Many academic studies have focused on investigating the influences of medieval European perspectivae on Renaissance theories of vision; however, the scholarly exploration of Graeco-Arabic legacies in optics that scientifically grounded these traditions is much rarer in connection with the Renaissance. It is hardly an exaggeration to note in this context that the most remarkable revolution in the classical science of optics from the second century till the seventeenth century (namely from the era of Ptolemy to that of Kepler) is embodied in the research of the Arab polymath al-Ḥasan Ibn al-Haytham (known in Latin as Alhazen; b. Basra 965 CE, d. Cairo c. 1041 CE).
Ibn al-Haytham’s groundbreaking studies in optics, including his research in catoptrics and dioptrics (respectively the sciences investigating the principles and instruments pertaining to the reflection and refraction of light), were principally gathered in his monumental opus: Kitāb al-manāẓir (The Optics; De Aspectibus or Perspectivae; composed between 1028 CE and 1038 CE).1 This classic corpus was divided into seven books that were grouped under three principal parts: Books I–III dealt with the problems of rectilinear direct vision, Books IV–VI focused on the science of catoptrics, and Book VII was dedicated to the science of dioptrics. In this revolutionizing oeuvre, Ibn al-Haytham devised a scientific solution to ancient controversies over the nature of vision, light, and color, which were disputed between the classical mathematicians (mainly the exponents of Euclid’s and Ptolemy’s legacies in optics) and the Aristotelian physicists. Ibn al-Haytham’s research in optics (including his studies in catoptrics and dioptrics) benefited from the investigations of his predecessors in the Archimedean-Apollonian tradition of ninth-century Arab polymaths, like the Banū Mūsā and Thābit ibn Qurra, and of tenth-century mathematicians, like al-Qūhī, al-Sijzī, and Ibn Sahl. Ibn al-Haytham’s Kitāb al-manāẓir (The Optics) was translated by Gerard of Cremona into Latin under the title: De Aspectibus (also known as Perspectivae), and it closely influenced the research of Franciscan scholars of optics in the thirteenth century, like Roger Bacon, John Peckham, and Witelo.2 Ibn al-Haytham’s tradition had also a direct impact on the investigations of fourteenth-century opticians, like Theodoric (Dietrich) of Freiberg (d. c. 1310 CE)3 and Kamāl al-Dīn al-Fārisī (d. c. 1319 CE); both scholars offered correct and experimentally oriented explications of the phenomenon of the rainbow and its coloration, while basing their studies on reformed revisions of Ibn al-Haytham’s theory of colors. Ibn al-Haytham’s and al-Fārisī’s tradition in Islamic civilization was subsequently continued through the investigations of the Syrian astronomer at the Ottoman court, Taqī al-Dīn Muḥammad ibn Maʿrūf (d. c. 1585 CE). Moreover, the Latin and Italian translations of Ibn al-Haytham’s Kitāb almanāẓir impacted scholars of the caliber of Biagio Pelacani da Parma,4 Francesco Maurolico, and Ettore Ausonio. Ibn al-Haytham’s opus was also assimilated in Renaissance mathematical and scholarly circles partly through Witelo’s Perspectivae, which closely paraphrased many of its sections, and acted through its Italian translation as the main theoretical source for Lorenzo Ghiberti’s Commentario Terzo.5 The Latin version of Ibn al-Haytham’s Kitāb almanāẓir was eventually printed under the title Opticae Thesaurus Alhazeni in the edition of Friedrich Risner, which was published in 1572 in Basle. This version of Ibn al-Haytham’s Optics, which became available in print, was read and consulted by scientists and philosophers of the caliber of Kepler, Galileo, Descartes, and Huygens.
An investigation of the historical and epistemic bearings of Ibn al-Haytham’s tradition in optics elucidates some of the dynamics that are at work in the emergence and development of novel scientific rationalities. His influential legacy established the principal scientific foundations of medieval perspectivae in the European traditions, and through them, it grounded the Renaissance theories of vision and perspective, while continuing furthermore to influence the unfolding of the science of optics up to the seventeenth century. In view of all of these historical and epistemic dimensions, I will mainly focus in this chapter on Ibn al-Haytham’s theories of vision, light, and color, and in the course of this inquiry I will also examine some of the applications of mathematics (principally geometry) in the science of optics, with a particular emphasis on Ibn Sahl’s dioptrics (fl. mid-tenth century CE). In addition, I will also consider Kamāl al-Dīn al-Fārisī’s (d. c. 1319 CE) explication of the phenomenon of the rainbow (qaws quzaḥ) in terms of his reforming commentary on Ibn al-Haytham’s Optics in Tanqīḥ al-manāẓir (The Revision of [Ibn al-Haytham’s] Optics).6 I will, moreover, account for selected historical developments in geometry that resulted in Ibn al-Haytham’s mathematical conception of place (al-makān) as extension qua space, which necessitated his refutation of the definition of topos as argued in Book Delta of Aristotle’s Physics, and also corresponded conceptually with his investigation of the visibility of spatial depth in the Optics.
While Ibn al-Haytham’s research in optics proved to be a revolutionizing tradition in the course of development of this scientific discipline up to the seventeenth century, other legacies in optics existed in the history of ideas in classical Islamic civilization. One of these principal traditions is attributed to the research of the Arab philosopher al-Kindī (d. c. 873), who partly influenced the optical investigations of Robert Grosseteste (d. c. 1253) through the Latin version of his De Aspectibus.7 However, this tradition in optics was primarily Euclidean and Ptolemaic, as was also later the case with the research of the Persian mathematician and philosopher Naṣīr al-Dīn Ṭūsī (d. c. 1274). It is also worth noting in this regard that the celebrated philosopher and physician Ibn Sīnā (Avicenna; d. 1037 CE) adopted a physical “intromission” theory of vision that is akin to that of Aristotle, and his contributions in optics were not as influential as those of Ibn al-Haytham. None the less, his research on the anatomy of the eye in his monumental Qānūn fī al-ṭibb (Canon of Medicine) impacted the evolution of ophthalmology up to the sixteenth century, and his research in meteorology inspired al-Fārisī’s revision of Ibn al-Haytham’s Optics. Furthermore, Ibn Sīnā’s theory of perception was ecumenically influential in Islamic civilization and European medieval scholarship, particularly in terms of elucidating philosophical meditations on the nature of the soul (al-nafs) and the bearings of its cognitive faculties.8
An investigation of classical optics necessitates the examination of the role of mathematics (geometry in particular) in the development of this scientific discipline. Mathematics permeated the intellectual history of medieval Islamic civilization. Besides accomplishments in the various domains of mathematical research, and the novel methods of inquiry devised in support of their advancement, the applications of mathematics sustained the progress of numerous scientific disciplines in medieval Islamic civilization. The mathematical sciences performed significant epistemic functions in grounding the systemic and technical development of optics, astronomy, mechanics, geography, and the applied arts of surveying and timekeeping. Mathematics also informed the theoretical and practical efforts invested in the perfection of scientific instruments like astrolabes, compasses, sundials, celestial globes, and the modeling of lenses, along with the investigation of the geometrical principles underlying their construction.
