KETĀB al-TAFHIM le-awāʾel ṣenāʿat al-tanjim, a book on the astral sciences by Abu Rayḥān Moḥammad b. Aḥmad Biruni (q.v.; b. near Kāṯ in Ḵvārazm, ca. 362/972; d. Ḡazna, 440/1048), written in 420/1029 in Ḡazna.

Ketāb al-tafhim is a textbook explaining the mathematical sciences through questions and answers for Rayḥāna, daughter of someone called either Ḥasan or Ḥosayn Ḵvārazmi, whose identity is not well known (Biruni, 1934, p. 2). The book exists in an Arabic and a Persian version. According to Robert Ramsay Wright and Jan P. Hogendijk, there are currently 27 manuscript copies of the Arabic version and 26 of the Persian version (Biruni, 1934, p. xii; The relationship between the two language sets has been differently evaluated, and various perspectives have been taken since the 18th century (Browne, II, p. 102; Biruni, 1934, pp. iii, xiv, 81 n.). Jalāl-al-Din Homāʾi appreciated Biruni’s Persian language skills very highly (Biruni, 1983, p. ku). But he also highlighted a number of substantial differences between the Arabic and the Persian texts in the manuscripts he had had access to (Biruni, 1983, pp. ku-m) In general, the Arabic text is more extensive and elaborate than the Persian text as well as more clearly formulated. The Persian text, in contrast, is often more condensed, semantically less stable, and at times somewhat cumbersome. These properties of the Persian text reflect the lack of a well-established scientific language. The very first definition of geometry already indicates this basic difference between the Arabic and the Persian versions of the Ketāb al-tafhim: the Arabic text uses expressions already well established in the scholarly idioms of the time; the Persian text applies a mixed language. At times it translates such technical terminology literally without regard for the grammar and textual units of the Arabic text into a non-technical, seemingly colloquial language. At other times, it continues using the Arabic terms. In a third set of cases, it replaces an Arabic expression by its explanation through another, non-standard Arabic terminology.(Biruni, 1934, p. 1b; Biruni, 1983, p. 3). This multiplicity of procedure remains fairly stable throughout the entire Persian text. Obviously, Biruni (if he was indeed the translator) had not designed a systematic approach to the task of translating and its linguistic challenges. Homāʾi believes that Biruni had authored both versions, thus allowing for the possibility that the Persian text is indeed not a translation (Biruni, 1983, p. kz).

The distribution of the extant manuscripts shows a wide spread of the work across the Islamicate world from South Asia to Morocco, as well as a relatively continuous interest in it since, at the latest, the 13th century. Remarks by a variety of scholars confirm that the book was not merely read by students, but also known and used by full-fledged scholars.

The book contains 530 questions and answers (Biruni, 1983, pp. 316-539). The book covers geometry, arithmetic including algebra, astral sciences, and geography. Other themes from the study of balances and optics are used as examples to explain a notion or a method. The presentation of the questions is thematically arranged. It starts with geometrical and arithmetical topics (questions 1-119; Biruni, 1983, pp. 2-55), followed by a survey on the Ptolemaic structure of the universe and its fundamental concepts and notions such as celestial movement, horizon, meridian, cardinal point, ecliptic, solar apogee, solar mean motion, solar equation, Zodiac, lunar mansions, epicycle, deferent, equant, node, anomaly, and so forth (questions 120-235; Biruni, 1983, pp. 56-284). Geography covers a mere six questions (236-41; Biruni, 1983, pp. 166-73). Then Biruni returns to astral themes for the rest of the book (questions 242-530), mixing astronomical with astrological topics, astrolabe construction, and calendrical themes such as eclipses, parallax, conjunctions of Jupiter and Saturn, the months and years of different people, the auspicious and inauspicious natures of planets, houses, or fortune telling.  All in all, the survey he delivers is very comprehensive.

The disciplinary sequence—geometry, arithmetic and number theory, the structure of the universe, and judicial astrology—suggests that he considered mathematical geography as part of the astral sciences and the configuration of the universe as an indispensable theoretical basis of judicial astrology, which he marks as the major professional qualification of the astrologer: “No one is worthy of the style and title of Astrologer who is not thoroughly conversant with these four sciences” (Biruni, 1934, p. 1, 1983, p. 2). Biruni’s position thus contradicts modern interpretations of the relationship between mathematical cosmography and astrology that sees the emergence of the science of the configuration of the universe as a process of separation from astrology and a measure of protection against attacks by religious scholars. Furthermore, the structure of the book shows similarities with and differences from later textbooks on mathematical cosmography, by preceding the explanation of astral themes with an introduction to geometry, not to natural philosophy. Geometry is identified as transforming arithmetic from the particular to the universal and elevating the astral sciences from conjecture and opinion to truth (Biruni, 1934, p. 1, 1983, p. 3).

