While in the service of the chief architect of the Ottoman Empire, Mehmed Agha (d. ca. 1622), in 1614, the scholar Caʿfer Efendi (d. after 1633) wrote *Risāle-i miʿmāriyye* (*A Book on Architecture*).^{1} He also wrote another book, or *risāle*, on geometry (*hendese*) from notes made during his conversations with Mehmed Agha. We know about this latter, now-lost manuscript only because of Caʿfer's mention of it in his introduction to *Risāle-i miʿmāriyye*:

Since we are associated with his [Mehmed Agha's] circle and have been near him for many years until the present time, whenever an issue related to the science of geometry [

ʿilm-i hendese] was discussed, this servant apprehended and took note of them. In accordance with this, arisāleon the science of geometry was written down and compiled.^{2}

Caʿfer's texts on architecture and geometry reveal the close relationship between these fields within his circle. Additionally, his reference to Mehmed Agha as a prime source for his book on geometry suggests that Caʿfer discerned this link through his interactions with the architect.

Ottoman literati like Caʿfer assumed that royal architects were highly skilled in geometry. The historian İntizâmi (d. after 1612), for instance, in a long ode written for imperial ceremonies in 1582, celebrated the architects' guild, whose members he called “the architects of the age” (*miʿmār-ı ruzgār*) and “the marvelous engineers of the revolving sphere.” He emphasized their knowledge of shapes, their skill in using compasses, and their ability to outline and divide a building's foundation—which was, in his view, enough to show that they could train “many Euclids.”^{3} The chief architect Mimar Sinan (1489–1588) described himself as highly skilled in geometry in his memoirs, and literati from Caʿfer Efendi in the early 1600s to Dāyezāde Mustafa Efendi in 1769 were well aware of Sinan's reputation in this area.^{4}

Scholars have often noted geometry's role in Islamic art and architecture from the medieval through the early modern periods.^{5} However, no one has yet explored the relationship between geometry and architecture against the backdrop of conversations and collaborations among Ottoman architects and scholars during the late sixteenth and early seventeenth centuries. My aim in this article is to reassess the levels of geometrical knowledge applied by these architects, engineers, and artists, avoiding generalizations by evaluating the linguistic and philosophical roots of geometric terms alongside sociocultural contexts and building conventions. What particular modes of geometry were most useful to architect-engineers and artists working in the early modern Ottoman world? What do the theoretical and practical bases—etymological, ontological, sociocultural, and technical—of the terms they used tell us about their habits of architectural thinking, knowing, and making?

I explore these questions through a close, critical reading of a unique source: Caʿfer's abridged section on geometry in his *Book on Architecture*. The section reveals the complex relationship between architectural practice and mathematics in this particular time and place. Although the various geometric terms and definitions that appear in Caʿfer's *Risāle* are related, they contain many subtle—yet crucial—differences in meaning. The sources of these terms range from medieval dictionaries and state records to geometry books. I examine the terms alongside architects' biographies, lexicons, encyclopedic works on the sciences, ethical texts on cognitive faculties, mathematical sources, and court records—and a few extant architectural drawings—to reveal their practical and intellectual contexts. My readings of this material show that the changing meanings of words used to describe geometry and its diverse practitioners reflect transformations in the conception of geometry and its application to architecture.

I begin by considering the kinds of books that Caʿfer and other scholars would have known—texts that would have stressed the connection between architecture and geometry. I then turn to the *Risāle* and analyze the etymology of key terms Caʿfer uses in relation to geometry. Next, I show how a thorough knowledge of geometry led to changes in the titles of practitioners, resulting in increased use of the term *architect-engineer* (*miʿmār-mühendis*). I focus on the figure of the Ottoman chief architect-engineer as one such practitioner. Following this, I investigate the practical applications of geometry—how architects took measurements, made estimates, and depicted geometric forms on paper. Finally, I explore how Caʿfer concludes his section with geometric forms used by mother-of-pearl inlay artists such as Mehmed Agha, which became foundational for his expertise in architecture. Caʿfer's text illuminates changing attitudes in the seventeenth-century Ottoman Empire about what an architect-engineer was and what such a person should know about theory and practice, from practical knowledge of building and surveying to mathematics. My examination underscores the union of hand and mind in early modern Ottoman architects' geometrical training, conceptualization, and practice.

## Geometry's Significance for Architecture

Caʿfer opens and closes the *Risāle* with two poems that stress geometry's significance for architecture. In his opening poem, “The Divine Creation,” he uses metaphors related to architecture, calling the universe a lofty mosque, the sky an exalted dome, and the sun an ornamented lamp—all well-known poetic images.^{6} But then Caʿfer asks, “What is this? Who made such an artful edifice? / Without drawings [*rüsūm*, singular *resm*] and without geometry [*hendese*], and without a model [*temsil*].”^{7} In his closing poem, Caʿfer says that he has “clothed his [architectural] book in the forms of geometry.”^{8} He thus bookends his text with poetic language that reveals the connection he sees between architecture and geometry.

Previous texts had made this connection. For example, an encyclopedia of sciences written by Taşköprüzade Ahmed Efendi (d. 1561)—translated from Arabic to Turkish by his son in 1587—gives us a window into the conception of architecture in the Ottoman Empire at that time, one that emphasized its geometric bases.^{9} Caʿfer arrived in Istanbul to study the sciences at a madrasa during the last decades of the sixteenth century, and he was likely familiar with Taşköprüzade's book.^{10} According to Taşköprüzade, who followed Aristotelian tradition, theoretical knowledge fell into three categories: metaphysics, mathematical sciences, and natural sciences.^{11} Mathematical sciences included geometry, arithmetic, astronomy, and music—that is, the classical quadrivium. Geometry concerned the knowledge of lines, surfaces, and solids, and was based on measurable proofs.^{12} Architecture, “the science of building,” was a subbranch of geometry.^{13} Its subjects included the construction and embellishment of fortifications, mansions, and bridges; the digging of canals; and the drilling of wells. Taşköprüzade's focus was on architecture's public use—that is, civic architecture—which related to his belief that knowledge should produce virtuous deeds, with human action building upon knowledge.^{14} Architecture's foundations in geometry enhanced the practice's standing among the sciences, and at the same time, geometry grew in social status because of its use in the cultivation of cities.

This connection between building and geometry was also made by the philosopher al-Farabi (d. 951), who classified the “art of supervising buildings” as related to geometry under the science of mechanical devices, one of the seven branches of the mathematical sciences.^{15} Al-Farabi's distinction between practical and theoretical geometry, however, implied that whereas building supervisors depended on more sophisticated, theoretical applications of arithmetic and geometry, carpenters and masons used practical geometry.^{16}

Caʿfer's poetic metaphors reveal a new approach to the role of geometry in building, emphasizing the architect's intellectual skills and the process of conceptualization. Caʿfer introduces new analogies between divine creation and drawings, models, and geometry in his opening poem. He modifies a well-known Qur'anic verse: “God erected the heavens without pillars.”^{17} In his memoirs, Sinan used this same verse and wrote that the seven firmaments of the universe were created “without architect, or builder and without column, or pier.”^{18} One of the four columns of the Sultan Ahmed Mosque (1609–17), built by Mehmed Agha in Istanbul, features the inscription, “God is he who raised the heavens without any pillars.” Caʿfer, in his comparison, does not refer to building parts as foundational elements, but underscores intellectual and visual tools such as geometry and drawing to emphasize architecture's mathematical basis.

In his memoirs Sinan also used metaphors related to architects' hand tools to describe divine creation. In his *Teẕkiterü'l-ebniye* (*Record of Buildings*), he expressed “thanks to that builder [*bennā*] of the palace of nine vaults who without balance [*mīzān*] or plumb line [*hencār*], without rule [*mıstar*] or compass [*perkār*], by his hand of creation, made firm its arched canopy.”^{19} These words, derived from Sinan's expertise in stonework and carpentry, indicate the importance of hand tools, such as the compass and balance, for giving shape to matter (Figure 1).^{20}

Caʿfer's metaphors shift the focus from building elements and hand tools toward geometry, drawings, and models.^{21} This underscores the architect's use of intellectual tools in the process of making. Earlier thinkers had already developed the analogy between God's three attributes—creator, maker, and form giver—and the acts of creation and building.^{22} Caʿfer's couplet alludes to al-Ghazali's (d. 1111) well-known analogy (relating God's attributes to building) to contrast God's creation from nothing with an architect's process of making.^{23} But Caʿfer's analogy emphasizes an architect's conceptual tools and highlights geometry as the most important science for architectural practice and for making drawings and models. Caʿfer's awareness of geometry's significance to architects leads him to introduce its different definitions in his book's first chapter, which discusses Mehmed Agha's artistic and architectural training.

## The Risāle 's Section on Geometry

In the *Risāle*'s first chapter, Caʿfer tells the story of Mehmed Agha's admission to the corps of royal architects. In 1569, while wandering inside the imperial gardens of the Topkapı Palace in search of his vocation, Mehmed Agha arrived at the workshop of the mother-of-pearl inlay artists (*ṣadefkāriler kārhānesi*), who operated under the corps of royal architects.^{24} The masters (*üstādlar*) listened to a “young fellow” reading and explaining geometry from a book:

Regarding that which is called the science of geometry [

ʿilm-i hendese], in this age, if the science of geometry is discussed among architects [miʿmārlar] and scholars [ulema], each one will answer, “yes we have heard of it, but in reality we have not heard what the science of geometry is and what it deals with.” Now this noble book fully describes that fine science [ʿilm-i latif]. As long as a person does not understand this agreeable science and art of alchemy [fenn-i kimyā], one cannot be fully adept in mother-of-pearl inlay, nor can one be expert and skilled in the art of architecture.^{25}

The words the young man read implied that although architecture (as an art, or *sānʿat*) depended on hands-on experience, it also required a body of knowledge (or *ʿilm*) for praxis. The visual, moral, and magical connotations of the phrases “agreeable science” and “art of alchemy” underscored geometry's transformative power to engender enchanting works.

When Mehmed Agha asked to join the brotherhood of architects and inlay artists, the masters suggested that he study the science of geometry. Against this backdrop, Caʿfer provides the reader with a window into architects' geometrical training. He first offers a technical definition of geometry, before describing Pythagoras as the compiler of a book on geometry and arithmetic after these were established by the prophet Enoch; finally, he discusses various other definitions of geometry, based on his consultation of written sources, including dictionaries and unspecified geometry books in Turkish, Persian, and Arabic (Figure 2).^{26} He also notes the titles used for architects and artisans depending on their particular types of work and their levels of expertise. He aligns the masters' speech and artisanal traditions with literary and court cultures. As a court official with expertise in geometry, Caʿfer serves as a sort of intermediary between architects and scholars; his text speaks the language of both.