Considerable contributions were advanced in geometry, algebra, arithmetic, and trigonometry. Novel solutions were devised to ancient geometrical problems, and new questions in mathematics were formulated, including the establishment of complex methodologies of mathematical research. The application of the newly founded discipline of algebra (ninth century),9 to arithmetic and geometry, and of these mathematical disciplines unto one another, all produced novel branches of research in mathematics, with their prolongations in optics, astronomy, mechanics, and physics. This also resulted in the flourishing of investigations in conics, stereographic projections, geometric transformations, infinitesimal mathematics, algebraic geometry, and combinatorial and numerical analysis.10
The application of algebra and geometry unto each other allowed mathematicians to render solid geometrical problems into algebraic equations, and to also resolve these by way of the intersection of conic curves.11 This mathematical development resulted also in the founding of the novel discipline of “geometrical algebra,” and the flourishing of research on geometrical transformations (like similitude, translation, homothety, and affinity). Geometers were no longer solely interested in studying figures, but they also closely investigated the relations uniting them. This new orientation in geometry resulted in the development of the “science of projection” (ʿilm al-tasṭīḥ),12 which was in part also inspired by research in mathematical astronomy. Another major consequence of the novel investigation of geometrical transformations was also the unprecedented introduction of motion (ḥaraka) in geometrical enunciations and demonstrations. All of these developments necessitated the epistemic reorganization of the elements of geometry in terms of motion, and a rethinking of the notion of “place” (al-makān) from a mathematical standpoint that overcomes the geometrical shortcomings of the Aristotelian physical conception of topos as “enveloping surface.”
Furthermore, the applications of conics informed the complex optical studies that were undertaken on the burning sphere, the spherical diopters, lenses, and burning mirrors. This tradition in Arabic sources impacted the investigations of Renaissance theorists of the Cinquecento on the use of conic sections as optical-geometrical tools to perfect the construction of perspectives, as was the case, for instance, with Francesco Barozzi (Franciscus Barocius, mainly in his Admirandum illud Geometricum Problema tredecim modis demonstratum).13 Moreover, phenomena that were originally treated as topics of meteorology were re-studied according to the new models of “reformed” optics (as was the case, for instance, with studying the rainbow and the halo). Fundamental epistemic questions were also debated in the “philosophy of mathematics” with reflections on their cognitive and methodological entailments. These endeavors were supported by elaborate critical and analytic commentaries on mathematical treatises that were translated, adaptively assimilated, and innovatively expanded in reference to the legacies of Euclid, Archimedes, Apollonius of Perga, Diophantus, Menelaus of Alexandria, Ptolemy, Heron of Alexandria, and Pappus of Alexandria.
Based on catoptrics and the study of burning mirrors and lenses, Abū Saʿd al-ʿAlāʾ Ibn Sahl (d. c. 1000 CE) devised a systemic elaboration of the fundamentals of dioptrics (the study of the geometrical principles of the refraction of light, and the geometrical properties of the configurations of diopters, and of the refractive characteristics of the passage of light through transparent media). It is believed that Ibn Sahl established a principle akin to the so-called “Snell’s Law” of refraction (a formula used in the calculation of the refraction of light between two media of differing refractive indexes, named after the seventeenth-century mathematician Snellius Willebrord, which was also investigated by Thomas Harriot and Descartes). Ibn Sahl shows that every transparent medium, including the “celestial sphere,” has a certain degree of opacity. Yet in his research on burning optical instruments he no longer determines transparency in terms of opacity, but rather characterizes it by a constant ratio, which acts as the basis of his study of the refraction of lenses. He postulated that this ratio is the inverse of the index of refraction (n) of a given transparent medium (for example, a crystal) with respect to air.14
Ibn Sahl’s research in dioptrics rested on geometrical enunciations and demonstrations based on the study of conic sections (ellipses, parabolas, hyperbolas). His studies on burning instruments (ḥarrāqāt) and the burning sphere (al-kura al-muḥriqa)15 were thoroughly analyzed by Ibn al-Haytham, and this resulted partly in the latter’s explication of spherical aberration in terms of studying the properties of lenses, including an investigation of the optical properties of the crystalline.
Ibn al-Haytham’s research on vision (al-baṣar), and particularly on the introduction of light into the eye, which is analyzed by way of examining the optical and geometrical properties of the anatomy of the crystalline, were all entangled with his research in dioptrics and conics. Ibn al-Haytham’s demonstrations with respect to the intromission of light via the outer surface of the crystalline rested on his research in geometrical optics and on his resultant determination of “spherical aberration.” The studies that were conducted on the optical properties of conics in Ibn Sahl’s dioptrics, and his demonstrations showing that conical sections are ideal aspheric shapes for lenses (planoconvex, bi-convex, plano-concave, bi-concave), were all further elaborated by Ibn al-Haytham in his research on spherical aberration,16 and on its application in the optical and anatomical study of the passage of light via and through the crystalline in explicating vision. Ibn al-Haytham’s commentaries on Ibn Sahl’s research in dioptrics eventually informed the reformative research in optics of Kamāl al-Dīn al-Fārisī in Tanqīḥ al-manāẓir (The Revision of [Ibn al-Haytham’s] Optics), who in his turn established an accurate explication of the rainbow’s coloring.
The anaclastic research on conics in terms of systematizing the science of dioptrics and the investigation of the properties of its instruments (diopters, lenses, and tools of measurement) was fundamental to solving geometrical problems that could not be resolved using an unmarked ruler and a compass. The effort to combine the geometries of conics, and those of projections and transformations, with neusis (namely, a verging geometric construction that adjusts a measured length on a marked straight edge to fit a diagram) all bestowed an epistemic legitimacy to the construction and use of scientific instruments (diopters, compasses, astrolabes, celestial globes, sundials).17 This mathematical procedure also evolved into methods deploying the intersections of conics, in addition to introducing novel directions in mathematics involving loci and quadratic surfaces (resonating with Pierre Fermat’s Loci in Surfaces; seventeenth century). Furthermore, conic sections were distinguished from curves that could be generated via mechanical motion (namely by deploying instruments [compasses] and mechanical procedures); geometry rather than mechanics was fundamental in founding dioptrics and expanding research in catoptrics.