The question-and-answer format that Biruni chose for Rayḥāna makes the book easily accessible and shows him as a proficient teacher, although no other teaching activities of his are known. Biruni justifies this choice as being “not only elegant, but (facilitating) the formation of concepts” (Biruni, 1934, p. 1, 1983, p. 2). In agreement with the second point, Biruni asks brief questions in a simple style such as “What is a body?” or “What is the line?” He answers the first question as “A solid body is that which can be felt by the sense of touch; standing by itself it occupies only its own share of space but entirely fills that to the extent of its dimensions, so that no other solid substance can occupy its place at the same time” (Biruni, 1934, p. 2, 1983, p. 3). With regard to the line, he wrote: “If a surface has boundaries, these are necessarily lines, and lines have length without breadth, therefore one dimension less than the surface, as that has one less than the solid; if it had breadth, it would be a surface, and we have assumed it to be the boundary of a surface. A line can be pictured through the line perceivable by the senses in a vessel filled with water and oil, or the line between sunshine and shadow, contiguous on the surface of the earth, or, also it is possible to picture all that to oneself from a thin sheet of paper, although it has thickness, until the familiar sense-perception leads gradually to the intellectual concept” (Biruni, 1934, p. 3, 1983, p. 6). The mixture of descriptive definitions in the style of Aristotle’s natural philosophy and Euclid’s geometry with empirical, sense-based imaginations characterizes Biruni’s entrance into the course. In some cases, he also added diagrams.

The topics defined by Biruni in the geometrical part of the text include plane, solid, and spherical geometry, as well as ratios of magnitudes. They cover some of the objects defined in Books I, II, III, and V of Euclid’s Elements.  Other definitions are not given in Euclid’s work, but define objects used in theorems of the Elements, such as the product of two lines, perpendicular, interior and exterior angles, or opposite angles or similar triangles. Since this introduction to geometry prepares the questions on astronomical and astrological themes, Biruni also presents definitions that do not belong to the Elements, but whose objects are tabulated in astronomical handbooks or employed in texts on surveying, or in the so-called “Middle Books.” Among them are the foot point of a perpendicular, the sine, versed sine (= 1-cosine [since Biruni put the radius = 1; otherwise R-Cosine]), and the so-called whole sine (= radius). In addition, he also draws on Archimedes’s “Measurement of the Circle and defines the ratio of the circumference of a circle to its diameter and explains how to calculate the area of the circle (Biruni, 1983, pp. 17-18). An interesting feature of this first part on geometry is Biruni’s use of balances (equal armed as well as unequal armed) and the behavior of their pointers when loaded for illustrating the formation of some of the objects. Question 55 on the inverse ratio makes the use of the steelyard explicit in word and drawing as the prime example for this kind of ratio (Biruni, 1934, p. 17, 1983, pp. 24-25). The geometrical questions end with definitions of solids, which include remarkably conic sections, and definitions of great and small circles on the sphere and the so-called secant figure (Biruni, 1934, pp. 19-23; 1983, pp. 28-32).

The second part on arithmetic is similarly broadly composed, bringing together definitions from Books VII and X of Euclid’s Elements with numbers defined by the 2nd-century philosopher Nicomachus of Gerasa and sexagesimal numbers used in the astral sciences, Indian decimal numbers, the basic arithmetical operations including the calculation of square and cubic roots, and quadratic equations. In addition to diagrams, Biruni explains here various concepts with the help of numerical examples (Biruni, 1934, pp. 23-43, 1983, pp. 32-55). The combined presentation of different number systems, calculation rules, themes, and sources is remarkable; most of them have become the focal point of individual treatises as well as the subject of chapters in scholarly encyclopedias thus remaining separate subject matters rather than being joined to form a coherent body of disciplinary knowledge.

