Ptolemy's Table Of Chords
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The table of chords, created by the
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astronomer, geometer, and geographer
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
in
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during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's ''Almagest'', a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy (an earlier table of chords by
Hipparchus Hipparchus (; , ;  BC) was a Ancient Greek astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hippar ...
gave chords only for arcs that were multiples of ). Since the 8th and 9th centuries, the sine and other trigonometric functions have been used in Islamic mathematics and astronomy, reforming the production of sine tables. Khwarizmi and Habash al-Hasib later produced a set of trigonometric tables.


The chord function and the table

A chord of a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
is a line segment whose endpoints are on the circle. Ptolemy used a circle whose diameter is 120 parts. He tabulated the length of a chord whose endpoints are separated by an arc of ''n'' degrees, for ''n'' ranging from to 180 by increments of . In modern notation, the length of the chord corresponding to an arc of ''θ'' degrees is : \begin & \operatorname(\theta) = 120\sin\left(\frac 2 \right) \\ = & 60 \cdot \left( 2 \sin\left(\frac \text \right) \right). \end As ''θ'' goes from 0 to 180, the chord of a ''θ''° arc goes from 0 to 120. For tiny arcs, the chord is to the arc angle in degrees as is to 3, or more precisely, the ratio can be made as close as desired to  ≈  by making ''θ'' small enough. Thus, for the arc of , the chord length is slightly more than the arc angle in degrees. As the arc increases, the ratio of the chord to the arc decreases. When the arc reaches , the chord length is exactly equal to the number of degrees in the arc, i.e. chord 60° = 60. For arcs of more than 60°, the chord is less than the arc, until an arc of 180° is reached, when the chord is only 120. The fractional parts of chord lengths were expressed in
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
(base 60) numerals. For example, where the length of a chord subtended by a 112° arc is reported to be 99,29,5, it has a length of : 99 + \frac + \frac = 99.4847\overline, rounded to the nearest . After the columns for the arc and the chord, a third column is labeled "sixtieths". For an arc of ''θ''°, the entry in the "sixtieths" column is : \frac. This is the average number of sixtieths of a unit that must be added to chord(''θ''°) each time the angle increases by one minute of arc, between the entry for ''θ''° and that for (''θ'' + )°. Thus, it is used for
linear interpolation In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known po ...
. Glowatzki and Göttsche showed that Ptolemy must have calculated chords to five sexigesimal places in order to achieve the degree of accuracy found in the "sixtieths" column.Toomer's translation of the Almaagest
1984, footnote 68, pages 57-59.
: \begin \hline \text^\circ & \text & & & \text & & \\ \hline \,\,\,\,\,\,\,\,\,\, \tfrac12 & 0 & 31 & 25 & 0 \quad 1 & 2 & 50 \\ \,\,\,\,\,\,\, 1 & 1 & 2 & 50 & 0 \quad 1 & 2 & 50 \\ \,\,\,\,\,\,\, 1\tfrac12 & 1 & 34 & 15 & 0 \quad 1 & 2 & 50 \\ \,\,\,\,\,\,\, \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 109 & 97 & 41 & 38 & 0 \quad 0 & 36 & 23 \\ 109\tfrac12 & 97 & 59 & 49 & 0 \quad 0 & 36 & 9 \\ 110 & 98 & 17 & 54 & 0 \quad 0 & 35 & 56 \\ 110\tfrac12 & 98 & 35 & 52 & 0 \quad 0 & 35 & 42\\ 111 & 98 & 53 & 43 & 0 \quad 0 & 35 & 29 \\ 111\tfrac12 & 99 & 11 & 27 & 0 \quad 0 & 35 & 15 \\ 112 & 99 & 29 & 5 & 0 \quad 0 & 35 & 1\\ 112\tfrac12 & 99 & 46 & 35 & 0 \quad 0 & 34 & 48 \\ 113 & 100 & 3 & 59 & 0 \quad 0 & 34 & 34 \\ \,\,\,\,\,\,\, \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 179 & 119 & 59 & 44 & 0 \quad 0 & 0 & 25 \\ 179\frac12 & 119 & 59 & 56 & 0 \quad 0 & 0 & 9 \\ 180 & 120 & 0 & 0 & 0 \quad 0 & 0 & 0 \\ \hline \end


How Ptolemy computed chords

Chapter 10 of Book I of the ''Almagest'' presents
geometric Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
theorems used for computing chords. Ptolemy used geometric reasoning based on Proposition 10 of Book XIII of Euclid's '' Elements'' to find the chords of 72° and 36°. That
Proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...
states that if an equilateral
pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
is inscribed in a circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is de ...
and the decagon inscribed in the same circle. He used Ptolemy's theorem on quadrilaterals inscribed in a circle to derive formulas for the chord of a half-arc, the chord of the sum of two arcs, and the chord of a difference of two arcs. The theorem states that for a
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
inscribed in a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
, the product of the lengths of the diagonals equals the sum of the products of the two pairs of lengths of opposite sides. The derivations of trigonometric identities rely on a
cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...
in which one side is a diameter of the circle. To find the chords of arcs of 1° and ° he used approximations based on Aristarchus's inequality. The inequality states that for arcs ''α'' and ''β'', if 0 < ''β'' < ''α'' < 90°, then : \frac < \frac\alpha\beta < \frac. Ptolemy showed that for arcs of 1° and °, the approximations correctly give the first two sexagesimal places after the integer part.