Besides his research in optics, Ibn al-Haytham advanced compelling investigations in mathematics. For instance, he assessed Apollonius of Perga’s Conica in his Maqāla fī tamām kitāb al-makhrūṭāt (On the Completeness of the Conics), and in his Shakl Banū Mūsā (The Proposition of the Banū Mūsā; namely, making reference to the erudite sons of Mūsā ibn Shākir; fl. Baghdad, ninth century). Moreover, his geometric Lemmas (muqaddimāt), which came to be known among seventeenth-century European mathematicians as: “Alhazen’s Problem,” became a classic in the history of science in describing a solution to the question: “How, from any two points opposite a reflecting surface (plane, spherical or cylindrical) can we find a point on that surface at which the light from one of these two points reflects unto the other?” Ibn al-Haytham also attempted to solve Euclid’s Fifth Postulate, and composed a commentary on the premises of the Elements (Sharḥ muṣādarāt kitāb Uqlīdis). He furthermore dedicated two tracts to studying “the quadrature of crescent figures/lunes” (alashkāl al-hilāliyya). He also systematized analytical geometry by deploying algebra in geometric constructions, and furthered the field of infinitesimal mathematics. In the domain of number theory, he built on the works of Thābit ibn Qurra (d. 901 CE), and on the findings of the latter’s grandson, Ibrāhīm ibn Sinān (d. 946 CE), by investigating amicable numbers (aʿdād mutaḥābba; namely a pair of numbers each of which equals the sum of the other’s aliquot parts) and perfect numbers (aʿdād tāmma; namely a numeral whose positive divisors, excluding it, sum up to itself). In addition, Ibn al-Haytham contributed engaging studies in mathematical astronomy, such as those included in his Shukūk ʿalā Baṭlāmiyūs (Dubitationes in Ptolemaeum; a critique of Ptolemy’s Almagest, Planetary Hypotheses, and Optics), or in his Ruʾyat alkawākib (The Appearance of the Stars), or Maqāla fī aḍwāʾ al-kawākib (Treatise on the Lights of the Stars). He also investigated the density of the atmosphere, the nature of the eclipse (Maqāla fī ṣūrat al-kusūf), the twilight and moonlight (Risāla fī ḍawʾ al-qamar; Treatise on Moonlight). Moreover, he inquired about a principle akin to “the first law of motion” in mechanics, according to which it was observed that a body would move perpetually unless arrested by an external agent. And, in his elucidation of phenomena associated with the attraction between masses, he made observations regarding the magnitude of acceleration, which resulted from a principle crudely analogous to a force of gravity. Ibn al-Haytham’s methodology consisted of combining mathematics with physics in the context of experimental demonstration, verification, and controlled testing (al-iʿtibār). This endeavor included the design and use of scientific instruments and installations (like al-bayt al-muẓlim/camera obscura,; which also constituted a vital notion in medieval and Renaissance perspectivae traditions).18
One of the principal aspects of Ibn al-Haytham’s reforming of the science of optics is encountered in his ingenious resolution of the long-standing ancient dispute between the mathematicians (aṣḥāb al-taʿālīm; the exponents of the traditions of Euclid and Ptolemy) and the physicists (aṣḥāb al-ʿilm al-ṭabīʿī; mainly of Aristotelian inspiration) over the nature of vision and light. Ibn al-Haytham showed that vision occurs by way of the introduction of physical light rays into the eye in a configuration that is geometrically determined in the form of a pyramid/cone (makhrūṭ) of vision, with its vertex at the center of the eye and its base on the visible and lit surfaces of the object of vision. He thus rejected the emission (“extramission”) theory of the mathematicians, which held that vision occurs by way of the emission of a subtle and non-consuming ray of light from the eye that meets the lit medium, which, as a physical phenomenon, is structured geometrically in the form of a cone/pyramid. In view of explicating the process of vision, Ibn al-Haytham retains the geometric modeling as presented by the ancient mathematicians (mainly as derived from the adaptations of Euclidean and Ptolemaic optics), while emphasizing that it was abstracted from matter, and that the lines determining this pyramidal/conical configuration were purely mathematical (postulated) rather than physical. Moreover, he refuted the physicists’ theory of vision (as inspired by Aristotle’s Physics and De anima), which ambivalently conjectured that sight results from the “intromission” into the eye of the form of the visible object without its matter when the transparent medium (al-shafīf; diaphanes) is actualized by physical illumination. Ibn al-Haytham demonstrated that vision occurs by way of the introduction of light into the eye, while showing that this physical phenomenon was geometrically structured in the shape of a virtual-mathematical cone of vision.19 Consequently, he distinguished vision from light, and devised novel methodological procedures that brought the certitude and invariance of geometrical demonstration to bear with isomorphism (instead of mere synthesis) on his research in physical optics.20 He moreover subjected the resultant mathematical-physical models and hypotheses to experimentation by way of controlled empirical procedures of testing, including the devising and use of the camera obscura (al-bayt al-muẓlim). Experimentation, as a notion that appeared in its rudimentary forms in the research of polymaths of the ninth century in medieval Islamic civilization, was known by the appellation: al-iʿtibār (experimentatio). This empirical process was further systematized in novel methodological applications by Ibn al-Haytham in the context of optics,21 and further refined by al-Fārisī after him. Ibn al-Haytham did not use experimentation as an element of empirical methodology, rather it was essentially (and theoretically) integral to his proofs, and it granted an apodictic value to his enquiries in optics.22
Ibn al-Haytham’s geometrical, physical, physiological, and meteorological studies in optics were also related to his psychology of visual perception and to his analysis of the faculties of judgment and discernment, of cognitive comparative measure, of (eidetic) recognition, imagination, and memory. He thus distinguished the immediate mode of perception by way of glancing, as idrāk bi-al-badīha (comprehensio superficialis or comprehensio per aspectum), from contemplative perception, as idrāk bi-al-taʾammul (comprehensio per intuitionem—Optics, II.4 [5, 20, 33]).23 In reflecting on the manner of perceiving particular visible properties (al-maʿānī al-mubṣara; intentiones visibiles—Optics, II.3 [43–8]),24 Ibn al-Haytham made a distinction between mujarrad alḥiss (pure sensation), which perceives light qua light and color qua color, and almaʿrifa (recognition) with the associated al-tamyīz wa-al-qiyās (discernment and comparative measure/inference)—the latter perceives properties in the form of a visible object that has been previously seen or remembered, and it involves inference (qiyās) in inspecting and discerning (tamyīz) the signs of that form (idrāk bi-al-amārāt; comprehensio per signum—Optics, II.3 [50–52], II.4 [22]). Visual perception engages al-quwwa al-mumayyiza (virtus distinctiva; faculty of discernment), which perceives all the properties (Optics, II.3 [1–25]), while being aided by imagination (al-takhayyul; imaginatio) and memory (aldhikr or al-tadhakkur; rememoratio), and is usually operative without deliberate or excessive effort (Optics, II.4 [12–15, 22]).
Although Ibn al-Haytham noted that pure sensation perceived light qua light, and color qua color, he did none the less argue that sensation was ultimately effected (psychologically-physiologically) by the last sentient (al-ḥāss al-akhīr; sentiens ultimum) and not by the eye alone (Optics, I.6 [74]), while also basing his conclusions on anatomical (al-tashrīḥ) examinations of the structure of the eye (fī ḥayʾat al-baṣar; Optics, I.5 [1–39]) and furthermore reinforcing them through investigations of binocular vision (Optics, I.6 [69–82]).25 The image formed on the crystalline (al-jalīdīyya) passes through the vitreous (al-zujājīyya) to the hollow optic nerve (alʿaṣaba al-jawfāʾ), which connects to the common nerve (al-ʿaṣaba almushtaraka) as a sensation leading to the last sentient in the anterior part of the brain (muqaddam al-dimāgh). Moreover, and in reference to binocular vision, the beholder, under normal circumstances of sight, perceives a single visible object with two sound eyes (salāmat al-baṣarayn). The form of that single visible object occurs on the surface of the crystalline of each of the eyes. Looking at that object, two of its forms are received, one in each of the eyes. Consequently, two forms, each occurring on the crystalline, pass via the vitreous to the hollow nerves, and (as sensations) they ultimately become unified in the common nerve, thus reaching the last sentient as an ordered single form of a sensible object (al-ṣūra al-muttaḥida li-al-mubṣar al-wāḥid).