The part on the structure of the universe contains a number of claims not shared by many other astral scientists in Islamicate societies. Neither are all of them upheld by Biruni in his own scholarly works. In question 122, he describes the universe as homocentric around the earth as if it was an onion with eight spherical layers (Biruni, 1934, pp. 43-44, 1983, p. 56). Later, in question 151, he introduces the concept of the epicycle (Biruni, 1934, p. 61, 1983, pp. 75-76). He returns to it again in question 178, followed by definitions of two further elements (deferent, equant) of the Ptolemaic system of the universe (Biruni, 1934, pp. 92-93, 1983, pp. 117-18). In question 122, he also identifies ether, the substance of the universe, with the whole body of the same rather than its material, attributing this idea to the philosophers in general (Biruni, 1934, p. 44, 1983, pp. 56). Informing Rayḥāna that some scholars believe in the existence of a ninth, immobile sphere, which the Brahmins called Baraham (Barahmānd), Biruni rejects this idea saying that this immovable mover neither had a body nor could be called an orb (falak; Biruni, 1934, p. 45, 1983, p. 57). He tells Rayḥāna that there were ancient scholars who believed in an infinite void beyond the eighth sphere, while others thought this was an infinite body contrary to what Aristotle had taught (Biruni, 1983, pp. 57-58). Wright misunderstood most of the answer to question 122. Biruni also teaches the doctrine of the four elements, making up the sub-lunar world, in a modified version. He puts earth and water together into a single sphere, fire being produced by the movement of the lunar sphere and its friction with the sphere of the air and being unequally hot due to the slower movement of the lunar sphere at the poles (Biruni, 1934, p. 46, 1983, pp. 58-59). The copyist of the Arabic text published in facsimile by Wright deviated from these statements when drawing the diagram of the sub-lunar world, adding a space for water outside earth (PLATE I).

Many other interesting and edifying topics are described, compared, explained, and at times visualized in the remaining 379 questions that are dealt with on different levels of knowledge and skills. All the constellations of the northern and southern hemispheres are named in Arabic or Persian and in a few cases in other languages. They are described and discussed in detail, including individual stars, but only a few manuscripts also contain the corresponding images. They are followed by a lengthy discussion of the different portrayals of celestial constellations by Arab nomads and the lunar mansions (Biruni, 1934, pp. 69-86, 1983, pp. 94 ff.).

The technical terms for the orbits of the sun, the moon, and the planets, needed for reading the tables in an astronomical handbook, are explained afterwards, such as the mean sun, the equation of the sun, the mean and the corrected longitude of a planet or the equation of the anomaly (Biruni, 1934, pp. 86-96, 1983, pp. 116-26).  When discussing the movements of the planets, Biruni even emphasizes the particularities of Mercury, the most challenging planet to model (Biruni, 1934, pp. 98-99, 1983, pp. 130-32). In other cases too, Biruni draws Rayḥāna’s attention to new observational results or contested theories. On issues where scholars were of different opinions, Biruni either offers his own reflections or admits that he is not aware of any solutions. The enumeration of the sizes of the planets, sun, moon, and earth and the distances between them are the last example from among the theoretical topics of Biruni’s discussion on the configuration of the universe (Biruni, 1934, pp., 116-19, 1983, pp. 150-65). For the values of the celestial bodies, Biruni relied on Ptolemy’s Almagest; for the circumference and diameter of the earth on the reports of Caliph al-Maʾmun’s (r. 813-33) astrologers.

The questions on the astrolabe provide Rayḥāna not only with explanations of the parts, engravings, names, and types of the instrument, but also with a number of practical exercises, which is rarely the case in the previous sections of the Book of the Stars (Biruni, 1934, pp. 194-210, 1983, pp. 286-315). The questions concerning astrology cover less than a third of the entire book. Although Biruni expresses his skeptical attitude towards astrological tasks and methods, he nonetheless presents them as the goal of his teaching (Biruni, 1934, p. 210). Biruni repeatedly adds information about Indian astrological doctrines and interpretations (e.g., Biruni, 1934, pp. 211-13, 216, 257-59, 261-63). The main authors he refers to time and again are Abu Maʿšar Balḵi (q.v.; d. 272/886) and Ptolemy.


Editions and translations.

Abu Rayḥān Moḥammad Biruni, Ketāb al-tafhim le-awāʾel ṣenāʿat al-tanjim, ed. Jalāl-al-Din Homāʾi, Tehran, 1940, rev. ed., Tehran, 1974, repr., Tehran, 1983; ed. and tr. Robert Ramsay Wright, as The Book of Instruction in the Elements of the Art of Astrology, London, 1934; ed. ʿAli Ḥasan Musā (Ar. version), Damascus, 2003, tr. Giuseppe Bezza, as L’arte dell’astrologia, Milan, 1992.


Edward G. Browne, A Literary History of Persia II, Cambridge, 1928, pp. 96-98, 101-2, 105.

Jan P. Hogendijk:

Moḥammad-Mehdi Rokni, “Farhang‑e ʿāmma dar Ketāb al-tafhim,” Majalla-ye Dāneškada-ye adabiyāt va ʿolum‑e ensāni‑e Dānešgāh‑e Mašhad 9, 1973, pp. 159-87.

(Sonja Brentjes)

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