Mathematics
Ptolemy's Table of Chords: Trigonometry in the Second Century
Contents
 Introduction
 Special angles
 Ptolemy's theorem
 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 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 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 halfdegree 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}.
 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
Statements  Reasons  

(1)  CF × DF + ED^{2} = EF^{2}  (1) 


(2)  CF × DF + ED^{2} = BE^{2}  (2)  BE = EF  
(3)  ED^{2} + DB^{2} = BE^{2}  (3)  Pythagoras' theorem  
(4) 

(4)  combine (3) with (2) and solve  
(5)  DF:DC is the golden ratio, a.k.a. the extreme and mean ratio

(5)  Ptolemy was rather unclear on how he arrived at this statement other than to cite Euclid (Euclid VI, 3). On inspection, however, if we let r = 2 then a = 1, x + a = √5, and x = √5 − 1. Thus DF:DC = x:r is indeed the golden ratio.  
∴  DC is the side of a regular inscribed hexagon  ∴  DC is a radius  
∴  DF is the side of a regular inscribed decagon  ∴  from Euclid, "the side of the hexagon and the side of a decagon which arre inscribed in the same circle… cut that line in the extreme and mean ratio (Euclid XII, 9)" (Ptolemy 20)  
∴  BF is the side of a regular inscribed pentagon  ∴  also from Euclid, "the square on the side of a pentagon is equal to the square on the side of a hexagon together with the square on the side of a decagon, all inscribed in the same circle (Euclid XII, 10)" (Ptolemy 20) 
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°^{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} − 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 

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° 
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
Statements  Reasons  

(1)  ∠ABD = ∠EBC  (1)  add ∠EBD to ∠ABE and ∠DBC  
(2)  ∠BDA = ∠BCE  (2)  since they subtend the same arc  
(3) 

(3)  △ABD and △BCE are similar  
(4)  ∠ABE = ∠DBC  (4)  given  
(5)  ∠BAE = ∠BDC  (5)  since they subtend the same arc  
(6) 

(6)  △ABE and △BCD are similar  
(7) 

(7)  crossmultiply (3) and (6) and add them  
∴  AC × BD = AB × CD + AD × BC  ∴  AE + CE = AC 
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]
Statements  Reasons  

(1)  AD is a diameter, AC and AB are known  (1)  given  
(2)  CD and BD can be found  (2)  Pythagoras' theorem  
∴  BC can be found  ∴  substitution into Ptolemy's theorem AB × CD + AD × BC = AC × BD  
(4) 

(4)  solve Ptolemy's theorem for BC and apply Pythagoras' theorem to BD and CD  
∴ 

∴  in chord notation (note θ = 2α and ϕ = 2β)  
(6) 

(6)  divide Ptolemy's theorem by AD^{2}  
(7)  sin β cos α + sin(α − β) = sin α cos β  (7)  definition of sine and cosine  
∴  sin(α − β) = sin α cos β − sin β cos α  ∴  in modern notation 
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]
Statements  Reasons  

(1)  BD = DE  (1)  △BAD ≅ △EAD by S.A.S.  
(2)  DC = DE  (2)  BD = DC since ∠BAD = ∠DAC  
(3)  △DEC is isosceles  (3)  definition of isosceles and (2)  
(4)  EF = CF  (4)  altitude to the base of an isosceles triangle bisects the base  
(5)  EC = AC − AB  (5)  AE = AB  
∴  CF = ½(AC − AB)  ∴  (4) and (5)  
(7)  △ACD and △DCF are similar  (7)  ∠ADC = ∠DFC = 90° ∠ACD = ∠FCD 

(8) 
CD^{2} = AC × CF 
(8)  similar triangles crossmultiplication 

(9)  CD^{2} = ½AC(AC − AB)  (9)  substitute (6) into (8)  
(10)  CD^{2} = ½AC[AC − √(AC^{2} − BC^{2})]  (10)  Pythagoras' theorem on chord AB  
∴  crd ½θ = √[7200° − 60°√(120°^{2} − crd^{2}θ)]  ∴  in chord notation (note θ = 2α)  
(12) 

(12)  divide (9) by AC^{2}  
∴ 

∴  in modern notation 
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]
Statements  Reasons  

(1)  AD is a diameter AB is known BD can be found from which DE can be found 
(1)  given given Pythagoras' theorem Pythagoras' theorem 

(2)  BE is a diameter BC is known CE can be found 
(2)  given given Pythagoras' theorem 

(3)  CD can be found  (3)  substitution into Ptolemy's theorem BC × DE + CD × BE = BD × CE  
∴  AC can be found  ∴  substitution into Ptolemy's theorem AB × CD + AD × BC = AC × BD  
(5)  BE = AD  (5)  both are diameters  
(6)  DE = AB  (6)  application of Pythagoras' theorem to △BDE and △ABD and (5)  
(7) 

(7)  combine Ptolemy's theorem in (3) and (4) with (5) and (6) then eliminate CD  
(8)  BD = √(AD^{2} − AB^{2}) CE = √(AD^{2} − BC^{2}) 
(8)  Pythagoras' theorem and (5)  
(9) 

(9)  substitute (8) into (7) and solve  
∴ 

∴  in chord notation (note θ = 2α and ϕ = 2β)  
(11) 

(11)  divide (7) by AD^{3}  
(12)  ∠BEC = ∠BDC = β  (12)  since they subtend the same arc  
(13) 

(13)  definition of sine and cosine Pythagoras' theorem (identity) divide both sides by cos α 

