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date: 06 December 2022

trigonometryfree

trigonometryfree

  • Nathan Camillo Sidoli

Subjects

  • Science, Technology, and Medicine

Updated in this version

Article and bibliography updated to reflect current scholarship.

The development of trigonometry as a branch of mathematics was a combined effort of mathematical scholars working in a number of different languages and cultures, over many centuries. The first texts containing trigonometric computations are found in Greek sources, although these do not contain the trigonometric functions we now use. The introduction of the trigonometric functions is found in Sanskrit sources, and scholars working in Arabic composed the first works devoted entirely to trigonometry, adopting and expanding on the work of their Greek and Sanskrit sources. The word trigonometry itself was a neologism of Latin scholars, whose treatises developed this field as an independent branch of mathematics, adopting and extending previous Arabic works.

Trigonometric Approaches in Antiquity

Trigonometry was not regarded as an independent branch of mathematics in the ancient period—the word itself is an early modern neologism and does not translate any ancient expression. Ancient Mesopotamian and Egyptian sources—which do not introduce angles—appear to have handled the mensuration of triangles, and slopes, through the ratios of the sides of normalized triangles. The preserved texts of these cultures contain some tables that might be regarded as trigonometric, but computations that are clearly trigonometric have not yet been found in these texts.

The earliest known trigonometric computations are transmitted in Greek sources. Although the surviving Greek mathematical sources are not sufficient for writing a complete history of the development of trigonometric methods, it is clear that Greek mathematicians were able to produce a general computational trigonometry based on the use of a chord table, which could theoretically be justified by purely geometrical theorems in Euclid's Data. There are three stages in the development of computational trigonometry in Greco-Roman authors: (a) a proto-trigonometry based on ratio inequalities relating sides and angles in a pair of right triangles, (b) the production of chord tables, and (c) the application of chord-table trigonometry, along with general arguments based in the methods of the Data, to the solution of plane and spherical problems.

Trigonometric Concepts in the Data

A number of theorems in Euclid's Data establish the theoretical possibility of trigonometry. Since the Data was composed in the early Hellenistic period, based on mathematical methods developed in the late Hellenic period, the entire history of trigonometry is understood as having developed within a context in which mathematicians were familiar with these theorems in the Data. For example, Data 40 shows that if the angles of a triangle are all given, or fixed, then the ratios of the sides are also fixed; Data 43 shows that if the ratio of a leg and the hypotenuse of a right triangle is given, or determined, then the angles of the triangle are also determined.1Data 87 and 88 show that, in a circle of a certain size, chords and the angles they subtend inversely determine one another.2 Finally, Data 93 can be used to show that if the ratios of the side of an equilateral triangle, square, and pentagon to the diameter of a circle in which they are inscribed are given, then the chords of angles at 1 1/2º intervals are determined, but not how they are calculated.

While all this material is conceptually relevant to the development of trigonometry, the mathematical methods of the Data are rather far removed from computational trigonometric practice. In fact, the proofs of the Data all rely on the purely constructive problem-solving methods of Euclid's Elements and provide little or no insight into computational methods. Nevertheless, Ptolemy's fully developed chord-table trigonometry employs a style of analytical reasoning that appears to have been justified by Data 40, 43, 87, and 88.3

Proto-Trigonometry

In the early Hellenistic period, mathematicians such as Aristarchus and Archimedes used ratio inequalities relating a pair of legs and a pair angles from two right triangles under the same height to establish computational methods for producing upper and lower bounds for angles and for ratios of lengths in right triangles.4 Although Aristarchus and Archimedes assumed these proto-trigonometric ratio inequalities as well-known lemmas, proofs of their validity are found in later texts and scholia.5 Using these lemmas, mathematicians were able to produce fairly decent upper and lower bounds on the values of sought angles or lengths. These methods, however, had two drawbacks: (a) the inequalities are only sufficiently precise for small angles, so that if large angles were involved this approach would produce unacceptably inaccurate results; and perhaps more importantly, (b) the computational procedures involved are quite cumbersome. Hence, although mathematicians were able to use these methods to effect the arduous calculation of a few significant results, they should not be regarded as a general trigonometry.

The Development of Chord-Tables

This formative period in the history of trigonometry has left the least evidence the least evidence in surviving sources. Nevertheless, the circumstantial evidence indicates that Greek mathematicians of the late Hellenistic period were inspired to introduce the use of tables into their computational practices through their growing familiarity with Babylonian mathematical astronomy and its promise of a numerically predictive astronomy.6 Although we cannot be certain who introduced the first chord-table, or for what end, it has often been argued that this was done by Hipparchus. Indeed, some scholars believe the table that Hipparchus used can be stated in full detail, while others are skeptical of this claim to certainty.7

The argument for Hipparchus' chord table is indirect. It involves attempting to explain the specific structure of numbers attributed in Alm. 4.11 by Ptolemy to Hipparchus for the parameters of the latter's solar model by means of a chord table that is similar to a sine table constructed on a radius of 3438p, which is found in certain Indian sources, such as the Āryabhaṭīya. The proposed table would have involved 24 rows, tabulated at 7 1/2º intervals set into one-to-one correspondence with chords of 450p to 6975p.

