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Ptolemy's Table of Chords: Trigonometry in the Second Century

Contents

28 June 1994

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

sin  θ  =  AM  =  2AM  =  AB  =  crd θ
2 OA 2OA diameter 120°

(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.

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
   
DE2 + DB2 =  BE2
30°2 + 60°2 =  BE2
BE =  67° 4' 55"
BE =  EF
DF =  EF − DE
DF =  67° 4' 55" − 30°
DF =  37° 4' 55"

thus crd 36° = 37° 4' 55"

BF is the side of a pentagon
   
DF2 + DB2 =  BF2
(37° 4' 55")2 + 60°2 =  BF2
BF =  70° 32' 3"

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
 
DC =  AC  =  120°  = 60°
2 2

thus crd 60° = 60°

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

crd 090° = √(2 × 60°) = 084° 51' 10"
crd 120° = √(3 × 60°) = 103° 55' 23"

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,

crd 108° = √(120°2 − crd2 72°) = 097° 4' 56"0
crd 144° = √(120°2 − crd2 36°) = 114° 7' 37"4

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

Table 1: Chords of the special angles
angle crd
036° 037° 04' 55"
060° 060°
072° 070° 32' 03"
090° 084° 51' 10"
108° 097° 04' 56"
120° 103° 55' 23"
144° 114° 07' 37"
180° 120°

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

crd 12° = crd(72°− 60°) = 12° 32' 36"
crd 06° = crd(18°− 12°) = 06° 16' 50"
and so on…

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…

crd 3° = crd(½ × 6°) = 3° 8' 28"
crd 1½° = crd(½ × 3°) = 1° 34' 15"
crd ¾° = crd(½ × 1½°) = 0° 47' 8"
and so on…

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…

crd 19½° = crd(18° + 1½) = 20°19'20"
crd 21° = crd(18° + 3°) = 21°52'6"
crd 22½° = crd(21 + 1½°) = 23°24'40"
and so on…

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
00.5 00 31 25 0 1 2 50   12.0 12 32 36 0 1 2 28
01.0 01 02 50 0 1 2 50   12.5 13 03 50 0 1 2 27
01.5 01 34 15 0 1 2 50   13.0 13 35 04 0 1 2 25
02.0 02 05 40 0 1 2 50   13.5 14 06 16 0 1 2 23
02.5 02 37 04 0 1 2 48   14.0 14 37 27 0 1 2 21
03.0 03 08 28 0 1 2 48   14.5 15 08 38 0 1 2 19
03.5 03 38 52 0 1 2 48   15.0 15 39 47 0 1 2 17
04.0 04 11 16 0 1 2 48   15.5 16 10 56 0 1 2 15
04.5 04 42 40 0 1 2 47   16.0 16 42 03 0 1 2 13
05.0 05 14 04 0 1 2 47   16.5 17 13 09 0 1 2 10
05.5 05 45 27 0 1 2 46   17.0 17 44 14 0 1 2 07
06.0 06 16 49 0 1 2 45   17.5 18 15 17 0 1 2 05
06.5 06 48 11 0 1 2 43   18.0 18 46 19 0 1 2 02
07.0 07 19 33 0 1 2 42   18.5 19 17 21 0 1 2 00
07.5 07 50 54 0 1 2 41   19.0 19 48 21 0 1 1 57
08.0 08 22 15 0 1 2 40   19.5 20 19 19 0 1 1 54
08.5 08 53 35 0 1 2 39   20.0 20 50 16 0 1 1 51
09.0 09 24 51 0 1 2 38   20.5 21 21 11 0 1 1 48
09.5 09 56 13 0 1 2 37   21.0 21 52 06 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 05 0 1 2 32   22.5 23 24 39 0 1 1 36
11.5 12 01 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
θ crd θ (crd θ)/120° sin (θ/2)
016.5° 017° 13' 09" 0.1434930556 0.1434926220 0.0000004336
049.0° 049° 45' 48" 0.4146944444 0.4146932427 0.0000012017
064.0° 063° 35' 25" 0.5299189815 0.5299192642 0.0000002827
083.5° 079° 54' 21" 0.6658819444 0.6658816660 0.0000002784
110.5° 098° 35' 32" 0.8216018519 0.8216469379 0.0000450860
126.0° 106° 55' 15" 0.8910069444 0.8910065242 0.0000004202
155.0° 117° 09' 20" 0.9762962963 0.9762960071 0.0000002892
176.5° 119° 56' 39" 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

Updates go here.

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 1/3,600th of the unit. Thus 0° 36' 39" represents 0 + 36/60 + 39/3,600 = 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 + 2/3,600 + 50/216,000 = 0.0174537037.

Sources