficiem speculi plani et super mediam longitudinis eius lineam.

3.66 ◉ Furthermore, the common section of the [top surface of the] bronze plaque and the panel, which is a line across the width of the panel, is parallel to the common section of the base-block and the panel, so the midline of the bronze plaque falls orthogonally to the common section of the panel and the bronze plaque. And the panel stands perpendicular to the [bottom] surface of the square [hollowed out of the base-block], and the [bottom] surface of the square [hollowed out of the base-block] is parallel to the [top] surface of the base-block [itself], so the [top] surface of the base-block is orthogonal to the surface of the panel.

3.68 ◉ Amplius, superficies tabule enee est equidistans superficiei descendenti per centra foraminum, quoniam longitudo centrorum a superficie tabule enee eadem—id est medietatis unius grani ordei—et dyameter foraminis est unius grani ordei. Similiter latitudo superficiei columpne est unius grani, que superficies descendens per centra foraminum secat columpnam per medium. Et ita axis columpne est in superficie illa, et columpna descensu suo tangit lineam in tabula enea cui quidem equidistat axis, quoniam axis est equidistans cuilibet linee superficiei columpne.

3.68 ◉ In addition, the surface of the bronze plaque is parallel to the plane passing through the centers of the holes [in the ring], for the centers of [all] the holes lie the same distance from the surface of the bronze plaque—i.e., half a grain of barley—and the diameter of [each] hole is 1 grain of barley. Likewise, the diameter of the surface of the [iron] tube is 1 grain [of barley], and the plane passing through the centers of the holes bisects the tube. Hence, the axis of the tube lies in that plane, and, as it extends [into the hollow of the ring], the tube touches a line [drawn] on the [surface of the] bronze plaque, that line of course being parallel to the [tube’s] axis, for that axis is parallel to any line [of longitude] on the surface of the tube.

3.69 ◉ Et axis columpne cadit in punctum superficiei regule, a quo puncto linea ducta ad centrum tabule enee est perpendicularis super tabulam eneam, quoniam, per quodcumque foramen descendat columpna, axis eius cadit super mediam longitudinis regule lineam, et omnes ille perpendiculares sunt equales.

3.69 ◉ Moreover, the axis of the tube falls to a point on the surface of the panel, and the line drawn from that point to the center of the bronze plaque is perpendicular to the bronze plaque, because, no matter what hole the tube extends through, its axis falls upon the midline along the length of the panel, and all of these perpendiculars are equal [in length].

3.70 ◉ Et linea protracta a puncto regule in quem cadit axis per centrum foraminum est equidistans linee protracte a centro tabule enee ad terminum dyametri foraminis. Quoniam linea inter punctum illud et centrum est ortogonaliter super superficiem tabule enee, cum sit pars linee medie longitudinis regule, et etiam super axem. Et huic linee interiacenti centrum tabule enee et punctum est equidistans linea anuli transiens per centra foraminum et perpendiculariter cadens in superficiem tabule enee, quare equidistantes erunt linee cadentes in terminos linee anuli et longitudinis regule equalium et equidistantium. [3.71] Pari modo in singulis foraminibus, quare linee a puncto regule in quem cadit axis producte ad centra duorum foraminum se respicientium sunt equidistantes duabus lineis a centro tabule enee ad extremitates dyametrorum eorumdem foraminum protractis, quare hee due linee equalem tenent angulum cum illis lineis.

3.70 ◉ Also, the line extended from the point where the axis falls on the panel through the center[s] of the holes is parallel to the line extended from the center of the bronze plaque to the endpoint of the hole’s diameter. For the line [drawn] between that point [on the panel’s midline] and the [bronze plaque’s] center is perpendicular to the surface of the bronze plaque, since it is a segment of the midline along the length of the panel, and [it is] also [perpendicular] to the axis [of the hole]. And [every] line on the [inner surface] of the ring that passes through the centers of the holes and falls perpendicular to the surface of the bronze plaque is parallel to this line, which extends between the centerpoint of the bronze plaque and that point [where the axes of the holes intersect the midline along the length of the panel]; hence, the lines extending from the [respective] endpoints of the [segment of the] line on the [inner surface of the] ring and [the endpoints of the segment of the midline] along the length of the panel will be parallel [since they connect the endpoints] of equal and parallel [lines].

3.72 ◉ Et si a termino axis erigatur linea ad centrum foraminis, erit in superficie per centra descendente, et erit equidistans medie linee tabule enee. Quoniam linea inferior interiacens capita eorum est perpendicularis super tabulam eneam et equalis superiori eadem capita interiacenti et super tabulam eneam perpendiculari. Et est equidistans ei, quare linea a centro foraminis medii ad terminum axis columpne est equidistans medie linee tabule enee, et illa est perpendicularis super regulam, quare et ista. Igitur hec linea et latera alterum angulum continentia sunt equidistantes medie linee tabule enee et alteri linearum in tabula enea angulum continentium, quare quasi partiales sibi oppositi sunt equales.

3.72 ◉ If a line is erected from the endpoint of the axis [where it intersects the midline along the length of the panel] to the center of the [middle] hole, it will lie in the plane passing through the centers [of all the holes], and it will be parallel to the midline of the bronze plaque. For the line connecting the inner ends of these lines is perpendicular to the bronze plaque, and it is equal to the line connecting their outer ends, which is perpendicular to the bronze plaque. And [this inner line] is parallel to that [outer one], so the line from the center of the middle hole to the endpoint of the tube’s axis [where it intersects the midline along the length of the panel] is parallel to the midline of the bronze plaque, and that line is perpendicular to the panel, so the other is too.⁑ Therefore, this line [extending from the center of the middle hole to the midline along the length of the panel] and the sides forming alternate angles [with it] are parallel respectively to the midline of the bronze plaque and to each of the [corresponding] lines on the [surface of the] bronze plaque forming the [same] angle, so the corresponding segments [of the entire angle formed by those respective sides] are equal.

3.73 ◉ Igitur linea media tabule enee dividit angulum suum per equalia, quare linea a centro foraminis medii dividit angulum suum per equalia. Et cum certum sit quod lux foramen declinatum intrans per illas lineas angulum continentes moveatur, planum quod lux omnis reflectitur per lineas que cum lineis descensus sunt in eadem superficie ortogonali super superficiem reflexionis et angulum equalem facientibus cum perpendiculari cum lineis descensus.

3.73 ◉ Hence, the midline of the bronze plaque bisects the [entire] angle in its plane, so the [axial] line [drawn] from the center of the middle hole bisects the angle in its plane.⁑ And since there is no doubt that the light entering one of the inclined holes proceeds along lines that form an angle, it is evident that all light is reflected along lines that lie on the same plane as the lines of incidence, that plane being orthogonal to the reflecting surface [formed by the panel’s face], and such lines [of reflection] form an angle with the normal that is equal to the angle formed by the lines of incidence with that normal.

3.74 ◉ Et lux perpendiculariter descendens reflectitur per perpendicularem. Et hoc generale in omni luce.

3.74 ◉ Furthermore, light that falls along the normal is reflected along the normal. And this holds universally for all light.

3.75 ◉ Si autem declinetur regula non in latus suum sed in caput ut axis foraminis medii non sit perpendicularis super regulam, reflectitur lux, et videbitur super lineam altitudinis anuli perpendicularem et per centrum foraminis transeuntem. Et quanto maior fuerit declinatio, maior erit lucis reflexe a foramine vel axe elongatio. Et si diminuatur declinatio, diminuetur elongatio, et ita donec situs regule ad rectitudinem regrediatur, super perpendicularem illam reflectitur lux.

3.75 ◉ However, if the panel is slanted not sideways but [lengthwise] from the top so that the axis of the middle hole is not perpendicular to the panel, the light is [still] reflected, and it will be seen on the line drawn lengthwise [on the inner surface] of the ring, this line being perpendicular [to the bronze plaque] and passing through the center of the hole. And the greater the slant, the farther the reflected light will fall from the hole or from its axis. However, if the slant is decreased, the distance [between the center of the hole and where the reflected light falls] will decrease, so when the panel is restored to a perfectly upright position, the light is reflected along the normal.

3.76 ◉ Quod autem in hac declinatione axis foraminis medii et linea reflexionis sunt in eadem superficie ortogonali super superficiem reflexionis planum per hoc quoniam axis foraminis medii est perpendicularis super latitudinem regule—id est super lineam communem superficiei regule et superficiei per centra foraminum descendentis—et media linea tabule anuli est equidistans huic axi et equidistans medie linee tabule enee.

3.76 ◉ Furthermore, it is evident from the following that in the case of such an orientation the axis of the middle hole and the line of reflection lie in the same plane, which is orthogonal to the reflecting surface [formed by the panel’s face]. For the axis of the middle hole is perpendicular to the [line along the] width of the panel—i.e., to the common section of the panel’s surface and the plane passing through the centers of the holes—and the midline of the block [at the bottom] of the ring is parallel to this axis and parallel to the midline of the bronze plaque.

3.77 ◉ Et media linea tabule enee est perpendicularis super latitudinem regule, et est super lineam communem superficiei regule et superficiei tabule enee, quare superficies in qua sunt media linea tabule enee et axis foraminis medii ortogonalis est super superficiem regule. Et in hac superficie est linea perpendicularis in altitudine anuli, quoniam transit per terminos equidistantium—scilicet medie tabule enee et axis foraminis medii.

3.77 ◉ Also, the midline of the bronze plaque is perpendicular to the [plane] of the panel along its width, and [so] it is perpendicular to the common section of the panel’s surface and the surface of the bronze plaque, so the plane in which the midline of the bronze plaque and the axis of the middle hole lie is orthogonal to the panel’s surface. Moreover, the line drawn lengthwise [and perpendicular to the bronze plaque on the inner surface] of the ring lies in this same plane, for it passes through the endpoints of parallel lines—i.e., the midline of the bronze plaque and the axis of the middle hole.

3.78 ◉ Palam igitur quod lux reflexa que apparet in perpendiculari altitudinis anuli reflectitur per lineam que cum axe per quem fit descensus est in superficie ortogonali super superficiem regule. Luce ergo descendente in speculum planum, fit reflexio secundum lineas quarum eadem declinatio super superficiem speculi, et ipse cum perpendiculari in superficie ortogonali super superficiem speculi.

3.78 ◉ It follows, therefore, that the reflected light appearing on the perpendicular drawn lengthwise [on the inner surface] of the ring is reflected along a line that lies in the same plane as the axial line along which it is incident, and that plane is orthogonal to the surface of the panel. Hence, when light is incident upon a plane mirror, its reflection occurs along lines that are identically inclined to the surface of the mirror, and those lines lie in the same plane as the normal, that plane being orthogonal to the surface of the mirror.⁑

3.79 ◉ In speculo columpnari exteriori eadem penitus probatio que est in plano—scilicet quod acumen tabule enee cadit super lineam longitudinis speculi ortogonalis, et similiter columpna descendens super eandem. Et pars illius linee super hos casus est ortogonaliter super tabulam eneam. Et semper, sive per foramen medium sive per declinatum descendet lux, reflexio eius cum descensu erit in eadem superficie ortogonali super superficiem contingentem lineam longitudinis speculi.

3.79 ◉ In the case of the cylindrical mirror polished on its outer [convex] surface, the proof is precisely the same as in the case of the plane mirror—i.e., because the point of the bronze plaque falls orthogonally to the [mid]line along the length of the mirror, and so does the [axial line through the iron] tube when it reaches it. And the segment of the [mid]line [of the mirror] that passes through those [points] is orthogonal to the bronze plaque. Invariably, then, whether the light shines through the middle hole or through one of the inclined holes, its [line of] reflection will lie in the same plane as its [line of] incidence, that plane being orthogonal to the plane that is tangent to the [mid]line along the length of the mirror.

3.80 ◉ In piramidali vero exteriori, cum superficies regule sit in eadem superficie cum linea media longitudinis piramidalis, sicut in columpnari, erit idem situs linearum superficierum et idem reflexionis modus, sicut in speculo plano, et eadem penitus probatio.

3.80 ◉ In the case of the conical mirror [polished] on its outer [convex surface], since the surface of the panel lies in the same plane as the midline along the length of the conical mirror, as it does in the case of the [convex] cylindrical mirror, the lines in the planes [of the axial lines and the bronze plaque] will be equivalently disposed, so reflection will occur in the same way as it does in the plane mirror, and the demonstration will be precisely the same.

3.81 ◉ In speculo autem columpnari concavo descendit acumen tabule enee usque ad lineam longitudinis eius mediam, et super eandem cadit axis cuiusque foraminis. Et linea pars illius inter hos casus est ortogonalis super superficiem tabule enee, et axis foraminis et media linee tabule enee sunt ortogonales super superficiem tangentem speculum illud in linea longitudinis, que est locus reflexionis, et equidistantes superficiei regule.

3.81 ◉ In the case of the concave cylindrical mirror, as well, the point of the bronze plaque falls on the midline along its length, and the axis of each hole falls on that same line. And the line-segment between these [two points] to which [the two lines] fall is orthogonal to the surface of the bronze plaque, and the axis of the [middle] hole and the midline of the bronze plaque are orthogonal to the plane tangent to that mirror upon the [mid]line along its length, which is where reflection occurs, and they are parallel to the surface of the [bronze] plaque.

3.82 ◉ Et ita idem modus probandi qui prius—quod scilicet reflexio et descensus sunt in eadem superficie ortogonali super superficiem loci reflexionis, et eiusdem sunt declinationis, et quod descensus per medium efficit reflexionem per ipsum. Et declinato capite regule, erit reflexio super perpendicularem anuli, sicut dictum est in plano.

3.82 ◉ Thus, as before, [we follow] the same method for proving that [the lines of] reflection and [the lines of] incidence lie in the same plane, which is perpendicular to the reflecting surface [formed by the panel’s face], that those lines are equivalently slanted [with respect to the normal dropped to the point of reflection], and that [light] incident through the central hole is reflected back to that hole. Also, when the top edge of the panel is slanted [backward], the light will be reflected to the perpendicular [drawn lengthwise on the inner surface] of the ring, just as was pointed out in the case of the plane mirror.

3.83 ◉ In speculo piramidali concavo eadem in omnibus probatio.

3.83 ◉ In the case of the concave conical mirror, the same proof holds in all respects.

3.84 ◉ In speculo sperico exteriori palam quod medius eius punctus est in superficie regule, et axis cadit in punctum illud, et erit in eo idem situs linearum. Et aliorum penitus quod in plano, et eadem demonstratio.

3.84 ◉ In the case of the [convex] spherical mirror [polished] on its outer surface, it is clear that its midpoint lies in the plane of the panel, while its axis falls on that point, so the lines [of incidence and reflection] will have the same disposition in that case. And what holds for the plane mirror holds in all respects for the other mirrors, and the demonstration is the same [for all].

3.85 ◉ In speculo sperico concavo iam determinatum est quod axis foraminis descendit ad punctum eius medium, et acumen tabule enee transit per foramen in speculo iam factum usque dum sit in eadem superficie cum puncto illo medio. Et linea a puncto illo ad acumen protracta est equidistans medie linee longitudinis regule, et ita descensus et reflexio sunt in superficie ortogonali super superficiem contingentem speculum in illo puncto medio et equidistantem superficiei regule. Et eadem probatio penitus ut in aliis.

3.85 ◉ In the case of the spherical concave mirror it has already been determined that the axis of the hole passes to the mirror’s midpoint, and the point of the bronze plaque passes into the hole in the mirror, which we made earlier, until it lies in the same plane as that midpoint. And the line drawn from that point to the point [of the bronze plaque] is parallel to the midline along the length of the panel, and so [the lines of] incidence and reflection lie in a plane that is orthogonal to the plane tangent to the mirror at that midpoint, [that latter plane being] parallel to the surface of the [bronze] plaque. And in this case the demonstration is precisely the same as it is for the other mirrors.

3.86 ◉ Palam ergo quod omnis lux in quodcumque speculum eorum cadit reflexio et descensus sunt in eadem superficie ortogonali. Hic autem modus reflexionis non accidit ex proprietate axis, vel puncti in quod cadit, vel foraminis per quod intrat, vel proprietates speculi. Accidit enim in quodlibet foramen quecumque sit lux, et per quamcumque lineam descendat, et in quodcumque speculi punctum cadat. Quoniam quocumque puncto speculi sumpto, si lux in ipsum descendat, cum idem sit ei situs respectu longitudinis speculi, et cuicumque alii erunt similiter idem respectu linearum ab eo protractarum, que eiusdem sunt declinationis cum lineis a puncto priori intellectis, sicut et puncto priori, vel cuicumque alii.

3.86 ◉ Hence, it is evident that if any light shines on any of those mirrors [just discussed], the [lines of] reflection and incidence lie in the same plane, which is orthogonal [to the surface of the mirror or the plane tangent to the point of reflection on the mirror]. But the reason reflection follows this rule is not specific to the axis, or to the point on which the light shines, or to the hole through which it shines, or to the mirror. Indeed, it holds for any hole, no matter what kind of light [shines through it], and it holds for any line of incidence as well as for any point on the mirror to which the light may fall. For, no matter what point on the mirror is taken, if light shines on it, since its disposition is the same with regard to the [midline along the] length of the mirror, and since any of the others will be equivalently disposed with regard to the lines extended from it, those lines are identically inclined with respect to the lines understood [to extend] from that point, just as from the point taken earlier, as well as from any other point.