Caʿfer emphasizes that geometry has both dictionary (*luġat*) and technical (*ıstılāh*) definitions. He notes that architects should be particularly attentive to his technical definitions, which were drawn from scholarly traditions. Caʿfer's translation of Persian and Arabic terms into contemporary Turkish also shows his desire to inform scholars about the words commonly used by practitioners. The artists of mother-of-pearl inlay were expected to be familiar with geometric terms as well. Caʿfer's section on geometry, thus, is neither a purely theoretical text on mathematics or scientific definitions nor a manual on practical geometry for architects-artists. Rather, the section investigates the linguistic and etymological origins of the term *hendese*, with attention to the artisanal and scientific milieus that used and produced these. Caʿfer, therefore, relies on language in inquiring about the meanings of words used in the practical sphere while deducing architectural knowledge.

## Linguistic and Etymological Roots of Geometry

Caʿfer studied grammar, rhetoric, and lexicography during the course of his madrasa education, which enabled him to explore the variant meanings and spellings of the word *hendese*, or geometry. Just as his metaphors reveal a connection between architecture and geometry, so too does his investigation into the connotations of this word. Caʿfer first explores *hendese*'s dictionary meanings and its use in everyday language.^{27} He writes that the term *hendese* derives from the Arabic word *hindāz*, meaning “measure.”^{28} According to the popular Persian-to-Turkish lexicon *Luġat-i Niʿmetu'llāh* (written by Niʿmetullāh, d. 1561), the root of *hindāz* is the Persian word *andāza*, which referred to a wooden or iron rod used for measuring fabric and other textiles.^{29} Caʿfer explains how *hindāz* later became *hendez* and, finally, *hendese*.^{30}

*Luġat-i Niʿmetu'llāh*'s reference to measuring fabric implies that the word *hindāz* referred to measurements employed in the marketplace.^{31} Caʿfer, however, directly relates the term to the measurement of surfaces. In the *Risāle*'s trilingual dictionary on architectural terms, he writes that the Turkish translation of *hindāz* is “to measure” (*ölçmek*) or “to give proportion” (*oranlamaḳ*).^{32} He places the word *hindāz* among his entries related to the tools of architects, engineers, and artisans (e.g., the cubit, ordering cord, plumb bob, and compass). Here, *hindāz* refers to the measurement of lands, building sites, and architectural surfaces. Its meaning “to give proportion” relates to the deduction of magnitudes through geometric ratios. But Caʿfer adds that *oranlamaḳ* was an ancient Turkish word, which by his time was not commonly used.

Caʿfer quotes the thirteenth-century Arabic lexicon *Mukhtār al-Ṣiḥāḥ* (written by al-Razi, d. 1267) to give a still more specific description of *hendese*. He explains in Turkish that, according to *Mukhtār al-Ṣiḥāḥ*, *hendese* means “to measure [*ölçmek*] and to give proportion [*oranlamaḳ*] to canals [*lağım*] and buildings [*binālar*].”^{33} At the time, this was one of the most common practical meanings of *hendese*, given that *Mukhtār al-Ṣiḥāḥ* was an abridged dictionary of widespread terms. The definition in *Mukhtār al-Ṣiḥāḥ*, however, does not reflect the traditional distinction between surveying (*mesāḥa*)—classified as a subbranch of *hendese*, like architecture—and geometry.^{34} Ottoman court records often used the term *mesāḥa* when describing the measurement of surfaces with a cubit. A court official like Caʿfer would have been aware of this distinction, as is indicated by his chapters on mensuration related to land surveying.^{35}

In assessing the etymological and linguistic transformations of the word *hendese*, Caʿfer relates the changing meanings to daily language and practice against the backdrop of earlier uses. At the same time, he shows how geometry's diverse uses led to alterations in the titles of practitioners.

## Practitioners: The Architect-Engineer

Just as the meanings of *hendese* changed, so too did the titles of practitioners, according to their levels of geometric expertise. Chief architects Sinan and Mehmed Agha were acknowledged for their high-level geometric skills. Taşköprüzade's book was written during a period when Sinan's public status and numerous prominent buildings made architecture the leading art form in the empire. Scholars' increasing awareness of the architect's geometric competence led to the classification of architecture alongside the more established mathematical sciences. In the mid-sixteenth century, the practical geometry of builders and the administrative class's “art of supervising buildings”—which al-Farabi differentiated in his encyclopedia of knowledge—were both part of the expertise of the Ottoman chief architect, whose civic duties encompassed construction and administration.^{36} In the account books of the Suüleymaniye and Sultan Ahmed Mosque complexes, Sinan and Mehmed Agha are mentioned together with the names of scribes and building superintendents (*binā emini*) such as Kalender Pasha, whose appointment as the building supervisor of the latter mosque in 1609 was initiated by his geometrical knowledge and experience in finance.^{37} By the early seventeenth century, the chief architect's elevated status, based on his mathematical skills, was widely recognized among scholars who collaborated with architects. In imperial Istanbul's centralized system of governance, scholars, officials, and architects frequently interacted during court cases concerning issues ranging from record keeping for building renovations to land division to the inspection of measures in the marketplace. Caʿfer was well aware of the exchanges of mathematical knowledge among these groups and of the varying degrees of geometrical knowledge expected of diverse practitioners.

Caʿfer notes that *Mukhtār al-Ṣiḥāḥ* used the term “geometer-engineer” (*mühendis*) for the active practitioner of *hendese*. This was because *mühendis* derived from *hendese*'s root word, *hindāz*, meaning “measure.”^{38} In his tenth chapter, on units of area measurements, Caʿfer mentions the ancient geometer-engineers (*mühendisīn*) who surveyed the earth's inhabited regions to determine latitudes and longitudes.^{39} This relates *mühendis* to geodesic surveying, which applied the knowledge of both practical geometry and mathematical geography.^{40} In the *Risāle*'s dictionary, after his entry on “architect” (*miʿmār*)—defined as a person who “cultivates” environments—Caʿfer lists the term *mühendis*, referring to a person who “measures with a cubit.”^{41} A *mühendis* was a practitioner who used geometry to survey, measure, or determine magnitudes, either by employing a tool or by making calculations based on a unit of measure. The word *cubit* (*arşın* or *ẕirāʿ*) refers to both the tool and the unit.

The listing of the word *mühendis* alongside architectural terms, however, indicates that the mathematical practitioners who were referred to in this way no longer surveyed only the natural world—they now measured the built environment, too. The tenth-century scholar al-Khwārizmī explained in his *Mafātiḥ al-ʿulūm* (*Keys of the Sciences*) that a *muhandis* (Arabic for *mühendis*) was someone who measured “the course of irrigation canals and the areas in which they are to be dug.”^{42} *Muhandis* also determined (*yuqaddiru*) the course of waterways, which suggests that they traced lines in the earth to delineate these areas.^{43} Likewise, *Mukhtār al-Ṣiḥāḥ*'s definition of geometry related the “geometer-engineer” to measuring hydraulic structures and determining measures.^{44}

Caʿfer's inclusion of this particular definition of *mühendis* in his telling of the life story of Mehmed Agha was not arbitrary. He knew that just before Mehmed Agha became the imperial chief architect, he had been the water channel superintendent (*suyolu nāzırı*), from 1598 to 1606.^{45} The tradition of royal architects having first managed water systems is evidenced by Sinan's waterworks experience as well as the subsequent appointments of Davud Agha, Dalgıç Ahmed Agha, and Mehmed Agha to the position of water superintendent prior to their becoming Ottoman chief architects.^{46} Caʿfer was aware that knowledge of hydraulic architecture was important for anyone aiming to become chief architect, as such knowledge proved an individual's geometric skills. A unique archival document dated 1598 verifies that while acting as a judge (*qadi*) in Istanbul, Caʿfer gave orders to the chief architect and the water superintendent for the distribution of water to fountains.^{47} This shows his direct observation of how building canals required the advanced geometric skill of an architect-engineer.

In the mid-sixteenth century, beginning with Sinan, architects' mastery of geometric skills led to the increased use of the title “architect-engineer” (*miʿmār-mühendis*). Caʿfer refers to Sinan, known as the Great Architect (*ḳoca miʿmār*), as both “the chief engineer of the world” (*ser-i mühendisān-ı cihān*) and “the world-famous” (*meşhūr-ı afāk ve devrān*) architect.^{48} The Ottoman chronicler Selaniki Mustafa Efendi called Sinan the “architect-engineer” of the Selimiye Mosque in Edirne.^{49} Selaniki gave Horos Memi, from Egypt, who built 360 domes around the Kaʿba, the title “the architect” and “the engineer of the age” and also called Sinan's successor, Davud Agha, “the engineer of the age” and “the master architect of the world.”^{50} The famous seventeenth-century traveler Evliyâ Çelebi used the title “engineer” when mentioning monumental construction projects in Istanbul, including aqueducts and fortifications.^{51}

In his travel account, Mehmed Âşık mentions a dream about the building of Hagia Sophia.^{52} He recounts that the emperor Justinian's architect Ignatius (called Agnados or Ignadyus in Turkish texts) directed the construction together with many architect-engineers.^{53} Ignatius is also named “the architect-engineer” in Sinan's memoirs.^{54} Just as Sinan expressed his desire to surpass Hagia Sophia's dome with his imperial mosques, so too did his appropriation of Ignatius's titles show his desire to outshine that architect's geometric skills. Whereas Greek texts called Hagia Sophia's builders “engineers” (*mechanikos*), Sinan's title, “the architect-engineer,” or *miʿmār-mühendis*, blurs the distinction between *mechanikos* and *miʿmār*. Sinan, in fact, represented himself as both a knowledgeable master and an expert practitioner, credentials that earned him his dual title.^{55} Accordingly, the role of the *mühendis* merged with that of the architect, and the attributes and duties of both were reshaped. It would be misleading, however, to use this dual title for all Ottoman or Islamic architects, because it is evident that Mimar Sinan, Davud Agha, and Mehmed Agha possessed geometric skills that distinguished them from common practitioners. The designation *mühendis* seems to have become a qualifying term that some Ottoman royal architects earned on the basis of their geometrical knowledge, knowledge that was indispensable for the success of monumental building projects.^{56}

## The Architect-Engineer's Use of Geometry

So far I have discussed the metaphorical and theoretical connections between geometry and architecture based on Caʿfer's linguistic explorations. But what of geometry's practical applications? Caʿfer relied not only on literary sources but also on real-life experiences and observations of geometry's use in architecture, which informed his definitions. The construction of waterworks as documented in visual and textual sources particularly offers a useful case study of practitioners' tools and methods.^{57} Royal architects and water inspectors were responsible for the equitable distribution of water throughout a city, via fountains and wells, and for the repair of canals and sewer systems.^{58} All of this required the ability to measure and estimate, which in turn demanded high levels of mathematical competence.

The Ottomans' use of the title “engineer” for architects involved with hydraulic works reflects their understanding of the mathematical knowledge required. An architect-engineer needed expertise in a wide range of areas: using surveying instruments in vast territories; measuring heights and distances over difficult topographies; estimating the dimensions of water courses and canals; calculating the surfaces of aqueducts in square cubits; supervising the construction and operation of mechanical devices such as waterwheels; laying out foundations to resist natural forces, such as water pressure; orienting and leveling tunnels between distant water sources and outlets; and estimating quantities of water. Eyyubī's detailed account of Sinan's struggles during the building of the Kırkçeşme waterways demonstrates the chief architect's responsibilities and challenges.^{59} Selaniki also mentions that, under the reign of Süleyman I, Sinan and Kiriz Nikola measured (*mesāḥa*), estimated (*taḥmīn-i sahih*), and delineated (*taṣmīm*) an area of 20,000 cubits to excavate a canal between Lake Sapanca and the bay of Iznik.^{60} This initial attempt failed, but the architect-engineers undertook the project again in 1591, this time with the aid of geometry.

