Sine-table computation in al-Battani's Zij

In chapter three of his Zij, Muhammad ibn Jabir al-Battani explains constructing a table of half-chords of doubled arcs and using it in both directions: from an arc to a length, and from a length to an arc ¹. We call these quantities sines today; this is a computational refinement within older traditions, not invention of sine or all trigonometry ². The chapter combines geometry, tabulation and approximation rather than presenting numbers without instructions for using them ³.

The table connects an arc with a calculated length in a circle of specified scale: al-Battani chooses diameter120 and radius60 ⁴. A “half-chord of the doubled arc” means doubling the arc's angle and taking half the corresponding chord, not halving the original angle ⁵. This makes the tabulated sine correspond directly to the user's input arc, avoiding repeated doubling during calculation ⁶.

The author explicitly follows Ptolemy in choosing diameter120 for convenient calculation; the scale is inherited, not a private invention ⁴. He begins with chord geometry and right-triangle relationships, then constructs chords of further arcs through halving, addition and subtraction ⁷. This does not establish that every operation first appeared in this book: the record concerns their practical arrangement and the table's use within the Zij ³.

Al-Battani's observing work was associated with Raqqa, and his Zij comprised fifty-seven chapters, not an isolated sheet of sines ^{8 9}. Computational foundations precede later chapters on using tables, making chapter three an introduction to a larger working tool ¹⁰.

The inspected pages supply no independent composition year for this chapter; neither a star-catalogue epoch nor the beginning of observations becomes an invention date ^{3 11}. The record's open-ended date terminates at his death in317 AH/929: a latest-lifetime bound, not a claim that he composed it specifically in929 ¹¹.

Al-Battani's recorded role is “improver”: the text documents a computational choice and working procedures on top of earlier geometry ². His Muslim identity here rests on the author's religious opening, read in Arabic, and Nallino's explicit argument from it—not his name or family affiliation ¹². The museum's Sunni frame is separately declared for the Abbasid Raqqa setting, not presented as a personal legal-school declaration; his biography retains named historical identity opinions without inferring religion from “al-Sabi” alone ^{11 13}.

The displayed right-triangle relation is planar; spherical problems in the zij are a separate astronomical application, not the same drawn triangle ^{14 15}.

The invented numerical example also shows why the table interval matters:6/30is not6/60, so using a full degree instead of half a degree changes the numerical increment. In reverse,20.4alone is insufficient without its neighboring rows and their difference2; an excess becomes minutes through the division and multiplication described by the chapter, not merely a new label ¹⁶. These are further explanations of the classroom example, not manuscript values.

The endpoint illustrates the geometric scale: at90degrees, half the chord of the doubled180-degree arc is half the diameter, or60. At30degrees, half the chord of60degrees is30, because that chord equals the radius ¹⁷. Exact geometric values at these points differ from interpolation between neighboring rows; a table can combine geometric results with intermediate estimates.

MacTutor gives a right-triangle relation attributed to al-Battani: b sin(A) = a sin(90 - A) ¹⁸. Its notation is the modern reference's presentation, not copied Arabic manuscript symbols. It combines a sine, the sine of a complementary angle and two side lengths, while the half-chord table supplies numbers for such computations: theoretical relation and numerical table perform different functions ^{18 19}.

A modern demonstration, not a manuscript problem, takes45degrees: its sine equals its complementary sine, so the compared sides are equal. For30degrees, the complement is60and those sines differ; being right-angled alone does not make the two sides equal. Writing both sides with sines scaled to radius60introduces the same factor on each side, which cancels, leaving the side ratio unchanged ^{18 20}. These calculations explain the reported relation without claiming the author used these particular examples.

Interpolation can likewise be shown with an explicitly invented classroom table, not Zij values: suppose neighboring rows separated by30minutes have values20and22. A position6minutes above the lower row is one fifth of the interval, so one fifth of the numerical increase2is added to20, giving20.4. In reverse, the excess0.4divided by the row difference2returns one fifth of the angular interval, or6minutes ¹⁶. Success in reversing the arithmetic does not prove the actual curve straight: it explains the proportional step between points, not exactness of the approximation ²¹.

TDV identifies other functions within the Zij: chapters33–48concern table use, chapter32calendar conversion, and chapter28the lengths of the four seasons based on observation ¹⁰. Not every tabular value is therefore a sine, nor every difference between dates an angle: the function table is part of work joining timekeeping, observation and geometry ¹⁰.

The encyclopedia also attributes ideas based on orthographic projection and the spherical cosine theorem to discussions in chapters5–26 ¹⁰. This secondary account is distinguished from the directly inspected half-chord chapter: a spherical triangle and projection are not the same object as a plane right triangle, and shared functions do not imply that every proof was read in chapter3 ^{10 19}.