Accuracy

Gerald J. Toomer in his translation of the Almagest gives seven entries where some manuscripts have scribal errors, changing one "digit" (one letter, see below). Glenn Elert has made a comparison between Ptolemy's values and the true values (120 times the sine of half the angle) and has found that the
root mean square In mathematics, the root mean square (abbrev. RMS, or rms) of a set of values is the square root of the set's mean square. Given a set x_i, its RMS is denoted as either x_\mathrm or \mathrm_x. The RMS is also known as the quadratic mean (denote ...
error is 0.000136. But much of this is simply due to rounding off to the nearest 1/3600, since this equals 0.0002777... There are nevertheless many entries where the last "digit" is off by 1 (too high or too low) from the best rounded value. Ptolemy's values are often too high by 1 in the last place, and more so towards the higher angles. The largest errors are about 0.0004, which still corresponds to an error of only 1 in the last
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
digit. Elert states that "the Table is accurate to three decimal places — not the five or six I stated in the main body of the paper", but in fact the "five or six" decimal places (after the decimal point) was for \sin(\theta/2) which is 120 times smaller.


The numeral system and the appearance of the untranslated table

Lengths of arcs of the circle, in degrees, and the integer parts of chord lengths, were expressed in a
base 10 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of t ...
numeral system A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent differe ...
that used 21 of the letters of the
Greek alphabet The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC. It was derived from the earlier Phoenician alphabet, and is the earliest known alphabetic script to systematically write vowels as wel ...
with the meanings given in the following table, and a symbol, "∠′, that means and a raised circle "○" that fills a blank space (effectively representing zero). Three of the letters, labeled "archaic" in the table below, had not been in use in the Greek language for some centuries before the ''Almagest'' was written, but were still in use as numerals and
musical notes In music, notes are distinct and isolatable sounds that act as the most basic building blocks for nearly all of music. This discretization facilitates performance, comprehension, and analysis. Notes may be visually communicated by writing them in ...
. : \begin \hline \alpha & \mathrm & 1 & \iota & \mathrm & 10 & \rho & \mathrm & 100 \\ \beta & \mathrm & 2 & \kappa & \mathrm & 20 & \sigma & \mathrm & 200 \\ \gamma & \mathrm & 3 & \lambda & \mathrm & 30 & \tau & \mathrm & 300 \\ \delta & \mathrm & 4 & \mu & \mathrm & 40 & \upsilon & \mathrm & 400 \\ \varepsilon & \mathrm & 5 & \nu & \mathrm & 50 & \varphi & \mathrm & 500 \\ \stigma & \mathrm & 6 & \xi & \mathrm & 60 & \chi & \mathrm & 600 \\ \zeta & \mathrm & 7 & \omicron & \mathrm & 70 & \psi & \mathrm & 700 \\ \eta & \mathrm & 8 & \pi & \mathrm & 80 & \omega & \mathrm & 800 \\ \theta & \mathrm & 9 & \koppa & \mathrm & 90 & \sampi & \mathrm & 900\\ \hline \end Thus, for example, an arc of ° is expressed as ''ρμγ''∠′. (As the table only reaches 180°, the Greek numerals for 200 and above are not used.) The fractional parts of chord lengths required great accuracy, and were given in
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
notation in two columns in the table: The first column gives an integer multiple of , in the range 0–59, the second an integer multiple of  = , also in the range 0–59. Thus in Heiberg'
edition of the ''Almagest'' with the table of chords on pages 48–63