3.87 ◉ Et generaliter idem est situs cuiuslibet puncto in quod cadit lux qui et in priori sumpto et respectu axis, et respectu acuminis tabule enee. Et eadem in omnibus probatio, et similis demonstratio, unde certum non esse hoc ex proprietate lucis vel figure alicuius speculi sed ex quadam proprietate communi omni rei polite et cuilibet luci. Si autem per diversa in quodcumque punctum descendit lux foramina, videbitur reflexio diversa et angulorum diversitas suo descensui consona, et sic in omnibus.

3.87 ◉ So it is invariably the case that the disposition of each [such line] at the point to which the light falls, which is the same as the previously taken point, is equivalent with respect both to the axis and to the point of the bronze plaque. And the same proof and the same demonstration hold for all cases, so it is certain that this is due not to a particular kind of light nor to the shape of any particular mirror, but it is a characteristic that is common to every polished body and to any kind of light. Moreover, when the light shines through various holes to a given point, the variation in reflection and in the angles of reflection will be observed to conform to the way the light is incident, and the same holds in all cases.

3.88 ◉ Manifestum ex superioribus quod, si corpus politum opponatur corpori luminoso, cadit in quodlibet punctum eius lux a quolibet luminosi puncto, unde super quodlibet politi punctum cadit piramis cuius acumen in eo, et superficies luminosi basis. Et a quolibet puncto luminosi procedit piramis cuius acumen in eo et basis superficies politi.

3.88 ◉ It is obvious from what we have established above that, if a polished body faces a luminous body, light from any point on the luminous body falls to any point on the [exposed surface of the] polished body, so at any point on the polished body there stands a cone whose vertex lies on that point and whose base is formed by the surface of the luminous body. Also, from any point on the luminous body there extends a cone whose vertex lies at that point and whose base is formed by the [exposed] surface of the polished body.

3.89 ◉ Si autem inter luminosum et politum intelligatur punctum aliquod, veniet quidem ad illud punctum lux luminosi in modum piramidis cuius acumen in puncto, et latera huius piramidis procedentia usque dum cadant in superficiem politi piramidem efficiunt. Unde in puncto intellecto erunt acumina duarum piramidum quarum bases sunt superficies luminosi et superficies politi, et si ad punctum quodcumque intermedium intelligatur piramis cuius basis superficies politi, et procedant huiusmodi piramidis linee, illud quod occupabunt ex superficie luminosi hoc est a quo procedebat lux ad politum secundum duas piramides quarum acumina in puncto intellecto.

3.89 ◉ Moreover, if some point is imagined between the luminous and polished bodies, the light from the luminous body will reach that point in the form of a cone whose vertex lies at that point, and when the edges of that cone are extended until they reach the surface of the polished body, they form a cone. Hence, the vertices of two cones will lie at the point [so] imagined, their bases consisting of the surface of the luminous body and the surface of the polished body, and if a cone is imagined at some intermediate point with its base formed by the polished surface, and if the lines [contained] by this sort of cone are extended, the portion of the luminous body’s surface that they will envelop forms the part from which the light has radiated to the polished body according to two cones whose vertices lie at the [intermediate] point [just] imagined.

3.90 ◉ Et quod procedit lucis in hiis duabus piramidalibus procedit et includitur in duabus primis piramidalibus, et a luminoso secundum lineas equidistantes procedit lux ad speculum, sed hee linee includuntur in duabus primis piramidibus. Et per quascumque lineas moveatur lux ad speculum, observant linee reflexionis eundem penitus situm quem habebant linee motus lucis; unde si moveatur lux per equidistantes, reflectitur per equidistantes, et lux cadens in modum politi ad modum piramidalis reflectitur observans modum eiusdem piramidis.

3.90 ◉ And the light that propagates within these two cones radiates within and is contained by the first pair of cones, and the light propagates from the luminous source to the mirror along parallel lines, but these lines are included within the first pair of cones. Moreover, whatever lines the light follows as it radiates to the mirror, the lines of reflection maintain precisely the same relative disposition as the lines of incidence the light originally followed, so if the light radiates along parallel lines, it is reflected along parallel lines, and light falling within the extent of the polished body in the form of a cone is reflected in such a way as to form an equivalent cone.⁑

3.91 ◉ Cum descendit lux a corpore luminoso per foramen aliquod ad corpus politum, si in superficie foraminis ex parte illuminosi intelligatur punctus a quo puncto intelligantur due piramides basis unius in luminosa alterius in polito, a sola base piramidis cuius luminosum basis venit lux ad politum super illud punctum. Similiter, si in superficie foraminis ex parte politi intelligatur punctum in quo acumina duarum piramidum unius ad speculum alterius ad luminosum, a sola base piramidis que basis est in luminoso accedit lux ad speculum super hoc punctum.

3.91 ◉ When light shines from a luminous body to a polished body through some hole, and if a point is imagined in the plane of the hole facing the luminous source, and from that point two cones are imagined, one with its base on the luminous body, the other on the polished body, light reaches the polished body through that point only from the base of a cone whose base is on the luminous body. Likewise, if a point is imagined in the plane of the hole facing the polished body, that point forming the vertex of two cones, one based on the mirror, the other on the luminous body, the light reaches the polished body through that point only from the base of a cone whose base is on the luminous body.

3.92 ◉ Et a parte luminosi hiis duabus piramidalibus communi accidit lux ad partem speculi commune duabus piramidalibus. Venit etiam lux a luminoso ad speculum per lineas equidistantes, sed per quascumque accedat, fit reflexio modo predicto. Et quelibet linee reflexionis conservant situm linearum descensus lucis eas respicientium, et in omni reflexione observatur ydemptitas forme lucis que fuerit in polito corpore, et hoc deinceps explanabimus explanatione evidenti.

3.92 ◉ So it is from the direction of the luminous body, which forms one common [base] of these two cones, that light shines toward the mirror, [which forms the other] common [base] of the two cones. Also, light reaches the mirror from the luminous source along parallel lines, and no matter what lines it follows, it is reflected in the way described earlier. And all of the lines of reflection maintain an equivalent disposition with respect to the corresponding lines of the light’s incidence, and in every reflection the light maintains the same precise form as it had in the polished body, and in what follows we will explain this in a clear manner.⁑

3.93 ◉ Amplius, patuit quod lux quanto plus ab ortu suo elongatur plus debilitatur. Patuit etiam quod lux continua fortior disgregata. Cum igitur ab aliquo puncto luminosi procedit lux ad superficiem speculi in modum piramidis, quanto magis elongatur a puncto illo tanto maior est eius debilitatio duplici de causa: et propter elongationem ab ortu suo, et propter disgregationem. Cum autem ab aliquo speculi puncto reflectitur lux, ista fit debilior tripliciter: et propter reflexionem que debilitat, et propter elongationem a loco reflexionis, et propter disgregationem.

3.93 ◉ Now it has been established that the farther light extends from its source, the weaker it gets. It has also been established that concentrated light is more intense than dispersed light.⁑ Thus, since light shines from any point on a luminous source to the surface of a mirror in the form of a cone, the farther it extends from that point, the weaker it gets for two reasons: because of its [increasing] distance from its source and because of its dispersal. Furthermore, when light is reflected from a point on a mirror, it is weakened in three ways: because of the reflection [itself], which weakens [the light]; because of its [increasing] distance from the point of reflection; and because of its dispersal.

3.94 ◉ Si vero lux reflexa a speculo agregetur in punctum aliquod, fiet quidem fortior propter agregationem, sed debilitatur per reflexionem et elongationem. Si igitur agregatio lucis tantum redit ei fortitudinis quantum subtrahunt reflexio et elongatio, erit lux reflexa agregata eiusdem fortitudinis cuius est in superficie speculi. Si vero agregatio minus addat fortitudinis quam diminuant illa duo, erit debilior, et si plus addat, erit fortior.

3.94 ◉ If, however, the light that is reflected from the mirror is concentrated at some point, it will be strengthened by that concentration, although it is weakened by the reflection [itself] as well as by its [increasing] distance [from its source].⁑ Therefore, if the concentration of the light intensifies it as much as the fact of reflection and the distance [it lies from its source] weaken it, the concentrated reflected light will be as intense as it is at the mirror’s surface. On the other hand, if the increase in intensity due to concentration is outweighed by the weakening due to the other two factors, the [reflected] light will be weaker [than it is at the mirror’s surface], whereas if that increase outweighs [the weakening], the light will be more intense.⁑

3.95 ◉ Similiter, si a superficie luminosi procedat piramis ad aliquod punctum speculi, erit lux procedens secundum hanc piramidalitatem debilior propter elongationem sed fortior propter agregationem. Si autem agregatio potest super elongationem, erit lux in puncto speculi agregata fortior luce unica a luminoso veniente per lineam unam. Unica dico, quia ad quodlibet punctum linee ex illis sumpte venit etiam piramis a luminoso, que quidem piramis cum similibus excluditur in hac consideratione.

3.95 ◉ Likewise, if the cone [of radiation] extends from the luminous surface [which forms its base] to some point on the mirror [which forms its vertex], the light radiating in the form of such a cone will be weakened by [increasing] distance but intensified by concentration. If, moreover, the [intensification caused by] concentration outweighs [the weakening caused by increasing] distance, the light concentrated at the point on the mirror will be more intense than a single [point of] light that radiates from the luminous source [to the mirror] along a single ray. I say »single,« because a cone radiates from the luminous source to any point on the line chosen among [all those radiating from the given point on the luminous source], but such a cone is excluded from this consideration along with others of its kind.

3.96 ◉ Si vero elongatio ponderet super agregationem, erit lux puncti politi minor luce sola unius linee sumpta, et si agregatio plus ponderet elongatione, erit fortior. Luces autem que a luminoso ad speculum accedunt super lineas equidistantes erunt debiliores quam modo alio accedentes, quoniam debilitate propter elongationem non agregentur in speculum, et in reflexione per lineas equidistantes moventur. Unde per reflexionem et elongationem debilitantur. Et si agregentur in reflexione, conferetur eis fortitudo comparata ad fortitudinem quam habuerint in speculo secundum posse agregans super reflexionem et elongationem.

3.96 ◉ But if [the weakening due to] distance outweighs [the intensification due to] concentration, the light at the point [of incidence] on the polished surface will be weaker than the light taken along a single line of radiation, whereas if [the intensification due to] concentration outweighs [the weakening due to] distance, the light [at the point of incidence] will be stronger [than the light taken along a single line of radiation].⁑ Moreover, light that extends from the luminous source to the mirror along parallel lines will be weaker than that extending in the other way [i.e., along a cone with its base in the luminous object and its vertex at the mirror], for such light does not become concentrated on the mirror to counterbalance the weakening due to distance [from its source], and it proceeds along parallel lines after reflection. Hence it is weakened both by reflection [itself] and by [increasing] distance. But if it is concentrated upon reflection, that concentration will intensify it to the degree that it was intense [when it shone] upon the mirror, insofar as such concentration can balance out [the weakening due to both] reflection [itself] and [increasing] distance.⁑

3.97 ◉ Amplius, omnis linea per quam movetur lux a corpore luminoso ad corpus oppositum est linea sensualis, non sine latitudine, lux enim non procedit nisi a corpore, quoniam non est nisi in corpore. Sed in minori luce que sumi possit est latitudo, et in linea processus eius est latitudo. Et in medio illius linee sensualis est linea intellectualis, et alie eius linee sunt equidistantes illic. Et si dividatur minor ex lucibus, neutra eius pars erit lux, sed utraque extinguetur nec apparebit. Si autem lux minima duplicetur, aut amplius multiplicetur per equalia, et compacta dividatur, erit lux utraque eius pars. Si vero per inequalia fiat divisio, erit altera pars eius lux, altera minime.

3.97 ◉ Now, every line along which light radiates from a luminous body to a body facing it is a sensible line, not one without breadth, for light proceeds only from bodies, since it can subsist only in bodies. But in the least light that can be imagined there is some breadth, and [so] there is breadth in the line [or shaft] along which it radiates. And in the middle of that sensible shaft there is an imaginary line, and [all] the other [such] lines within that sensible shaft are parallel to it. So if the least [possible amount of visible] light is divided, neither part of it will constitute [actual, sensible] light; rather, both will be [effectively] extinguished and will [therefore] not be visible. On the other hand, if the least [possible amount of visible] light is doubled, or further multiplied according to equal increments, and if the resulting compound [light] is divided [equally], both of its portions will constitute [visible] light. But if [that compounded light] is divided unequally, one portion of it will constitute [effective] light, and the other [will be diminished] to the smallest [possible amount of visible light].⁑

3.98 ◉ Lux autem minima procedit ad minimam corporis partem quam lux occupare possit, et processus eius est secundum lineam intellectualem linee sensualis mediam, et extremitates ei equidistantes. Et cadit lux minima non in punctum corporis intelligibilem, sed sensibilem, et refertur per lineam sensibilem cuius latitudo est equalis latitudini linee sensibilis venientis. Et si intelligatur in linea sensibili linea reflexa intellectualis media, eundem habet situm super reflexionis locum quem habet linea intelligibilis media linee sensibilis venientis, et quelibet linea intellectualis in linea reflexa sensibili eundem penitus observat situm cum linea intelligibili alterius sensibilis ipsam respiciente. Observatur ergo in omni luce reflexio linearum et punctorum intellectorum, licet ab eis aut per ipsas non procedat lux, et in hunc modum erit reflexio lucis.

3.98 ◉ Now the smallest [possible amount of visible] light radiates upon the smallest portion of the [surface of a] body that light can occupy, and it radiates along an imaginary line centered within the sensible shaft [of radiation] whose edges are parallel to it. So the least quantum of light falls not upon an imaginary point on a [facing] body, but upon a sensible spot, and it is reflected along a sensible shaft that is the same breadth as the sensible shaft along which it reached [that body]. So if one conceives of an imaginary line in the center of the sensible shaft of reflection, it maintains the same disposition with regard to the point of reflection as the imaginary line in the center of the sensible shaft of incidence, and every line imagined in the sensible shaft of reflection maintains precisely the same disposition with respect to the imaginary line corresponding to it in the other sensible shaft [of incidence]. Thus, in the case of any light, the reflection of lines and of imaginary points follows this rule, even though light does not [actually] proceed from such [points] along such [lines], and this is how the reflection of light will occur.⁑

3.99 ◉ Amplius, quare ex politis corporibus non ex asperis fiat reflexio est quoniam lux, ut diximus, non accedit ad corpus nisi per motum citissimum, et cum pervenit ad politum, eicit eum politum a se. Corpus vero asperum nec potest eam eicere, quoniam in corpore aspero sunt pori quos lux subintrat; in politis autem poros non invenit. Nec accidit hec eiectio propter corporis fortitudinem vel duriciem, quia videmus in aqua reflexionem; sed est hec repulsio propria politure, sicut de natura accidit quod aliquod honerosum cadens ab alto super lapidem durum revertitur in altum, et quanto minor fuerit duricies lapidis in quam ceciderit, regressio cadentis debilior erit. Et semper regredietur cadens versus partem a qua processit. Verum in arena, propter eius mollitiem, non fit regressio que quidem accidit in corpore duro.

3.99 ◉ Moreover, the reason that reflection occurs from polished rather than from rough bodies is because, as we have [already] pointed out, light approaches [any given] body with only the swiftest of motions, and when it reaches a polished body, that polished body causes it to rebound from it.⁑ A rough body, on the other hand, cannot cause it to rebound because there are pores in the rough body into which the light enters; in polished bodies, though, it encounters no pores. But this rebound is not due to the [physical] resistance or hardness of the body, for we see reflection [occur] in water; on the contrary, this [kind of] repulsion is a function of polish, just as happens in nature when some heavy object falls from high above onto a hard stone [surface] and bounces back up, and the less hard the stone [surface] upon which it falls, the weaker the rebound of the falling object will be.⁑ Furthermore, the falling object will invariably rebound in the opposite direction from that along which it [originally] traveled. The rebound that occurs in the case of a hard body does not, however, occur in the case of sand because of its softness.

3.100 ◉ Si autem in poris asperi corporis sit politio, tamen lux intrans per poros non reflectitur, et si eam reflecti accidit, dispergitur, et propter dispersionem a visu non percipitur. Pari modo, si in aspero corpore partes elatiores fuerint polite, fiet reflexa dispersio, et ob hoc occultabitur visui. Si vero eminentia partium adeo sit modica, ut sit eius quasi idem situs cum depressis, tunc comprehendetur eius reflexio tamquam in polito non aspero, licet minus perfecte.

3.100 ◉ Moreover, [even] if there is some polish in the pores of a rough body, the light entering into those pores is nevertheless unreflected, and if it does happen to reflect, it is scattered and is not perceived by sight because of this scattering. By the same token, if the elevated portions of the rough body are polished, they will cause the reflected light to scatter, and on that account the reflection will be unnoticeable to sight. If, however, the height of these portions is slight, so that it is roughly the same as that of the lower portions, then the light reflecting from it will be perceived as if it came from a polished rather than from a rough body, even though [it will be perceived] less clearly [than light reflected from a perfectly polished surface].⁑

3.101 ◉ Quare autem fiet reflexio lucis secundum lineam eiusdem situs cum linea per quam accedet ad speculum ipsa lux est quoniam lux motu citissimo movetur, et quando cadet in speculum, non recipitur; sed ei fixio in corpore illo negatur. Et cum in ea perseveret adhuc prioris motus vis et natura, reflectitur ad partem a qua processit, et secundum lineas eundem situm cum prioribus habentes.