Historical accounts convey little of how practitioners operated on-site, but a passage in Sinan's *Teẕkiretü'l-bünyān* (*Record of Construction*) sheds light on the procedures and instruments used to measure and level canals. When there was a water shortage in Istanbul, Süleyman I ordered an investigation of the plain of Kağıthane and its ancient waterways to inform repairs of the Kırkçeşme waterways.^{61} While searching for aqueducts and water, Sinan inspected “the heights and depths of the valleys with an aerial balance [*havāyi terāzū*].”^{62} This description of Sinan's method resonates with sections of the mathematician al-Karaji's (d. 1029) eleventh-century *Inbāṭ al-miyāh al-khafīyah* (*Book on Extracting Hidden Waters*).^{63} Al-Karaji gave detailed instructions for measuring heights and distances with different types of balances on-site, along with geometrical proofs for the design of instruments he made, which mostly targeted mathematicians.^{64} The work of scholars therefore overlapped with that of practitioners in his account. The balance that Sinan used to measure differences in heights and level the land to open watercourses resembled the “triangular balance” that al-Karaji described for use in leveling water canals.^{65} This instrument consisted of a triangular plate made of “chips or solid wood,” a plumb bob, and a cord of 30 cubits divided by knots that was used together with two measuring rods (Figure 3). Al-Karaji explained its fabrication and use, and his drawing of it resembles the eighteenth- and nineteenth-century balances preserved in collections, which relate to Sinan's own surveying device (Figure 4). A similar device was described and depicted in al-ʿAmilī's (d. 1622) renowned seventeenth-century book *Khulāṣat al-ḥisāb* (*The Essentials of Arithmetic*) and its late eighteenth-century commentary (Figure 5 and 6).^{66}

Architects used similar balances to lay out foundations. In his account of the 1609 foundation ceremonies of the Sultan Ahmed Mosque, Ahmed I's spiritual adviser, Mustafa Safi, praises architect Mehmed Agha as an expert in geometry and describes him using an instrument called “an aerial balance.”^{67} In the *Risāle*'s dictionary, the entry for the balance, called *terāzū* (*mizān* in Arabic), is listed together with tools used for measuring, laying out, and leveling foundations and buildings. These include the architect's cubit, the ordering cord or plumb line (*düzen ipi*), the plumb bob (*şākūl*), the handle (*imām*) for tying the cord with the plumb bob, and a pulley/reel (*makara*) used for the cord (Figures 7 and 8).^{68} Sources demonstrate that the balance—called “aerial” because it was suspended between two poles—was a surveying device used by Ottoman architect-engineers for measuring distances, heights, and depths, and for leveling and partitioning surfaces. The overlap between an architect's manner of measuring hydraulic works and al-Karaji's instructions shows that methods in mathematical books sometimes coincided with architect-engineers' procedures. Moreover, the similarities between historical instruments held in museum collections and the illustrations in widely read mathematical books indicate that scholars knew the importance of these tools for practical operations. Such overlap took place most often when architects conducted works on topographically difficult sites, jobs requiring a high degree of geometrical knowledge and numeracy, which often meant collaborating with mathematician-practitioners.

*kārnāme*) of the Kırkçeşme water supply system—made after a flood damaged the aqueducts in 1563—is a rare visual source for understanding the level and type of mathematics architects used to measure or repair waterworks (Figure 9).

^{69}The lengths of canals and aqueducts (

*kemer*), as well as their heights, are annotated in cubits in the drawing, and straight lines were drawn with the aid of a ruler. Architects probably used an aerial balance, ordering cord, measuring rod, and plumb bob to make linear site measurements before making such drawings. After finding the widths of aqueducts, which are annotated on Sinan's drawing, architects probably calculated heights by applying the rule of proportion in similar right-angled triangles (i.e., the triangulation method).

^{70}

The representation of aqueducts as trapezoids—in response to the shape of river valleys—was a visual abstraction that enabled the conceptualization and calculation of vast lengths and areas by means of geometry (Figure 10).^{71} For example, Sinan's survey drawing features lines dividing the trapezoids into right-angled triangles on the aqueducts Cebeciköy Kemeri (known as Güzelce) and Mağlova Kemeri. Proportions between annotated lengths on perpendicular lines and hypotenuses (representing the slopes of the aqueducts) on all three triangles approximate the ratio of 6:9:15.^{72} This proportion—derived from a right-angled triangle with angles measuring 36 degrees, 54 degrees, and 90 degrees—was used in geometric constructions. Notably, there was a set square defined according to this triangle.^{73} Sinan probably used this tool (along with a ruler) to draw the triangles' lines. Although architects might not have known angle measurements, the utilization of this set square in masonry and carpentry must have familiarized them with its geometric proportions. In his dictionary, Caʿfer lists the set square (*gönye*) among the tools used by architects and artists, after which he describes the three types of corners, or angles (*zāviye*): right, obtuse, and acute.^{74} Here, *gönye* (*zāviye* in Arabic) refers to the tool used to verify or define these angles (Figure 11). Although Caʿfer does not relate the set square's degrees and applications, its inclusion in his dictionary indicates that architects used this tool for various practices and on diverse surfaces, from building right-angled corners to drawing perpendicular or diagonal lines on wood or paper. The fact that a ratio in Sinan's drawing approximates the ones used in set squares suggests that architects knew at least some geometric ratios and used them to conceptualize the forms of architectural elements and to estimate the dimensions of otherwise immeasurable areas. It is not clear, however, whether they also used the Pythagorean theorem to deduce the lengths of the sides.

The architect could then calculate the aqueducts' surface areas either by dividing the trapezoid into triangles and rectangles or by using an exact formula, although the method used is indeterminable from Sinan's drawing. Annotations on the drawing, however, convey that areas of curtain walls (*germe*) were calculated in square cubits. The results were called the “builder's calculation” (*ḥisāb-ı bennā*). Another note on the curtain walls states that “this was the total cubits according to the ‘chessboard’ method.” The “chessboard calculation” (*ḥisāb-i ṣāṭrāncī*), which Caʿfer references in his chapter on surveying, involved determining areas in square units by multiplication.^{75} Similarly, the builder's method indicates the calculation of areas in square units to estimate the amount of materials and costs per square cubit.^{76} Such measurements and estimates were crucial to architects as they sought to secure the amounts of materials and numbers of workers they would need while preventing unnecessary expenditures.^{77}

Because Sinan's Kırkçeşme drawing is not fully scaled, it displays some discrepancies between annotated numbers, real lengths, and angles. Still, textual and visual sources pertaining to the Kırkçeşme water supply system indicate that architect-engineers overcame the difficulty of building and repairing monumental structures by mastering particular skills and tools: measuring and leveling elaborate waterworks with instruments; using geometric ratios and mathematical calculations to estimate lengths and areas; and illustrating their surveys and estimates on paper by using drafting tools to create geometric schemes. Caʿfer's subsequent definitions of geometry derive from the necessity of such mathematical skills as measuring and estimation for architects-engineers in a diverse array of works.

## Measurements

Caʿfer's second definition of *mühendis* shows that he distinguished among the types of measurements undertaken by practitioners. Caʿfer writes that the twelfth-century Arabic-Persian dictionary *Sāmī fi al-Asāmī* (written by al-Maydani, d. 1124) defined a *mühendis* as someone who “determined [*takdīr edici*] the measure or value” of a thing.^{78} Caʿfer notes that when Arabic people requested favors, they said, “Oh, God! Please allow us measureless and numberless grants.”^{79} His association of numbers, measures, and abundance with a religious saying links the act of “determining” to theological reasoning. The metaphoric connection between the verb “to determine” (*takdīr*) and the notion of fate might explain this seemingly unrelated reference in his section on geometry. Al-Ghazali maintained that God determined what was needed, just as a “geometer-engineer” determined a building's layout, materials, and dimensions.^{80} The analogy originated from the contrast between God's ability to determine a person's fate or grant measureless favors and the limits of human knowledge and actions in determining a thing's fortune. These linguistic and semantic links between “measure” (*takdīr*), “fate” (*kader*), and “magnitude/amount” (*miḳdār*) are traceable in their shared Arabic root *q-d-r*, indicating a quantity or measure. Because of these associations, *takdīr* could denote the allocation of values, measures, or fates by geometer-engineers, yet without any claim to certainty or limitlessness. The importance of having a thing's destiny (or measure) determined by a *mühendis* shows how foreseeing things through geometry could be analogous to an omnipotent act. Caʿfer's reference to *takdīr* in regard to granting favors discloses the sociocultural value of this geometric term regarding the just distribution of measures and values for public prosperity.

Although *Sāmī fi al-Asāmī* associated *takdīr* with the deeds of a *mühendis*, in sixteenth- and seventeenth-century court records, the term *takdīr* most often referred to the act of determining an amount of water or the value of goods.^{81} Caʿfer was aware of these usages, hence he hesitates to link the term to architectural measurements. An imperial decree addressed to the *qadi* and water superintendent in 1617–18 requested inspection (*keşf*) of a water source in Cebeci on land controlled by the vizier Mehmed Agha (not to be confused with the architect Mehmed Agha). The practitioners had to determine (*takdīr*) the amount of water that would be added to the canals leading to the Süleymaniye Mosque, so that it could be distributed to the vizier's residence and public fountain from this water supply system.^{82} In another court decree of 1620, court officials (experts in religious law), the water superintendent, and royal water inspectors were again ordered to determine (*takdīr*) the amount of water added to the Süleymaniye waterways by Hasan Pasha so that the water could be allocated equitably to his palace.^{83} These cases suggest that the practice of surveying waterways and estimating the amounts of water they carried was regulated by the court and undertaken by experts. The differences between the vocabularies used by practitioners also show that officials and architects became attentive to specific terms to distinguish the types of measurements required.

## Estimation

With his final definition of *hendese*, Caʿfer evaluates geometry's contemporary use and meaning, particularly as applied to architectural practice. *Hendese* traditionally had two meanings, he notes: “to measure [*ölçmek*] with a cubit [*ẕirāʿ*]” and “to give proportion [*oranlamaḳ*].”^{84} The word *taḥmīn* (estimation), he underlines, had replaced *hendese* in everyday usage, and meant only “to give proportion.”^{85} Caʿfer claims that the word *oranlamaḳ* itself did not have a contemporary application, but he includes it nonetheless to validate estimation's relation to proportion. His definition of geometry as “estimation” disassociates geometry from measuring with a cubit and gives it a different connotation, suggesting a change in mental processes and a new level of abstraction. *Taḥmīn* here means “to explain with reason [ʿ*aḳl*], that is, to give proportion.”^{86} This definition in Caʿfer's dictionary follows those for *miʿmār* and *mühendis*—words indicating physical action on a building site (Figure 12).

