Mathematics

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

Contents

• Introduction
  • Equivalence of the Table of Chords and a table of sines
• 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
  • Approximation of small chords
  • Sixtieths
• The Table of Chords
  • Table 2: Page 1 from the Table of Chords
  • Table 3: A comparison of the Table of Chords with a ten-place calculator
• Conclusion
• Postscripts
  • How accurate is the Table of Chords?
• Footnotes
• Sources

28 June 1994

Introduction

Although certainly not the first trigonometric table¹, Ptolemy's On the Size of Chords Inscribed in a Circle (2nd century CE) is by far the most famous. Based largely on an earlier work by Hipparchus (ca. 140 BC) it was included in Ptolemy's definitive Mathematike Syntaxis, 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 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.

Given

circle ABC with center D
BD ⟂ ADC
DE = EC and EF = BE

Prove

CD is the side of a regular inscribed hexagon
DF is the side of a regular inscribed decagon
BF is the side of a regular inscribed pentagon

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

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°²) = 084°51'10"
crd 120° = √(3 × 60°²) = 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.

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.

Given

the inscribed quadrilateral ABCD
pick a point E such that ∠ABE = ∠DBC

Prove

AC × BD = AB × CD + AD × BC

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

Given

semicircle ABCD with diameter AD
AC and AB chords of known length
[α and β as shown but also
let AC = crd θ and AB = crd φ]

Prove

BC can be found
[find a formula for the chord of the difference of two arcs and show its equivalence to the identity for the sine of the difference of two angles]

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 6° = crd(18° − 12°) = 6°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

Given

semicircle ABCD with diameter AC
BC chord of known length
arc BC bisected at D
DF ⟂ AC
let AE = AB
[½α as shown, but also let BC = crd θ so that BD = DC = crd ½θ]

Prove

CF = ½(AC − AB)
[find a formula for the chord of half an arc and show its equivalence to the identity for the sine of half an angle]

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

Given

circle ABCDE with center F
AFD and BFE diameters
AB and BC chords of known length
[α and β as shown, but also let AB = crd θ and BC = crd φ]

Prove

AC can be found
[give a formula for the chord of the sum of two arcs and show its equivalence to the identity for the sine of the sum of two angles]

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

Given

circle ABCD
BA and BC chords of unequal length (BA < BC)
∠ABC bisected by BD
DFH ⟂ AC at F
DFH = DE = DG
[let α and β be angles on separate inscribed right triangles such that chord BC and AB are opposite angle α and β respectively, thus α > β]

Prove

BC	<	arc BC
BA		arc BA

⎡ ⎢ ⎣	and that	sin α	<	α	⎤ ⎥ ⎦
		sin β		β

Approximation of small chords

Given

circle ABC
two chords AB and AC such that AC > AB

Find

crd 1°
crd ½°

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.

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.

As the table shows, Ptolemy's results agree with modern calculator values to five or six decimal places. (See the Postscript for more on the accuracy of the Table of Chords.)

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

How accurate is the Table of Chords?

All 360 values from Ptolemy's Table of Chords were compared to their "actual" values calculated in Google Sheets.

The graph below shows that the values in the Table are generally a tiny bit larger than they should be. The root-mean-square of the error calculated this way is 0.000136°, implying that the Table is accurate to three decimal places — not the five or six I stated in the main body of the paper. Since we don't compute chords anymore and we don't use the sexagesimal convention to divide the diameter into 120° parts, this number may not be that helpful. The graph below also shows an increasing trend in the absolute error caused, no doubt, by the increase in the value of the entries as one reads down the table. As the angle gets bigger, the chords get bigger. As the chords get bigger, the error gets bigger.

Relative error may be a more useful way to test Ptolemy's work.

The graph below shows that the relative error is greatest for small angles and seems to stay nearly constant after 60°. This makes sense as the precision of Ptolemy's degree-minute-second notation is always the same — to the nearest second. A one second error is bigger relative to a small angle than a large one. The root-mean-square of the error calculated this way is 0.00000737 or 7.37 parts per million.

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 Mathematike Syntaxis (mathematical composition or compilation) by the superlative "ta megiste" (the greatest). This was subsequently transliterated by Arab scholars as al-magiste, then Almagestum by Latin speaking European scholars, then eventually Almagest by English speaking scholars.
3. In the sexagesimal notation used by Ptolemy, the degrees symbol (°) refers to a unit of measure, the minutes symbol (') to ¹/₆₀ of the unit, and the seconds symbol (") to ¹/_3,600 of the unit. Thus 0°37'5" represents 0 + ³⁷/₆₀ + ⁵/_3,600 = 0.618055556. 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

• Bunt, Lucas N.H., Jones, Philip S. and 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.
• Ptolemy, Claudius. Ptolemy's Almagest. Translated and annotated by G.J. Toomer. London: Gerald Duckworth and Co., 1984.