3.101 ◉ Now, the reason that the reflection of light will occur along a line that has an equivalent disposition with the line along which that same light will reach the mirror is because light moves extremely swiftly, so when it strikes the mirror it is not absorbed by the mirror; instead, its being trapped in that body is prevented. And since, on that account, it conserves the force and nature of its previous motion, it is reflected back in the direction along which it arrived and along lines that have the same [relative] disposition as the original lines [of incidence].

3.102 ◉ Huius autem rei simile in naturalibus motibus videre possumus et etiam accidentalibus. Si corpus spericum ponderosum ab aliqua altitudine descendere permittamus perpendiculariter super politum corpus, videbimus ipsum super perpendicularem reflecti per quam descenderat. In accidentali motu, si elevetur speculum secundum aliquam altitudinem hominis, et firmiter in pariete figuratur, et in acumine sagitte consolidetur corpus spericum, et proiciatur sagitta per arcum in speculum hoc modo ut elevatio sagitte sit equalis elevationi speculi (et sit sagitta equidistans orizonti), planum quod super perpendicularem accedit sagitta ad speculum, et videbis super eandem perpendicularem eius regressum. Si vero motus sagitte ad speculum fuerit super lineam declinatam in ipsum, videbitur reflecti non per lineam per quam venerat sed per aliam non equidistantem orizonti, sicut et alia erat, et eiusdem situs respectu speculi cum ea et respectu perpendicularis in speculo. Quod autem ex prohibitione politi corporis accidat luci motus reflexionis palam quia, cum fortior fuerit repulsio vel prohibitio, fortior erit lucis reflexio.

3.102 ◉ We can observe the same thing in the case of natural as well as accidental motions.⁑ If we drop a heavy spherical body perpendicular to a polished body from some height, we will see it reflected back along the perpendicular it followed in dropping. In the case of accidental motion, if the mirror is raised to the height of a man and is attached firmly to a wall, and if a spherical body is attached to the point of an arrow, and the arrow is shot from a bow at the mirror in such a way that the arrow is at the same height as the mirror (and the arrow should be parallel to the horizon), it is obvious that the arrow reaches the mirror orthogonally, and you will see it rebound along the same orthogonal. If, on the other hand, the arrow flies along an oblique line with respect to the mirror, it will be seen to reflect not along the line according to which it arrived but along another one that is not parallel to the horizon, as was the case with the earlier path, but it will maintain the same disposition with respect to the mirror as that [original path] and with respect to the normal to the mirror.⁑ Moreover, that the motion of light in reflection is due to the resistance of the polished body is evident from the fact that the more intense the repulsion or resistance [of that body], the more intense the reflection of the light will be.⁑

3.103 ◉ Quare autem accidit idem motus reflexionis et eius accessus hec est ratio. Cum descendit corpus ponderosum super perpendicularem, reflexio corporis politi et motus descendentis ponderosi directe sibi sunt oppositi, nec est ibi motus nisi perpendicularis. Et prohibitio fit per perpendicularem, quare repellitur corpus secundum perpendicularem, unde perpendiculariter regreditur. Cum vero descendat corpus super lineam declinatam, cadit quidem linea descensus inter perpendicularem superficiei politi per ipsum politum transeuntem et lineam superficiei eius ortogonalem super hanc perpendicularem.

3.103 ◉ Here is why the motions of incidence and reflection turn out to be identical. When a heavy object falls orthogonally, the rebound from the polished body and the motion of the heavy body’s incidence are perfectly opposed, so in this case there is motion only along the perpendicular. And the resistance occurs along the perpendicular, which is why the body is repelled along the perpendicular, so it rebounds orthogonally. When, however, that body impinges along an inclined path, the path of incidence lies between the normal that passes through the surface of that polished body and the line on its surface that is orthogonal to that normal.

3.104 ◉ Et si penetraret motus ultra punctum in quem cadit, ut liberum inveniret transitum, caderet quidem hec linea inter perpendicularem transeuntem et lineam superficiei ortogonalem super perpendicularem. Et observaret mensuram situs respectu perpendicularis transeuntis et respectu linee alterius que ortogonalis est super illam perpendicularem. Compacta enim est mensura situs huius motus ex situ ad perpendicularem et situ ad ortogonalem. [3.105] Repulsio vero per perpendicularem incedens, cum non possit repellere motum secundum mensuram quam habet ad perpendicularem transeuntem, quia nec modicum intrat, repellit ergo secundum mensuram situs ad perpendicularem quam habet ad ortogonalem. Et quando motus regressio eadem fuerit mensura situs ad ortogonalem que fuit prius ad eandem ex alia parte, erit similiter ei eadem mensura situs ad perpendicularem transeuntem que fuit prius. [3.106] Sed ponderosum corpus in regressu, cum finitur repulsionis motus, ex natura sua descendit et ad centrum tendit. Lux autem eandem habens reflectendi naturam, cum ei naturale non sit ascendere aut descendere, movetur in reflexione secundum lineam inceptam usque ad obstaculum quod sistere faciat motum, et hec est causa reflexionis.

3.104 ◉ But if the motion [of the body] were to penetrate through the point [on the polished body’s surface] that it strikes so as to find free passage, then the resulting line [of transit] would fall between the normal passing [through the surface of the polished body at the point of incidence] and the line on [the polished body’s] surface that is orthogonal to that normal. Furthermore, it would maintain the same degree of orientation with respect to the normal passing [through the surface] as well as with respect to the other line that is orthogonal to that normal. For the measure of the disposition of this motion depends on the disposition [of its motion with respect] to the normal and the disposition [of its motion with respect] to the orthogonal [to that normal].

3.107 ◉ Patet etiam ex superioribus quod colores simul moventur cum lucibus, unde erit reflexio coloris sicut et lucis. Et si probationem eius videre volueris secundum modum in parte secunda assignatum, poteris iterum per instrumentum. Ad hanc denotandam reflexionem non plene videbis propter debilitatem coloris, debilitatur enim color per elongationem, per reflexionem, per foramen in quod intrat. Quod autem foramen debilitat planum per hoc quod lux apparet maior post foramen magnum quam parvum. Pari modo, cum foramina stricta sint, color post reflexionem aut nullus apparebit aut valde modicus. Tamen, si in predicto instrumento videre volueris, facias speculum argenteum, in ferreo enim speculo color apparet debilior, quoniam in reflexione misceretur cum luce reflexa mixta ex luce descendente et luce speculi ferrei modica, et color ferreus colori reflexo mixtus debilitaret ipsum.

3.107 ◉ It is also clear from [what we have established] above that colors radiate in tandem with light, so the reflection of color will be like that of light. And if you want to determine this fact observationally in the way described in chapter 2, you can do so by again using the [ringlike] apparatus. In attempting to observe such reflection accurately, you will not see [the relevant phenomena] clearly because of the weakness of color, for color is weakened by distance, by reflection [itself], and by the [restriction posed by the] hole through which it passes. That the [restriction posed by the] hole has a weakening effect is evident from the fact that light appears more intense after [passing through] a large hole than [after passing through] a small one. By the same token, when the holes [through which the color passes] are narrow, no color at all will be seen after reflection, or it will appear very faint. Nevertheless, if you wish to observe [the reflection of color] in the aforementioned apparatus, then you should make the mirror out of silver, for color appears very weak in an iron mirror, because it would be mingled with the reflected light, which is composed of the incident light and the dim light in the iron mirror, and the color of the iron mingled with the reflected color would weaken it.

3.108 ◉ Iterum in domo unici foraminis tantum habeatur instrumentum predictum cui domui paries albus opponatur. Et instrumentum foramini domus aptetur cuius foraminis latitudo sit ut duo instrumenti foramina occupare possit per quorum alterum inspiciatur paries albus domui oppositus. Et parti comprehense parietis opponatur corpus coloris fortis, et per aliud instrumenti foramini videatur pars parietis. Cum ergo lux intraverit per foramina instrumenti, videbitur color reflecti per foramen illud respiciens, quod est oppositum corpori colorato, per aliud minime. Et ita accidet quocumque opposito corpori foramine, et que dicta sunt in reflexione lucis considerari poterunt in reflexione coloris. Occupavit autem latitudo foraminis parietis duo instrumenti foramina ei adhibita ut maior descendat in speculum lux et melior apparet color reflexus. Et quoniam color debilitatur per foramen directus, et similiter reflexus, cum in corpus ceciderit visui oppositus percipietur secundus, unde, si post reflexionem cadat in corpus album foraminis colorationis adhibitum, forsan propter debilitatem non comprehendet eum visus. Adhibito autem secundo visu foramini colorationis, forsan comprehendetur, quoniam primus non secundus videbitur.

3.108 ◉ Once again, place the aforementioned apparatus in the room with only one window facing a white wall inside the room. Set the apparatus toward the room’s window, which is narrow enough that [the space between] two of the holes in the apparatus can block it, [and let it be set up] so that the white wall facing it in the room can be viewed [in the inserted mirror] through either of the holes. Place a brightly colored body in front of the visible portion of the wall, and look at that portion of the wall through either hole in the apparatus. Accordingly, when the [colored] light enters through the holes in the apparatus, the color will be seen reflected through the hole corresponding to the one facing the colored body, [but it will be] invisible through the other hole. And this will happen with any [colored] body facing the hole, and what has been claimed about the reflection of light can be extended to the reflection of color. Moreover, the opening in the wall was as wide as the two holes in the apparatus exposed to it so that the light might shine more intensely upon the mirror, and the reflected color is [thereby] more apparent. And because color is weakened when it shines directly through an opening and [is] likewise [weakened when] reflected, then, when it shines on a body facing the eye, it will be perceived as secondary, so if, after reflection, it shines on a white body that is exposed to the color at the opening, it may not be perceived by the eye because of the [resulting] weakening. When, however, the color at the second hole is exposed to the eye, it may be perceived because it will be seen as primary rather than secondary.⁑

◉[Capitulum 4]

◉[Chapter 4]

Pars quarta: quod comprehensio forme in corporibus fit per reflexionem

4.1 ◉ Super modum comprehensionis forme in politis corporibus dissentiunt plurimi. Unde quidam eorum radios a visu exire ad speculum, et a speculo redire, et formam rei in reditu comprehendere. Alii affirmant formam corporis speculo ei opposito imprimi, et proinde in eo videri sicut in corporibus fit comprehensio formarum naturalium eius.

4.1 ◉ A number [of authorities] disagree about how the [visible] form is perceived in polished bodies. Accordingly, some of them [suppose] that rays emanate from the eye to the mirror, return from the mirror, and perceive the form of an object [seen in the mirror] upon its return.⁑ Others claim that the form of the object is impressed upon a facing mirror, so it is seen in the mirror the same way that natural forms of objects are perceived in objects.⁑

4.2 ◉ Verum quod aliter sit palam per hoc: quoniam si quis se viderit in aliqua speculi parte motum in partem aliam, non videbit se in parte prima, sed in secunda, quod non accideret si in parte prima infixa esset eius forma. Pari modo, si ad tertiam mutetur partem, mutabitur locus apparentie forme, nec apparebit in prima vel in secunda parte.

4.2 ◉ That in fact [the perception of images in mirrors] happens in another way [than this] is clear from the following. If someone sees himself moving in a given direction in some portion of a mirror, he will no longer see himself in the original place, but in a subsequent place, and this would not happen if his form were impressed on the original portion [of the mirror]. By the same token, if he moves to a third location, the place where his form appears [in the mirror] will shift, and his image will not be seen in the first or second location.

4.3 ◉ Amplius, viso corpore aliquo, et vidente ab eo situ remoto, poterit accidere quod non videat corpus illud in speculo illo, licet videat totam speculi superficiem, quod quidem non esset si imprimeretur forma in speculo, cum videatur speculum et non mutetur locum, et corpus similiter sit immotum, et forma eius inficiat speculum, sicut et prius.

4.3 ◉ Moreover, when some object is viewed [in a mirror] and the viewer draws away from it, it may happen that the object cannot be seen in that mirror, even though he may see the entire surface of that mirror, and this would certainly not be the case if the form were impressed on the mirror[‘s surface] while the mirror remains in view and does not change its location, and while the object, too, remains stationary as its form is impressed on the mirror, just as before.

4.4 ◉ Ut plane appareat non accidere hoc ex comprehensione forme, obturetur medietas foraminum instrumenti, et in aliquo obturatorum sit scriptura aliqua. Si inspiciatur speculum regule per foramen scripturam respiciens, comprehendetur in speculo scriptura, per quodcumque aliud minime. Quod si scripture forma speculo esset impressa, per quodcumque foramen instrumenti posset percipi. Simili modo, in speculis columpnaribus per foramen respiciens tantum comprehendetur scripture situs. Verum in speculis piramidalibus et spericis situs et magnitudo scripture mutabitur.

4.4 ◉ In order to make it obvious that this is not how the form is perceived, block up half of the holes of the [ringlike] apparatus,⁑ and place something with a letter on it at one of the blocked holes. If [the plane] mirror [inserted] in [its] panel is viewed through the hole corresponding to the hole where the letter lies, the letter will be perceived in the mirror, but it will not be [perceived] through any other hole. If the form of the letter were impressed on the [surface of the] mirror, though, it could be perceived through any hole in the apparatus. Likewise, in the cylindrical mirrors, the situation of the letter will be perceived only through the corresponding hole. However, in the conical and spherical mirrors, both the situation and the size of the writing will be changed.⁑

4.5 ◉ Amplius, speculo columpnari extracto, regula super bases suas directe sita apparebit facies hominis in eo directa. Si vero erigatur regula aut multum declinetur, videbitur distorta. Palam ergo quod non accidet comprehensio ex forma fixa in speculo, cum non comprehendatur res visa in speculis nisi fuerit visus in situ reflexionis. Palam etiam quod distortio faciei apparentis non est ex forma rei sed dispositione speculi.

4.5 ◉ Furthermore, if the cylindrical mirror is taken out [of the apparatus] and the panel [in which it is inserted] is placed [sideways] with its top and bottom edges standing vertical, the face of a person will appear frontal in it. If, however, the panel is placed upright or is placed at a considerable slant, [the viewer’s face] will appear distorted.⁑ So it is obvious that what is perceived will not be due to the form’s being impressed on [the surface of the] mirror, since the visible object is not perceived in mirrors unless the eye is at the location of reflection. It is also obvious that the apparent distortion of the face is not due to the [actual] form of the object but to the disposition of the mirror [within which that form is seen].

4.6 ◉ Amplius, viso corpore in speculo et post elongato, comprehendetur corpus magis intra speculum quam prius, quod non erit si forma corporis in superficie speculi sit et ibi comprehendatur. Comprehensionem igitur forme in speculo efficit reflexio.

4.6 ◉ Moreover, if an object is viewed in a mirror and then drawn away, that object will be perceived [to lie] farther behind the mirror than before, and this will not happen if the form of the object lies on the surface of the mirror and is perceived there. Hence, reflection causes the form to be perceived [as it is] in a mirror.⁑

◉[Capitulum 5]

◉[Chapter 5]

Pars quinta: in modo comprehensionis formarum in corporibus politis

5.1 ◉ Iam patuit in parte superiori quod, si opponatur speculo corpus coloratum lucidum, a quolibet eius puncto procedit lux cum colore ad totam speculi superficiem, et reflectitur per lineas reflexionis proprias. Igitur a puncto sumpto in corpore opposito speculo procedit lux cum colore ad speculum in modum piramidis continue, cuius basis est superficies speculi, et forma illa reflectitur per lineas eiusdem situs cum lineis accessus, et erit post reflexionem continuitas sicut in accessu. Et si lineis reflexis occurrat superficies corporis, propter continuitatem earum tota occupabitur ut nichil intersit vacuum. Si ergo forma illius corporis moveatur ad speculum per lineas illas (scilicet per reflexas) et ad basem piramidis pervenerit, quoniam linee piramidis eiusdem sunt situs cum lineis reflexis, reflectitur forma per lineas piramidis, et agregabitur tota in puncto sumpto.

5.1 ◉ It has just been shown in [the third] chapter above that, when a luminous colored body faces a mirror, light, along with color, radiates from any point on it to the entire surface of the mirror and is reflected along the appropriate lines of reflection. Thus, from any given point on a body facing the mirror, light, along with color, radiates to the mirror in the form of a continuous cone with its base on the surface of the mirror, and its form is reflected along lines whose disposition is equivalent to that of the lines of incidence, so the [cone of radiation] after reflection will be continuous with the cone of incidence. And if the form reaches the surface of an object [that faces the mirror as it radiates] along reflected lines, then, because those lines form a continuum, the entire form will fill the [facing] surface [of that object], leaving no area on it untouched. Accordingly, if the form of that body were to radiate to the mirror along these [same] lines (i.e., the [original] lines of reflection), and if it were to reach the base of the cone [on the mirror’s surface], then, since the lines forming this same cone [of incidence] are disposed in the same way as [subsequent] lines of reflection, the form is reflected along lines that form an [equivalent] cone, and the entire form will be concentrated on the point [previously] chosen [on the original luminous body].⁑

5.2 ◉ Quotiens ergo forma alicuius corporis per lineas aliquas ad speculum venerit, si linee ille eiusdem sunt situs cum lineis piramidis a puncto sumpto ad speculum intellecte, cum eas respicientibus, movebitur forma per piramidem illam ad punctum sumptum. Et si in puncto sumpto fuerit visus, videbit corpus cuius est forma illa. Et superius determinatum est quod in situ determinato fit adquisitio forme in speculo. Situs igitur proprius et naturalis adquisitio visus per reflexionem hic est ut linee accessus forme ad speculum eundem habeant situm cum lineis piramidis a centro visus ad capita illarum linearum intellecte unaqueque cum ea respiciente, nec accidit forme reflexe comprehensio nisi in situ isto.