*Taḥmīn* is the only term here associated with the architect's inner faculties or mental powers, and Caʿfer's reference to “proportion” links it to mathematical deductions. This entry shifts the focus from physical measurements to cognitive operations. Estimation's reliance on intellectual abilities—for deducing an unknown from a known measure—in architectural practice makes it key to understanding an architect-engineer's geometric conceptualization.

Estimation was an important concept for Ottoman scholars and their European contemporaries.^{87} In his book on ethics, *Ahlāk-ı alâi* (1564), the Ottoman jurist Kınalızade (d. 1572) described human beings' “external” and “internal” senses.^{88} The ability to estimate (*ḳuvvat al-wahm*) was the third internal sense.^{89} According to Kınalızade, people deduced particular meanings or intentions from the physical world through estimation, relying on both corporeal experiences and mental powers.^{90} Originating from the material world, estimation required practical reasoning, which differentiated it from theoretical reasoning.^{91} Judgments deduced from estimation were still recognized as true, because estimation “perceived the particulars which are derived from objects of sense.”^{92} Kınalızade called the practical intellect “reason” (ʿ*aḳl*) because people depended on it to reach happiness through ethical decisions and deeds.^{93} Caʿfer's references to “reason” in regard to architectural practice similarly link it to the Aristotelian concept of practical wisdom (*phronesis*), combining experience, knowledge, and action. Moreover, Avicenna (d. 1037) wrote that the “estimative faculty”—uniting “imagination and judgment”—was the source of “mathematical thinking.”^{94} People estimated the dimensions, amounts, quantities, or costs of objects by relying on pragmatic judgments. Architectural estimations, which informed action, were necessarily based on measurable units, not mere abstractions.^{95} In court cases and account books, amounts of building materials were first estimated in cubits and then their cost was converted into aspers—both measures rooted in local customs and physical conditions.^{96} To respond to fluctuating conditions at a time when building codes were not fixed, practitioners had to rely on precedents and adjust their estimates according to changing regulations by establishing proportional relationships among various types of units.

It is difficult to deduce from the *Risāle* how exactly architectural estimations were made, but building conventions, court records, and imperial decrees shed light on this. Many court documents from this time refer to “estimation” (*taḥmīn*) and “accurate estimates” (*taḥmīn-i sahih*) in describing mathematical operations related to the renovations of buildings endowed to charitable/pious foundations (*waqfs*). Additionally, “examination” (*nazar*), “contemplation” (*müşahede*), and “inspection” (*keşf*) refer to the actions of officials and architects that informed these estimations.^{97} In this regard, court records support Caʿfer's claim that the contemporary meaning of geometry was *taḥmīn*, which meant mathematical operations used in architectural practice.

Early seventeenth-century judicial decrees on *waqfs* show the steps for calculating measures and estimating costs. When a request was made to the court for estimates of the dimensions, amounts, and costs of materials pursuant to renovations, an architect was employed, along with experts on law, like Caʿfer Efendi. In most cases, *mesāḥa* refers to measuring with a cubit, whereas “calculation” (*ḥisāb*) indicates addition, subtraction, or multiplication. These decrees commonly use the term *taḥmīn* to describe the method for finding an unknown value based on measurements.^{98} If the renovation was already complete, a tenant might ask to have the new structure appraised. Officials then estimated the monetary value of repairs according to the amounts of materials used.

In one case of 1618, the royal architect and legal experts evaluated Mehmed Halife's expenses after he renovated a property he was renting from a *waqf*.^{99} After the architects measured (*mesāḥa*) a stone wall's length and height (*tūlun ve arẓen*) in cubits, they calculated the surface area in square cubits by multiplying sides—the chessboard calculation mentioned previously. Next, they subtracted the areas of two windows measured in square cubits from this wall surface. In the second phase, they estimated (*taḥmīn*) the total cost in aspers by using the rule of proportion: the cost of one square cubit of this type of masonry wall was multiplied by the total surface area.^{100} In another case, when Caʿfer was working at the Galata court in 1605, a man called Penapot asked to register his costs after repairing his house in Galata, which he rented from a *waqf*.^{101} The royal architects inspected the newly constructed cellar (*mahzen*), the lower-story room (*tahtānī oda*), and the latrine (*kenīf*) on the first floor and measured the sizes of rooms on the second floor. Next, they estimated the costs of materials for columns, flooring, beams, roofing, walls, windows, and doors in addition to the labor fees for the carpenters and plasterers. These examples show the process of estimation, during which architects inferred amounts or costs from the known measures of building elements and materials.

The phrase “accurate estimate” was also used during the initial inspection before renovations. In 1612, the *waqf* inspector informed the court that Fatma Hatun's house—which she had bestowed on her charitable foundation—needed repairs.^{102} The court scribe, legal experts, and architect surveyed the building's area and measured its dimensions in cubits. They then accurately estimated (*taḥmīn-i sahih ile taḥmīn*) the cost of a 38-square-cubit brick wall, eleven beams, and forty-four timbers for flooring, along with the value of the 191-square-cubit plot of land. In other cases, total costs were estimated in proportion to the amounts of materials and labor required per square cubit of a building's surface area.^{103} Decrees suggest that despite attempts by the state to maintain standards, dimensions and prices for materials could fluctuate.^{104} Similarly, units of length, area, weight, and monetary value were not fixed; rather, they depended on changing social and cultural contexts and regional customs. Estimations required the prudence—the “practical reason”—of expert architects and officials working on-site. To make accurate estimates, practitioners had to align their judgments with local customs and precedents and follow official regulations.

Court records rarely included illustrations, but drawings were sometimes made for the repair of buildings. One such working drawing (*kārnāme*), made for the renovation of the Abdal Ata convent in Çorum (which includes this dervish's mausoleum), provides evidence of the estimation process (Figure 13).^{105} In the layout of a single module with sarcophagi, the length and width of the room are annotated: the area is 10 by 10 cubits. The paper is subdivided with uneven blind grids, which equate each drawn square with one square cubit. Annotations show that the total areas of two other proposals were estimated based on this first module. There are no numbers for other building elements, although domes above each room are mentioned on the drawing. Subsequently, the cost for each proposal was estimated based on these area measurements. The visible remains of the convent and the dimensions of the existing sarcophagi must have guided the survey drawing, measures, and estimations.^{106} This drawing provides visual evidence of the estimation process described above, in which practitioners surveyed buildings before and after renovations. Drawings like this were preserved as models to be consulted on future projects. Given that comprehension of such images required geometric proficiency and numeracy, the drawings indicate the presence of expert practitioners—namely, royal architects.

These written and visual records lead us toward two conclusions: (1) the calculation of areas was achieved through multiplication of measurements of the sides of what were approximated as rectangular shapes, despite possible surface irregularities; and (2) the total amounts and costs of materials were estimated in proportion to unit values valid at the time. It is possible that mathematical proofs were included to ensure accuracy. Caʿfer's definition of *taḥmīn* demonstrates that calculations and estimations could be made mentally, on paper, or on a drawing board—in contrast to the method of making direct measurements on physical surfaces with a cubit. These architectural estimations originated from the measures of geometric entities, whether a building site or a building's remains. For that reason, *taḥmīn* specifically refers to geometric proportions. One decree uses the term *taḥmīn* for the building of a new mosque (*cāmi-i şerif*) in Gökören, Filibe, which is estimated to have cost 10,000 aspers.^{107} The reference to the word *estimation*, rather than the word *imagination* (*taṣavvur*) in the conceptualization of a new building underscores that amounts and costs of materials and labor were based on precedents as models rather than on purely abstract ideas. These examples demonstrate how *estimation* became a proper term in architectural practice for deducing approximate sums through practical intellect and geometric proportion.

Caʿfer himself, acting as both a legal expert and an expert in geometry, inspected a collapsed brick arch on the building of one pious foundation.^{108} Despite emphasizing the importance of architectural measurements and including a trilingual dictionary of architectural terms, Caʿfer conveys no mathematical instructions for measuring the built environment in his chapters in the *Risāle* on the Kaʿba, the Sultan Ahmed Mosque, or surveying. He probably knew of mathematical texts, such as al-Kāshī's *Miftāḥ al-ḥisāb* (*The Key of Arithmetic*, 1427), that described measuring the surface areas of complex geometrical shapes, such as those used for *qubba*, *muqarnas*, and vaults (Figure 14).^{109} There is little evidence in his writings, however, that architects studied or applied formulas or computational tables pertaining to these complex shapes when using geometry to find approximate amounts during estimations; nor is there evidence that scholars interested in architecture's geometric basis could easily translate composite architectural volumes into numerical values. Al-Kāshī's book captured the direct attention of architect-engineers only at the end of the eighteenth century, when a new school for military architect-engineers was finally founded, and the book's section on mensuration was translated into Turkish (Figure 15).^{110} This shows that practitioners of Caʿfer's era still relied mostly on the established techniques of geometry and practical reasoning rather than on arithmetic formulas or statics for measuring or constructing architectural elements such as arches or vaults. Caʿfer was well aware that architects did not learn geometry and measurement methods primarily from books.

His emphasis on estimation's difference from direct measurements and its connection to reason, nevertheless, represents a new awareness of the architect's cognitive and mathematical skills. Architects and scholars collaborated on increasing numbers of renovations during the seventeenth century, work intended mainly to prevent the deterioration of the buildings of the pious foundations. These endeavors required both accurate calculations and hands-on experience. Working together, scholars like Caʿfer observed diverse geometrical operations applied to architectural surfaces, while architects improved their geometrical knowledge. Early modern chief architect-engineers were already skilled in many aspects of geometry, but such collaborations—particularly beneficial in legal cases—made their increased proficiency highly sought-after.

## Geometric Forms

Caʿfer concludes his section on geometry with a list of geometric forms, and he notes that mother-of-pearl inlay artists like Mehmed Agha used these shapes in their work.^{111} After twenty years of experience working with cabinetry and mother-of-pearl inlay, having learned the science of geometry and the art of architecture from chief architect Mimar Sinan, Mehmed Agha became a skilled artist and master architect.^{112} Caʿfer's emphasis suggests that his training in fine wood and inlay work, along with his knowledge of geometric forms, prepared him for architecture, and for his life as an artist-architect.

Before describing forms, Caʿfer provides two related technical definitions for geometry. He first states that “the amount of a thing's [*eşya*] essence and the magnitudes of shapes [*eşkal*, plural of *şekil*] are known, according to the precepts of the science of arithmetic [ʿ*ilm-i ḥisāb*] and the science of extracting roots [*cezr-i ʿilm-i ḥisāb*].”^{113} Opening his section on geometry, this definition is first quoted in Arabic and then translated into Turkish, suggesting that it was probably extracted from an earlier scholarly work. The second definition of geometry, he writes, is “to know the amount/magnitude of a thing's essence or shapes, according to the science of arithmetic.”^{114} Caʿfer believes it necessary to transmit learned traditions to artisans, yet he is unable to determine with certainty what uses they will make of them. Calculating the quantity of a shape or a thing through arithmetic means that formulas and proofs have been used to find or determine the lengths, areas, and volumes of geometric shapes, including more complex forms such as conics. His inclusion of these definitions might give the impression that such theoretical knowledge was necessary for all practitioners, but the relevance of these terms for architects and artists requires a closer examination.

