Ptolemy's Table of Chords
Trigonometry in the Second Century

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© 1992-2008 by Glenn Elert
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28 June 1994


Table of Contents


Introduction

Although certainly not the first trigonometric table1, Ptolemy's On the Size of Chords Inscribed in a Circle (2nd Century AD) is by far the most famous. Based largely on an earlier work by Hipparchus (ca. 140 BC) it was included in Ptolemy's definitive Syntaxis Mathematica, better known by its Arabic name Almagest2. 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.

Equivalence of the Table of Chords and a table of sines

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. Given, in the diagram to the right that

(where crd theta is the length of the chord described by the central angle subtending an arc of theta 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.

Special Angles

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. He determined the first three of these chords using the figure below with the following proof3.

 

Using these results, Ptolemy then calculated the chord lengths for the central angles.

DF is the side of a decagon

thus crd 36° = 37°4'55"
 
BF is the side of a pentagon

thus crd 72° = 70°32'3"
(this was reported as 70°32'4" in the Table of Chords)
DC is the side of a hexagon

thus crd 60° = 60°
Table 1
Chords of the special angles
 angle   crd 
36°  37°  4'  55" 
60°  60°     
72°  70°  32'  3" 
90°  84°  51'  10" 
108°  97°  4'  56" 
120°  103°  55'  23" 
144°  114°  7'  37" 
 180°   120°   000°   000° 

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

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°. Therefore, by application of Pythagoras theorem,

4

The chords of the special angles are summarized in Table 1 to the right. For the remaining chords we need to create new mathematical tools.

Ptolemy's Theorem

In a cyclic quadrilateral the product of the diagonals is equal to the sum of the products of the pairs of opposite sides.

 

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. I will now present these corollaries and the subsequent proofs given by Ptolemy. I will also derive a formula from each corollary that can be used to calculate the additional chords. (Ptolemy did not supply any formulae.) Furthermore, I will show that the three corollaries are equivalent to the trigonometric identities for the sine of the difference of two angles, the sine of half an angle, and the sine of the sum of two angles respectively.

Corollary 1: Chord of the difference of two arcs

 

By successive application of this theorem to the chords summarized in Table 1, it is possible to calculate all the chord lengths for the angles between 6° and 180° in 6° intervals. Thus

These values are within 1" of those found in the Table of Chords. When there is a discrepancy, it is usually due to rounding errors. It appears that either Ptolemy's computers (persons hired to do the menial calculations) did not carry their work out beyond the seconds place or they did not believe in rounding up ever. This was true for many of the values I calculated.

Corollary 2: Chord of half an arc

 

This theorem makes it possible to calculate chords in ever smaller increments. Thus

Corollary 3: Chord of the sum of two arcs

 

By successive application of this theorem to the chords found with the first two corollaries it is possible to calculate all the chord lengths for the angles between 0° and 180° in 1½° increments. Thus

Again these values are within 1" of those calculated by Ptolemy.

With things as they stand now, we still cannot calculate the chords for two-thirds of the values in our intended table. However, if we knew the values of crd ½° and crd 1° we could then apply corollary 3 repeatedly to the chords already known and finish the table. If the trisection of an angle were geometrically possible, we could use crd 1½° to find crd ½° algebraically and then apply corollary 2 to find crd 1°. Given the well-known impossibility of this trisection, Ptolemy decided instead to approximate the value of crd 1° by means of "a little lemma which, even if it may not suffice for determining chords in general, can yet in the case of very small ones, keep them indistinguishable from chords rigorously determined" (Ptolemy 28). This lemma, attributed to Aristarchus, appears with its proof below.

Aristarchus' Inequality

 

Approximation of small chords

 

Sixtieths

Ptolemy carried his work out further by dividing the interval between successive chords into thirtieths. This effectively allows for the calculation of any chord between 0° and 180° in one second intervals. While not rigorously produced, the values of the sixtieths are, in Ptolemy's words, "accurate as far as the sense are concerned" (Ptolemy 32).

The Table of Chords

A section of the Table of Chords is shown in Table 2 below.