5.2 ◉ So, whenever the form of a given body reaches the mirror along some given lines, then, since those lines are disposed in the same way as the lines forming a cone that can be imagined projected to the mirror from a given point [on the other side], and since those lines correspond, the form [of the given body] will reach the given point according to that [imagined] cone. And if the eye lies at that given point, it will see the body that is represented by that form. And it has been established earlier that a form is apprehended in a mirror at a specific location.⁑ Hence, the appropriate location and the natural way in which sight perceives [an object] in reflection are such that the lines [of radiation] along which the form reaches the mirror are disposed in the same way as the lines comprising the cone that is imagined to extend from the center of sight to the endpoints of each of those corresponding lines, and the reflected form cannot be perceived anywhere else but at this point.

5.3 ◉ Palam ergo quod secundum hanc dispositionem linearum tantum fit comprehensio formarum. Et palam quod ex corpore colorato luminoso procedit lux cum colore ad speculum et reflectitur, nec procedit aliquid ex corpore preter colorem et lucem. Patet igitur quod ex luce et colore tantum huiusmodi forma comprehenditur, et cum moveatur forma ex colore et luce compacta secundum predictam situs observationem, superfluum est dicere quod ab oculo exeant radii ad speculum et reflectantur super situm predictum, sicut a pluribus dictum est. Hic igitur est reflexionis modus geometrarum doctrine non adversus sed consonus, cum in eo geometrice radiorum exeuntium opinione observetur situs, et hic modus michi soli usque nunc patuit.

5.3 ◉ It is thus evident that the perception of forms [in reflection] occurs according to this arrangement of lines alone. It is also evident that light, along with color, radiates to the mirror from a luminous colored body and is [then] reflected [from it], and nothing but color and light radiate from that body. So it follows that the form of such a body is perceived on the basis of light and color alone, and since the form consisting of light and color radiates in such a way as to maintain the disposition described above, it is superfluous to claim, as many authorities have done, that rays pass from the eye to the mirror and then reflect to the place just described. So this [account of] how reflection occurs does not contradict, but accords with, the theory proposed by the geometers, for it preserves the disposition of the visual rays posited in the geometrical theory, and up to now this way [of accounting for reflection] has been evident only to me.⁑

5.4 ◉ Verum cum a corpore luminoso procedat forma ad speculum secundum varietatem situum propter lineas a quolibet puncto corporis ad totam speculi superficiem intellectas, erit forme eiusdem reflexio per diversas piramides quarum capita sunt diversa puncta et bases speculi superficies situm linearum motus forme observantes. Ob hoc accidit quod eadem hora speculo fixo eadem percipitur corporis forma a diversis super quorum intuitis cadunt capita piramidum reflexarum. Similiter, si idem visus moveatur super illa piramidum capita, apparebit ei speculo immoto a locis diversis eadem forma. Sed diversis in speculo eandem formam comprehendentibus in diversa speculi loca cadunt eorum intuitus, quoniam ab eodem speculi puncto diversorum punctorum corporis formas comprehendere eandem non possunt. [5.5] Iam dictum est quod a quolibet corporis puncto procedit lux ad quodlibet punctum speculi, unde super quodlibet punctum corporis est acumen piramidis cuius superficies speculi basis. Et quodlibet superficiei speculi punctum est acumen piramidis cuius basis superficies corporis. Tota ergo forma corporis erit in quolibet speculi puncto per lineas procedentes in partes diversas, nec concurrere possibiles. Et forma a corpore ad quodcumque speculi punctum accedens per piramidem reflectetur per piramidem. Et licet in speculi superficie super numerum multiplicetur eadem iteratio forme, cum concurrat forma totalis cum qualibet parte et in quolibet puncto. Et non sit in formis illis discretio, sed continuitas inseparabilis in reflexione. Tamen, quia forma totalis non cadit in diversas speculi partes, secundum ydemptitatem situs dirigitur ad loca diversa in quibus eam comprehendit visus.

5.4 ◉ However, since the form radiates from the luminous body to the mirror according to various dispositions along lines that are imagined [to extend] from each point on the body to the entire surface of the mirror, the same form will be reflected according to various cones with their vertices at various points and their bases on the surface of the mirror, [and they will be reflected in such a way as] to maintain a [corresponding] disposition with the lines along which the form radiates. From this it follows that at any given instant, with the mirror fixed in place, the form of a body is perceived from various viewpoints where the vertices of the cones of reflection lie. By the same token, when a given center of sight passes over the vertices of such cones while the mirror remains stationary, the same form will appear at different locations in the mirror. But when the same form is perceived at different places in the mirror, the lines of sight are directed to different spots on the mirror, for they cannot grasp the forms of different points on the body identically at the same spot on the mirror.

5.6 ◉ Cum ergo similis sibi fuerit forma speculi figure corporis, erit in speculo complementum forme corporis et figure. Quoniam in speculo eiusdem figure cum corpore forma puncti primi dirigitur ad primum punctum speculi, secundi ad secundum, et sic in omnibus se respicientibus. Et ita erit in speculi superficie figura totalis figure, quod non accidit in speculo alterius figure. Similiter, sumpta quacumque speculi parte cui eadem cum corpore figura, erit complementum figure corporis in ea. Et cum infinite sint tales speculi partes, infinite erunt forme corporis reflexionis sed ad puncta diversa procedentes ex quibus formam comprehendit visus.

5.6 ◉ Thus, if the shape of the mirror is the same as the shape of the object [seen in it], the complement of the form and shape of the body will be [seen] in the mirror. For in the case of a mirror that is the same shape as the object, the form of the first point [on that object] is directed to the first point on the mirror, the second to the second, and so forth among all the corresponding points. Hence, the shape of the entire object will be [seen] on the mirror’s surface, which does not happen in the case of a mirror with a different shape.⁑ Likewise, if some portion of the mirror is taken, and if it is the same shape as the object [seen in it], the complement of the shape of the object will be [seen] in it. Moreover, since such parts on the mirror are infinite, the forms of the body that are reflected will be infinite, but they will radiate to different points where the viewer perceives the form.

5.7 ◉ Cum igitur secundum hanc linearum dispositionem fiat forme comprehensio, non erit forme procedentis a corpore in speculi superficie fixio. Et in hunc modum accidit in omnibus speculis, sed in planis certius; in aliis autem accidit quedam diversitas ex errore visus secundum modum predictum. Et quilibet visus secundum modum predictum ab uno speculi puncto non percipit nisi unum corporis punctum, nec a duobus percipitur in eodem speculi puncto idem corporis punctus.

5.7 ◉ Thus, since the form is perceived according to this arrangement of lines, the form radiating from the body will not be impressed on the surface of the mirror. And this is the way reflection occurs for all mirrors, but most evidently in plane mirrors; in the other [sorts of mirror], however, some variation occurs because of visual error, which occurs in the way previously mentioned.⁑ So, according to the way just described, any [given] center of sight perceives only one of the object’s points at one point on the mirror, and the same point on the object is not perceived by two centers of sight at the same point on the mirror.

5.8 ◉ Amplius, si opponatur speculum visui, et intelligatur a centro visus ad speculi superficiem piramis, et basis illius piramidis si sumatur punctum, et intelligatur linea piramidis a centro visus ad illud punctum, cum a puncto illo infinite possunt produci linee, si aliqua earum cum latere piramidis eundem habeat situm et equalem cum perpendiculari teneat angulum, et ita accidat quolibet puncto speculi sumpto, planum quod a quolibet puncto speculi potest fieri reflexio. Dico ergo quod inter lineas a puncto sumpto productas est linea eundem habens situm cum latere piramidis, et equalem tenet angulum cum perpendiculari super illud punctum. Et est illa latus piramidis intellecte a puncto illo superficiei rei occurrentis, et quod super terminum illius linee ceciderit, cum per eam ad punctum sumptum venerit, reflectetur ad visum per latus piramidis eius iam dictum. Et hoc piramidis latus cum linea a puncto illo producta erit in eadem superficie ortogonali super superficiem speculum in illo puncto tangentem. Et hoc dico, cum lateris piramidis super punctum sumptum fuerit declinatio. Si enim ortogonaliter cadat super superficiem speculum in puncto sumpto tangentem latus piramidis productum a centro visus, reflectetur in se et redibit in visum ad originem sui motus.

5.8 ◉ Furthermore, suppose that a mirror faces the eye, and imagine a cone [extending] from the center of sight to the surface of the mirror. Let a point be taken on the base of this cone, and imagine a line [extending] within this cone from the center of sight to that point. Then, given that an infinite number of lines can be extended from that point, since each of them maintains an equivalent disposition with respect to its cone’s edge and forms an equal angle with the normal, and since this holds for every point taken on the mirror, it is clear that reflection can occur from every point on the mirror. I say, therefore, that among [each of] the lines extended from the given point there is a line that is equivalently disposed with respect to the edge of the cone, and it forms an equal angle with the normal to that point. And this [line] forms the edge of a cone imagined to extend from that point to the surface of the [visible] object, and whatever lies at the endpoint of that line, since [its form] reaches the given point [on the mirror] along that line, [that form] will be reflected to the center of sight along the edge of the cone just described. Furthermore, this edge of the cone will lie in the same plane as the line projected from that point, and that plane is perpendicular to the plane tangent to the mirror at that point. And I say [that] this [is so] when the edge of the cone is oblique to that point. If, in fact, the [line along the] edge of the cone extending from the center of sight falls orthogonally to the plane tangent to the surface of the mirror at the given point, it will be reflected back onto itself and will return to the source of its propagation in the eye.

5.9 ◉ In speculo plano planum est quod diximus, quoniam in quodcumque punctum superficiei plane cadat radius a puncto illo potest erigi linea ortogonalis super superficiem illam, et a centro visus potest intelligi linea perpendiculariter cadens in superficiem planam predicte continuam, aut in eandem. Et hee due perpendiculares erunt in superficie eadem, quoniam sunt equidistantes, et linea a termino unius usque ad terminum alterius protracta in superficie plana tenebit angulum acutum cum utraque, et erit in eadem superficie cum utraque. Et radius qui a linea illa elevatur tenebit acutum angulum cum perpendiculari speculi, et similiter cum perpendiculari visus. Et intelligatur linea in alteram partem superficiei plane transiens ortogonaliter per terminos perpendicularium. Tenebit ex parte alia cum perpendiculari speculi angulum rectum, unde ex illo recto poterit abscindi angulus acutus equalis angulo acuto quem cum eadem perpendiculari tenet radius. Et hii duo anguli in eadem superficie, quare radius exiens et reflexus in eadem superficie et in superficie perpendicularium dictarum. Inspecto autem alio puncto, idem situs accidet radiorum cum perpendicularibus quarum una linea a puncto viso, alia a centro visus.

5.9 ◉ What we have claimed is obvious in the case of a plane mirror, for at whatever point on the [mirror’s] plane surface the ray from that point [on the given object] falls, a normal can be erected to that surface. And from the center of sight a line can be imagined to fall orthogonally to the surface continuous with the aforementioned surface, or to the surface itself. And these two perpendiculars will lie in the same plane, because they are parallel, and the line extending in that plane surface from the endpoint of one to the endpoint of the other will form an acute angle with both and will lie in the same plane with both. So the ray projected according to that line will form an acute angle with the normal to the mirror [at the point of reflection], and likewise [it will form an acute angle] with the perpendicular drawn from the center of sight.⁑ Then imagine a line drawn on the other side of the plane surface passing orthogonally through the endpoints of the perpendiculars. On that other side it will form a right angle with the normal to the mirror, so from that right angle an acute angle can be cut that is equal to the acute angle the ray forms with that same normal. These two angles [will lie] in the same plane, so the incident and reflected rays lie in the same plane as the two aforementioned perpendiculars. Moreover, if some other point [on the visible object] is viewed [in the mirror], the [two] rays will be equivalently disposed with respect to the perpendiculars, one of those rays extending from the point viewed, the other from the center of sight.

5.10 ◉ In omni ergo superficie reflexionis accidit quatuor punctorum concursus—scilicet centrum visus, et punctus comprehensus, et terminus perpendicularis a centro visus, et punctus reflexus. Et omnes reflexionis superficies secant se in perpendiculari a puncto reflexionis intellecta, et ipsa est communis omnibus superficiebus reflexionis. Et cum idem accidat quolibet puncto superficiei plane inspecto, erit ex omnibus punctis similis reflexio, et eodem modo.

5.10 ◉ Hence, in every plane of reflection four points occur in a related group—i.e., the center of sight, the point [on the visible object] that is perceived, the endpoint of the perpendicular [dropped] from the center of sight [to the mirror’s surface], and the point of reflection.⁑ And all the planes of reflection [for the given point of reflection] intersect along the normal imagined to extend to the point of reflection, and this line is common to all the planes of reflection. And since the same thing happens when any point on the plane surface is viewed, there will also be an identical reflection from all points, and it will occur in the same way.

5.11 ◉ In speculis autem spericis palam erit quod diximus. Opposito visui speculo sperico—et est oppositio ut visus non sit in superficie illius sperici aut in superficie continua et sperica—et inspecto hoc speculo, pars eius comprehensa erit pars spere circulo inclusa quam efficit motu suo radius tangens superficiem spere, si per girum moveatur contingendo speram donec redeat ad punctum primum a quo sumpsit motus principium. Et si intelligantur superficies se secantes super dyametrum spere a polo circuli predicti intellectum, quilibet arcuum superficiei spere et hiis superficiebus communium a polo circuli ad ipsum circulum intellectorum erit minor quarta circuli magni, quoniam linea a centro spere ad terminum radii speram contingentis protracta—et est ad circulum predictum—tenet cum radio angulum rectum ratione contingentie. Tenet ergo angulum acutum cum semidyametro a polo circuli producto, et hunc angulum respicit arcus interiacens polum circuli et circulum, quare quilibet horum arcuum erit minor quarta circuli.

5.11 ◉ Furthermore, what we have claimed will be clear in the case of [convex] spherical mirrors. If a [convex] spherical mirror faces the eye—and such a facing situation means that the eye does not lie on the surface of the spherical mirror itself or on the spherical surface continuous [with that surface]⁑—and if one looks into this mirror, the portion of it that is perceived will be the portion of the sphere defined by the circle cut off by a ray tangent to the surface of the sphere that rotates [axially] about that circle of tangency until it returns to the original point where the rotation began. And if one imagines planes intersecting along the diameter that is imagined to extend [through the sphere] from the pole of the aforesaid circle, any of the arcs upon the surface of the sphere that are imagined to be the common sections formed by these planes on the surface of the sphere between the pole and the [defining] circle will be less than a quarter of the great circle [of the sphere], because the line from the center of the sphere to the endpoint of [any] ray drawn tangent to the sphere—that is, to the aforementioned [defining] circle—forms a right angle with the ray on account of its tangency. Hence, that tangent ray forms an acute angle with the radius drawn from the circle’s pole, and this angle corresponds to the arc between the circle’s pole and the [defining] circle itself, so every one of these arcs is less than a quarter of the [great] circle.⁑

5.12 ◉ Dico ergo quod a quolibet huius portionis puncto poterit fieri reflexio, quoniam, sumpto aliquo eius puncto, dyameter spere ab illo puncto intellectus erit perpendicularis super superficiem planam tangentem speram in puncto illo. Et huius rei probatio est: Intellectis duabus superficiebus speram super dyametrum a puncto sumpto intellectum secantibus, linee communes superficiei spere et hiis superficiebus sunt circuli spere transeuntes per punctum sumptum. Et intellectis duabus lineis tangentibus hos circulos in puncto sumpto, erit dyameter perpendicularis super utramque lineam, quare super superficiem in qua sunt ille linee. Et cum descenderit radius super punctum sumptum, erit in eadem superficie cum dyametro spere cuius terminus est punctus sumptus, et linea a centro visus ad centrum spere intellecta, que quidem transit per polum circuli, et est radius ortogonaliter cadens super superficiem spere. Et ex hiis tribus lineis erit triangulus, et radius super punctum sumptum incidens tenet acutum angulum cum dyametro spere ab exteriori parte, quoniam, cum elatior sit iste radius radio speram contingente, secabit speram cum productus intelligitur. Et superficies tangens speram in puncto sumpto dimissior erit hoc radio, et secabit inter speram et visum dyametrum—id est lineam a centro visus ad centrum spere intellectam per polum circuli transeuntem.

5.12 ◉ I say, then, that reflection can occur from any point on this portion [of the sphere], for, if any point on it is taken, the diameter of the sphere imagined [to extend] from this point will be perpendicular to a plane tangent to the sphere at that point. The proof of this claim is as follows. If two planes are imagined to cut the sphere along the diameter imagined [to extend] through the given point, the common sections of the sphere’s surface and these planes form [great] circles on the sphere that pass through the given point. And if two lines are imagined tangent to these circles at the given point, the diameter will be orthogonal to both of the lines, so [it is orthogonal] to the plane in which they lie. And when a ray is incident to the given point, it will lie in the same plane as the sphere’s diameter, whose endpoint is the given point, and [it also lies in the same plane as] the line imagined [to extend] from the center of sight to the center of the sphere, which of course passes through the circle’s pole, and this radial line also falls orthogonally to the surface of the sphere. So a triangle will be formed by these three lines, and the ray that is incident to the given point forms an acute angle with the diameter of the sphere [extended] beyond [its surface], for, since this ray lies above the ray that is tangent to the sphere [at the given point], it will cut the sphere when it is imagined to extend [into the sphere]. And the plane tangent to the sphere at the given point will lie below this ray, and it will extend between the sphere and the visual axis—i.e., the line that is imagined to pass from the center of sight to the center of the sphere through the pole of the [defining] circle [formed by the rotating tangent].