The object of vision is seen by way of the introduction into the eye of the light rays that are emitted from the visible and lit surfaces of the seen object, which propagate rectilinearly across the transparent medium that is between the observer and the observed, while the reception of these rays in the eye is structured geometrically in the shape of a virtual visual cone (makhrūṭ al-shuʿāʿ), with its vertex at the center of the eye and its base on the seen and lit surfaces of the visible object (Optics, I.2, I.3). The light rays that are structured within this mathematical model travel from every point on the lit and appearing surfaces of the visible object in a punctiform-corpuscular configuration, with a spherical irradiation that is emitted through transparent media in all directions. This phenomenon reflects also a point-by-point correspondence between each point on the lit and visible surface of the object of vision and each correlative point on its retinal image, which secures the ordering of the visible aspects of the seen object. This is also the case given that only the light rays that meet the outer surface of the crystalline humor (al-ruṭūba al-jalīdīyya) perpendicularly are admitted into the eye. As for peripheral visible aspects of objects that fall outside the virtual cone of vision, they may be sensed laterally but not in terms of direct and clear vision. This phenomenon is also analyzed in terms of studying the geometrical properties of the outer surface of the crystalline, which is treated as an optical lens (spherical section), with its analysis relating to Ibn al-Haytham’s studies on spherical aberration that were partly based on Ibn Sahl’s dioptrics (as we have also indicated above).
To illustrate the optical phenomenon of the one-to-one correlation between the points on a lit visible surface and those that correspond with them on its retinal image, let us consider the case of a given point A on a lit and visible surface S. The light emitted from every point of the lit and visible surface S, as is the case here with point A, would irradiate in a spherical configuration; hence innumerable light-rays are emitted from A in all directions, and propagate rectilinearly across transparent media. If a transparent medium like air exists between A and the eye of the observer, then among all the rays that are rectilinearly emitted from A in all directions, only one is introduced through the outer surface of the crystalline as it meets it perpendicularly. This ray of light then passes through the focal centre of the crystalline, and it correlates point A on the visible surface S, from which it is emitted, with a corresponding point A1 on the retinal image of this seen surface S. In terms of binocular vision, A1, as the correlative point on the retinal image, which corresponds with the point A on surface S, would itself be a unified point of two separate impressions made by A on each of the eyes; hence A has an impression A1R on the right eye, and an impression A1L on the left eye, and both A1R and A1L appear as one and the same unified impression A1. Each light ray that causes impressions A1R and A1L passes via the crystalline, through the vitreous and up to the hollow nerve of each of the eyes. Both impressions A1R and A1L are unified as “sensations” in the common nerve, reaching the last sentient as a single impression A1. Ultimately, the surface S is seen as one surface, and not as two or many (namely, under normal conditions of vision: healthy eyes, adequate illumination, and transparency in the medium between the observer and the object of vision, optimal distance and position of observer and observed, suitable size of the visible properties of the seen entity, and so on).
Light was divided by Ibn al-Haytham in his Risāla fī al-ḍawʾ (Treatise on Light)26 into two forms: one that is substantial and is the first light (ḍawʾ jawharī awwal), and the other that is accidental and is the second light (ḍawʾ ʿaraḍī thānī; not to be confused with reflected light).27 Substantial light is emitted from luminous sources (in themselves), and accidental light is emitted from lit opaque surfaces that are illuminated by substantial lights radiating on them from luminous sources (in themselves). Ibn al-Haytham deploys the classical distinction between forma substantialis (ṣūra jawharīyya) and forma accidentalis (ṣūra ʿaraḍīyya) in the context of his reform of optics. He thus uses a traditional vocabulary in a novel methodological and conceptual context, and in support of his reforming of optics. The division between both types of light played a central function in terms of his conception of colors as being ontologically/existentially independent from light, but always accompanying accidental lights and appearing with them, as well as following the same principles of rectilinear propagation and spherical irradiation that govern them. Colors were consequently postulated by him as being objects of geometry as well as being physical phenomena akin to light, but ontologically separate from it (Optics, I, 3 [134–9]).
It is worth highlighting that Ibn al-Haytham rejected the Aristotelian theory of the diaphanes (the transparent medium). After all, Aristotle noted that the object of vision is the color (khrôma) that is on the surface of objects that are visible propter (De Anima, 418a 26–31; 422a 14), and is on the surface (epiphaneia) of the body (Physics 210a 29–31, 210b 4–5). Moreover, the transparent body (diaphanes), which is invisible (or at least barely visible; De Anima, 418b 26–27) and is uncolored, would be the milieu receiving colors. Furthermore, light (phôtos) is conceived as the actualization of that which is transparent (De Anima, 418a 31–32; 418b 9–11): the more a medium is transparent, the more it becomes colored by way of its actualization in terms of illumination; color thus appears as the limit of transparency (eskhatos).28
In connection with the question concerning the ontological status of colors, the investigation of the rainbow (qaws quzaḥ) occupied a focal place in the history of meteorological optics.29 Ibn al-Haytham’s thesis that colors had an existential objective reality that is separate from that of light (I, 3 [134–9]), even though their apparition was always intermixed with lighting and presupposed illumination, was subsequently rejected by al-Fārisī, who showed (experimentally and through mathematical means) in his Tanqīḥ that: light manifests colors out of itself as a result of the differential aspects of its refraction and reflection when passing from a transparent medium into another (min shafīf ilā ākhar) that has a varying refractory disposition (hence having heterogeneous indexes of refraction).30 The rainbow appears to observers situated in optimal places when sun rays are refracted and reflected on raindrops resulting from cloud condensations. Two arcs forming concentric color bands of the spectrum of light appear in the sky while being separated by a grayish (subtly obscure) zone.31
Al-Fārisī had recourse to experimentation (iʿtibār) in his explication of the rainbow, while sustaining his analysis in terms of geometric constructs. He deployed a synthetic (artefact) manufactured object (as a large spherical glass vessel filled with water) to experimentally constitute the modeling of a natural phenomenon (a raindrop). Geometry, along with a controlled experimental series, using this model within a camera obscura, and the deployment of measured illuminations, all allowed him to study the phenomenon of the propagation of light (in refraction and reflection). His research in this domain benefited from the investigations in dioptrics of Ibn Sahl and Ibn al-Haytham on the refractive properties of spherical transparent vessels. He was also inspired by Ibn Sina’s meteorological studies and investigations in catoptrics. Al-Fārisī ultimately showed that colors resulted from the formation of two or more images (forms) of light that are projected on a screen (or received by the eye), which are caused by refractions and reflections through a spherical transparent vessel modeling the raindrop.32 Moreover, while Ibn al-Haytham observed that the speed of light, which is enormous, is none the less finite, al-Fārisī affirmed this proposition, and furthermore argued that the velocity of light was inversely proportional to the optical density of the media it traverses. This perspective on the nature of light allowed him to present a “wave” theory of light that is connected with his explication of colors, and led him to reject the “corpuscular” theory of light that was proposed by Ibn al-Haytham.