Table 2
Page 1 from the Table of Chords5
 Arcs       Chords        Sixtieths | Arcs       Chords        Sixtieths
  0.5      0  31  25     0  1  2  50 | 12.0     12  32  36     0  1  2  28 
  1.0      1   2  50     0  1  2  50 | 12.5     13   3  50     0  1  2  27 
  1.5      1  34  15     0  1  2  50 | 13.0     13  35   4     0  1  2  25 
  2.0      2   5  40     0  1  2  50 | 13.5     14   6  16     0  1  2  23 
  2.5      2  37   4     0  1  2  48 | 14.0     14  37  27     0  1  2  21 
  3.0      3   8  28     0  1  2  48 | 14.5     15   8  38     0  1  2  19 
  3.5      3  38  52     0  1  2  48 | 15.0     15  39  47     0  1  2  17 
  4.0      4  11  16     0  1  2  48 | 15.5     16  10  56     0  1  2  15 
  4.5      4  42  40     0  1  2  47 | 16.0     16  42   3     0  1  2  13 
  5.0      5  14   4     0  1  2  47 | 16.5     17  13   9     0  1  2  10 
  5.5      5  45  27     0  1  2  46 | 17.0     17  44  14     0  1  2   7 
  6.0      6  16  49     0  1  2  45 | 17.5     18  15  17     0  1  2   5 
  6.5      6  48  11     0  1  2  43 | 18.0     18  46  19     0  1  2   2 
  7.0      7  19  33     0  1  2  42 | 18.5     19  17  21     0  1  2   0 
  7.5      7  50  54     0  1  2  41 | 19.0     19  48  21     0  1  1  57 
  8.0      8  22  15     0  1  2  40 | 19.5     20  19  19     0  1  1  54 
  8.5      8  53  35     0  1  2  39 | 20.0     20  50  16     0  1  1  51 
  9.0      9  24  51     0  1  2  38 | 20.5     21  21  11     0  1  1  48 
  9.5      9  56  13     0  1  2  37 | 21.0     21  52   6     0  1  1  45 
 10.0     10  27  32     0  1  2  35 | 21.5     22  22  58     0  1  1  42 
 10.5     10  58  49     0  1  2  33 | 22.0     22  53  49     0  1  1  39 
 11.0     11  30   5     0  1  2  32 | 22.5     23  24  39     0  1  1  36 
 11.5     12   1  21     0  1  2  30 | 

A random sample of sines produced from the Table of Chords were compared with those generated by a pocket calculator accurate to ten places. The results are summarized in Table 3 below.

Table 3
A comparison of the Table of Chords with a ten-place calculator
theta crd theta (crd theta)/120° sin (theta/2) difference
16.5°  17°  13 ' 9"  0.1434930556 0.1434926220 0.0000004336
49.0°  49°  45'  48"  0.4146944444 0.4146932427 0.0000012017
64.0°  63°  35'  25"  0.5299189815 0.5299192642 0.0000002827
83.5°  79°  54'  21"  0.6658819444 0.6658816660 0.0000002784
110.5°  98°  35'  32"  0.8216018519 0.8216469379 0.0000450860
126.0°  106°  55'  15"  0.8910069444 0.8910065242 0.0000004202
155.0°  117°  9'  20"  0.9762962963 0.9762960071 0.0000002892
 176.5°   119°   056'   039"   0.9995347222   0.9995335908   0.0000011314 

As the table shows, Ptolemy's results agree with the "exact" values to five or six decimal places. The unusually high deviation of crd 110½° is probably due to a typographical error.

The remainder of the Almagest consists of astronomical calculations: the position of the sun, moon, and planets at various times relative to the fixed stars. The Table of Chords played an important role in their compilation.

Conclusion

Hipparchus' earlier 12-book treatise on the construction of a table of chords disappeared sometime after the fourth-century because it was superseded by the far more comprehensive Almagest. The Almagest reigned supreme as the treatise in practical trigonometry for approximately one-thousand years. During the tenth-century, the Islamic mathematician Abû'l-Wefâ computed the values for the sines and tangents of an angle in quarter-degree intervals and essentially reproduced the Table of Chords in contemporary form. In the sixteenth-century, the Teutonic mathematician George Joachim Rhaeticus had, over the course of twelve years and with the help of hired computers, calculated the values of all six trigonometric functions to ten places and the sine function to fifteen places in ten second intervals. With the ubiquity of programmable calculators and personal computers, computational ability has advanced to the point where it is within the economic means of large segments of the earth's population to reproduce the life work of the ancients on demand. Technology has rendered the work of such mathematicians superfluous in much the same way the Almagest obliterated all twelve volumes of Hipparchus.


Postscripts

External Links


Footnotes

  1. The earliest reputed trigonometric table, fifteen secants from 30° to 45°, can be found in the famous Babylonian tablet, Plimpton 322 (ca. 1900-1600 BC).

  2. Commentators identified the Syntaxis Mathematica (mathematical composition or compilation) by the superlative "magiste" (greatest). This was subsequently transliterated by the Arabs as al-Magiste from whence came the name Almagest.

  3. In the sexagesimal notation used by Ptolemy, the degrees symbol (°) refers to a unit of measure, the minutes symbol (') to one-sixtieth of the unit, and the seconds symbol (") to 1/3600th of the unit. Thus 0°36'39" represents 0 + 36/60 + 39/3600 = 0.6108333333. The notation applies equally to the lengths of arcs (angular measure) and line segments (linear measure). Only the angular usage of this notation has survived to the present.

  4. For some reason, crd 144° was reported as 114°7'47" in the Table.

  5. The sexagesimal system is carried out one place further in the sixtieths column. Thus 0°1'2"50''' represents 0 + 1/60 + 2/3600 + 50/216,000 = 0.0174537037.

Sources


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