5.13 ◉ Unde cum dyameter spere sit ortogonalis in superficie punctum tangente, tenebit angulum recto maiorem ex interiori parte cum radio in punctum descendente, unde in exteriori parte tenebit cum eo angulum minorem recto. Et productus ortogonalis erit super superficiem contingentem exterius, quare ex angulo recto quem tenebit cum superficie ex alia radii parte poterit abscindi acutus equalis ei quem includit radius cum illo dyametro. Et erunt linee tres. hos duos angulos includentes in eadem superficie, quare a puncto portionis sumpto potest produci linea in eadem superficie cum radio in punctum illud cadente et linea ortogonali in superficie punctum contingente et ad paritatem angulorum cum perpendiculari illa. Et illi linee occurret forma puncti mota ad superficiem speculi per radium illum. Igitur eiusdem est situs cum linea que poterit reflecti, et erit superficies in qua sunt hee linee ortogonalis super superficiem speram in puncto contingentem, et ita in quolibet puncto portionis intelligendum.

5.13 ◉ Therefore, since the diameter of the sphere is perpendicular to the plane that is tangent [to the sphere] at the [given] point, it will form an angle greater than a right angle with the [incident] ray inside the circle at the point of incidence, so outside the circle it will form with it an angle less than a right angle. And when [this diameter is] extended beyond [the sphere’s surface], it will be orthogonal to the plane tangent [to that surface at the given point], so within the right angle that it will form with that plane on the other side of the ray an acute angle can be cut off equal to that [original acute angle] formed by a ray with that diameter. And the three lines forming these two angles will lie in the same plane, so a line can be drawn from the point on that portion [of the surface] that lies in the same plane as the ray that is incident to that point, as well as in the same plane as the normal to the plane tangent to the surface at that point, and it can form an equal angle with that normal. And the form of a point propagated to the mirror’s surface along that ray will reach that line. Thus, it is equivalently disposed with respect to the line along which it can be reflected, and the plane in which these lines lie is orthogonal to the surface of the sphere at the point of tangency, and this should be understood to hold for every point on the [visible] portion [of the spherical mirror].⁑

5.14 ◉ Ergo in omni superficie reflexionis erunt centrum visus, centrum spere, punctus reflexionis, et punctus reflexus, et omnes hee superficies secabunt se super lineam a centro visus ad centrum spere protractam. Et cuilibet reflexionis superficiei et superficiei spere communis linea erit circulus spere, et omnes circuli secabunt se super punctum spere in quem cadit dyametrum visus, et est super circuli portionis polum. Cum autem radius ceciderit in speculum ortogonaliter super superficiem in punctum in quem cadit radius speram tangentem—et est radius ille dyameter visus per polum circuli portionis ad centrum spere—fiet reflexio ad visum per eundem radium ad motus radii ortum.

5.14 ◉ Therefore, the center of sight, the center of the sphere, a point of reflection, and a point [whose form is] reflected will lie in every plane of reflection, and all such planes will intersect along the line that extends from the center of sight to the center of the sphere. And the common section for each plane of reflection and the surface of the sphere will be a [great] circle on that sphere, and all [these] circles will intersect at the point on the [surface of the] sphere where the visual axis falls, and this [point] lies on the pole of the circular section [of visibility defined by the tangent rays].⁑ Moreover, when the ray strikes the mirror along the perpendicular to the surface where the plane tangent to the sphere touches the sphere’s surface at the point where the ray strikes—and this ray constitutes the visual axis [that passes] along the pole of the circular segment to the center of the sphere—the reflection will take place along the same radial line to the eye according to the source of the ray’s propagation.

5.15 ◉ In speculis autem columpnaribus patebit quod diximus. Opponatur speculum columpnare exterius politum oculo—et est oppositio ut non sit visus in superficie columpne aut superficie ei continua—et intelligemus superficiem a centro visus ad columpne superficiem secantem columpnam super circulum equidistantem basibus columpne. Et in hac superficie sumantur due linee tangentes circulum sectionis in duobus punctis oppositis. Ab utroque illorum punctorum producatur linea secundum longitudinem columpne, et intelligantur due superficies in quibus sunt hee due linee longitudinis et due linee a centro visus ducte contingentes circulum sectionis. Dico quod hee superficies tangent columpnam.

5.15 ◉ Now, what we have said [above] will also be clear in the case of [convex] cylindrical mirrors. Place a cylindrical mirror that is polished on the outer [convex] surface directly facing the eye—and such a facing situation means that the eye does not lie on the surface of the cylindrical mirror itself or on the continuation of that surface—and we shall imagine a plane [passing] through the center of sight that cuts the surface of the cylinder along a circle that is parallel to the bases of the cylinder. Then, in this plane take two lines that are tangent to the circle of intersection at two opposite points. From each of those points extend a line along the length of the cylinder, and imagine two planes that contain these two lines [extending] along the [cylinder’s] length as well as the two lines that are drawn from the center of sight to the circle of intersection and tangent to it. I say that these [two] planes are tangent to the cylinder.

5.16 ◉ Si enim dicatur quod altera secat illam, planum quod sectio erit super lineam longitudinis columpne in quam superficies cadit, et similiter erit sectio super lineam longitudinis columpne huic oppositam. Et circulus sectionis transit per has duas lineas longitudinis. Et linea contingens circulum sectionis, cum sit in superficie aliqua, secat columpnam super aliquas longitudinis lineas sibi invicem equidistantes, et si transit per unam earum, transibit per alteram, et ad paritatem angulorum. Cum ergo transeat per punctum in quo circulus sectionis secat primam longitudinis lineam, transibit etiam per punctum in quo alia longitudinis linea tangit hunc circulum. Et ita secat circulum, quare non erit contingens, quod est contra ypothesim. Palam ergo quod ille due superficies contingunt speculum, et quod inter illas cadit ex superficie speculi est quod apparet visui.

5.16 ◉ For if it is claimed that either of them cuts it, then it is obvious that an intersection will occur upon a line that extends along the length of the cylinder where the plane falls on its surface, and likewise an intersection will occur along a line of longitude on the opposite side. And the circle of intersection passes through these two lines of longitude. In addition, since the line tangent to the circle of intersection lies in some plane, it cuts the cylinder according to some [given] lines along the length that are parallel to one another, and if it intersects one of them it will intersect the other, both cuts being at equal angles. Therefore, if it passes through the point where the circle of intersection touches the first line along the [cylinder’s] length, it will also pass through the point where the other line along its length touches that circle. Accordingly, it cuts the circle, so it will not be tangent to it, and this is counter to what we supposed. It is clear, then, that these two planes are tangent to the mirror and that whatever lies between them upon the surface of the mirror is visible to sight.⁑

5.17 ◉ Cum autem illarum duarum superficierum sit concursus in centro visus, secabunt se, et linea sectionis communis transibit per centrum visus, et est equidistans axi columpne, quoniam axis columpne ortogonalis est super eundem circulum sectionis. Et linee longitudinis columpne ortogonales super eundem circulum, et superficies tangentes columpnam secundum lineas has sunt ortogonales super circulum eundem. Quare super superficiem secantem columpnam in illo circulo, quare linea communis harum superficierum est ortogonalis super eandem superficiem, quare equidistans axi columpne.

5.17 ◉ Furthermore, since these two planes meet at the center of sight, they will intersect, and their common section will pass through the center of sight, and it is parallel to the axis of the cylinder, because the cylinder’s axis is perpendicular to the circle of intersection. The lines along the length of the cylinder are also perpendicular to that circle, and the planes tangent to the cylinder along these lines are perpendicular to that same circle. So they are perpendicular to the plane cutting the cylinder along that circle, and the common section of these planes is perpendicular to that same plane, so they are parallel to the cylinder’s axis.⁑

5.18 ◉ Dico ergo quod quocumque puncto in sectione speculi apparente sumpto, linea a centro visus ad punctum producta secabit speculum. Quoniam intellecta linea longitudinis columpne a puncto sumpto, transibit per circulum sectionis, et tanget ipsum in puncto ad quem punctum, si ducatur linea a centro visus, secabit speculum quod cadit inter lineas contingentes hunc circulum. Et superficies a centro procedens in qua fuerit hec linea secabit speculum. Cum ergo in eadem superficie fuerit linea hec et linea a centro ad punctum sumptum ducta, secabit linea illa speculum, et ita quelibet linea a centro visus ad portionem speculi intellecta secat speculum. Eodem modo quelibet linea a linea communi per centrum visus intellecta ad hanc portionem ducta secat speculum, unde quelibet superficies tangens speculum in aliqua portionis apparentis linea secat superficies que contingunt portionis extremitates. Et nulla omnium superficierum portionem tangentium pervenit ad visus centra; sed inter visum extendetur et speculum.

5.18 ◉ I say, then, that whatever point is taken on the visible portion of the mirror, the line extended from the center of sight to that point will cut the mirror. For if the line extended from that point along the length of the cylinder is imagined, it will pass through the circle of intersection and will touch it at a point where a line drawn from the center of sight will intersect the portion of the mirror that lies between the lines tangent to this circle. Also, the plane passing through the center [of sight] that contains this line will cut the cylinder. Therefore, since this line lies in the same plane as the line drawn from the center of sight to the point chosen [on the mirror’s surface], it will intersect the mirror, and so, any line that is imagined [to extend] from the center of sight to the [visible] portion of the mirror cuts the mirror. By the same token, any line that is imagined to extend from the common section [of the two tangent planes passing] through the center of sight to that portion [of the mirror] cuts the mirror, so whatever plane is tangent to the mirror along any line on the visible portion cuts the planes that are tangent at that portion’s edges. And not one of all the planes tangent to that portion passes through the center of sight; instead, it will extend between the eye and the mirror.

5.19 ◉ Dico ergo quod a quolibet puncto portionis huius potest fieri reflexio lucis. Dato enim puncto, fiat super ipsum circulus equidistans columpne basibus. Si ergo superficies a centro visus procedens et columpne superficiem equidistantem basibus secans, secet eam super hunc circulum, et linea a centro visus ad circuli centrum ducta transeat per punctum datum. Fiet reflexio forme illius puncti per eandem lineam ad linee ortum, quia linea illa est axis visus super axem columpne perpendicularis. Sumpto autem quocumque per quem transeat axis perpendiculariter super axem columpne, fiet reflexio illius puncti per eundem axim.

5.19 ◉ Accordingly, I say that light can be reflected from any point on this portion [of the mirror]. For, given such a point, let a circle parallel to the bases of the cylinder be drawn through it. Hence, if the plane passing through the center of sight cuts the cylinder along a plane parallel to the bases, it should cut the cylinder along that circle, and the line extended from the center of sight to the center of the circle should pass through the given point. The form of that point will be reflected back along the same line to its origin-point, because that line is the visual axis [which is] perpendicular to the axis of the cylinder. Furthermore, if any point through which the [visual] axis passes orthogonally to the axis of the cylinder is taken, the reflection of that point will take place along that very [visual] axis.

5.20 ◉ Si vero pretereat axim punctus sumptus, quecumque sit linea a centro circuli super ipsum equidistantis basibus punctum ducti, ad superficiem in linea longitudinis columpne per punctum illud transeuntis contingentem erit ortogonalis super axem, quare super lineam longitudinis per punctum illud transeuntem. Et quoniam visus est altior superficie punctum contingente, linea a centro visus ad punctum sumptum ducta tenebit acutum angulum cum perpendiculari illa a puncto ad centra circuli ducta. Et hoc ex parte exteriori, quia obtusum ex interiori. Et ex angulo recto quem illa perpendicularis tenet cum linea superficiei contingente circulum poterit abscidi acutus huic equalis. Et perpendicularis illa cum centro visus in eadem superficie, quare cum linea a centro ad punctum ducta. Et erit linea reflexa in eadem superficie, et erit hec superficies ortogonalis super superficiem contingentem speculum in puncto illo, quoniam perpendicularis ortogonaliter cadit super hanc superficiem. Et huiusmodi erit reflexionis superficies.

5.20 ◉ But if the chosen point lies outside this axis, then any line extended to that point from the center of the circle drawn parallel to the bases [of the cylinder]—that line also being extended to the plane tangent to the line along the length of the cylinder passing through that point—will be orthogonal to the [cylinder’s] axis, so it is orthogonal to the line along the length [of the cylinder] passing through that point. And since the center of sight lies above the plane tangent [to the cylinder] at that point, the line drawn from the center of sight to the chosen point will form an acute angle with the normal extended to that point from the center of the circle. This is the case outside [the cylinder], so [the angle formed] inside [the cylinder] is obtuse. And from the right angle that the normal forms with the line on the plane that is tangent to the circle [of intersection] an acute angle the same size [as the aforementioned angle] can be cut. And that normal lies in the same plane as the center of sight, so it lies in the same plane as the line drawn from the center [of sight] to the [chosen] point. So the reflected ray will lie in the same plane [as this line], and this plane will be orthogonal to the plane tangent to the mirror at that [chosen] point, for the normal falls orthogonally to this plane. And this is how the plane of reflection will be defined.⁑

5.21 ◉ Est autem diversitas inter lineas superficiebus reflexionis et superficiei columpne communes, cum enim reflexio erit per eundem radium, cadet idem radius ille ortogonaliter super axem. Et linea communis superficiei columpne et superficiei reflexionis erit linea recta—scilicet latus columpne—cum in superficie reflexionis sit dyameter columpne. Et hoc planum, quoniam columpne compositio est ex motu superficiei equidistantium laterum super unum latus immotum. Unde superficiei columpnam secanti in qua sit axis—id est latus immotum—communis linea ei et superficiei columpne erit latus motum. Et dico quod ex omnibus reflexionis superficiebus una sola est cui et columpne superficiei sit linea communis recta, quoniam unica potest intelligi superficies in qua sit axis columpne et centrum visus, et non plures.

5.21 ◉ Furthermore, the common sections of the planes of reflection and the surface of the cylinder vary among each other, for, when reflection occurs [back] along the same ray [along which the form reaches the mirror], that same ray will fall orthogonally to the axis [of the mirror]. And the common section of the cylinder’s surface and the plane of reflection will be a straight line—i.e., an edge of the cylinder—since the diameter of the cylinder lies in the plane of reflection. And this is clear because a cylinder is generated by the rotation of a plane with parallel sides about one of those sides, which remains stationary [to form the axis of rotation]. Accordingly, the common section of a plane that cuts the cylinder through its axis—i.e., the stationary side [of the generating surface]—and the surface of the cylinder will be the side that moves [in the process of generation]. And I claim that only one among all the planes of reflection forms a rectilinear common section with the surface of the cylinder, for only one plane, and no more, can be imagined to contain the axis of the cylinder and the center of sight.

5.22 ◉ Si vero superficies reflexionis sit equidistans basibus columpne, erit linea communis circulus, et hec sola est superficies que cum columpne superficie lineam communem habeant circularem, quoniam in omni reflexione perpendicularis super superficiem contingentem punctum reflexionis est dyameter circuli basibus columpne equidistantis. Et non potest esse in columpne superficie nisi unus circulus equidistans basibus qui cum centro visus sit in eadem superficie. Omnes alie reflexionis superficies secant columpnam et axem columpne, quoniam perpendicularis ducta a puncto reflexionis secat axem columpne, et linee communes hiis superficiebus et superficiei columpne sunt sectiones quas in columpnis et piramidalibus assignant geometre.

5.22 ◉ On the other hand, if the plane of reflection is parallel to the bases of the cylinder, the common section will be a circle, and this is the only plane that can form a circular common section with the surface of the cylinder, for in the case of every reflection, the normal to the plane that is tangent to the point of reflection forms the diameter of a circle parallel to the bases of the cylinder. And on the surface of the cylinder there can be but one circle that is parallel to the bases and that lies in the same plane as the center of sight. All the remaining planes of reflection cut the [surface of the] cylinder and the cylinder’s axis, for the normal dropped to the point of reflection cuts the cylinder’s axis, but the common sections of these planes and the surface of the cylinder form the sections that geometers attribute to cylinders and cones [i.e., ellipses].⁑

5.23 ◉ Cum superficiebus columpne et reflexionis linea recta fuerit communis, quodcumque punctum illius linee intueatur visus, fit reflexio in superficie eadem in qua scilicet est axis, quoniam est superficies unica contingens columpnam in linea illa longitudinis. Et quocumque puncto huius linee sumpto, perpendicularis ab eo ad axem ducta erit in eadem superficie cum axe, et hec longitudinis linea ortogonalis est super superficiem contingentem superficiem columpne. Sed centrum visus est in superficie ortogonali, ut super eandem, et sit in ea axis columpne et linea communis, et una sola est superficies ortogonalis super illam superficiem in eadem, quare omnes reflexiones a punctis huius linee facte sunt in eadem reflexionis superficie. [5.24] Verum cum linea communis superficiei reflexionis et columpne fuerit circulus, quocumque puncto illius circuli viso, fiet in una et eadem superficie reflexio, quoniam quecumque perpendicularis a puncto viso ducta erit dyameter huius circuli, quare in superficie huius circuli, et punctum visus similiter. Et superficies huius ortogonalis est super superficiem quodcumque punctum huius circuli sumptum contingentem, quare in hac sola superficie erit cuiuslibet puncti predicti circuli reflexio. Quacumque vero alia linea communi sumpta, non fiet in eadem reflexionis superficie reflexio nisi ex uno tantum huius linee puncto, quoniam perpendicularis ducta a puncto reflexionis ortogonalis est super lineam longitudinis columpne per punctum illud transeuntis, quare et super axem. Et perpendicularis illa est dyameter circuli equidistantis basibus columpne. Et superficies reflexionis et circulus ille secant se, et linea eis communis est dyameter illius circuli, et est illa perpendicularis, et superficies reflexionis secans est, et est declinata super ipsum. Et in superficie super lineam aliquam declinatam non potest intelligi nisi una linea ortogonaliter cadens in illam. Si a duobus reflexionis superficiei punctis fieret reflexio in eadem superficie, essent due linee illius superficiei ortogonales super axem, quod esse non potest, cum ipsa sit delinata super eum.

