Taqi al-Din's decimal trigonometric tables

In Jaridat al-durar wa-kharidat al-fikar, historians of science identify a particular computational development by Taqi al-Din ibn Ma'ruf: applying decimal fractions to trigonometric functions, astronomical tables and fractional parts of degrees. This does not mean that he invented decimal fractions or trigonometry. Rather, he extended an inherited arithmetic into an astronomer's working instrument. Ihsan Fazlioglu connects this achievement with the continuation of al-Kashi's arithmetic and describes decimal tables while retaining an exception for the fixed-star table. We do not remove that exception to produce a more spectacular headline ².

The contribution is a refinement in organizing calculation and representing numbers. It differs from al-Uqlidisi's decimal-fraction operations, and from invention of a calculator or modern decimal-point symbol.

A zij is a work of astronomical computation and tables, not merely a list of stars. Fazlioglu describes the Jarida and Tashil zij al-ashariyya as expressing fractional parts of degrees decimally and carrying out calculations accordingly; he also mentions tangent and cotangent tables. TDV describes the Jarida as a small zij containing decimal tables ^{1 3}.

Three things must remain distinct: the angle being measured, the function connecting it with a ratio or a length at a chosen scale, and the notation used to write the resulting number. Moving a fraction into base ten does not change the triangle. Readability and checking depend on consistent conventions and on knowing what every place represents; “decimal” alone cannot explain an entire page.

Sexagesimal arithmetic, dividing a unit into sixty parts and each part into sixty again, already belonged to astronomical practice, as did traditions of sines and shadows. Fazlioglu connects Taqi al-Din with the Samarqand school and the works of al-Kashi, Qadizade and Ali Qushji, and describes his continuation of decimal arithmetic in theory and practice, not its appearance from nothing ⁴.

Conversely, an earlier author is not automatically a directly documented predecessor whose book lay on his desk. Al-Uqlidisi provides a museum comparison; we do not draw a personal transmission edge without evidence. A single-inventor story would erase centuries of teaching, writing and correction. Calling every predecessor's contribution an invention of this particular zij would instead confuse general arithmetic with its specific application.

Taqi al-Din received religious and mathematical education, taught and served as a judge in Egypt, and returned to Istanbul in1570 before becoming head astronomer in1571, according to Fazlioglu. The biography describes books and instruments reaching him and encouragement to pursue mathematics and astronomy. The tables thus belong to a scientific life involving institutions and relationships rather than an idea detached from work ⁵.

The passages used here do not establish an exact composition date for the Jarida. Our open interval ends at1585, the death year in the biography, not an assertion that the invention occurred then. A modern edition's publication date cannot become the old book's composition date, and no unverified Hijri equivalent is added. Observatory chronologies also differ: Tekeli gives1575, TDV describes use from1577, and Fazlioglu reports a building edict in early1579. We do not turn these into one date for the table ^{6 7 5}.

His recorded role is “improver” of this application. The individual basis for Muslim identity is the biographies' account of religious education, madrasa teaching and judicial service, not his name or birthplace. Fazlioglu's heading identifies him as Hanafi, but that is not the sole basis used. Sunni is separately the museum's declared frame for the documented Ottoman setting, not an invented certificate of his detailed personal theology ^{8 7}.

Nor does a list of teachers and patrons become a list of joint inventors of the Jarida. Providing support, access to books, instruments, observations and teaching are different roles. The narrative preserves the environment's importance while the record attributes the contribution identified by scholarship.

We use explicitly modern teaching examples, not numbers purportedly transcribed from the Jarida. Half a degree is0.5degree and30arcminutes; half an arcminute is instead one120th of a degree. A degree contains60arcminutes and an arcminute60arcseconds. Changing notation never licenses dropping the unit. This explains the meaning of the biography's discussion of decimal angular parts and calculations, but does not establish a particular manuscript row ¹.

Likewise, the same teaching quantity can be written0.25 or one quarter, and0.75 or three quarters. The notation here is contemporary. TDV says that Taqi al-Din expressed fractional places verbally without the modern decimal separator. We must not add a point in a new drawing and then present it as his invention. Decimal structure and the printed symbol familiar today are different matters ⁹.

In a right triangle, our modern image explains the ratio between opposite side and hypotenuse. This is a classroom explanation of sine, not a reconstruction of the Jarida's table scale. We do not assume its radius was one, ten or sixty: those details require the relevant text. The inspected passages of Tekeli describe diameter120 and radius60 in the Sidra, and define sine as half the chord subtending twice the arc. But the Sidra is not the Jarida ¹⁰.

That distinction is essential. Fazlioglu explicitly describes sexagesimal trigonometric functions in the Sidra before discussing the Jarida and Tashil in a decimal context. Not every book by Taqi al-Din was decimal, and the existence of the project does not mean every star-table entry was replaced. We do not fill absent rows using a modern computer and label the output as the author's numbers ¹¹.

TDV records Remzi Demir's study of the Jarida in a1991 dissertation and publication of that study with a Turkish translation in2000. This establishes modern scholarly reception, not circulation of every copy in every country during the sixteenth century. That edition was not read for this record; it is identified as a known route to the text, not primary evidence personally inspected here ³.

