# Mathematics

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

### Contents

- Introduction
- Special angles
- Table 1: Chords of the special angles

- Ptolemy's theorem
- Corollary 1: Chord of the difference of two arcs
- Corollary 2: Chord of half an arc
- Corollary 3: Chord of the sum of two arcs

- Aristarchus' inequality
- The Table of Chords
- Conclusion
- Postscripts
- Footnotes
- Sources

28 June 1994

### Introduction

Although 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. 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 *Almagest*^{2}.
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 proof^{3}.

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

DE^{2} + DB^{2} = |
BE^{2} |

30°^{2} + 60°^{2} = |
BE^{2} |

BE = | 67° 4' 55" |

BE = | EF |

DF = | EF − DE |

DF = | 67° 4' 55" − 30° |

DF = | 37° 4' 55" |

thus crd 36° = 37° 4' 55"

DF^{2} + DB^{2} = |
BF^{2} |

(37° 4' 55")^{2} + 60°^{2} = |
BF^{2} |

BF = | 70° 32' 3" |

thus crd 72° = 70° 32' 3"

(This was reported as 70° 32' 4" in the *Table of Chords*.)

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} − crd^{2} 72°) = 097° 4' 56"^{0}

crd 144° = √(120°^{2} − crd^{2} 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.

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 of Chords*

^{5}

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 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

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

- Bunt, Lucas N. H., Jones, Philip S. & Bedient, Jack D.
*The Historical Roots of Elementary Mathematics*. New York, NY: Dover, 1988. - Eves, Howard.
*An Introduction to the History of Mathematics*. sixth edition. Fort Worth, TX: Harcourt Brace Jovanovich, 1990. - Eves, Howard.
*Great Moments in Mathematics (Before 1650): The Dolciani Mathematical Expositions, Number Five*. Washington, DC: The Mathematical Association of America, 1980. - Kline, Morris.
*Mathematical Thought from Ancient to Modern Times*. Oxford, UK: Oxford University Press, 1972. - Ptolemy, Claudius.
*Mathematical Composition (Almagest)*. translated from the Greek text of Heiberg by R. Catesby Taliaferro. Annapolis, MD: St. John's University, 1938.