Caʿfer notes that the science of geometry identifies several shapes.^{115} If a person can master these, the rest will be easy. He offers a list of forms that he considers essential for mother-of-pearl inlay artists:

The first is the round circumference [

devr-i muḥīṭ], which is a perfect circle [kāmil bir dāire]. The second is the arc [ḳavs] of a round circumference, which is a half circle [nıṣf-ı dāire]. The third is the small arc of a round circumference that was less than a half circle. The fourth is the large arc of a round circumference that is more than a half circle. The fifth are the triangular forms [eşkāl-i müsellesāt]. The equilateral triangle [müselles-i muṭlaḳ] is simple, because its sides [ażlāʿ], that is, its ribs [eyegüler], are equal. Similarly, its shape is like a triangular trivet [ṣacayaḳ], three sides and ribs of which are equal. Trivets are made in the shape of equilateral triangles.^{116}

Geometric forms begin with the perfect circle. Triangles come next, and they are differentiated by their angles and sides; other than the equilateral triangle, all have unequal sides and obtuse or acute angles (*zāviye*). Caʿfer next lists quadrangles (*murabbʿāt*), pentagons, hexagons, heptagons, octagons, nonagons, and decagons. He implies that these forms are illustrated in geometry books, and probably in his own book on geometry, which highlights the importance of visual models for learning geometry.^{117}

Caʿfer understood that these shapes were foundational for artists like Mehmed Agha. For example, Mehmed Agha obtained permission to perform his art and receive a stipend (*gedik*) within the corps of royal architects only after he proved his geometric expertise.^{118} He did this by making an exquisite reading desk for Sultan Murad III. This geometrically complex object had “from top to bottom the sides of triangles and quadrangles and the forms of pentagons and hexagons, and heptagons interlocked together on the sides.”^{119} Caʿfer's description of the figures on this desk matches his list of geometric shapes. Yet he mentions no mathematical formulas used for generating or measuring these forms.^{120} He defines shapes according to their sides and angles, without mention of their areas, and he does not refer to solids or volumes, which indicates that Caʿfer's scientific definition did not align with artists' actual uses of geometry. In all likelihood, he recognized that artists generally did not need advanced mathematics, although, as a scholar, he expected them to know basic mathematical operations. His writings suggest that the primary mathematical knowledge transmitted to artists involved the definition of geometric terms and essential shapes. Illustrated books, inherited from previous generations and read aloud to apprentices, must have preserved and transmitted such elementary knowledge. These texts would have been supplemented with oral communication and hands-on instruction in the use of manual tools.

Mehmed Agha's training likely included exercises involving the ruler, compass, and set square—the main tools used to construct geometric figures. Caʿfer's emphasis on sides and angles accords with the artist's method of constructing geometric figures by using tools to draw and divide lines, as evidenced in al-Buzjani's tenth-century text *Kitāb fī mā yaḥtāju ilayhi al-ṣāniʿ min aʿmāl al-handasa* (*Book on the Geometrical Constructions Necessary for the Artisan*).^{121} According to Caʿfer, architect-artists' tools were made from shapes defined by the science of geometry.^{122} For example, there were three types of set squares, just as there were three types of triangles (Figure 16).^{123} Master artists either used patterns on paper or directly traced geometric constructions with tools before fashioning figures on their materials.^{124} Mehmed Agha probably used similar methods to generate artwork in his early years. However, as a master architect-engineer, he obtained a more advanced knowledge of geometry and arithmetic in later years.

As a master artist, Mehmed Agha learned to draw geometric patterns at full scale on flat surfaces. As an architect-engineer, he learned to depict vast areas at small scale using survey drawings. He became capable of imagining buildings' forms at full scale and depicting them through drawings and models.^{125} Caʿfer mentions Mehmed Agha's knowledge of geometric forms when praising his drawing for the Sultan Ahmed Mosque.^{126} He underlines Mehmed Agha's modesty, saying that it was only through his works that people could see his knowledge of geometry.^{127} His most detailed examples of this expertise are the architect's mother-of-pearl inlaid woodwork. Caʿfer was able to scrutinize the configurations of these small-scale works from different angles during his visits to Mehmed Agha's workshop. He probably lacked the craft knowledge to understand the art's inner workings, but he was aware of the importance of geometry along with embodied knowledge, which could be acquired only through making. As a friend of Mehmed Agha, Caʿfer recognized the importance of long-term, hands-on experience, as well as the limitations of books when it came to artistic training. In his technical definitions, however, he tried to correlate artistic making with scientific knowledge, revealing art's geometric basis while emphasizing the primacy of praxis. Through these experiences, Caʿfer realized that Mehmed Agha's skill in making geometrically intricate artwork was foundational for his expertise in generating two- and three-dimensional architectural images, such as drawings and models, as he implied by his opening poem.

## Conclusion: Limits and Levels of Geometrical Knowledge

Although Caʿfer underscores the geometric qualities of Mehmed Agha's artworks and buildings, he does not provide instructions for fabricating similar work.^{128} His concern was that artists and architects obtain a higher level of mathematical competence so that they could better align their works with geometric rules—knowledge of which, according to Ottoman scholars like Caʿfer, was first received by prophets and philosophers directly from God. The firm and noble basis of the science of geometry provided solid foundations for buildings and infrastructure that enhanced the cultivation of cities; it enabled equitable distribution of natural resources (e.g., water) and material goods among the populace, guaranteed just decisions during legal cases concerning the rejuvenation of pious foundations, and allowed for the production of intricate artworks that were sources of wonder and wisdom for viewers.

In contemplating artworks and architecture as these were being made, Caʿfer recognized their dependence on geometry, and this relationship occupies an important place in his book. His friendship with Mehmed Agha, his scholarly activities, and his personal experiences as a court official afforded him the practical and intellectual sources for his geometrical definitions.^{129} His participation in estimation processes and his observations on the role of reason led him to recognize architects' potential cognitive powers. He was not alone. Following Sinan, the learned Ottomans began to distinguish more sharply between the architect-engineer and the mere builder: the latter required little beyond manual skill; the former, a substantial knowledge of mathematics. Scholars were aware of the different types of geometry needed by various practitioners, although some, such as the architect-engineer-artist Mehmed Agha, seemed to master them all.

It is no coincidence that Caʿfer wrote his book in 1614, while the Sultan Ahmed Mosque was under construction. He might have even acted as a consultant to Mehmed Agha, who was responsible for tracking building materials and expenses, as the account books suggest. In his chapter on the Sultan Ahmed Mosque, Caʿfer urges his readers to visit the building under construction and contemplate the wondrous qualities of its geometric forms.^{130} In the mosque, he recognizes architecture's noble foundations in geometry, as well as the architect's geometric skill and intellectual capabilities. By pointing to the building's embeddedness in the science of geometry—which attested equally to engineering skills and artistic ingenuity—Caʿfer advocates for the chief architect, indicating to the sultan and his officials, such as Kalender Pasha, who valued the intersections between geometry and arts, that this monumental mosque would be completed successfully.

The *Risāle*'s definitions of geometry illuminate continuities and changes in the relationship between architecture and the mathematical sciences during the early seventeenth century. The absence of theoretical mathematics (such as complex arithmetic equations) from the training of early modern architect-engineers and artists does not mean they lacked knowledge of geometry. Diverse types and uses of geometry and arithmetic existed throughout early modern Ottoman architectural and artistic practice. In addition to chief architects, some distinguished royal architects knew arithmetic sufficiently well to calculate lengths, areas, amounts, and costs—calculations that were probably aided by expert scholars, who provided final proofs. Some imperial architects and artists knew how to draw geometric forms to represent surveys, estimations, and interlocking figures—which demanded a high degree of geometric conceptualization—although they rarely included detailed or precise calculations for complex structural or formal elements. An understanding of the limits and levels of the early modern architect's geometrical knowledge is important for the evaluation of later developments, types, and uses of geometry in eighteenth-century Ottoman architecture—a subject still awaiting exploration.

Interactions between architects and scholars in public and private settings—as with Caʿfer Efendi and Mehmed Agha—provided ample opportunities for the exchange of knowledge. Although the relationship between mathematical theories and architectural practice was not subject to systematic analysis, at least not before the late eighteenth century, the wide-ranging practical uses of geometry in diverse sociocultural settings led to broader recognition of the geometric basis of architecture, and of the architect-engineer's geometric and arithmetic skills. Greater focus on abstract thinking, awareness of the mathematical abilities of architect-engineers, and the emergence of texts like the *Risāle* suggest that architects were now widely acknowledged for their intellectual abilities, even though manual dexterity was still highly valued. The *Risāle* emphasizes the importance of cognitive skills for an architect-engineer's practice and demonstrates an increasing interest in theories of praxis during the early seventeenth century. This led scholars like Caʿfer Efendi to establish architecture's foundations in noble and firm sciences, such as geometry, and to codify the uses of those sciences for wider circulation among scholars, architects, and artists.

## Notes

I would like to thank the *JSAH*'s editor, Keith Eggener, for his thoughtful suggestions during the editing and revision of this article. I also thank the anonymous reviewer for the helpful comments.

The title “chief architect” was an official designation in the Ottoman Empire, akin to “court architect” in Europe. Mehmed Agha was also a renowned mother-of-pearl inlay artist. For the English translation of Caʿfer's manuscript, the only known autograph copy of which is held at the Topkapı Palace Museum Library, see Caʿfer Efendi, *Risāle-i miʿmāriyye: An Early-Seventeenth-Century Ottoman Treatise on Architecture—Facsimile with Translation and Notes*, trans. and ed. Howard Crane (Leiden: Brill, 1987). For the Turkish text (which I refer to hereafter as the *Risāle*), see Caʿfer Efendi, *Risâle-i miʿmâriyye* (in romanized Ottoman Turkish; with facsimile of MS. Yeni Yazma 339, ca. 1614, Topkapı Palace Museum Library), ed. I. Aydın Yüksel (Istanbul: İstanbul Fetih Cemiyeti, 2005). All translations from Turkish are mine unless otherwise noted.

Caʿfer, *Risāle*, trans. Crane, 22–23; Caʿfer, *Risâle*, 9. I have slightly modified Howard Crane's English translation to better convey the specific meanings of some Turkish terms, such as *mensūb olmak*, *aḫẕ ve żabṭ*, and *tahrīr*.

İntizâmi, *İntizâmi Sûrnâmesi*, ed. Mehmet Arslan, 2 vols. (Istanbul: Sarayburnu, 2009), 2:286–87.

Mimar Sinan and Mustafa Sai Çelebi, *Sinan's Autobiographies: Five Sixteenth-Century Texts*, critical ed. and trans. Howard Crane and Esra Akın, ed. Gülru Necipoğlu (Leiden: Brill, 2006), 62, 118, 145. For references, see Caʿfer, *Risâle*, 9; Dâyezâde Mustafa Efendi, *Edirne Sultan Selim Camii risalesi*, ed. Oral Onur (Istanbul: Kuşak, 2002), 7–8. On the circulation of Mimar Sinan's texts, see Gülru Necipoğlu, *The Age of Sinan: Architectural Culture in the Ottoman Empire* (Princeton, N.J.: Princeton University Press, 2005), 146; Gülru Necipoğlu, “Preface: Sources, Themes, and Cultural Implications of Sinan's Autobiographies,” in Sinan and Sai Çelebi, *Sinan's Autobiographies*; Howard Crane and Esra Akın, “Introduction,” in Sinan and Sai Çelebi, *Sinan's Autobiographies*, 1–44.