It was not uncommon in the history of scientific ideas in classical Islam that selected problems in theoretical philosophy were solved with the assistance of mathematics. This was the case, for instance, with Abū Sahl Wayjan ibn Rustam al-Qūhī’s (fl. tenth century) geometrical demonstration of the possibility of achieving “an infinite motion in a finite time,”33 which aimed at contesting Aristotle’s views on this matter as they were delineated in Book VI of the Physics. Moreover, we attest similar endeavors to deploy mathematics in solving selected problems of speculative onto-theology, as was the case with Naṣīr al-Dīn al-Ṭūsī’s use of mathematical combinatorial analysis in explicating the Neo-Platonic ontological-cosmological process of “emanation from the One.”34 It is in this spirit that Ibn al-Haytham presented his geometrical conception of place as a solution to a long-standing problem that remained philosophically unresolved, which, to our knowledge, also constituted the first viable attempt to geometrize “place” in the history of science. Ibn al-Haytham aimed to promote a geometrical conception of place that is akin to extension in view of addressing selected mathematical problems that resulted from the unprecedented developments in geometrical transformations (similitude, translation, homothety, and affinity), the introduction of motion in geometry, the anaclastic research in conics, and dioptrics in the Apollonian-Archimedean Arabic legacy, as also shown in the earlier sections of this chapter.35
Besides the penchant to offer mathematical solutions to problems in theoretical philosophy that were challenged by long-standing historical obstacles and epistemic impasses, Ibn al-Haytham’s remarkable and successful endeavor in geometrizing place was undertaken in view of sustaining and grounding his research in mathematical analysis and synthesis (al-taḥlīl wa-al-tarkīb),36 and in response to the needs associated with the unfurling of his studies on knowable mathematical entities (al-maʿlūmāt), and in order to reorganize most of the notions of geometry and rethink them anew in terms of motion.37 Consequently, he had to critically reassess the dominant philosophical conceptions of place in his age, which were encumbered by inconclusive theoretical disputes that were principally developed in response to Aristotle’s Physics.38 So, in his Qawl fī al-makān (Discourse on Place)39 Ibn al-Haytham presented his mathematical refutation of the Aristotelian physical conception of topos as “enveloping surface.”
Even though Aristotle affirmed that topos has the three dimensions of length, width, and depth (Physics, IV, 209a 5), he rejected the theories that posited place as being the form (eidos), the matter (hulê) or the interval (diastêma) between the extremities of the body that it contains (Physics, IV, 212a 3–5). He ultimately defined topos as: “the innermost primary surface-boundary of the containing body that is at rest, and is in contact with the outermost surface of the mobile contained body” (Physics, IV, 212a 20–21). Based on this thesis, one would add that a place could be grasped as a vessel that is immovable. Moreover, when something moves inside another that is also in motion, like a boat in a river, it uses the containing body as a vessel, while the river basin acts as the motionless place (Physics, IV, 212a 15–20)—“A place is together with the [contained] thing, since the limit [of that which contains] coincides with that which is limited” (Physics, IV, 212a 29–30); and this is the case given that the inner boundary of a containing body coincides with the shape of the container.
Topos, as “the inner surface of the containing body that is in contact with the outer surface of what it contains,” is an enveloping surface of containment, which resulted in the grasp of al-makān by “Aristotelian” physicists in classical Islam as a saṭḥ muḥīṭ or saṭḥ ḥāwī (surrounding or containing surface). This definition refers principally to what we may call a “local place,” which is the specific containing body that a given thing occupies, in contrast with the “cosmic qua natural place,” namely, the one toward which things tend to go back to due to their own nature (phusei) if not prevented from doing so; as heavy bodies travel by their nature downwards toward the Earth in a fall in the direction of the center of the Universe (kosmos), or light bodies travel by their nature upwards toward the heavens (Physics, IV, 4, 212a24).
In contesting the long-standing Aristotelian physical conception of topos (specifically in its “local” containment sense), Ibn al-Haytham posited al-makān as an imagined void (khalāʾ mutakhayyal; postulated void) whose existence is secured in the imagination, like it is the case with invariable geometrical entities. He moreover held that the “imagined void” qua “geometrized place” consisted of imagined immaterial distances that are between the opposite points of the surfaces surrounding it (al-abʿād al-mutakhayyala al-latī lā māda fīhā, al-latī hiya bayna al-nuqaṭ al-mutaqābila min al-saṭḥ al-muḥīṭ).40 He furthermore noted that the imagined distances of a given body, and those of its containing place, get superposed and united in such a way that they become the same distances (qua dimensions) as mathematical lines having lengths without widths/breadths. Consequently, it is worth noting in this regard that Ibn al-Haytham’s geometrization of place was “ontologically” neutral. This is the case given that his mathematical notion of al-makān was not simply obtained through a “theory of abstraction” as such, nor was it derived by way of a “doctrine of forms,” nor was it grasped as being the (phenomenal) “object” of “immediate experience” or “common sense.” It is rather the case that his geometrized place resulted from a mathematical isometric “bijection” function between two sets of relations or distances.41 Nothing is thus retained of the properties of a body other than extension, which consists of mathematical distances that underlie the geometrical and formal conception of place. Accordingly, the makān of a given object is a “region of extension that is defined by the distances between its points, on which the distances of that object can be applied bijectively.”42
To give an example of Ibn al-Haytham’s mathematical refutation of Aristotle’s physical definition of topos, we could consider the case of his geometric demonstration based on the properties of a parallelepiped (mutawāzī al-suṭūḥ; namely, a geometric solid bound by six parallelograms). If this given parallelepiped were to be divided by a rectilinear plane that is parallel to one of its surfaces, and is then recomposed, the cumulative size of its parts would be equal to its magnitude prior to being divided, while the total sum of the surface areas of its parts would be greater than its surface-area prior to being partitioned. Following the Aristotelian definition of topos, and in reference to this divided parallelepiped, one would conclude that an object divided into two parts occupies a place that is larger than the one it occupied prior to its division. Hence, the place of a given body increases, while that body does not (makān al-jism yaʿẓim wa-al-jism lam yaʿẓim); consequently, an object of a given magnitude is contained in unequal places, which is an untenable proposition.43 Likewise, if we consider the case of a parallelepiped that is carved, then its bodily magnitude is diminished while the total sum of its surface area would increase. Following the Aristotelian definition of topos, and in reference to this carved parallelepiped, one would conclude that: an object that diminishes in magnitude occupies a larger place, which is untenable.
Moreover, using mathematical demonstrations, in terms of geometrical solids of equal surface areas, and figures that have equal perimeters, Ibn al-Haytham showed that the sphere is the largest in (volumetric) size with respect to all other primary solids that have equal surface-areas (al-kura aʿẓam al-ashkāl al-latī iḥāṭatuhā mutasāwiya).44 So, if a given sphere has the same surface area as a given cylinder, they occupy equal places according to Aristotle, and yet the sphere would have a larger (volumetric) magnitude than the cylinder; hence unequal objects occupy equal places, which is not the case.