The chapter first constructs known chords, including the60-degree chord equal to the radius, using squared-side relationships in right triangles and chords whose arcs complete a semicircle ⁷. It then turns to half-chords of doubled arcs, tabulated every half-degree through90 degrees ⁵. The construction includes an approximation for the one-degree chord, so not every stage should be described as exact without approximation ²².

In modern explanatory notation—not copied manuscript symbols—S(theta) = 60 sin(theta); the entries for30 and90 degrees are therefore30 and60, not0.5 and1 until divided by60 ²³. A full chord is recovered by doubling the sine of half its angle, not by doubling the sine of that same full angle ²⁴.

For an angle between rows, take the difference between adjacent table values, multiply by the extra arcminutes above the lower row, divide by30 arcminutes, and add the result to the lower value ²⁵. This is linear interpolation: a teaching example derived from it places30 degrees15 minutes halfway between30 degrees and30 degrees30 minutes, taking half the tabular increase without inventing historical digits for the two entries ²⁶.

The reverse procedure locates the lower table value, converts the excess length into arcminutes proportionally to the adjacent-row difference, and adds them to the starting arc ²⁴. Nallino calls this “arcuation”; his footnote explains choosing Latin sine/cosine terminology to render the half-chord expressions. The translator's Latin words should not be mistaken for literal Arabic phrasing ²⁷.

The Zij reached Latin and Castilian readers, and the biographies discuss the book's reception in later astronomy ^{10 28}. Transmission of a work containing computation and tables does not alone prove that each chapter-three step reached every reader unchanged; this record therefore draws no precisely dated worldwide itinerary for the procedure alone ²⁸.

The explanation's immediate foundation is chord geometry, the diameter120 convention, and procedures constructing arcs and their corresponding lengths; Ptolemy's name is retained where the author explicitly acknowledges his convention ²⁹. Moving to half-chords of doubled arcs changes the input used in table work, but does not erase the preceding geometry or make the circle a new invention ²².

In the following chapter four, sine and inverse-arc computation serve a declination problem at the same radius: an identifiable application inside the work, rather than a merely modern resemblance ³⁰. The spherical sine law is not silently bundled into this result; a table and its use are not every theorem attributed to the Zij's author and require their own investigation ³¹.

The immediate benefit concerns the calculator's work: treating an unlisted angle and solving the reverse problem without reconstructing the geometry from scratch ³². The inspected pages provide no census of market or navigation users, so a computational benefit does not become an undocumented story about every household or ship ²⁶.

MacTutor describes al-Battani's trigonometric methods as an advance in astronomical calculation; the inspected chapter exposes some of the work behind that broad statement ^{33 2}. Its value lies in connecting length construction, organized values and instructions for finding a quantity without its own printed row—not declaring one father of all modern digital computation ³.

The narrower answerable question is who explains this system in this Zij: al-Battani, with an explicit debt to older geometry ³. The first sine, first table and first interpolation across every civilization are different questions, not settled by these pages or by a scholar's name on a book ².

Readers can compare the historical radius60 scale with modern unit normalization and understand interpolation as an estimate between entries, not an exact replacement for the underlying function at every input ²⁶. The image is a modern right-triangle teaching diagram released by TheOtherJesse to the public domain in2007, not al-Battani's table; its side names are relative to a particular angle ³⁴.

The inspected textual witness is Nallino's edition: Arabic printed13–17 corresponds to scans1139 down to1135, and Latin printed9–11 to scans98–100; the Arabic part is reverse-ordered in this bound file ¹. This is reading specified pages of a published edition, not physically examining the manuscript or claiming to have read all the book's tables ³⁵.

The biography of “al-Battani” or “Albaten(i)us” belongs to the transmission of an Arabic work into other languages, but variant names among readers do not justify giving him credit for every later method ²⁸. The museum distinguishes the old text, a translator's Latin terminology, and a modern teaching image accompanying it ^{36 37}.

Imagine a table that cannot print every possible angle: what happens when yours falls between two rows? Al-Battani explains estimating a share of the difference, then reversing from length to angle; the lesson is a way of reading, not only memorizing a long string of numbers ³². Start with the halfway idea, remembering that it is an approximation, not a life-size drawing of the sky ²⁶.

• Nallino's edition/translation: chapter three Arabic13–17/scans1139–1135, Latin9–11/scans98–100; beginning of chapter four Latin12; Arabic opening6/scan1146 and Latin identity argumentXVI/scan25 ³⁵.
• Ferruh Müftüoğlu, “BETTÂNÎ,” TDV,6 (1992),9–10 ³⁸.
• O'Connor and Robertson, MacTutor, November1999; conflicting translation dates and name-only religious inference not adopted ³⁹.
• TheOtherJesse's original archived2007 triangle and author-release metadata, not the revised2024 rendering ⁴⁰.