On the use of geometry in medieval Islamic artistic practice with reference to Caʿfer, see Gülru Necipoğlu, “Theory and Praxis: Uses of Practical Geometry,” in *The Topkapı Scroll: Geometry and Ornament in Islamic Architecture—Topkapı Palace Museum Library MS H. 1956* (Santa Monica, Calif.: Getty Center for the History of Art and the Humanities, 1995), 150–81. See also Alpay Özdural, “Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World,” *Historia Mathematica* 27, no. 2 (2000), 131–66; Alpay Özdural, “On Interlocking Similar or Corresponding Figures and Ornamental Patterns of Cubic Equations,” *Muqarnas* 13, no. 1 (1996), 191–211; Alpay Özdural, “Omar Khayyam, Mathematicians, and ‘Conversazioni’ with Artisans,” *JSAH* 54, no. 1 (Mar. 1995), 54–71. For recent discussions of this relationship, see the essays by Gülru Necipoğlu, Elaheh Kheirandish, and Jan P. Hogendijk in *The Arts of Ornamental Geometry: A Persian Compendium on Similar and Complementary Interlocking Figures*, ed. Gülru Necipoğlu (Leiden: Brill, 2017).

On cosmic metaphors, see Walter G. Andrews, *Poetry's Voice, Society's Song: Ottoman Lyric Poetry* (Seattle: University of Washington Press, 1985), 44–45.

Caʿfer, *Risâle*, 5. Although Crane translates *temsil* as “analogy,” Gülru Necipoğlu has suggested that it should be “model.” See Gülru Necipoğlu, “Review: *Risāle-i Miʿmāriyye: An Early Seventeenth-Century Ottoman Treatise on Architecture* by Caʿfer Efendi, Howard Crane (ed.),” *JSAH* 49, no. 2 (June 1990), 210–13; Gülru Necipoğlu-Kafadar, “Plans and Models in 15th- and 16th-Century Ottoman Architectural Practice,” *JSAH* 45, no. 3 (Sept. 1986), 224–43. The word *temsil* generally means “representation” in modern usage. In the context of the *Risāle*, I translate *temsil* as “model.”

Caʿfer, *Risâle*, 123.

Aḥmad ibn Muṣṭafā Taşköprüzāde, *Mevzuat'ül ulûm: İlimler ansiklopedisi* [Encyclopedia of the sciences], ed. Mümin Çevik, trans. Kemâluddin Muḥammed Efendi, 2 vols. (Istanbul: Üçdal, 1966), 1:305. The premodern use of the word *science* refers to a general body of knowledge, called *ʿilm*, that is difficult to relate to modern notions of science. It is more in accord with *scientia*, or “knowledge,” as conveyed in Taşköprüzāde, *Mevzuat*, 1:57–70.

Caʿfer, *Risâle*, 37.

Taşköprüzāde, *Mevzuat*, 1:249.

Taşköprüzāde, 1:302.

Taşköprüzāde, 1:305. Geometry included optics, the science of lenses and mirrors, the science of balances, the science of surveying, the science of extracting water, the science of military tools and archery, and the science of mechanical devices. Taşköprüzāde's section on architecture and its medieval sources, such as the classification of sciences by al-Farabi (d. 950) and al-Akfani (d. ca. 1348), are discussed in Necipoğlu, *Topkapı Scroll*, 140–41.

Taşköprüzāde, *Mevzuat*, 1:34–37.

On an earlier example of this link, see al-Farabi, *İhsâ-ül-ulûm* [Enumeration of the sciences], ed. Ahmet Ateş (Istanbul: Milli Eğitim Bakanlığı, 1990), 109; George Saliba, “The Function of Mechanical Devices in Medieval Islamic Society,” *Annals of the New York Academy of Sciences* 441, no. 1 (Apr. 1985), 146.

On artisans' practical geometry, see al-Farabi, *İhsâ-ül-ulûm*, 94; Necipoğlu, *Topkapı Scroll*, 131–40.

Qur'an, al-Rad [The thunder]: 13; 2.

Sinan and Sai Çelebi, *Sinan's Autobiographies*, 58, 61, see also 64, 77, 88, 102.

Sinan and Sai Çelebi, 91, 104, see also 114, 140. I have changed the translation of the word *mīzān* from “measure” to “balance,” according to the dictionary entry in Caʿfer, *Risâle*, 111.

The type of compass shown in Figure 1 was called *devvāre* in Arabic and *pergel* in Turkish. As the root meaning of the name implies, it was used for dividing and for drawing arcs and circles (as indicated by the fact that the shorter leg of the compass has a pointed edge, while the longer leg has a square edge). The resistant iron structure and dimensions of this compass suggest that it was used on hard surfaces, such as stone or wood.

For Ottoman drawings and models, see Necipoğlu, *The Age of Sinan*, 166–76; Necipoğlu-Kafadar, “Plans and Models.” For plans in Islamic architecture, see Necipoğlu, *Topkapı Scroll*, 3–29; Jonathan M. Bloom, “On the Transmission of Designs in Early Islamic Architecture,” *Muqarnas* 10 (1993), 21–28; Renata Holod, “Text, Plan and Building: On the Transmission of Architectural Knowledge,” in *Theories and Principles of Design in the Architecture of Islamic Societies*, ed. Margaret Bentley Ševčenko (Cambridge, Mass.: Aga Khan Program for Islamic Architecture, 1988), 1–12.

For similar references, see Necipoğlu, *Topkapı Scroll*, 4, 9, 117–18.

On this figural, rather than literal, analogy, see al-Ghazali, *Ninety-Nine Names of God in Islam*, ed. Robert Charles Stade (Ibadan: Daystar Press, 1970), 30–35. Al-Ghazali's focus on the process of creation is noted in Samer Akkach, *Cosmology and Architecture in Premodern Islam* (Albany: State University of New York Press, 2005), 51–52.

Caʿfer, *Risâle*, 18; see also Caʿfer, *Risāle*, trans. Crane, 28.

Caʿfer, *Risāle*, trans. Crane, 28; I have slightly modified Crane's translation. For the Turkish text, see Caʿfer, *Risâle*, 18.

Caʿfer, *Risâle*, 20–22, 85.

Caʿfer considers both linguistic rules and contemporary uses related to the terms he explores. On essentialist, prescriptive, and linguistic types of definitions, see Kiki Kennedy-Day, *Books of Definition in Islamic Philosophy* (New York: Routledge, 2003), 14–15.

Caʿfer, *Risâle*, 20.

Caʿfer, *Risâle*, 20. For the *Luġat-i Niʿmetu'llāh* entry, see Ni*ʿ*metu'llāh Ahmed, *Lügat-i Niʿmetu'llâh*, ed. Adnan İnce (Ankara: Türk Dil Kurumu Yayınları, 2015), 60. See also H. Suter, “Handasa,” in *Encyclopaedia of Islam, First Edition (1913–1936)*, ed. M. Th. Houtsma, T. W. Arnold, R. Basset, and R. Hartmann (Leiden: Brill, 2012), https://referenceworks.brillonline.com/entries/encyclopaedia-of-islam-1/handasa-SIM_2686 (accessed 2 Jan. 2020).

Caʿfer, *Risâle*, 20–21.

Caʿfer, *Risâle*, 20–21.

Caʿfer, *Risâle*, 111.

Caʿfer, *Risâle*, 21.

See H. Schirmer, “Misāḥa,” in *Encyclopedia of Islam, Second Edition*, ed. P. Bearman, Th. Bianquis, C. E. Bosworth, E. van Donzel, and W. P. Heinrichs (Leiden: Brill, 2012), https://referenceworks.brillonline.com/entries/encyclopaedia-of-islam-2/misaha-COM_0753?s.num=124&s.start=120 (accessed 2 Jan. 2020); İhsan Fazlıoğlu, “Mesaha,” in *İslâm ansiklopedisi* (Ankara: TDV, 2004), 261–64; İhsan Fazlıoğlu, *Uygulamalı geometrinin tarihine giriş* (Istanbul: Dergah, 2004), 18–19, 37–54. In Islamic sciences, the Greek term *geometria* often referred to “surveying.” Surveying was differentiated as a particular kind of geometry when fields such as land division needed more specificity.

For discussion of Caʿfer's chapters on surveying, see Gül Kale, “Intersections between the Architect's Cubit, the Science of Surveying, and Social Practices in Caʿfer Efendi's Seventeenth-Century Book on Ottoman Architecture,” *Muqarnas* 36, no. 1 (Nov. 2019), 131–77.

On administrators' practical geometry, see al-Farabi, *İhsâ-ül-ulûm*, 107–10; Saliba, “Function of Mechanical Devices,” 146.

See Ömer Lütfi Barkan, ed., *Süleymaniye Cami ve imareti inşaatı (1550–1557)*, 2 vols. (Ankara: Türk Tarih Kurumu Basımevi, 1972), 1:15–16, 2:276. For Kalender's album preface and skills in album making as well as his patronage of arts, see Emine Fetvacı, *The Album of the World Emperor: Cross-Cultural Collecting and the Art of Album-Making in Seventeenth-Century Istanbul* (Princeton, NJ: Princeton University Press, 2019), 61—62, 90.

Caʿfer, *Risâle*, 21.

Caʿfer, *Risâle*, 92.

For the distinction between the *muhandis* (geometer), who knew geometrical proofs, and the surveyor (*māṣah*), who applied geometry on physical surfaces, see the description by Abu'l-Wafaʾ al-Buzjani (d. 998) in Özdural, “Mathematics and Arts,” 173–74. In Caʿfer's definition, the two aspects merge, making the *mühendis* the practitioner of geometry, like the surveyor.

Caʿfer, *Risâle*, 108.

Cited in Dimitri Gutas, *Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early ʻAbbāsid Society (2nd–4th/8th–10th Centuries)* (New York: Routledge, 1998), 112–13. See also Clifford Edmund Bosworth, “A Pioneer Arabic Encyclopedia of the Sciences: Al-Khwārizmī's *Keys of the Sciences*,” *Isis* 54, no. 1 (Mar. 1963), 97–111.

Gutas, *Greek Thought, Arabic Culture*, 112–13. Based on al-Khwārizmī's definition, Gutas concludes that the *muhandis* was someone who “prepared plans” (*yuqaddiru*). However, this translation might overlook the architectural context of the time, because there are no extant plans (in the modern sense) from this early period. The word *yuqaddiru* shares the same root as the word *taqdīr*, which refers to determining the value or measure of something.