Ultimately, Ibn al-Haytham’s critique of Aristotle’s definition of topos, and his own geometrical positing of al-makān as an “imagined void” (khalāʾ mutakhayyal), both substituted the grasping of the body as being a totality bound by physical surfaces to construing it as a set of mathematical points that are joined by geometrical line-segments. Hence, the qualities of a body are posited as an extension that consists of mathematical lines, which are invariable in magnitude and position, and that connect points within a region of the three-dimensional space independently of the physical body.
The geometrical place of a given object is posited as a “metric” of a region of the so-called “Euclidean” qua “geometrical space,” which is occupied by a given body that is in its turn also conceived extensionally, and corresponds with its geometrical place by way of “isometric bijection.” Consequently, Ibn al-Haytham’s geometrization of place points to what was later embodied in the conception of the “anteriority of spatiality” over the demarcation of a metric of its regions by means of mathematical lines and points, as explicitly implied by the notion of a “Cartesian space.”45 The scientific and mathematical significance of the geometrization of place was confirmed in the maturation of mathematics and science in the seventeenth-century conceptions of place as extension (as a volumetric, three-dimensional, uniform, isotropic, and homogeneous space), particularly in reference to the works of Descartes (on extensio) and Leibniz (analysis situs).
There is no doubt that the maturation of Euclidean geometry and its prolongations benefited immensely from the geometrization of place, which among other developments resulted in the emergence of what came to be known in periods following Ibn al-Haytham’s age as being a “Euclidean space”: namely, an appellation that is coined in relatively modern times, and describes a notion that is historically posterior to the geometry of figures as embodied in Euclid’s Stoikheia (The Elements; Kitāb Uqlīdis fī al-Uṣūl).46 After all, the expression deployed by Euclid that is closest to a notion of “space” as expressed in the Greek term khôra, is the appellation khôrion, which designates “an area enclosed within the perimeter of a specific geometric abstract figure”—as noted, for instance, in Euclid’s Data (Dedomena; al-Muʿṭayāt) Prop. 55 (related to: Elements, VI, Prop. 25): “if an area [khôrion] be given in form and in magnitude, its sides will also be given in magnitude.”
While Ibn al-Haytham’s geometrization of place corresponds with later developments in history of science and mathematics, one of the principal last attempts (rather “unsuccessful”) on the part of physicist-philosophers in medieval Islamic civilization to rescue Aristotle’s definition of topos is encountered in the “refutation of Ibn al-Haytham’s makān” by ʿAbd al-Laṭīf al-Baghdādī (in his fourteenth-century treatise Fī al-radd ʿalā Ibn al-Haytham fī al-makān [A refutation of Ibn al-Haytham’s place]).47
Although Ibn al-Haytham’s Optics influenced theories of vision and perspectivae in European scholarship up to the seventeenth century, there is no documented evidence that his geometrization of place had a wider reception beyond the history of scientific ideas in Islam. However, his geometrical conception of place as space qua extension corresponded with his affirmation of the visibility of spatial depth in the Optics, which also displayed a coherent and rigorous eschewing of the Aristotelian definition of topos.
Ibn al-Haytham’s reflections on the notion of space in his Kitāb al-manāẓir (Optics) were commensurable with his mathematical conception of place in his Qawl fī al-makān (Discourse on Place), which also carried resonances with the epistemic evolution of Renaissance and Early-Modern conceptions of spatiality and extendedness. The definition of place as “space,” rather than grasping it as an “enveloping surface,” corresponded also with the manner in which architecture and perspective shared a sense of coherent spatiality as embodied in the notion of the “room.” This “idealized representation” acquired in the history of its conceptual development the characteristics of the “isotropic space of geometry” that was perhaps “anticipated” in the “perspectivity” of architecture with the “parallelism” of its structuring components (columns, pillars, walls) and the “axial regularity” of its spatial articulations,48 which ultimately integrated the impetus of geometry and optics within the structuring of the pictorial order and its relatable forms of organizing space.
1 Ibn al-Haytham, Kitāb al-manāẓir, ed. A. I. Sabra (2 vols., Kuwait: National Council for Culture, Arts and Letters, 1983); Ibn al-Haytham, The Optics, Books I–III, On Direct Vision, trans. A. I. Sabra (2 vols., London: Warburg Institute, 1989).
2 In a critical analysis of A. C. Crombie’s thesis that “modern” scientific methodology is attributable to the tradition of R. Grosseteste, and to thirteenth-century opticians like R. Bacon, J. Peckham, and Witelo, A. Koyré rather argued that the legacy of Alhazen (Ibn al-Haytham) in optics resulted in this flourishing of the perspectivae tradition of Franciscan scholars in the European Middle Ages, in addition to the applications of its experimental methods. See Alexandre Koyré, Etudes d’histoire de la pensée scientifique, 2nd edn. (Paris: Gallimard, 1973), 59–86, esp. 69–71. Refer also to Alistair C. Crombie, Robert Grosseteste and the Origins of Experimental Science, 1100–1700 (Oxford: Clarendon Press, 1953); Gérard Simon, “La psychologie de la vision chez Ptolémée et Ibn al-Haytham,” in A. Hasnaoui, A. Elamrani-Jamal, and M. Aouad, eds., Perspectives arabes et médiévales: sur la tradition scientifique et philosophique grecque (Leuven and Paris: Peeters-Institut du Monde Arabe, 1997), 189–207; Graziella Federici Vescovini, “La fortune de l’Optique d’Ibn al-Haytham: le livre De aspectibus (Kitāb almanāẓir) dans le Moyen Age latin,” Archives d’histoire des sciences 40 (1990), 220–38.
3 See also: Graziella Federici Vescovini, “La nozione di oggetto secondo la Perspectiva di Teodorico di Freiberg,” in Graziella Federici Vescovini and Orsola Rignani, eds., Oggetto e spazio. Fenomenologia dell’oggetto, forma e cosa dai secoli XIII–XIV ai post-cartesiani, Micrologus 24 (Florence: SISMEL, Edizioni del Galluzzo, 2008), 81–9.
4 This is particularly the case with Pelacani’s Quaestionis perspectivae. See Biagio Pelacani, Quaestionis perspectivae, ed. Graziella Federici Vescovini (Paris: Vrin, 2002).
5 Graziella Federici Vescovini, “Alhazen vulgarisé. Le De li aspecti d’un manuscrit du Vatican (moitié du XIVe siècle) et le troisième commentaire sur l’optique de Lorenzo Ghiberti,” Arabic Sciences and Philosophy 8 (1998), 67–96.
6 Kamāl al-Dīn al-Fārisī, Kitāb tanqīḥ al-manāẓir (2 vols., Hyderabad: Osmania Press, 1928–29).
7 Abū Yūsuf Yaʿqūb b. Isḥāq al-Kindī, Kitāb fī ʿilal ikhtilāf al-manāẓir (De Aspectibus), ed. and trans. from Latin Roshdi Rashed, Œuvres philosophiques et scientifiques d’al-Kindī, vol. 1, L’optique et la catoptrique (Leiden: Brill, 1997); Abū Yūsuf Yaʿqūb b. Isḥāq al-Kindī, Rasāʾil al-Kindī al-falsafīyya, ed. Muḥammad ʿAbd al-Hādī Abū Rīda (Cairo: Dār al-fikr al-ʿarabī, 1950–53), vol. II, including his epistle “On Reforming Euclid’s Optics” (Fī iṣlāḥ Kitāb Uqlīdis).