Although it is not certain that Hipparchus' table had this particular structure, there are reasons to believe that it did. Nevertheless, whatever the actual structure of the table, it is clear that Hipparchus had a chord table and that chord-table methods were used by him and other mathematicians before Ptolemy, such as Diodorus and Menelaus. These mathematicians were able to address computational problems in both plane and spherical trigonometry using the chord function, which is mathematically equivalent to the sine function, since Crd(2α‎) = 2 sin α‎.

Plane and Spherical Chord-Table Trigonometry

The best evidence for the fully developed chord-table trigonometry of Greek practitioners comes from the mathematical writings of Ptolemy, such as the Planisphere, the Analemma, and particularly the Almagest, as well as the late ancient commentators on these works. All of these texts contain an intermixture of chord-trigonometric computation and analytical arguments by givens that can be justified by theorems of the Data.

The only development of a chord table that we have in ancient sources is that provided by Ptolemy in Alm. 1.1.8 Ptolemy uses theorems, based on propositions from Euclid's Elements, analytical arguments, based on theorems of Euclid's Data, and computations using sexagesimal fractions to argue as follows: (a) if the diameter of a circle is assumed as given, as say 120p, then the sides of an equilateral triangle, square and pentagon are computable; (b) if Crd(α‎) is given, then Crd(α‎/2) is given and computable; and (c) if Crd(α‎) and Crd(β‎) are given, then Crd(α ± β‎) are given and computable. On this basis, he shows that the chords subtending the angles at intervals of 1 1/2º are given and computable. He then uses one of the proto-trigonometric lemmas to compute upper and lower bounds on Crd(1º), so that he can fill out a table listing arcs and chords in one-to-one correspondence at 1/2º intervals from 1/2º to 180º—stating the chords in sexagesimal fractions, along with a factor corresponding to the mean increment to the chord for 1/60th of 1º, which can be used for interpolation.

Ptolemy does not give a detailed explanation of the use of his chord-table, and how it was meant to be used can be understood only by following the details of the computations he provides in his Almagest.9 By assuming that the hypotenuse of a right triangle is 120p, Ptolemy is able to use the chord table to compute the angles of a right triangle in which the ratios of the sides are known, or to compute the ratios of the sides of a right triangle in which the angles are known. Although these procedures are sometimes laborious, they are sufficient for solving problems equivalent to those amenable to the techniques of plane trigonometry that are learned in school. As well as performing computations, which are often lengthy, Ptolemy uses analytical reasoning, ostensibly founded on the theorems of the Data discussed above, to show that if certain values are assumed as given, then the sought values can be shown to be given and computable. These analytical passages function as demonstrations of computability—in some ways similar to our use of general symbolic equations.10

Although spherical trigonometry was developed to a high degree by Ptolemy's predecessor Menelaus, this approach did not take hold in the ancient period. In the later sections of his Spherics—which only survives in medieval translations—Menelaus showed how a theorem that asserts a proportion as a compound ratio for the chords of the double arcs of a convex quadrilateral—known as the sector theorem, or Menelaus' theorem —can be used to develop a spherical trigonometry based on the spherical triangle as the fundamental element, which can elegantly solve many problems in spherical astronomy.11 Menelaus' text, however, only provides general theorems that will be of use in spherical trigonometry and does not give any actual computation.

When Ptolemy presented his approach to spherical astronomy in Alm. 2, 8.5-6, for whatever reason, he chose not to follow Menelaus' approach through spherical triangles and instead based everything on the sector theorem, which Menelaus had also used as the foundation of his theory.12 Although Ptolemy's approach strikes us—as it also struck many mathematicians of the classical Islamic period—as cumbersome, it has the advantage of requiring only that we learn and apply a single lemma—namely the sector theorem. The sector theorem asserts that a ratio of the chords of double arcs is equal to a compound ratio of the same, so that there are six terms involving the chords of double arcs. In this way, if five of these arcs are known, one can use the chord table to compute their chords, operate on the compound ratio to produce the final chord, and use the chord table again to convert back to find the sought arc. Because all of the chords in Ptolemy's spherical trigonometry can be asserted in terms of an assumed diameter of the sphere, it is computationally simpler than his plane trigonometry. In practice, Ptolemy chooses his terms such that a majority of them are quadrants, which is key to finding a sector figure in which five of the terms are known. As in his plane trigonometry, Ptolemy sometimes introduces into his trigonometric presentation short analytical arguments—which act as general claims about computability and are justifiable by theorems in the Data—to argue that if certain values are given then the sought values must also be given and computable.

Bibliography

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Notes