Caʿfer's dictionary definition of *muhandis* derives from earlier examples. On the *muhandis* in Mamluk Egypt and Syria (1250–1517), associated with building bridges, canals, and aqueducts, and the architect (*miʿmār*), responsible for repairs and restorations, see Doris Behrens-Abouseif, “*Muhandis*, *Shād*, *Muʿallim*: Note on the Building Craft in the Mamluk Period,” *Der Islam* 72, no. 2 (1995), 293–309. See also Nasser Rabbat, “Architects and Artists in Mamluk Society: The Perspective of the Sources,” *Journal of Architectural Education* 52, no. 1 (Sept. 1998), 30–37.

Caʿfer, *Risâle*, 36.

For an overview of the water superintendent's duties, see Necipoğlu, *Age of Sinan*, 140–42, 171–72; Cengiz Orhonlu, *Osmanlı İmparatorluğunda şehircilik ve ulaşım üzerine araştırmalar*, ed. Salih Özbaran (Izmir: Ege Üniversitesi, 1984), 78–82.

E. 7471/4, Topkapı Palace Museum Archives.

Caʿfer, *Risâle*, 27.

Selaniki Mustafa Efendi, *Tarih-i Selânikî*, ed. Mehmed İpşirli, 2 vols. (Istanbul: Edebiyat Fakültesi Basımevi, 1989), 2:95.

Selaniki, 2:95–96, 1:244–45.

Evliyâ Çelebi, *Evliyâ Çelebi Seyahatnâmesi:1. Kitap,* ed. Robert Dankoff, Seyit Ali Kahraman, and Yücel Dağlı, 2 vols. (Istanbul: Yapı Kredi Yayınları, 2011), 1:19, 24, 27.

Mehmed Âşık, *Menâzırü'l-‘avâlim*, ed. Mahmut Ak, 3 vols. (Ankara: Türk Tarih Kurumu Basımevi, 2007), 3:1055–57. For Byzantine stories on buildings circulating in Ottoman written and oral cultures, see Gülru Necipoğlu, “The Life of an Imperial Monument: Hagia Sophia after Byzantium,” in *Hagia Sophia from the Age of Justinian to the Present*, ed. Robert Mark and Ahmet S. Çakmak (Cambridge: Cambridge University Press, 1992), 195–255.

Mehmed Âşık, *Menâzırü'l-‘avâlim*, 3:1057. Evliyâ Çelebi also mentions Ignatius (Agnados) as the architect (*miʿmār*) of Hagia Sophia and calls him a *mühendis*. See Evliyâ, *Seyahatnâmesi*, 1:49. Ignatius is mentioned as Hagia Sophia's master builder and engineer (*mechanikos*) in the *Narratio de St. Sophia*. See Cyril A. Mango, ed., *The Art of the Byzantine Empire, 312–1453: Sources and Documents* (Englewood Cliffs, N.J.: Prentice Hall, 1972), 96–97. However, the text *De aedif* by Procopius notes Anthemius of Thalles and Isidore of Miletus as its architects. Mango, *Art of the Byzantine Empire*, 72–75. Ignatius was one of the legendary architects associated with the building by the Ottomans. See Necipoğlu, “Life of an Imperial Monument,” 198–202.

Sinan and Sai Çelebi, *Sinan's Autobiographies*, 66. On Sinan's competition with the builders of Hagia Sophia, see Gülru Necipoğlu, “Challenging the Past: Sinan and the Competitive Discourse of Early Modern Islamic Architecture,” *Muqarnas* 10 (1993), 169–80.

On Sinan's engineering skills similar to those of a Roman military architect-engineer called a *mechanicus* (Latin for *mechanikos*), see Necipoğlu, *Age of Sinan*, 132–33.

In his section on mosques and masjids built in Istanbul, Evliyâ Çelebi lists some of Sinan's buildings under the title “Masjids built by Koca Sinan with the aid of the science of geometry.” See Evliyâ, *Seyahatnâmesi*, 1:148.

On Ottoman waterway maps and hydraulic systems, see Kazım Çeçen, *İstanbul'un vakıf sularından Halkalı suları* (Istanbul: İstanbul Büyükşehir Belediyesi, 1991); Kâzım Çeçen, *Mimar Sinan ve Kırkçeşme tesisleri* (Istanbul: İstanbul Büyükşehir Belediyesi, 1988). On Ottoman waterway maps of the late seventeenth and early eighteenth century and their social contexts, see Deniz Karakaş, “The Social Impact of Water in Ottoman Times, Water Resources Management and Development in Ottoman Istanbul: The 1693 Water Survey and Its Aftermath,” in *Istanbul and Water*, ed. Paul Magdalino and Nina Ergin (Leuven: Peeters, 2015), 177–204. On the social context of hydraulic works executed by rural engineers in Ottoman Egypt, see Alan Mikhail, *Under Osman's Tree: The Ottoman Empire, Egypt, and Environmental History* (Chicago: University of Chicago Press, 2017), 93–108.

For court decrees on repairing waterworks, see nos. 496 and 497, Istanbul 3 (1618), in Timur Kuran, ed., *Mahkeme kayıtları ışığında 17. yüzyıl İstanbul'unda sosyo-ekonomik yaşam, 1617–61* [Social and economic life in seventeenth-century Istanbul: Glimpses from court records, 1617–61], 10 vols. (Istanbul: Türkiye İş Bankası, 2011), 6:16–22. For imperial decrees, see Ahmet Refik Altınay and Abdullah Uysal, eds., *Onuncu asr-ı hicrîde İstanbul hayatı* (Ankara: Kültür ve Turizm Bakanlığı, 1987), 23, 25–31; Ahmed Refik, ed., *Hicrî on birinci asırda İstanbul hayatı (1592–1688)* (Istanbul: Devlet, 1931), 34, 49–50.

Eyyubî, *Menâḳıb-ı Sultan Süleyman*, ed. Mehmet Akkuş (Ankara: Kültür Bakanlığı, 1991), 156–209.

Selaniki, *Tarih*, 1:232–33; also quoted in Necipoğlu, *Age of Sinan*, 141, 166.

Sinan and Sai Çelebi, *Sinan's Autobiographies*, 117–22, 145–48. On the Kırkçeşme waterways, see Çeçen, *Mimar Sinan*.

Sinan and Sai Çelebi, *Sinan's Autobiographies*, 118, 145.

Taşköprüzāde mentions books by Ibn al-Haytham (Alhazen, d. 1039) and al-Karaji (ca. 1025) under the subfield of architecture in *Mevzuat*, 1:105. See also Necipoğlu, *Topkapı Scroll*, 140–41; George Saliba, “Artisans and Mathematicians in Medieval Islam,” *Journal of the American Oriental Society* 119, no. 4 (Oct.–Dec. 1999), 637–45.

See al-Karaji, “Surveying and Surveying Instruments, Being Chapters 26, 27, 28, 29, and 30 of the Book on Finding Hidden Water by Abu Bakr Muhammad Al-Karaji,” trans. Frans Bruin, *Biruni Newsletter* 31 (1970), 1–43.

See al-Karaji, 2–3.

On the popularity of Bahā' al-Dīn al-ʿAmilī's *Khulāṣat al-ḥisāb* among Ottoman scholars, see Cevat İzgi, “Osmanlı medreselerinde aritmetik ve cebir,” *Osmanlı Bilimi Araştırmaları* 1 (1995), 129–58.

Mustafa Sâfî, *Mustafa Sâfî'nin Zübdetü't-Tevarîh'i*, ed. İbrahim Hakkı Çuhadar, 2 vols. (Ankara: Türk Tarih Kurumu Basımevi, 2003), 1:51–52.

Caʿfer, *Risâle*, 111.

See Necipoğlu, *Age of Sinan*, 171–72. Sinan's survey drawing was first published and examined in Aygen Bilge, “Mimar Sinan hakkında araştırmalar,” *Mimarlık* 67 (1969), 18–34. See also Çeçen, *Mimar Sinan*, 53.

For discussion of the use of a similar triangulation method based on proportions to measure inaccessible heights in Abu Bakr al-Khalil's fourteenth-century book on surveying, see Gülru Necipoğlu, “Ornamental Geometries: A Persian Compendium at the Intersection of the Visual Arts and Mathematical Sciences,” in Necipoğlu, *Arts of Ornamental Geometry*, 14–19.

The trapezoids shown here represent the Cebeciköy Kemeri on the right and the Mağlova Kemeri on the left, with the annotations giving their lengths and heights in cubits. The total length of the canal from the Cebeciköy aqueduct to the Mağlova aqueduct is also given in cubits, which suggests that measurements were taken from the direction of the Cebeciköy aqueduct.

The ratio between the dimensions 30 and 72 cubits (*ẕirāʿ*), annotated on the perpendicular line and the hypotenuse of the first aqueduct, Cebeciköy Kemeri, is approximately 6:15. The ratio between 45 and 113 cubits on the second aqueduct, Mağlova Kemeri, is again approximately 6:15.

For “the *gūnyā*-5: set square [number] 5,” which corresponded to angles of 36 degrees, 54 degrees, and 90 degrees (with ratio 6:9:15) and was used to facilitate the construction of regular pentagons and decagons, as mentioned in the *Anonymous Compendium*, see the glossary of technical terms in Necipoğlu, *Arts of Ornamental Geometry*, 183.

Caʿfer, *Risâle*, 111.

Caʿfer, *Risâle*, 86.

For estimates made per square cubit of surfaces, see Necipoğlu, *Age of Sinan*, 166, 168; Necipoğlu-Kafadar, “Plans and Models,” 231.

Architects executed these large-scale projects with the aid of expert practitioners under the supervision of high officials, such as the chief commander of the navy and the chief of the janissaries. See Eyyubî, *Menâḳıb-ı Sultan Süleyman*, 168–213; Bilge, “Mimar Sinan,” 22–26.

Caʿfer, *Risâle*, 21. Caʿfer first gives the saying in Persian, followed by a Turkish translation. See also Kees Versteegh, “Taqdīr,” in *Encyclopedia of Arabic Language and Linguistics*, ed. Lutz Edzard and Rudolf de Jong (Leiden: Brill, 2011), https://referenceworks.brillonline.com/search?s.f.s2_parent=s.f.book.encyclopedia-of-arabic-language-and-linguistics&search-go=&s.q=Taqd%C4%ABr (accessed 2 Jan. 2020).

Caʿfer, *Risâle*, 21.

See al-Ghazali, *Ninety-Nine Names of God*, 31. Akkach translates *taqdīr* as “to design” and *muhandis* as “architect.” See Akkach, *Cosmology and Architecture*, 51. Within the epistemological and cultural context of the eleventh century, it seems more appropriate to translate these terms as “to determine” and “geometer-engineer.”

In legal cases, *taqdīr* indicates the rendering of sound judgments. See Versteegh, “Taqdīr.” On water registers, see Gülru Necipoğlu, “ ‘Virtual Archaeology’ in Light of a New Document on the Topkapı Palace's Waterworks and Earliest Buildings, circa 1509,” *Muqarnas* 30 (2013), 315–50; Çeçen, *Mimar Sinan*, 165–76.

Decree no. 51, in *82 numaralı Mühimme Defteri (1617–18)*, ed. Murat Şener et al., vol. 47 (Ankara: Başbakanlık Devlet Arşivleri Genel Müdürlüğü, 2000), 34.

