I will now present these corollaries and the subsequent proofs given by Ptolemy. With this theorem, Ptolemy produced three corollaries from which more chord lengths could be calculated: the chord of the difference of two arcs, the chord of half of an arc, and the chord of the sum of two arcs. In a cyclic quadrilateral the product of the diagonals is equal to the sum of the products of the pairs of opposite sides. Table 1: Chords of the special angles angle For the remaining chords we need to create new mathematical tools. The chords of the special angles are summarized in Table 1 below. Given these angles, Ptolemy then showed how it was possible to derive other chord lengths using the fact that the inscribed angle that subtends the diameter of a circle is 90°.
Likewise since the square of the side of an inscribed square is twice the square of the radius and the square of the side of an inscribed equilateral triangle is three times the square of the radius, we get Using these results, Ptolemy then calculated the chord lengths for the central angles. He determined the first three of these chords using the figure below with the following proof 3. Ptolemy began his discourse by calculating the chord lengths for the central angles corresponding to the sides of a regular inscribed decagon, hexagon, pentagon, square, and triangle. (where crd θ is the length of the chord described by the central angle subtending an arc of θ parts of the circumference), the Table of Chords as compiled by Ptolemy is equivalent to a table of sines for every angle up to 90° in quarter degree intervals. Given, in the diagram to the right that sin Given a circle whose diameter and circumference are divided into 120 and 360 parts respectively, Ptolemy was able to calculate the corresponding chord length for every central angle up to 180° in half-degree intervals. Equivalence of the Table of Chords and a table of sines In this paper I will describe the geometric theorems used in the construction of this table and attempt to relate them to their contemporary trigonometric counterparts. 140 BC) it was included in Ptolemy's definitive Syntaxis Mathematica, better known by its Arabic name Almagest 2. Based largely on an earlier work by Hipparchus (ca. Table 3: A comparison of the Table of Chords with a ten-place calculatorĪlthough certainly not the first trigonometric table 1, Ptolemy's On the Size of Chords Inscribed in a Circle (2nd Century AD) is by far the most famous.Corollary 3: Chord of the sum of two arcs.Corollary 1: Chord of the difference of two arcs.Equivalence of the Table of Chords and a table of sines.Ptolemy's Table of Chords: Trigonometry in the Second Century Contents
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