5.23 ◉ When the common section of the surface of the cylinder and the plane of reflection is a straight line [along the length of the cylinder], then, to whatever point on that line one directs his sight, reflection occurs in the same plane as the axis, for there is only one plane tangent to the cylinder along that line of longitude. So, given some point on this line, the perpendicular extended from it to the axis will lie in the same plane as the axis, and this extended line is perpendicular to the plane that is tangent to the surface of the cylinder [along that line of longitude]. But the center of sight lies in a plane orthogonal [to the plane of tangency], and so it lies in the same plane [as just described], and the axis of the cylinder and the common section lie in that plane, and there is only one plane orthogonal to that surface along the same common section, so all reflections that take place from points on this line lie in the same plane of reflection.

5.25 ◉ Amplius, perpendicularis a puncto reflexionis cadit in circulum equidistantem basibus columpne et in puncto axis communis circuli et superficiei reflexionis. Si ergo ab alio linee communis puncto in eadem superficie fieret reflexio, alia perpendicularis ab alio puncto ducta esset dyameter alterius circuli columpne huic equidistantis, et caderet in punctum axis in quod non cadit superficies reflexionis. Et ita in omnibus reflexionis superficiebus est intelligendum quod ab uno tantum puncto linee communis fiat reflexio in eadem superficie respectu eiusdem visus. Quoniam respectu duorum visuum potest fieri a duobus punctis circuli dyametri terminus, id est perpendicularis. Respectu vero unius visus non accidit, quoniam illa duo puncta non simul ab eodem visu possunt comprehendi, semper enim necesse est partem columpne medietate minorem videri.

5.25 ◉ Moreover, the normal through the point of reflection falls on a circle that is parallel to the bases of the cylinder, and [it falls on] a point on the axis that is common to the circle and the plane of reflection. If, therefore, reflection were to occur from any other point on the common section [of the cylinder and the plane of reflection] within the same plane, there would be another normal dropped from the other point, and it would form the diameter of another circle on the cylinder parallel to the first one, and it would fall on a point on the axis where the plane of reflection does not fall. And so it must be understood that in every [oblique] plane of reflection, reflection occurs from only one point on the common section [of the cylinder and the plane of reflection] in the same plane with respect to the same center of sight. Hence, with respect to both eyes [reflection] can take place from the two endpoints of the circle’s diameter, which constitutes the normal. Yet with respect to one eye this does not occur, because those two points cannot be perceived at the same time by the same eye, for [with one eye] it is necessarily the case that less than half of the cylinder is seen.⁑

5.26 ◉ Palam ex predictis perpendicularem super punctum reflexionis intellectam exterius intus transeuntem dyametrum circuli efficere, quia, si non, cum constet dyametrum circuli super punctum illud transeuntem perpendicularem esse super superficiem contingentem columpnam in puncto illo, et perpendicularem exterius similiter, erit continuitas inter has perpendiculares, et unam efficient lineam. Quia, si non est quod dyametrum extra productum perpendiculare sit super illam superficiem, accidet ex eodem superficiei puncto duas erigi perpendiculares. In omni ergo superficie reflexionum patet quatuor punctorum concursus: centri visus, puncti axis in quem cadit perpendicularis, puncti visi in speculo, puncti a quo forma corporis procedit.

5.26 ◉ From the foregoing it is evident that the perpendicular imagined to extend to the point of reflection from outside [the cylinder] forms the diameter of the circle as it passes inside [the cylinder], for, if such were not the case, then, since it is agreed that the diameter of the circle, which passes to that point, is perpendicular to the plane tangent to the [surface of the] cylinder at that point, and [since it is agreed that it is] also perpendicular [to that plane] on the outside [surface of the cylinder], there will be continuity between these perpendiculars, and they will form one line. For if the diameter is not perpendicular to that plane [of tangency] when it is extended beyond [the cylinder’s surface], it will follow that two perpendiculars could be drawn to the same point on a [given] plane. Hence, it is clear that in every plane of reflections four points occur in a related group: the center of sight, the point on the axis where the normal falls, the point that is seen on the mirror [i.e., the point of reflection], and the point from which the form of the [visible] object radiates.

5.27 ◉ In speculis piramidalibus super bases suas ortogonalibus politis exterius est oppositio visus ut non sit visus in superficie speculi aut in ei continua, et secundum visus situm respectu speculi piramidalis erit quantitas comprehense in eo partis.

5.27 ◉ In the case of conical mirrors that are perpendicular to their bases [i.e., formed from right cones] and that are polished on their outer [convex] surfaces, the eye is considered to face them if it does not lie on the surface of the mirror or on the continuation of that surface, and how much of [the surface of the] mirror is seen will depend on how the eye is disposed with respect to the mirror.

5.28 ◉ Igitur, si radius ab oculi centro ad terminum axis piramidis—id est ad acumen intellectus—faciat cum axe angulum acutum ex parte piramidis, intelligemus a centro visus superficiem secantem piramidem super circulum equidistantem basi piramidis. Et intelligemus duas lineas a centro quidem visus tangentes illum circulum in punctis oppositis, a quibus punctis protrahemus lineas secundum longitudinem piramidis. Superficies ergo ex una harum linearum longitudinis et altera contingentium circulum continget piramidem, si enim secaverit, continget aliud punctum quam punctum contingentie circuli. Super illud punctum producatur linea longitudinis piramidis, et illud punctum et acumen piramidis sunt sicut in hac superficie, quare illa linea erit in hac superficie et transibit per aliquod punctum circuli. Illud ergo punctum est in hac superficie et in circulo, quare est in linea communi circulo et superficiei. Sed illa est contingens circulum, quare contingens transit per duo puncta circuli quem contingit, quod est impossibile. Restat ergo quod superficies illa tangat piramidem.

5.28 ◉ Accordingly, if the ray from the center of sight to the endpoint of the cone’s axis—i.e., [the ray that is] understood [to extend to] the vertex—forms an acute angle with the axis on the [visible] side of the cone, we will imagine a plane [passing] through the center of sight and cutting the cone’s surface along a circle that is parallel to the base of the cone. And we will imagine two lines from the center of sight that are tangent to that circle at opposite points, and from those points we will draw lines along the length of the cone. Hence, the plane containing either of these lines along the length [of the cone] as well as the [corresponding line that is] tangent to the circle will be tangent to the cone, for if it were to cut the cone, it would touch some other point besides the point of tangency on the circle. On that [other] point draw a line along the length of the cone, and that point and the vertex of the cone lie in this plane, so that line will lie in this plane and will pass through some point on the circle. That point, therefore, lies in this plane and on the circle, so it lies on a line that is common to the circle and to the surface [of the cone]. But it is [also] tangent to the circle, so this tangent passes through two points on the circle that it touches, which is impossible.⁑ It follows therefore that this plane is tangent to the [surface of the] cone.

5.29 ◉ Et generaliter omnis superficies reflexionis in qua concurrunt linea tangens aliquod punctum piramidis et linea longitudinis per illum punctum transiens tangit piramidem super lineam longitudinis. Habemus ergo duas superficies ab oculi centro procedentes piramidem contingentes inter quas est portio piramidis apparens visui in hoc situ, et est minor medietatum piramidis, quoniam linee contingentes circulum includunt eius partem medietatum minorem.

5.29 ◉ And it is invariably the case that any plane of reflection in which the line that is tangent to some point on the cone and the line of longitude that passes through that point intersect is tangent to the cone along the line of longitude. Hence, we have two planes passing through the center of the eye and tangent to the [surface of the] cone, and in this situation the portion of the cone between them is visible to sight, and it constitutes less than half the cone, because the lines that are tangent to the circle encompass less than half of it.⁑

5.30 ◉ Si vero linea a centro ad acumen piramidis ducta tenet angulum rectum cum axe, intelligatur circulus secans piramidem equidistans basi. Linea communis huic circulo et superficiei in qua sunt axis piramidis et centrum visus erit ortogonalis super axem piramidis, quoniam axis est ortogonalis super superficiem circuli. Et super lineam communem protrahatur per centrum circuli dyameter ortogonaliter super hanc lineam, et a terminis huius dyametri protrahantur due contingentes circulum, et etiam due linee usque ad acumen piramidis. Due superficies in quibus erunt hee due linee cum contingentibus contingunt piramidem secundum modum predictum. Et quoniam linea communis circulo et superficiei in qua sunt centrum visus et axis piramidis est equidistans linee a centro visus ad terminum axis producte, et huic linee communi sunt equidistantes linee circulum in predictis punctis contingentes, erunt ille contingentes equidistantes linee a centro visus ad terminum axis ducte, quare erunt in eadem superficie cum illa. Igitur utraque superficierum circulum contingentium transit per centra visus, et communis illarum superficierum sectio est linea a centro visus ad terminum axis ducta. Et quod inter illas superficies cadit ex piramide apparet visui, et est medietas piramidis, quoniam lineas has contingentes circulum interiacet medietas circuli. Et ita palam quod in hoc situ apparet medietas piramidalis speculi.

5.30 ◉ If, on the other hand, the line drawn from the center [of sight] to the vertex of the cone forms a right angle with the axis, then imagine a circle that cuts the cone so as to be parallel to the base. The common section of this circle and the plane that contains the axis of the cone and the center of sight will be [a line] perpendicular to the axis of the cone, because the axis is perpendicular to the plane of the circle. To this common section draw a diameter through the center of the circle perpendicular to this line [that forms the common section], and from the endpoints of this diameter draw two [lines] tangent to the circle, and also draw two lines to the vertex of the cone. The two planes in which these latter two lines lie, along with the lines tangent [to the circle], are tangent to the cone in the way we described before. And since the common section of the circle and the plane in which the center of sight and the cone’s axis lie is parallel to the line drawn from the center of sight to the endpoint of the axis, and since the lines drawn tangent to the circle at the aforementioned points are parallel to this common section, those tangents will be parallel to the line drawn from the center of sight to the endpoint of the axis, so they will [each] lie in the same plane as that line. Hence, both planes that are tangent to the circle pass through the center of sight, and the common section of those planes is the line drawn from the center of sight to the endpoint of the axis. So whatever lies on the cone between those planes is visible to sight, and it constitutes half the cone, because half the circle lies between those lines of tangency. And so it is evident that in this situation half the conical mirror is visible.⁑

5.31 ◉ Verum si linea a centro visus ducta ad terminum axis piramidis teneat cum axe angulum obtusum ex parte superiori apparenter, et fiat circulus secans piramidem equidistantem basi, linea communis huic circulo et superficiei in qua est centrum visus et axis est perpendicularis super axem piramidis. Et hec linea communis extra producta concurret cum linea a centro visus ad terminum axis ducta propter angulum acutum quem facit hec linea cum axe ex inferiori parte. A puncto concursus linearum protrahantur due linee contingentes circulum in duobus punctis oppositis, et producantur linee ab hiis punctis ad acumen piramidis. Superficies in quibus sunt linee contingentes cum hiis longitudinis lineis contingunt piramidem, et in utraque harum superficierum sunt duo puncta linee a centro visus ad terminum axis ducte—scilicet terminus axis et terminus perpendicularis in quo scilicet concurrunt linea illa et perpendicularis—quare linea illa est in utraque superficie. Igitur utraque superficies transit per centrum visus, et includunt hee superficies ex interiori in inferiori parte minorem partem piramidis, quia linee contingentes circulum includunt partem eius minorem medietatem. Unde ex parte superiori interiacet superficies piramidem contingentes pars medietatum maior, et illa est que apparet visui, quare in hoc situ comprehendit visus piramidis partem medietatum maiorem.

5.31 ◉ However, if the line extended from the center of sight to the endpoint of the cone’s axis forms an obtuse angle with the axis from a point of view above the cone, and if a circle is formed to cut the cone parallel to the base, the [line forming the] common section of this circle and the plane containing the center of sight and the axis is perpendicular to the cone’s axis. When this common section is extended beyond [the back edge of the cone], it will intersect the line drawn from the center of sight to the endpoint of the axis according to an acute angle, [and] an acute angle is [also] formed by this line [of sight] with the axis below [the point of intersection]. From the point of intersection of these lines, draw two lines tangent to the circle at two opposite points, and draw the lines [of longitude] from these points to the vertex of the cone. The planes containing the lines tangent [to the circle] and these lines along the length [of the cone] are tangent to the [surface of the] cone, and in both of these planes two points on the line extended from the center of sight to the endpoint of the axis lie—i.e., the endpoint of the axis and the endpoint of the perpendicular where that line [drawn from the center of sight to the endpoint of the axis] and the perpendicular intersect—so that line lies in both planes. Therefore, both planes pass through the center of sight, and on the inner, lower side [of the cone] these planes encompass a portion of the cone that is less than half, because the lines tangent to the circle encompass a portion [of the circle] that is less [than half]. Hence, from [the point of view] above [the cone], more than half [the cone] lies between the planes that are tangent to [the surface of the] cone, and that is what is visible to the eye, so in this situation the eye perceives more than half the [surface of the] cone.⁑

5.32 ◉ Si autem linea a centro visus ad terminum axis producta cadat super latus piramidis, ut ex ea et latere unum efficiatur continuum, dico quod non latebit visum ex hac piramide preter lineam quandam intellectualem, quoniam omnis superficies in qua est linea a centro visus ad terminum axis ducta et secundum lateris longitudinem prolongata secat piramidem una tantum excepta que contingit piramidem in latere quod est pars linee. Et hoc solum latus intellectuale in tota piramidis superficie super hoc situ visum preterit.

5.32 ◉ On the other hand, if the line extended from the center of sight to the endpoint of the axis falls on an edge of the cone so that this line and the edge form a continuous line, I say that none of the cone will be hidden from sight except for the given line imagined [to extend along the edge], for every plane that contains the line extended from the center of sight to the endpoint of the axis and a line extended along the length of the cone’s edge cuts the cone except where it touches the cone along the edge that forms part of the line [drawn from the center of sight to the axis]. And [of all the lines] imagined to lie on the surface of the cone in this situation, this edge-line alone passes [unseen] by the center of sight.

5.33 ◉ Et huius rei veritas patet ex hoc quod, quocumque superficiei piramidis puncto sumpto, si ad ipsum ducatur linea a centro visus et ab eo linea longitudinis piramidis ad terminum axis, efficient hee due linee triangulum cum linea lateri applicata. Et est triangulus in superficie a centro intellecta piramidem secante, et ex lineis huius superficiei non nisi due cadunt in superficie piramidis—scilicet linea lateris et linea longitudinis a puncto sumpto ad acumen piramidis. Et linea a centro ad punctum sumptum ducta secat lineam longitudinis reflexionis in puncto sumpto et lineam lateris in centro, quare huic linee non accidet concursus de centro cum aliqua harum linearum. Cum igitur non posset sumi punctus alius ad quem linea a centro accedat et in hoc punctum transeat, non occultatur punctus iste ab alio puncto. Et ita apparet visui, cum ei et visui non intercidat corporis solidi obiectio. Et eadem probatio in quolibet superficiei piramidis puncto.

5.33 ◉ The truth of this point is evident from the fact that, whatever point is chosen on the surface of the cone, if a line is drawn to it from the center of sight, and if from that same point a line is drawn along the length of the cone to the endpoint of the axis, these two lines will form a triangle with the line drawn [from the center of sight] that coincides with the edge [of the cone]. And this triangle lies in a plane that is imagined to pass through the center [of sight] so as to cut the cone, and among [all] the lines lying in this plane only two fall on the surface of the cone—i.e., the line [that coincides with the line along] the edge and the line along the length [that extends] from the chosen point to the vertex of the cone. But the line extended from the center [of sight] to the chosen point cuts the line along the length [of the cone] at the chosen point of reflection, and it cuts the line that coincides with the edge of the cone at the center of sight, so none of these lines [of longitude] will intersect that line [along the edge of the cone] outside the center of sight. Therefore, since no other point can be chosen to which the line from the center of sight reaches and through which it passes, this point is not blocked by any other point. And so it is visible to the eye, since there is no solid object lying between it and the eye. The same can be demonstrated for any [other] point on the surface of the cone.⁑

5.34 ◉ Et si linea a centro visus in terminum axis cadens intret piramidem, dico quod nullus occultatur visui punctus in tota piramidis superficie. Sumpto enim quocumque puncto in piramidis superficie, intelligatur ad ipsum linea a centro et alia ab eo usque ad acumen piramidis. Hee due includunt superficiem triangularem cum linea a centro visus ad terminum axis ducta piramidem intrante, et est iste triangulus in superficie piramidem secante. Cum omnis superficies in qua fuerit linea intrans piramidem secet eam, linea a centro ad punctum sumptum ducta secat in illo puncto lineam longitudinis ab eo ad acumen piramidis ductam. Et ex lineis superficiei in qua sunt hee due linee non sunt nisi due linee in superficie piramidis—scilicet hec linea longitudinis a puncto ad acumen ducta et alia opposita secans angulum quem includit hec cum linea piramidis intrante. Igitur linea illa opposita extra piramidem producta secat lineam a centro ad punctum sumptum ductam; quare linea hec secat duas lineas que sole ex lineis huius superficiei sunt in piramidis superficie, unam extra piramidem aliam in puncto sumpto, quare producta in infinitum non concurret cum altera illarum linearum. Unde non occultatur visui punctum sumptum secundum modum supradictum.