Decree no. 583, in *İstanbul Kadı Sicilleri: Eyüb Mahkemesi 19 (1619–1620)*, ed. Yılmaz Karaca et al., vol. 24 (Istanbul: İslâm Araştırmaları Merkezi, 2011), 436.

Caʿfer, *Risâle*, 21.

Caʿfer, *Risâle*, 21.

“ʿAḳl ile söylemege derler, oranlamaḳ demekdir.” Caʿfer, *Risâle*, 108, MS. Yeni Yazma 339, fol. 77v.

In Latin translations of Avicenna's *Canon*, *wahm* became *aestimatio*. See Harry Austryn Wolfson, “The Internal Senses in Latin, Arabic, and Hebrew Philosophic Texts,” *Harvard Theological Review* 28, no. 2 (Apr. 1935), 115–16; David Summers, *The Judgment of Sense: Renaissance Naturalism and the Rise of Aesthetics* (Cambridge: Cambridge University Press, 1987), 206–11.

Kınalızâde Ali Çelebi, *Ahlâk-ı alâî*, ed. Mustafa Koç (Istanbul: Klasik, 2007), 134–38. The five external senses are touch, smell, taste, hearing, and sight. The five internal senses are the sense that collects all images apprehended by the external senses; the faculty of fantasy or imagination, which stores all that is apprehended by the first sense; the faculty of estimation; the faculty of memory; and the faculty of creative imagination.

Kınalızade, 136–38. Kınalızade's book drew from the philosophical traditions of Galen, Aristotle, and Naṣīr al-Dīn al-Ṭūsī. On the faculty of estimation, see Deborah L. Black, “Estimation (*Wahm*) in Avicenna: The Logical and Psychological Dimensions,” *Dialogue* 32, no. 2 (Spring 1993), 219–58; David B. Macdonald, “Wahm in Arabic and Its Cognates,” *Journal of the Royal Asiatic Society of Great Britain and Ireland* 54, no. 4 (Oct. 1922), 505–21. In Greek-into-Arabic translations, *wahm* replaced *phantasia*. See Shlomo Pines, *Collected Works of Shlomo Pines* (Jerusalem: Magnes Press, 1986), 2:366.

Black, “Estimation (*Wahm*) in Avicenna,” 220. Kınalızade's classic example is that of the sheep that escapes from the wolf. Kınalızade, *Ahlâk-ı alâî*, 138.

On types of “reason,” see Majid Fakhry, “Rationality in Islamic Philosophy,” in *A Companion to World Philosophies*, ed. Eliot Deutsch and Ronald Bontekoe (Malden, Mass.: Blackwell, 1997), 504–16.

Macdonald, “Wahm in Arabic,” 509.

On theoretical intellect and the faculty of practical intellect, see Kınalızade, *Ahlâk-ı alâî*, 140–42. See also al-Farabi, *Alfarabi's Philosophy of Plato and Aristotle*, ed. Muhsin Mahdi (New York: Free Press of Glencoe, 1962), 123–29.

See Pines, *Collected Works*, 2:366–68. People judged the sun to be bigger than it appeared through this faculty.

For the difference between the type of number in “number theory” and that in the “science of reckoning,” which had Greek origins, see A. I. Sabra, “ʿIlm al-Ḥisāb,” in Bearman et al., *Encyclopedia of Islam, Second Edition*, https://referenceworks.brillonline.com/search?s.f.s2_parent=s.f.book.encyclopaedia-of-islam-2&search-go=&s.q=%CA%BFIlm+al-%E1%B8%A4is%C4%81b (accessed 2 Jan. 2020).

See Gülru Necipoğlu, “The Account Book of a Fifteenth-Century Royal Kiosk,” in *Raiyyet Rusumu: Essays Presented to Halil İnalcık on His Seventieth Birthday*, ed. Şinasi Tekin and Gönül Alpay Tekin (Cambridge, Mass.: Harvard University Press, 1987), 31–44.

On the general uses of the words *taḥmīn* and *keşf* in cost estimates, see Necipoğlu, *Age of Sinan*, 166–70; Necipoğlu, *Topkapı Scroll*, 159.

A register would typically first mention the mensuration of the roof structure, flooring, and walls, then describe the calculation of surface areas, and finally give the estimations of costs per square cubit. See, for example, decree no. 517, 1618 Istanbul 3, in *Mahkeme kayıtları (1617–61)*, 6:47–49. For similar tripartite processes, see the following decrees, also in *Mahkeme kayıtları (1617–61)*: no. 561, 1618 Istanbul 3, 6:105–6; no. 574, 1618 Istanbul 3, 6:123–25; no. 627, 1618 Istanbul 3, 6:186–87; no. 640, 1618 Istanbul 3, 6:206–7.

Decree no. 97, in *İstanbul Kadı Sicilleri: İstanbul Mahkemesi 3 (1618)*, ed. Yilmaz Karaca et al., vol. 13 (Istanbul: İslâm Araştırmaları Merkezi, 2010), 112–13.

The same operation was applied to the rubble wall (*dolma duvar*) above the masonry one. Costs for timber roof supports, timber for flooring, and a stove were estimated according to the principles of proportion based on unit costs.

Decree no. 176, 1605 Galata 27, in *Mahkeme kayıtları (1602–17)*, 5:249–51.

Decree no. 295, 1612 Istanbul 1, in *Mahkeme kayıtları (1602–17)*, 5:400–402.

Decree no. 60, 1604 Galata 25, in *Mahkeme kayıtları (1602–17)*, 5:117–19.

A decree dated 1565 suggests that the official responsible for controlling the quality, price, and dimension of the tradesmen's works preserved a model (*numune)* of the mold used for bricks made in Istanbul's workshops. See Altınay and Uysal, *Onuncu asr-ı hicrîde İstanbul hayatı*, 158–59. On the regulation of construction materials under the chief architect, see Necipoğlu, *Age of Sinan*, 165.

Necipoğlu has dated this drawing to the early sixteenth century. See Necipoğlu, *Age of Sinan*, 168–71; Necipoğlu-Kafadar, “Plans and Models,” 229–31. Zarif Orgun first published it in “Hassa mimarları,” *Arkitekt* 12, no. 96 (1938), 333–42.

See Necipoğlu, *Age of Sinan*, 169–70. On the endowments of this convent, see Huri İslamoğlu-İnan, *State and Peasant in the Ottoman Empire* (Leiden: Brill, 1994), 114–19.

Decree no. 52, in *82 numaralı Mühimme Defteri (1617–18)*, 34.

Decree no. 202, 1605 Galata 27, in *Mahkeme kayıtları (1602–17)*, 5:286–88.

See Yvonne Dold-Samplonius, “Calculating Surface Areas and Volumes in Islamic Architecture,” in *The Enterprise of Science in Islam: New Perspectives*, ed. Jan P. Hogendijk and Abdelhamid I. Sabra (Cambridge, Mass.: MIT Press, 2003), 235–65; Necipoğlu, *Topkapı Scroll*, 158–59; Yvonne Dold-Samplonius, “Practical Arabic Mathematics: Measuring the Muqarnas by al-Kashi,” *Centaurus* 35, no. 3 (Oct. 1992), 193–242; Yvonne Dold-Samplonius, “The XVth Century Timurid Mathematician Ghiyāth al-Dīn Jamshīd al-Kāshī and His Computation of the Qubba,” in *Amphora: Festschrift für Hans Wussing zu seinem 65. Geburtstag*, ed. Sergei S. Demidov et al. (Basel: Birkhäuser, 1992), 171–81.

This treatise is the Turkish translation of the fourth chapter on surveying in al-Kāshī's *Miftāḥ al-ḥisāb*; it addresses engineers-architects and military troops in the newly founded engineering school.

Caʿfer, *Risâle*, 21–22.

Caʿfer, *Risâle*, 27.

Caʿfer, *Risâle*, 20.

Caʿfer, *Risâle*, 21.

Caʿfer, *Risâle*, 21.

Caʿfer, *Risāle*, trans. Crane, 30–31. For the Turkish text, see Caʿfer, *Risâle*, 21–22.

Naṣīr al-Dīn Ṭūsī's (d. 1273) commentary on the Euclidean postulates, *Taḥrīr usūl al-handasa* (*Exposition of the Elements of Geometry*, ca. 1248), illustrated geometric forms and was widely read by scholars. Naṣīr al-Dīn Muḥammad ibn Muḥammad Ṭūsī, *Tahrîru usûli'l-hendese ve'l-hisâb*, ed. İhsan Fazlıoğlu (Istanbul: Türkiye Yazma Eserler Kurumu Başkanlığı, 2012). It is hard to determine exactly how artists and architects used such books, however.

Caʿfer, *Risâle*, 27–28.

Caʿfer, *Risâle*, 27; Caʿfer, *Risāle*, trans. Crane, 34; I have slightly modified Crane's translation.

For a similar conclusion on the minor role of arithmetic in medieval and early modern artistic practices, see Necipoğlu, *Topkapı Scroll*, 41–44; Gülru Necipoğlu, “Geometric Design in Timurid/Turkmen Architectural Practice,” in *Timurid Art and Culture: Iran and Central Asia in the Fifteenth Century*, ed. Lisa Golombek and Maria Subtelny (New York: Brill, 1992), 48–66.

Al-Buzjani alluded to artisans' need for practical geometry and demonstrated different methods for drawing and subdividing figures, but he differentiated between the approximations of artisans and the exactitude of mathematicians. See Necipoğlu, “Theory and Praxis” and “Manuals of Practical Geometry and the Scroll Tradition,” in *Topkapı Scroll*, 131–81; Özdural, “On Interlocking Similar or Corresponding Figures,” 191–211; Özdural, “Omar Khayyam.” For recent assessments of books on geometric constructions such as al-Buzjani's, see Necipoğlu, “Ornamental Geometries,” 11–78; Jan P. Hogendijk, “A Mathematical Classification of the Contents of an Anonymous Persian Compendium on Decorative Patterns,” in Necipoğlu, *Arts of Ornamental Geometry*, 145–62.

Caʿfer, *Risâle*, 117.

Caʿfer, *Risâle*, 111.

On the translatability of two- and three-dimensional geometric patterns into woodwork, see Necipoğlu, *Topkapı Scroll*, 22–27.

Mehmed Agha's drawings and models for the Kaʿba and the Sultan Ahmed Mosque are discussed in Gül Kale, “Unfolding Ottoman Architecture in Writing: Theory, Poetics, and Ethics in Caʿfer Efendi's ‘Book on Architecture’ ” (PhD diss., McGill University, 2014), chap. 5.

Caʿfer, *Risâle*, 66.

Caʿfer, *Risâle*, 26.

Caʿfer's lost book on geometry most likely would not have included the geometric constructions found in al-Buzjani's book or the “cut-and-paste methods” mentioned by Özdural in “On Interlocking Similar or Corresponding Figures,” 192.

Caʿfer's expertise pertained to that branch of practical geometry concerned with the uses of surveying in civic affairs, such as land division for inheritance shares. For new findings on Caʿfer's identity and judicial duties, see Kale, “Intersections between the Architect's Cubit.”

Caʿfer, *Risâle*, 69.