8 See, for instance, Dag Nikolaus Hasse, Avicenna’s De anima in the Latin West: The formation of a Peripatetic Philosophy of the Soul, 1160–1300 (London and Turin: Warburg Institute and Nino Aragno Editore, 2000). I have also investigated related themes in: Nader El-Bizri, “Avicenna’s De Anima between Aristotle and Husserl,” in Anna-Teresa Tymieniecka, ed., The Passions of the Soul in the Metamorphosis of Becoming (Dordrecht: Kluwer Academic Publishers, 2003), 67–89.
9 As originally systematized by al-Khwārizmī in Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-al-muqābala (c. 820 CE); see the Arabic critical edition and annotated French translation in Roshdi Rashed, Al-Khwārizmī: Le commencement de l’algèbre (Paris: Blanchard, 2007).
10 Roshdi Rashed, The Development of Arabic Mathematics: Between Arithmetic and Algebra (Dordrecht: Kluwer Academic Publishers, 1994).
11 This refers to the research of Abū ʿAbd’Allāh al-Mahānī (fl. 9th cent. CE) and Abū Jaʿfar al-Khāzin (fl. 10th cent. CE) in rendering geometrical problems into algebraic equations, based on al-Khwārizmī’s algebra, or the innovative classification of cubic equations with geometric solutions through intersections of conics. For further insights into these traditions in mathematical research, see Roshdi Rashed, Les mathématiques infinitésimales du IXe au XIe, vol. III, Ibn al-Haytham. Théories des coniques, constructions géométriques et géométrie pratique (London: al-Furqān Islamic Heritage Foundation, 2000); ibid., vol. IV, Méthodes géométriques, transformations ponctuelles et philosophie des mathématiques (London: al-Furqån Islamic Heritage Foundation, 2002), and Roshdi Rashed, Oeuvre mathématique d’al-Sijzi, vol. I, Géométrie des coniques et théorie des nombres au Xe siècle, special issue of Les Cahiers du MIDEO, vol. 3 (Leuven: Peeters, 2004).
12 Roshdi Rashed, Geometry and Dioptrics in Classical Islam (London: al-Furqān Islamic Heritage Foundation, 2005).
13 Regarding some aspects of the adaptive assimilation by Renaissance theorists of Arabic mathematical sources on conics and their applications in optics, I refer the reader to Dominique Raynaud, “Le tracé continu des sections coniques à la Renaissance: applications optico-perspectives, héritage de la tradition mathématique arabe,” Arabic Sciences and Philosophy 17 (2007), 299–345. See also Paul L. Rose, “Renaissance Italian Methods of Drawing the Ellipse and Related Curves,” Physis 12 (1970), 371–404.
14 Rashed, Geometry and Dioptrics in Classical Islam, 58–63; Roshdi Rashed, “A Pioneer in Anaclastics: Ibn Sahl on Burning Mirrors and Lenses,” Isis 81 (1990), 464–91.
15 For references to studies in dioptrics (Arabic critical editions and annotated English translations with commentaries), see Rashed, Geometry and Dioptrics in Classical Islam, as follows: Ibn Sahl’s research on the parabolic and ellipsoidal mirrors, on plano-convex and biconvex lenses in Kitāb al-ḥarrāqāt (Burning Instruments), 73–143; Ibn al-Haytham, al-Kāsir al-kurī (The Spherical Dioptre; Optics Book VII), 184–215; Ibn al-Haytham, The Spherical Lens, 216–23, and his Treatise on the Burning Sphere, 224–55, with al-Fārisī’s commentaries, 256–93.
16 Spherical aberration occurs when beams of light, which are parallel to the axis of the lens (as a spherical section) yet also vary in terms of their distance from it, become focused in different places, which results in the blurring of the resultant image.
17 For references to mathematical studies on scientific instruments (Arabic critical editions and annotated English translations with commentaries), see Rashed, Geometry and Dioptrics in Classical Islam, as follows: Al-Qūhī’s research, 726–97; al-Sijzī’s, 798–807, with commentaries from Kamāl al-Dīn ibn Yūnus and Athīr al-Dīn al-Abharī on the perfect compass (al-birkār al-tāmm) as a compass of conics (birkār al-makhrūṭ); see also al-Qūhī’s study on the astrolabe, 878–939, and Ibn Sahl’s commentary on 940–67.
18 I have also noted some of these aspects of Ibn al-Haytham’s research in: Nader El-Bizri, “Ibn al-Haytham,” in Thomas F. Glick, Steven J. Livesey, and Faith Wallis, eds., Medieval Science, Technology, and Medicine: An Encyclopedia (London: Routledge, 2005), 237–40.
19 Regarding selected studies on Ibn al-Haytham’s theory of visual perception, see A. I. Sabra, “Sensation and Inference in Alhazen’s Theory of Visual Perception,” in Peter K. Machamer and Robert G. Turnbull, eds., Studies in Perception: Interrelations in the History of Philosophy and Science (Columbus, OH: Ohio State University Press, 1978), 160–85; A. I. Sabra, “Form in Ibn al-Haytham’s Theory of Vision,” in Fuat Sezgin, ed., Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, vol. 5 (Frankfurt am Main: Institut für Geschichte der Arabisch-Islamischen Wissenschaften, 1989), 115–40; Muṣṭafā Naẓīf, al-Ḥasan bin al-Haytham, buḥūthahu wa-kushūfahu al-baṣarīyya (2 vols., Cairo: Maṭbaʿat al-nūrī, 1942–43); Roshdi Rashed, “Optique géométrique et doctrine optique chez Ibn al-Haytham,” in Optique et mathématiques: Recherches sur l’histoire de la pensée scientifique en arabe (Aldershot: Variorum Reprints, 1992), ch. II; Graziella Federici Vescovini, Studi sulla prospettiva medievale (Turin: Pubblicazioni della facolta di lettere e filosofia, Università di Torino, 1965).
20 Rashed, “Lumière et vision” (Optique et mathématique, ch. IV), 27, 42. I have examined associated themes in Nader El-Bizri, “A Philosophical Perspective on Alhazen’s Optics,” Arabic Sciences and Philosophy 15:2 (2005), 189–218.
21 Refer to: Saleh Beshara Omar, Ibn al-Haytham’s Optics: A Study of the Origins of Experimental Science (Minneapolis, MN: Bibliotheca Islamica, 1977), and Matthias Schramm, Ibn al-Haythams Weg zur Physik (Wiesbaden: Franz Steiner Verlag, 1963).
22 Rashed, Geometry and Dioptrics in Classical Islam, 181.
23 Subsequent references to Ibn al-Haytham’s Optics in the body of the text indicate the numbering of the book with its sections/chapters.
