Ptolemy’s table of chords

The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy’s Almagest,[1] a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy (an earlier table of chords by Hipparchus gave chords only for arcs that were multiples of

7+1/2° = π/24 radians).[2] Centuries passed before more extensive trigonometric tables were created. One such table is the Canon Sinuum created at the end of the 16th century.

. . . Ptolemy’s table of chords . . .

Example: The length of the chord subtending a (109+1/2)° arc is approximately 98.

A chord of a circle 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 1/2 to 180 by increments of 1/2. In modern notation, the length of the chord corresponding to an arc of θ degrees is

chord(θ)=120sin(θ2)=60(2sin(πθ360 radians)).{displaystyle {begin{aligned}&operatorname {chord} (theta )=120sin left({frac {theta ^{circ }}{2}}right)\={}&60cdot left(2,sin left({frac {pi theta }{360}}{text{ radians}}right)right).end{aligned}}}

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 π is to 3, or more precisely, the ratio can be made as close as desired to π/3  1.04719755 by making θ small enough. Thus, for the arc of 1/2°, 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 60°, 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 (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+2960+5602=99.48472¯,{displaystyle 99+{frac {29}{60}}+{frac {5}{60^{2}}}=99.4847{overline {2}},}

rounded to the nearest 1/602.[1]

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

chord(θ+12)chord(θ)30.{displaystyle {frac {operatorname {chord} left(theta +{tfrac {1}{2}}^{circ }right)-operatorname {chord} left(theta ^{circ }right)}{30}}.}

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 + 1/2)°. Thus, it is used for linear interpolation. 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.[3]

arcchordsixtieths12031250125011250012501121341501250109974138003623109129759490036911098175400355611012983552003542111985343003529111129911270035151129929500351112129946350034481131003590034341791195944000251791211959560009180120000000{displaystyle {begin{array}{|l|rrr|rrr|}hline {text{arc}}^{circ }&{text{chord}}&&&{text{sixtieths}}&&\hline {},,,,,,,,,,{tfrac {1}{2}}&0&31&25&0quad 1&2&50\{},,,,,,,1&1&2&50&0quad 1&2&50\{},,,,,,,1{tfrac {1}{2}}&1&34&15&0quad 1&2&50\{},,,,,,,vdots &vdots &vdots &vdots &vdots &vdots &vdots \109&97&41&38&0quad 0&36&23\109{tfrac {1}{2}}&97&59&49&0quad 0&36&9\110&98&17&54&0quad 0&35&56\110{tfrac {1}{2}}&98&35&52&0quad 0&35&42\111&98&53&43&0quad 0&35&29\111{tfrac {1}{2}}&99&11&27&0quad 0&35&15\112&99&29&5&0quad 0&35&1\112{tfrac {1}{2}}&99&46&35&0quad 0&34&48\113&100&3&59&0quad 0&34&34\{},,,,,,,vdots &vdots &vdots &vdots &vdots &vdots &vdots \179&119&59&44&0quad 0&0&25\179{frac {1}{2}}&119&59&56&0quad 0&0&9\180&120&0&0&0quad 0&0&0\hline end{array}}}

. . . Ptolemy’s table of chords . . .

This article is issued from web site Wikipedia. The original article may be a bit shortened or modified. Some links may have been modified. The text is licensed under “Creative Commons – Attribution – Sharealike” [1] and some of the text can also be licensed under the terms of the “GNU Free Documentation License” [2]. Additional terms may apply for the media files. By using this site, you agree to our Legal pages . Web links: [1] [2]

. . . Ptolemy’s table of chords . . .