Comparison with Simon Stevin in the history of decimals does not prove that either read the other. Chronological priority, similarity of a system and transmission of a particular book are separate questions. This record neither invents a route to Europe nor denies exchanges it has not investigated. It identifies what the sources actually read establish.

Fazlioglu explicitly identifies continuation of al-Kashi's decimal arithmetic, placing Taqi al-Din within mathematical and astronomical traditions of sines, shadows and tables. Tekeli compares proofs and trigonometric relationships in the Sidra and Copernicus and discusses tangent and cotangent near the article's end. This context makes clear that representing a function differently need not mean discovering that function anew ^{12 13}.

An editor must not bundle every theorem in a book into a single invention. Sine calculation, geometric proof, table scale, fractional places and celestial motion are connected questions, but they are not interchangeable.

The documented application is computation in tables within an astronomical project, not a list of modern inventions directly dependent on the Jarida. Fazlioglu distinguishes the Sidra's project of correcting Ulugh Beg's zij from the decimal zij works, and notes that the observing programme was not completed after the observatory's demolition. Incompleteness must not automatically be transferred to every book Taqi al-Din wrote ¹⁴.

Later dependency edges therefore remain empty in the data. An honest blank is preferable to an unsupported genealogy leading to phones or satellite navigation. Readers can compare representations across time without turning comparison into a historical transmission tree.

The direct setting supported by the sources is an astronomical computer's work: organizing fractions and table calculations, not a demonstrated immediate transformation of every household budget. Consistency in arithmetic helps identify where a mistake lies, but we supply no measured time saving or percentage reduction in errors. The sources read here do not provide such a performance experiment ¹.

The teaching comparison shows why units and place values matter. If a student copies0.5degree as0.5arcminute, the error concerns the unit, not invention of sine. This is a lesson available today, not a documented incident involving Taqi al-Din or one of his students.

The contribution reveals decimal computation entering Ottoman astronomical work, and reminds us that the history of trigonometry concerns more than names of functions or drawings of triangles. Organizing computational tools and explaining their use also belong to scientific knowledge. Fazlioglu's conclusion identifies the application of decimals to astronomy and trigonometry among Taqi al-Din's important contributions alongside his observatory activity ¹⁵.

That significance is not a claim that he alone caused modern science, or that every later school adopted his system. Decimal notation does not automatically produce more accurate observation: measurement quality, model correctness, number of places and calculation method require separate investigation.

“Improver” is more accurate than “inventor of decimal numbers.” Scholarship attributes a particular application to him while naming predecessors in arithmetic and trigonometry. Where an account describes the application as first, auditing all earlier tables would be a wider investigation not accomplished by manuscript inspection here. Appreciating the contribution does not require making a priority statement into an unlimited verdict ¹⁶.

The point or separator in our examples is not evidence of symbolic priority. TDV describes verbal places; consequently the numerical system, its separator and its application in a table remain distinct. Each may have a different history ⁹.

The topic can teach source reading and mathematics together. What does the zij description actually say? Which book is meant by “decimal table”? Is the number an angle or a function value? The museum uses TheOtherJesse's modern2007 public-domain drawing to explain triangle sides, not as evidence of an Ottoman page's appearance ¹⁷.

The accompanying chain is conceptual: earlier arithmetic and trigonometry followed by Taqi al-Din's application. It is not a manuscript plate, restored facsimile or precise instrumental measurement. A modern image cannot recover the Jarida's missing row count or accuracy.

Our directly inspected evidence consists of scholarship and its descriptions, not examination of every surviving copy. Fazlioglu and TDV were read, together with specified passages of Tekeli; neither the Jarida manuscript nor Demir's edition was inspected. We therefore do not give a purported historical cell value or invent a folio reference ^{1 3}.

Surviving text and surviving physical apparatus are different questions. A study describing tables does not make the accompanying drawing an Ottoman artifact, and a title in a works list cannot prove that every stage of the observatory programme was completed.

Science stories often become competitions over who came first. This record offers a more precise question: what changed in computational practice, and what supports the claim? Comparing the Sidra and Jarida also prevents the misleading picture of an author using one numerical system throughout his entire output ¹¹.

Visitors deserve the limits of our knowledge beside the achievement. Naming predecessors does not diminish the application, and we need not draw an imagined route to every computer for it to merit study.

Write “half a degree,” then0.5degree and30arcminutes beside it. Circle the unit each time. Next compare0.25 with one quarter, and0.75 with three quarters. This is a number-representation game, not an astronomical experiment or reconstruction of the Jarida.

Imagine two lists arranging the same quantities differently. Before copying, what information is necessary? The unit, place meaning and numerical scale. Do not look at the Sun or build an observing instrument, and do not attribute your paper's numbers to the author. Pencil and paper are enough to understand the question.

• Fazlioglu's complete BEA biography hosted at McGill, especially the Jarida, Sidra and arithmetic discussions ¹⁸.
• Topdemir, TDV biography, arithmetic discussion and works list ¹⁹.
• Tekeli1986, selected passages of the English study only; not an edition of the Jarida ²⁰.
• Modern image metadata and licence ¹⁷.