24 While Ibn al-Haytham enumerated 22 particular visible properties (Optics, II.3, [44]), Ptolemy restricted their number to seven. See: A. I. Sabra, “Ibn al-Haytham’s Criticisms of Ptolemy’s Optics,” Journal of the History of Philosophy 4 (1966), 46; Albert Lejeune, Euclide et Ptolémée, deux stades de l’optique géométrique grecque (Leuven: Bibliothèque de l’Université de Louvain, 1948), 12.
25 See also: Dominique Raynaud, “Ibn al-Haytham sur la vision binoculaire, un précurseur de l’optique physiologique,” Arabic Sciences and Philosophy 13 (2003), 79–99.
26 Roshdi Rashed, “Le Discours de la lumière d’Ibn al-Haytham (Alhazen): Traduction française critique,” Revue d’histoire des sciences 21 (1968), 197–224. See also Ibn al-Haytham, Majmūʿ rasāʾil Ibn al-Haytham (Hyderabad: Osmania Press, 1938–39).
27 Rashed, Discours de la lumière (Optique et mathématique, ch. V), 198.
28 This Aristotelian conception of colors and transparencies seems to correspond with the definition of topos in Book Delta of the Physics, which was refuted by Ibn al-Haytham in his Treatise on Place (Qawl fī al-makān). See Ibn al-Haytham, Qawl fī al-makān (Traité sur le lieu), Arabic critical edition and annotated French translation, in Roshdi Rashed, Les mathématiques infinitésimales du IXe au XIe siècle, vol. IV (London: al-Furqān Islamic Heritage Foundation, 2002), 666–85.
29 Roshdi Rashed, “Le modèle de la sphère transparente et l’explication de l’arc-en-ciel: Ibn al-Haytham, al-Fārisī,” Revue d’histoire des sciences 23 (1970), 109–40; Rashed, Optique et mathématiques, ch. III. See also Muṣṭafā Naẓīf, Kamāl al-Dīn al-Fārisī wa baʿḍ buḥūthih fī ʿilm al-ḍawʾ (La société égyptienne de l’histoire des sciences, 1958), vol. 2, 63–100.
30 Al-Fārisī, Tanqīḥ, vol. II, 258–409.
31 Ibid., 340–42. We find parallels with De iride et radialibus impressionibus of Theodoricus Teutonicus, ed. J. Würschmidt, in Beiträge zur Geschichte der Philosophie des Mittelalters XII:5–6 (1914).
32 Rashed, “Le modèle de la sphère transparente et l’explication de l’arc-en-ciel” (Optique et mathématique, ch. III), 110–13, 131–3, 135–40; al-Fārisī, Tanqīḥ, vol. II, 337.
33 Rashed, Geometry and Dioptrics in Classical Islam, 986; Roshdi Rashed, “Al-Qūhī vs. Aristotle: On Motion,” Arabic Sciences and Philosophy 9 (1999), 7–24. It is worth noting that al-Qūhī praised the epistemic and foundational value of mathematics in comparison with the merits of the other sciences in the preamble of his treatise On the Trisection of a Known Angle (Istikhrāj qismat al-zāwiya al-maʿlūma bi-thalāthat aqsām mutasāwiya): Rashed, Geometry and Dioptrics in Classical Islam, 494–5.
34 Ibid., 975; Roshdi Rashed, “Metaphysics and Mathematics in Classical Islamic Culture: Avicenna and His Successors,” in Ted Peters and Muzaffar Haq, eds., God, Life, and the Cosmos: Christian and Islamic Perspectives (Aldershot: Ashgate, 2002), 151–71; Roshdi Rashed, “Combinatoire et métaphysique: Ibn Sīnā, alṬūsī et al-Ḥalabī,” in Roshdi Rashed and Joël Biard, eds., Les doctrines de la science de l’antiquité à l’âge classique (Leuven: Peeters, 1999), 61–86.
35 I have investigated related aspects elsewhere: Nader El-Bizri, “Le problème de l’espace: Approches optique, géométrique et phénoménologique,” in Vescovini and Rignani, eds., Oggetto e spazio, 59–70; Nader El-Bizri, “In Defence of the Sovereignty of Philosophy: al-Baghdādī’s Critique of Ibn al-Haytham’s Geometrisation of Place,” Arabic Sciences and Philosophy 17 (2007), 57–80; Nader El-Bizri, “La perception de la profondeur: Alhazen, Berkeley et Merleau-Ponty,” Oriens-Occidens: sciences, mathématiques et philosophie de l’antiquité à l’âge classique (Cahiers du Centre d’Histoire des Sciences et des Philosophies Arabes et Médiévales, CNRS), vol. 5 (2004), 171–84.
36 The Arabic critical edition (based on four mss.) and the annotated French translation of this treatise (Fī al-taḥlīl wa-al-tarkīb; L’Analyse et la synthèse) are established in Rashed, Les mathématiques infinitésimales, vol. IV (2002), 230–391.
37 The Arabic critical edition (based on two mss.) and annotated French translation of this treatise (Fī al-maʿlūmāt; Les connus) are established in ibid., 444–583. See also Roshdi Rashed, “La philosophie mathématique d’Ibn al-Haytham, II: Les Connus,” Les Cahiers du MIDEO, vol. 21 (Leuven: Peeters, 1993), 87–275.
38 Aristotle, Physics, ed. W. D. Ross (Oxford: Oxford University Press, 1936).
39 The Arabic critical edition (based on five mss.) and annotated French translation of this treatise (Fī al-makān; Traité sur le lieu) are established in Rashed, Les mathématiques infinitésimales, vol. IV (2002), 666–85. See also note 28 above.
40 Ibid., 669.
41 “Bijection” refers to an equivalence relation or function of mathematical transformation that is both an “injection” (“one-to-one” correspondence) and “surjection” (also designated in mathematical terms as onto) between two sets.
42 Rashed, Les mathématiques infinitésimales, vol. IV (2002), 658, 901.
43 Ibid., 670–73.
44 A similar line of inquiry is encountered in the research of Abū Jaʿfar Muḥammad ibn al-Ḥusayn al-Khāzin. See ibid., vol. I, 776, 828; see also ibid., vol. II, 381–2, 451–7.
45 Ibid., vol. IV, 661–2.
46 Euclid, The Thirteen Books of Euclid’s Elements, vols. 1–3, trans. with introduction and commentary Thomas L. Heath (New York: Dover Publications, 1956); for the Greek edition in 8 vols. with a supplement, see: Euclides opera omnia, ed. J. L. Heiberg and H. Menge (Leipzig: Teubner Classical Library, 1883–1916).
47 The Arabic edition (based on one manuscript) and annotated French translation of this treatise (Fī al-radd ʿalā Ibn al-Haytham fī al-makān; La réfutation du lieu d’Ibn al-Haytham) are established in Rashed, Les mathématiques infinitésimales, vol. IV (2002), 908–53.
48 Dalibor Vesely, Architecture in the Age of Divided Representation: The Question of Creativity in the Shadow of Production (Cambridge, MA: MIT Press, 2004), 113, 140–41. I have also investigated related themes in Nader El-Bizri, “Imagination and Architectural Representations,” in Marco Frascari, Jonathan Hale, and Bradley Starkey, eds., From Models to Drawings: Imagination and Representation in Architecture (London: Routledge, 2007), 34–42.