5.34 ◉ Moreover, if the line [extended] from the center of sight to the endpoint of the axis enters the cone, I say that no point on the entire surface of the cone is hidden from the eye. For, given some point on the surface of the cone, imagine a line drawn to it from the center [of sight], and imagine another drawn from it to the vertex of the cone. These two lines enclose a triangular surface with the line extending from the center of sight to the endpoint of the axis that enters the cone, and this triangle lies in a plane that cuts the cone’s surface. Since every plane containing a line that enters the cone cuts the cone, then the line extended from the center [of sight] to the chosen point cuts at that point the line along the length of the cone that extends from it to the cone’s vertex. And among [all] the lines lying in the plane that contains these two lines, there are only two that lie on the cone’s surface—i.e., the one line that is extended along the length of the cone from the [chosen] point to the [cone’s] vertex and the other line that is [directly] opposite, which cuts the angle formed by the first line with the line that enters the cone. Thus, when it is extended beyond the cone, the line on the opposite side cuts the line extended from the center [of sight] to the chosen point; hence, within the plane [formed by these lines] the latter line [extending from the center of sight to the chosen point] cuts only two lines that lie on the surface of the cone, [cutting] one beyond the cone and the other at the chosen point, so if it is extended to infinity it will intersect none of the other lines [along the length of the cone]. According to the earlier account, then, the chosen point is not blocked from sight.

5.35 ◉ In hoc ergo situ nulla superficierum piramidis tangentium transibit per centrum visus, sed quelibet secabit lineam visus super terminum axis piramidem intrantis inter visum et piramidem, et est in termino axis. Cum vero linea visus linee longitudinis piramidis applicatur, nulla superficierum piramidis tangentium perveniet ad centrum preter illam que in predicta linea contingit piramidem. Et omnes superficies contingentes secabunt lineam illam inter visum et piramidem.

5.35 ◉ Therefore, in this situation none of the planes tangent to the cone will pass through the center of sight; rather, each of them will intersect the line of sight that enters the cone through the endpoint of the cone’s axis between the center of sight and the cone, and the [point of intersection] lies at the endpoint of the axis. But when the line of sight coincides with a line along the length of the cone, none of the planes tangent to the cone will reach the center of sight except the one that is tangent to the cone along the aforementioned line. So all the tangent planes will cut that line between the center of sight and the cone.⁑

5.36 ◉ Similiter, in situ in quo superficies due contingentes piramidem per centrum transeunt, quelibet superficies tangens piramidem in portione piramidis apparente que duas contingentes interiacet a centro visus divertit. Super quodcumque punctum illius portionis cadat linea visualis, secabit piramidem, cum intercidat duas contingentes visuales. Et superficies in qua fuerit hec visualis et linea longitudinis piramidis secabit piramidem, et erit hec visualis superficies cuicumque superficiei piramidis in hac portione contingat, quare et visus.

5.36 ◉ Likewise, in the case when two planes that are tangent to the cone pass through the center of sight, any plane that is tangent to the cone in the visible portion of the cone that lies between the two tangent planes misses the center of sight. No matter what point of that portion the line of sight reaches, that line will cut the cone, because that portion lies between the two lines of sight that are tangent [to the cone]. And the plane containing this line of sight [i.e., the one reaching the given point on the cone’s surface] and the line along the cone’s length will cut the cone, and this will be the visual plane for whatever place on the cone’s surface it touches within that portion, and so [that place on the cone’s surface] is also visible.⁑

5.37 ◉ Dico ergo quod in quolibet situ a quolibet puncto potest fieri reflexio. Sumatur enim punctus, et intelligatur circulus per punctum transiens basi piramidis equidistans. Dyameter huius circuli ab hoc puncto incipiens erit perpendicularis super axem, cum axis sit perpendicularis super circuli superficiem, quare linea longitudinis a puncto ad acumen piramidis ducta tenet angulum acutum cum dyametro et acutum cum axis termino in eadem superficie. Sit linea visualis super punctum cadens in superficie in qua est linea longitudinis et axis, in qua superficie ducatur perpendicularis super lineam longitudinis in puncto illo. Concurret hec quidem perpendicularis cum axe, et ex ea, et axe, et linea longitudinis efficietur triangulus. Super punctum illud intelligatur linea contingens, et super dyametrum circuli quem fecimus intelligatur dyameter alius ortogonalis super ipsum, qui erit ortogonalis super ipsum axem, et ita super superficiem in qua axis et dyameter primus. Et hic dyameter secundus est equidistans contingenti, quoniam contingens est perpendicularis super dyametrum primum. Et ita linea contingens ortogonalis est super superficiem in qua axis et dyameter primus, quare erit ortogonalis super perpendicularem quem primo fecimus. Et ita illa perpendicularis ortogonaliter cadit super superficiem contingentem piramidem in qua punctus sumptus.

5.37 ◉ I maintain, therefore, that in any [such] situation reflection can occur from any given point [on the visible portion of the mirror]. Choose a point, and imagine a circle passing through that point and parallel to the base of the cone. The diameter originating at that point on this circle will be perpendicular to the axis, since the axis is perpendicular to the plane of the circle, so the line along the length [of the cone] extended from the [chosen] point to the vertex of the cone forms an acute angle with the diameter as well as with the endpoint of the axis within the same plane [i.e., the plane containing the diameter, the axis, and the line of longitude]. Let a line of sight fall to the [chosen] point within the plane containing the line along the length [of the cone] and the axis, and in that plane draw a normal to the line along the length [of the cone] at that point. This normal will of course intersect the axis, and a triangle will be formed by this line with the axis and the line along the length [of the cone]. Imagine a line tangent [to the circle] at that [chosen] point, and imagine another diameter in the circle we drew that is orthogonal to the original one and orthogonal to the axis as well, so it will be orthogonal to the plane containing the axis and the first diameter. This second diameter is parallel to the line of tangency [just drawn], because that line of tangency is perpendicular to the first diameter. Hence, the line of tangency is perpendicular to the plane containing the axis and the first diameter, so it will be orthogonal to the normal that we just drew. And so that normal falls orthogonally to the plane tangent to the cone at the point we chose.

5.38 ◉ Igitur, si linea visualis cadens in punctum sumptum transeat secundum processum perpendicularis, erit quidem ortogonalis super superficiem piramidis illam in puncto contingentem, et fiet reflexio forme per eandem lineam. Si autem deviet a processu perpendicularis, faciet quidem angulum cum perpendiculari acutum in puncto sumpto. Et poterit produci in superficie huius linee visualis alia linea a puncto illo que equalem angulum huic teneat cum perpendiculari, cum perpendicularis ortogonalis sit super superficiem contingentem. Linea autem quacumque super superficiem contingentem in puncto sumpto ortogonaliter cadente, transit ad axem. Et si ab axe ducatur ortogonalis ad hanc superficiem, efficient perpendicularis interior et exterior lineam unam. Quod si non cum perpendicularis interior extra producta sit etiam perpendicularis super superficiem, accidet ab eodem puncto super illam superficiem erigi duas perpendiculares in eandem partem. [5.39] Palam igitur quod quocumque puncto superficiei piramidis viso potest fieri reflexio ad paritatem angulorum. Et cum linea reflexionis occurrerit forma, veniet ad speculum super lineam hanc et reflectetur ad visum super aliam, et sunt hee due linee in eadem superficie ortogonali super superficiem contingentem piramidem in puncto reflexionis. Et hec est superficies reflexionis in qua semper fit comprehensio quatuor punctorum: scilicet centri visus, puncti visi, puncti reflexionis, terminus perpendicularis.

5.38 ◉ Accordingly, if the line of sight that falls upon the chosen point passes to it along the normal, it will be orthogonal to the plane tangent to the cone at the chosen point, and the [visible] form will be reflected back along that same line of sight. On the other hand, if the line of sight does not pass along the normal, it will form an acute angle with that normal at the chosen point. And in the plane of this line of sight another line can be drawn from that point to form an angle equal to that which it forms with the normal, because that normal is orthogonal to the plane of tangency. Moreover, any line that falls orthogonally to the plane of tangency at the chosen point passes to the axis. And if a perpendicular is drawn to that plane from the axis, the inner and outer segments of that perpendicular will form a single line. For, if the interior segment of the perpendicular is not also perpendicular to the plane when it is extended outside [the cone], it will follow that from the same point on that [same] plane two perpendiculars can be erected on the same side [which is impossible].

5.40 ◉ Diversificantur autem linee communes superficiebus reflexionis et superficiei piramidis. Cum enim radius visualis continuus fuerit axi piramidis—scilicet cum in qualibet superficie reflexionis sit totus axis et perpendicularis ad axem transiens—erit cuilibet superficiei reflexionis et superficiei piramidis communis linea linea longitudinis in hoc situ. Quoniam quelibet superficies in qua est totus axis hanc habet lineam communem cum superficie piramidis.

5.40 ◉ The common sections of the planes of reflection and the surface of the cone [can] vary, however. For when the line of sight coincides with the axis of the cone—i.e., when the entire axis of the cone and the normal [to the cone’s surface] that passes to the axis lie in every plane of reflection—then in this case the common section of each plane of reflection and the surface of the cone will be a line along the length [of the cone]. For every plane that contains the entire axis [of the cone] has this [particular] common section with the surface of the cone.

5.41 ◉ Et in omni alio situ unica longitudinis piramidis linea erit communis illa—scilicet que fuerit in superficie centrum visus et axem continente. Et quia centrum visus non erit in directo axis, una tantum erit superficies talis, et omnis alia communis linea erit sectio piramidalis non circulus. Si enim fuerit circulus, erit superficies illius circuli in superficie reflexionis, et quia axis est ortogonalis super illum circulum (cum quilibet circulus piramidis sit equidistans basi), erunt latera piramidis declinata super circulum, et ita super superficiem reflexionis. Quare in superficie illa non potest duci perpendicularis super lineam longitudinis piramidis. Sed perpendicularis ducta super superficiem contingentem locum reflexionis est in superficie reflexionis, et perpendicularis super lineam longitudinis, cum quelibet superficies tangens piramidem tangat in linea longitudinis.

5.41 ◉ But in every other case only one line along the length of the cone will form the common section—i.e., the common section lying in the plane that contains the center of sight and the axis. And because the center of sight does not lie in a direct line with the axis, there will be only one such plane [of section], and every other common section will be a conic section, not a circle. For if it were a circle, the plane of that circle would lie in the plane of reflection, and since the [cone’s] axis is orthogonal to that circle (because any circle [of section] on the cone is parallel to its base), the edges of the cone will be inclined to the circle, as well as to the plane of reflection. In that plane [of the circle], therefore, a normal cannot be drawn to the line along the length of the cone. But the normal extended to the plane that is tangent [to the cone] at the point of reflection lies in the plane of reflection, and it is perpendicular to the line along the length [of the cone], because any plane that is tangent to the cone is tangent [to it] upon a line along its length.

5.42 ◉ Accidit igitur impossibile, quare restat omnes alias communes reflexionis lineas sectiones piramidales esse, et cum fuerit linea communis linea longitudinis, ex quocumque puncto illius linee fiat reflexio erit in eadem superficie cum cuiusque alterius puncti reflexione. Quoniam a quolibet huius linee puncto ducta perpendiculari continget axem; et erunt in superficie reflexionis centrum visus, et punctum reflexionis, et punctum axis; et huius reflexionis est superficies in qua sunt linea longitudinis et axis, quare in hac superficie fit reflexio a quocumque puncto.

5.42 ◉ It therefore follows [that it is] impossible [for the plane of reflection to form a circular section with the cone], so all the other common sections of [the plane of] reflection [and the surface of the cone] are conic sections, and when the common section is a line along the length [of the cone], reflection can occur from any point on that line in the same plane as reflection from any other point of reflection. For the perpendicular extended from any point on that line will meet the axis; and the center of sight, the point of reflection, and the point on the axis [where the normal intersects it] will lie in the plane of reflection; and the line along the length [of the cone] and the plane within which this reflection occurs is the one formed by the line along the length [of the cone] and the axis, so reflection takes place from any point in this plane.⁑

5.43 ◉ Si vero communis linea non fuerit linea longitudinis, dico quod vel ab uno communis linee puncto in eadem superficie fiat reflexio, vel a duobus tantum. Quoniam ducta perpendiculari a puncto reflexionis, perveniet ad axem, et cadet in aliquod punctum eius. Intellecto circulo super punctum reflexionis, ortogonaliter secabit circulus axem, et cum perpendicularis secat axem equidistans basi, erit perpendicularis declinata super circulum. Et circumquaque ducta, semper erit equalis, unde fiet piramis cuius basis circulus, acumen punctus axis in quem cadit perpendicularis. Igitur superficies reflexionis aut tanget hanc piramidem aut secabit.

5.43 ◉ If, however, the common section is not a line along the length [of the cone], I say that reflection may take place from either one point, or at most two points, on the common section within the same plane. For, if a normal is extended from the point of reflection, it will reach the axis and will fall on one of its points. When a circle is imagined [projected] through the point of reflection, that circle will cut the axis orthogonally, and since the perpendicular [in the plane of the circle] cuts the axis [in a plane] parallel to the [cone’s] base, the normal [to the cone’s edge] will be inclined to the [plane of the] circle. And when it is rotated about [this circle] that normal will always be equal[ly disposed with respect to the circle], so it will form a cone whose base is the circle and whose vertex is the point at which the normal intersects the axis. Therefore, the plane of reflection will either be tangent to, or will cut, this [second] cone.

5.44 ◉ Si tangat, dico quod a puncto reflexionis sumpto possit tantum fieri in eadem superficie reflexio. Planum quod superficies reflexionis continget hanc piramidem super perpendicularem, que est linea ortogonalis in superficie reflexionis, et si ab acumine totalis piramidis ducantur linee ad sectionem communem superficiei reflexionis et piramidi magne prius, cadent in circulum qui est basis piramidis intellecte quam in sectione preter unam que in punctum reflexionis cadit. Si ergo ab alio sectionis communis puncto fieret reflexio, linea ab illo puncto ad acumen intellecte ducta erit perpendicularis super lineam longitudinis piramidis per punctum illud transeuntem. Sed linea ab acumine piramidis intellecte ad punctum circuli per quem transit illa linea longitudinis absque dubio est perpendicularis super eam, quare alia angulum tenet acutum cum hac linea, non rectum.

5.44 ◉ If it is tangent to it, I say that reflection can occur in the same plane from only one given point of reflection. It is evident that the plane of reflection will be tangent to this [second] cone along the normal, which is an orthogonal line in the plane of reflection, and if lines are drawn from the vertex of the entire [second] cone to the common section of the plane of reflection and the larger, original cone, they will fall on the circle that forms the base of the imagined cone, but only one falls on that common section at the point of reflection. Therefore, if reflection were to take place from another point on this common section, the line imagined to extend from that point to the [second cone’s] vertex would be perpendicular to the line along the length passing through that point on the [original] cone. But the line [extending] from the vertex of the imagined cone to the point on the circle through which the line along the length of the [mirror’s] cone passes is certainly perpendicular to this latter line, so any other line forms an acute angle, not a right angle, with it.⁑

5.45 ◉ Si vero superficies reflexionis secet intellectualem piramidem, secabit circulum qui est eius basis in duobus punctis. Dico quod hec sola sunt puncta in tota sectione communi a quibus fieri possit reflexio in eadem superficie, quoniam ab utroque istorum punctorum linea ducta ad acumen intellecte piramidis est perpendicularis super lineam longitudinis super punctum suum transeuntem. A quocumque enim sectionis alio puncto ducatur linea ad acumen illius piramidis, tenebit angulum acutum cum linea longitudinis per ipsum transeuntem, cum perpendicularis cum eadem longitudinis linea angulum rectum teneat in circulo. Et linee ducte ab acumine intellecte piramidis ad puncta sectionis que intercidunt speculi acumen et circulum facient angulos obtusos cum lineis longitudinis versus partem acuminis piramidis totalis. Et que ducuntur ad puncta circulum et basem speculi interiacentia faciunt cum linea longitudinis angulos acutos ex parte acuminis speculi, obtusos ex parte basis.