∴  ∴  in modern notation 
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 twothirds 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 wellknown 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
Statements  Reasons  

(1)  CD = AD  (1)  ∠ABD = ∠DBC  
(2)  CE > EA  (2)  (Euclid VI, 3)  
(3)  DE > DF AD > DE DH > DF 
(3)  (Euclid III, 3 and 26) DH = DE 

(4) 

(4)  area △DEF < area sector DE area △DEA > area sector DEG 

(5) 

(5)  similar triangles  
(6) 

(6)  area of a sector ∝ angle describing it both sectors are on the same circle 

(7) 

(7)  (4), (5), and (6) "componendo… doubling the antecedents… separando" (Ptolemy 29–30) 

(8) 

(8)  (Euclid VI, 3 and 33)  
∴ 

∴  (7) and (8)  
∴ 

∴ 

Approximation of small chords
 Given
 circle ABC
two chords AB and AC such that AC > AB  Find
 crd 1°
crd ½°
Statements  Reasons  

(1)  let AB = crd ^{3}/_{4}° and AC = crd 1° then arcAC = ^{4}/_{3}arcAB AC < ^{4}/_{3}AB = ^{4}/_{3}(0°47'8") thus crd 1° < 1°2'50" 
(1)  substitution into and solution of Aristarchus' inequality 
(2)  let AB = crd 1° and AC = crd 1^{1}/_{2}° then arcAC = ^{3}/_{2}arcAB AB > ^{2}/_{3}AC = ^{2}/_{3}(1°34' 15") thus crd 1° > 1°2'50" 
(2)  repeat (1) with different values 
∴  crd 1° ≈ 1°2'50"  ∴  (1) and (2) 
∴  crd ½° ≈ 0°31'25"  ∴  corollary 2 
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.
Arcs  Chords  Sixtieths  Arcs  Chords  Sixtieths  

00½  00 31 25  0 1 2 50  12½  12 32 36  0 1 2 28  
01½  01 02 50  0 1 2 50  12½  13 03 50  0 1 2 27  
01½  01 34 15  0 1 2 50  13½  13 35 04  0 1 2 25  
02½  02 05 40  0 1 2 50  13½  14 06 16  0 1 2 23  
02½  02 37 04  0 1 2 48  14½  14 37 27  0 1 2 21  
03½  03 08 28  0 1 2 48  14½  15 08 38  0 1 2 19  
03½  03 38 52  0 1 2 48  15½  15 39 47  0 1 2 17  
04½  04 11 16  0 1 2 48  15½  16 10 56  0 1 2 15  
04½  04 42 40  0 1 2 47  16½  16 42 03  0 1 2 13  
05½  05 14 04  0 1 2 47  16½  17 13 09  0 1 2 10  
05½  05 45 27  0 1 2 46  17½  17 44 14  0 1 2 07  
06½  06 16 49  0 1 2 45  17½  18 15 17  0 1 2 05  
06½  06 48 11  0 1 2 43  18½  18 46 19  0 1 2 02  
07½  07 19 33  0 1 2 42  18½  19 17 21  0 1 2 00  
07½  07 50 54  0 1 2 41  19½  19 48 21  0 1 1 57  
08½  08 22 15  0 1 2 40  19½  20 19 19  0 1 1 54  
08½  08 53 35  0 1 2 39  20½  20 50 16  0 1 1 51  
09½  09 24 51  0 1 2 38  20½  21 21 11  0 1 1 48  
09½  09 56 13  0 1 2 37  21½  21 52 06  0 1 1 45  
10½  10 27 32  0 1 2 35  21½  22 22 58  0 1 1 42  
10½  10 58 49  0 1 2 33  22½  22 53 49  0 1 1 39  
11½  11 30 05  0 1 2 32  22½  23 24 39  0 1 1 36  
11½  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.
θ  crd θ  (crd θ)/120°  sin (θ/2)  ∆ 

16½°  17°13'9"  0.1434930556  0.1434926220  0.0000004336 
49°  49°45'48"  0.4146944444  0.4146932427  0.0000012017 
64°  63°35'25"  0.5299189815  0.5299192642  0.0000002827 
83½°  79°54'21"  0.6658819444  0.6658816660  0.0000002784 
110½°  98°35'32"  0.8216018519  0.8216469379  0.0000450860 
126°  106°55'15"  0.8910069444  0.8910065242  0.0000004202 
155°  117°9'20"  0.9762962963  0.9762960071  0.0000002892 
176½°  119°56'39"  0.9995347222  0.9995335908  0.0000011314 
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 12book treatise on the construction of a Table of Chords disappeared sometime after the fourthcentury because it was superseded by the far more comprehensive Almagest. The Almagest reigned supreme as the treatise in practical trigonometry for approximately onethousand years. During the tenthcentury, the Islamic mathematician Abû'lWefâ computed the values for the sines and tangents of an angle in quarterdegree intervals and essentially reproduced the Table of Chords in contemporary form. In the sixteenthcentury, 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.
absolute error  =  Ptolemy's value − 
The graph below shows that the values in the Table are generally a tiny bit larger than they should be. The rootmeansquare 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.
relative error  =  Ptolemy's value − 
spreadsheet value 
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 degreeminutesecond 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 rootmeansquare of the error calculated this way is 0.00000737 or 7.37 parts per million.
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 Mathematike Syntaxis (mathematical composition or compilation) by the superlative "ta megiste" (the greatest). This was subsequently transliterated by Arab scholars as almagiste, then Almagestum by Latin speaking European scholars, then eventually Almagest by English speaking scholars.
 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.
 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. 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.