the beginning of the table, corresponding to arcs from to looks like this: : \begin \pi\varepsilon\rho\iota\varphi\varepsilon\rho\varepsilon\iota\tilde\omega\nu & \varepsilon\overset\upsilon\theta\varepsilon\iota\tilde\omega\nu & \overset\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\ \begin \hline \quad \angle' \\ \alpha \\ \alpha\;\angle' \\ \hline\beta \\ \beta\;\angle' \\ \gamma \\ \hline\gamma\;\angle' \\ \delta \\ \delta\;\angle' \\ \hline\varepsilon \\ \varepsilon\;\angle' \\ \stigma \\ \hline\stigma\;\angle' \\ \zeta \\ \zeta\;\angle' \\ \hline \end & \begin \hline\circ & \lambda\alpha & \kappa\varepsilon \\ \alpha & \beta & \nu \\ \alpha & \lambda\delta & \iota\varepsilon \\ \hline \beta & \varepsilon & \mu \\ \beta & \lambda\zeta & \delta \\ \gamma & \eta & \kappa\eta \\ \hline \gamma & \lambda\theta & \nu\beta \\ \delta & \iota\alpha & \iota\stigma \\ \delta & \mu\beta & \mu \\ \hline \varepsilon & \iota\delta & \delta \\ \varepsilon & \mu\varepsilon & \kappa\zeta \\ \stigma & \iota\stigma & \mu\theta \\ \hline \stigma & \mu\eta & \iota\alpha \\ \zeta & \iota\theta & \lambda\gamma \\ \zeta & \nu & \nu\delta \\ \hline \end & \begin \hline \circ & \alpha & \beta & \nu \\ \circ & \alpha & \beta & \nu \\ \circ & \alpha & \beta & \nu \\ \hline \circ & \alpha & \beta & \nu \\ \circ & \alpha & \beta & \mu\eta \\ \circ & \alpha & \beta & \mu\eta \\ \hline\circ & \alpha & \beta & \mu\eta \\ \circ & \alpha & \beta & \mu\zeta \\ \circ & \alpha & \beta & \mu\zeta \\ \hline \circ & \alpha & \beta & \mu\stigma \\ \circ & \alpha & \beta & \mu\varepsilon \\ \circ & \alpha & \beta & \mu\delta \\ \hline \circ & \alpha & \beta & \mu\gamma \\ \circ & \alpha & \beta & \mu\beta \\ \circ & \alpha & \beta & \mu\alpha \\ \hline \end \end Later in the table, one can see the base-10 nature of the numerals expressing the integer parts of the arc and the chord length. Thus an arc of 85° is written as ''πε'' (''π'' for 80 and ''ε'' for 5) and not broken down into 60 + 25. The corresponding chord length is 81 plus a fractional part. The integer part begins with ''πα'', likewise not broken into 60 + 21. But the fractional part, \tfrac4 + \tfrac , is written as ''δ'', for 4, in the column, followed by ''ιε'', for 15, in the column. : \begin \pi\varepsilon\rho\iota\varphi\varepsilon\rho\varepsilon\iota\tilde\omega\nu & \varepsilon\overset\upsilon\theta\varepsilon\iota\tilde\omega\nu & \overset\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\ \begin \hline \pi\delta\angle' \\ \pi\varepsilon \\ \pi\varepsilon\angle' \\ \hline \pi\stigma \\ \pi\stigma\angle' \\ \pi\zeta \\ \hline \end & \begin \hline \pi & \mu\alpha & \gamma \\ \pi\alpha & \delta & \iota\varepsilon \\ \pi\alpha & \kappa\zeta & \kappa\beta \\ \hline \pi\alpha & \nu & \kappa\delta \\ \pi\beta & \iota\gamma & \iota\theta \\ \pi\beta & \lambda\stigma & \theta \\ \hline \end & \begin \hline \circ & \circ & \mu\stigma & \kappa\varepsilon \\ \circ & \circ & \mu\stigma & \iota\delta \\ \circ & \circ & \mu\stigma & \gamma \\ \hline \circ & \circ & \mu\varepsilon & \nu\beta \\ \circ & \circ & \mu\varepsilon & \mu \\ \circ & \circ & \mu\varepsilon & \kappa\theta \\ \hline \end \end The table has 45 lines on each of eight pages, for a total of 360 lines.


See also

* Aryabhata's sine table * Exsecant * '' Fundamentum Astronomiae'', a book setting forth an algorithm for precise computation of sines, published in the late 1500s *
Greek mathematics Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during Classical antiquity, classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities ...
* Madhava's sine table *
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
* Scale of chords * Versine


References

* * * *
Olaf Pedersen Olaf Pedersen (8 April 1920 – 3 December 1997) was a Danish historian of science who was "leading authority on astronomy in classical antiquity and the Latin middle ages."Michael Hoskin (October 1998Obituary: Olaf Pedersen Astronomy and Geoph ...
(1974) ''A Survey of the Almagest'', Odense University Press *


External links

* J. L. Heibergbr>''Almagest''
Table of chords on pages 48–63. * Glenn Eler
Ptolemy's Table of Chords: Trigonometry in the Second Century

''Almageste''
in Greek and French, at the internet archive. {{Ancient Greek mathematics Trigonometry
Trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
History of astronomy Elementary special functions Ptolemy Mathematical tables Greek mathematics