5.45 ◉ On the other hand, if the plane of reflection cuts the imagined cone, it will cut the circle at its base in two points. I say that these are the only points on the entire common section [formed by the cutting plane] from which reflection can take place in the same plane, for the line extended from either of these points to the vertex of the imagined cone is perpendicular to the line along the length [of the mirror’s cone] at the point through which it passes. From any other point on the section [formed by the cutting plane], the line extended to the vertex of this [imagined] cone will form an acute angle with the line along the length [of the mirror’s cone] through which it passes, since the normal to the same line along the length [of the mirror’s cone] forms a right angle in the circle. So the lines drawn from the vertex of the imagined cone to points on the section lying between the vertex of the mirror and the circle will form obtuse angles with the lines along the cone’s length on the side opposite the vertex of the entire cone [of the mirror]. And the lines drawn to the points lying between the circle and the base of the mirror form acute angles with the line along the length [of the cone] on the side opposite the vertex while forming obtuse angles on the side opposite the base.⁑

5.46 ◉ In speculis spericis concavis, si fuerit intra concavitatem speculi tota speculi superficies apparebit ei. Quod si extra fuerit visus, poterit comprehendere portionem eius maiorem medietatum quam scilicet fecerit circulus spere quem contingunt duo radii a centro visus ducti.

5.46 ◉ In the case of spherical concave mirrors, if the center of sight is located within the hollow of the mirror, the entire surface of the mirror will be visible to it. However, if the eye lies outside [that hollow], it can perceive more than half of the mirror, that portion being defined by the circle on the sphere upon which two rays drawn from the center of sight are tangent.⁑

5.47 ◉ Visu autem in centro huius speculi existente, non fiet ab aliquo puncto speculi reflexio nisi in se, quoniam quelibet linea a centro spere ad speram ducta perpendicularis est super superficiem speram in puncto illo tangentem. Unde in hoc situ non comprehendet visus per reflexionem nisi se tantum.

5.47 ◉ Now, if the eye lies at the center of this mirror, reflection will not take place from any point other than [that according to which the form of the eye reflects back] to itself, for any line extended from the center of the sphere to the [surface of the] sphere is perpendicular to the plane that is tangent to the sphere at that point. Hence, in this situation the eye will see nothing but itself through reflection.

5.48 ◉ Si vero statuatur visus extra centrum spere, poterit fieri reflexio in aliud corpus a quocumque speculi puncto preter quam ab eo in quem cadit dyametrum a centro visus ad speram per centrum spere ductus, quoniam dyameter cadit super superficiem contingentem speram ortogonaliter. Sumpto autem alio puncto, ducatur ad ipsum dyameter a centro spere et linea a centro visus. Ex hiis ergo lineis acutus includetur angulus, quoniam linea visualis cadit inter dyametrum et superficiem contingentem punctum, que scilicet est extra speram. Et sive sit oculus intra speculum sive extra, cadit hec visualis linea intra speculum, quia cadit inter lineas visuales contingentes circulum portionis spere cum visus fuerit extra.

5.48 ◉ If, however, the eye is located outside the center of the sphere, reflection can occur to another body from any point on the mirror other than the point where the diameter drawn from the center of sight to the sphere passes through the sphere’s center, because the diameter falls perpendicularly to a plane that is tangent to the sphere. Now, having chosen another point [on the surface of that sphere], draw a diameter [normal] to it from the center of the sphere, and [draw] a line from the center of sight [to it]. Accordingly, an acute angle will be formed by these lines, for the line of sight lies between the diameter [that forms the normal] and the plane tangent to that point, and that plane lies [on the] outside [surface of] the sphere. And whether the center of sight lies inside or outside the sphere, this line of sight falls inside the mirror because it falls between the lines of sight that are tangent to the circle [defining] the [visible] portion of the sphere when the center of sight lies outside [the sphere].

5.49 ◉ Oculo cadente intra, planum quod intra cadit linea. Cum igitur dyameter angulum rectum teneat cum contingente, secetur ex eo acutus equalis predicto in eadem superficie. Dico ergo quod linea reflexionis cadit intra speculum, quoniam linea communis superficiei speculi et superficiei reflexionis est circulus tenens cum dyametro angulum acutum maiorem omni rectilineo acuto, et in singulis punctis erit hic modus reflexionis.

5.49 ◉ If the center of sight lies inside the sphere, it is obvious that the line of sight falls inside the mirror. Accordingly, since the diameter [that forms the normal] forms a right angle with the tangent [to the chosen point], cut off from that [right angle] an acute angle equal to the aforementioned acute angle in the same plane. I say, then, that the line of reflection falls inside the mirror, because the common section of the mirror’s surface and the plane of reflection is a circle that forms with the diameter [constituting the normal] an acute angle greater than any acute rectilinear angle, and in each such point there will be reflection of this kind.

5.50 ◉ Palam ex hiis quod in omni superficie reflexionis erunt centrum visus, centrum speculi, punctus reflexionis, punctus visus, terminus dyametri a centro visus per centrum spere ad speram ducti. Et communis omnium linea cum superficie speculi est circulus, et a quolibet linee communis puncto potest fieri in eadem superficie reflexio.

5.50 ◉ It is evident from these observations that in every plane of reflection there will be a center of sight, the center of the mirror, a point of reflection, a point that is seen, and the endpoint of the diameter extended from the center of sight to the [surface of the] sphere through the sphere’s center. Furthermore, the common section of all these planes with the surface of the mirror is a circle, and reflection can occur in the same plane from any point on this common section.⁑

5.51 ◉ In speculis columpnaribus concavis potest totum comprehendi speculum si fuerit visus intra ipsum. Sed eo extra sito, videbitur maior medietatum speculi, portio que scilicet interiacet duas superficies a centro visus procedentes columpnam contingentes.

5.51 ◉ In the case of concave cylindrical mirrors the entire mirror can be perceived if the center of sight lies inside it. However, if it lies outside, more than half the mirror will be visible, the portion, that is, that lies between the two planes projected from the center of sight and tangent to [the surface of] the cylinder.⁑

5.52 ◉ Intelligemus autem superficiem a centro visus procedentem basibus columpne equidistantem. Hec superficies aut cadet in columpnam aut non. Si ceciderit, linea communis huic superficiei et columpne erit circulus, et linea visualis transiens per centrum huius circuli cadet ortogonaliter super superficiem contingentem columpnam in puncto in quem cadit linea. Et fiet reflexio per eandem lineam ad eius originem.

5.52 ◉ We will imagine a plane projected from the center of sight and parallel to the bases of the cylinder. This plane will either intersect the cylinder or it will not. If it intersects, then the common section of this plane and the cylinder will be a circle, and the line of sight passing through the center of this circle will fall orthogonally to the plane tangent to the cylinder at the point where the line of sight intersects it. So reflection will take place along the same line to its source.

5.53 ◉ Quicumque alius sumatur punctus. Linea perpendiculariter ab hoc puncto ducta cadet in axem, et linea visualis in punctum illud cadens faciet angulum acutum cum linea perpendiculari, cum sit inter perpendicularem et contingentem. Et quia hec linea cadet intra speculum, planum ex hoc quod cadit inter superficies portionem apparentem contingentes. Poterimus igitur in eadem reflexionis superficie ex angulo quem facit perpendicularis cum contingente excipere angulum acutum equalem angulo predicto acuto. Et cadet linea reflexionis hunc angulum continens intra columpnam, quoniam cadet inter perpendicularem et lineam longitudinis per terminum perpendiculariter transeuntem. Erunt igitur in superficie reflexionis centrum visus, punctum reflexionis, punctum visum, punctum axis in quem cadit perpendicularis.

5.53 ◉ Let some other point be chosen. The line extended normal [to the cylinder’s surface] from this point will intersect the axis, and the line of sight dropped to this point will form an acute angle with the [aforementioned] normal, since it lies between that normal and the [line] tangent [to the cylinder’s surface at that point]. And that this line will fall inside the mirror is obvious from the fact that it falls between the planes that are tangent to the visible portion [of the cylinder]. Hence, in the same plane of reflection we can cut off from the angle formed by the normal and the tangent an acute angle equal to the aforementioned acute angle. Moreover, the line of reflection forming this angle will fall inside the cylinder, because it will fall between the normal and the line along the length [of the cylinder] passing orthogonally through the endpoint [of the flanking, tangent line of sight]. Thus, the center of sight, the point of reflection, the point that is seen, and the point on the axis to which the normal falls will [all] lie in the plane of reflection.⁑

5.54 ◉ Et si hoc modo statuatur visus ut communis linea superficiei reflexionis et superficie columpne sit linea longitudinis, a quocumque puncto communis linee fiat reflexio. In una et determinata erit superficie omnibus hiis reflexionibus communi—ea scilicet in qua centrum visus et axis columpne totus—sicut dictum est superius in columpnari speculo non concavo.

5.54 ◉ Moreover, if the center of sight is located in such a way that the common section of the plane of reflection and the surface of the cylinder constitutes a line along [the cylinder’s] length, reflection may occur from any point on that common section. The plane common to all these reflections will be in one specific plane—i.e., the one in which the center of sight and the entire axis of the cylinder lie—as was said earlier in the case of the non-concave cylindrical mirror.

5.55 ◉ Similiter, si linea communis fuerit circulus, omnes reflexiones a punctis illius circuli facte procedent in eadem superficie, sicut in aliis circulis patuit.

5.55 ◉ Likewise, if the common section is a circle, all of the reflections occurring from points on that circle will proceed in the same plane, as was shown earlier for the other circles.

5.56 ◉ Et si sectio columpnaris fuerit linea communis, a duobus quidem eius punctis tantum fiet reflexio in eadem superficie, licet in superioribus columpnis circulus tantum ab uno puncto in unica superficie fieret reflexio, unico visu adhibito, quoniam supra latebant visum puncta sectionis se respicientia per que scilicet transit circulus columpne equidistans basibus. Viso enim uno latebat alius propter minoris columpne portionis apparentiam, sed in hiis apparet maior columpne portio, unde ab unico visu percipiuntur puncta circuli equidistantis basibus et sectionis communis.

5.56 ◉ But if the common section is a cylindric section [i.e., an ellipse], reflection will occur from only two of its points within the same plane, although in the case of [convex] cylinders discussed above, [when the points of reflection lie at the intersection of the ellipse and a] circle, reflection would occur from only one point in a single plane when one eye is looking, because in the previous case the corresponding points of intersection through which the circle parallel to the bases passes were invisible to the eye. For when one [such point] was seen [with one eye], the other was invisible because a portion smaller [than half] of the cylinder is visible, but in these cases [when the cylinder is concave], a greater portion [than half] of the cylinder is visible, so the [corresponding] points [of intersection] of the circle parallel to the bases and the common section are perceived by one eye.⁑

5.57 ◉ In speculis piramidalibus concavis, si fuerit visus intra speculum, videbit ipsum totum. Si vero extra, et linea a centro visus ad acumen piramidis ducta intret piramidem aut applicetur linee longitudinis piramidis, nichil videbitur ex speculo. Quoniam quecumque alia linea ab oculo ad piramidem ducta cadet in piramidis superficiem exteriorem, unde occultabitur interior superficies.

5.57 ◉ In the case of concave conical mirrors, if the center of sight lies inside the mirror, it will see all of it. If, however, it lies outside, and if the line extended from a center of sight [located above the mirror] to the vertex of the cone enters the cone or coincides with a line along the length of the cone, nothing of the mirror will be seen. For any other line extended from the eye to the cone will fall on the cone’s outer surface, so the inner surface will be hidden.⁑

5.58 ◉ Si autem auferatur portio a piramide, poterit videri pars piramidis cadens inter contingentes superficies a centro ductas, scilicet maior, et si linea a centro visus sit perpendicularis super superficiem contingentem piramidem et continuetur axi, erunt linee communes, sicut dictum est in aliis piramidalibus, aut linee longitudinis piramidis aut sectiones. Et in hiis a duobus punctis sectionis poterit reflexio in eadem superficie respectu eiusdem visus; et in superficie reflexionis erunt centrum visus, punctus visus, punctus reflexionis, punctus axis. [5.59] Sed speculum piramidale integrum, si apponatur visui, et sit visus ex parte basis, non percipiet nisi hoc quod fuerit intra speculum, quoniam perpendicularis tenet angulum acutum cum linea ab oculo ad ipsam ducta ex parte basis. Unde fit reflexio ex parte acuminis, et cadent omnes linee reflexe intra piramidem, et videri poterit quod intra piramidem positum sit.

5.58 ◉ If, however, a segment is removed from the cone, the portion of the cone lying between the planes projected from the center of sight and tangent to [the cone’s] surface—i.e., more than half—can be seen, and if the line from the center of sight is perpendicular to the plane tangent to the cone and reaches to the axis, then, as was said in the case of the other cones [i.e., convex cones], the common sections [of the plane of reflection and the mirror’s surface] will be either [straight] lines along the length of the cone or [conic] sections.⁑ Moreover, in these cases [i.e., when the plane of reflection is a conic section and more than half the surface of the mirror is visible], reflection can occur from two points on the [conic] section in the same plane with respect to the same center of sight; and the center of sight, the point seen, the point of reflection, and the point on the axis will lie in the same plane of reflection.

5.60 ◉ Si autem auferatur ex eo portio secundum longitudinem, poterunt quidem comprehendi exteriora, cum pateat exitus lineis reflexionis. Similiter, si secetur piramis ad modum anuli ut auferatur conus, liberum habebunt linee gressum, et exteriora apparebunt. Et si fuerit visus ex parte concavi, plura poterit comprehendere exteriora quam ex parte basis, quia latior reflexis lineis datur ad egrediendum via.

5.60 ◉ On the other hand, if a segment of the cone is cut off lengthwise, things outside the mirror can be perceived, since what lies beyond may be open to the lines of reflection. Likewise, if the cone is truncated in the form of a ring with the vertex removed, lines [of incidence] will have free entry [through that upper opening], and things outside the mirror will be visible. And if the center of sight lies inside the concave [inner surface of the mirror], more things that lie outside can be seen than [if the the center of sight lies] on the side of the base, because it provides a wider scope for the lines of reflection to project outward.⁑

5.61 ◉ Amplius, sumpto uniuscuiusque speculi puncto, non est possibile in eo percipi formam nisi formam unius puncti ab eodem visu. Quoniam super perpendicularem et centrum visus unica transit superficies, et una sola est linea a centro visus ad punctum, et unicus angulus ex linea et perpendicularis acutus, et unicus angulus in eadem superficie acutus equalis huic, unde unica linea angulum equalem huic cum perpendiculari faciens. Et cum linea pervenit ad punctum corporis, non potest forma alterius puncti per ipsam vehi, cum punctum precedens occultet postpositum. Sed duobus visibus possunt in eodem speculi puncto comprehendi due punctales forme, quoniam infinite possunt sumi superficies super perpendicularem secantes in quarum qualibet circa perpendicularem sumi poterunt duo anguli equales acuti.

5.61 ◉ Furthermore, given a point on any mirror, it is possible for the form of only one point to be perceived at it by the same center of sight. For only one plane passes through the normal and the center of sight, and there exists only one [straight] line between the center of sight and the [given] point, and there exists only one acute angle formed by that line and the normal, and in the same plane there is only one acute angle equal to this one, so there is only one line that forms an angle with the normal that is equal to this one. So when the line [of incidence] is extended to a point on a [visible] object, the form of another point [on that object] cannot be conveyed along it, since the [form of the] first point passing along it will block any following it. But two point-forms can be perceived at the same point on the mirror by both eyes, because an infinite number of cutting planes can be imagined along the normal, and in each of them two equal acute angles can be imagined with respect to the normal.

5.62 ◉ Iam ergo proprietatem reflexionis declaravimus, et similiter cuiuslibet speculi proprium. Visus, cum per reflexionem formas comprehendit, non advertit quod hec adquisitio per reflexionem sit. Non enim accidit ex proprietate visus reflexio, quoniam, visu remoto, procedit non minus forma a corpore ad speculum et reflectetur secundum modum predictum. Et si accidit visum esse in loco in quem linearum reflexarum fit agregatio, comprehendet visus formam illam in capitibus harum linearum, et est in speculo tamquam non adveniens sed naturalis esset forma in speculo. Amplius, aliquando adquirit visus formas in speculis in sola superficie, aliquando intra speculum, aliquando ultra. Et erit apparens locus forme secundum figuram speculi et secundum situm rei vise, et semper comprehenditur forma in loco proprio, mutato situ visus et speculi. Et erit diversitas elongationis loci forme ad speculi superficiem secundum diversitatem figure speculi. Et locus forme dicitur locus ymaginis, et forma dicitur ymago. Visus autem comprehendit rem visam in loco ymaginis, et nos dicemus locum illum et eius proprium in quolibet speculorum, que numerabimus et assignabimus causas comprehendi res visas in loco illo, et hoc in sequenti libro, si deus voluerit.

5.62 ◉ We have therefore now explained the character of reflection as well as the character of each [type of] mirror. When sight perceives forms by means of reflection, it is unaware that this perception is due to reflection. For reflection is not a function of sight, because, when the eye is removed, the form nonetheless radiates from the [visible] object to the mirror, and it will be reflected in the way just described. But if the eye happens to lie where the lines of reflection come together, then sight will perceive that form at the endpoints of these lines, and that form [appears to] exist in the mirror, not as if it were adventitious but as if it were a natural form [actually] in the mirror. Moreover, sight sometimes perceives forms in mirrors on the surface alone, sometimes in front of the mirror, sometimes beyond it. And the place where the form appears will depend on the shape of the mirror and the location of the visible object, and the form is always perceived in its appropriate place when the [relative] situation of the center of sight and the mirror changes. And the distance of the form’s location from the mirror will vary with variation in the shape of the mirror. And the location of the form is called image-location, and the form is called an image.⁑ Moreover, the eye perceives the visible object at the image-location, and we shall discuss this location and its essential characteristic for each [type of] mirror, and we will enumerate and explain the reasons visible objects are perceived at that location, and we will do so in the following book, God willing.