Ptolemy's Table of Chords


Ptolemy's table of chords greatly aided in the advancement of trigonometry. In this section of this website we will explore how Ptolemy created this table.
The table of chords is based off of a circle with radius 60. The interval between entries in the table is 1/2°. To begin, Ptolemy made a proposition, known today as, Ptolemy's theorem: In this he was able to prove that if ABCD is a quadrilateral inscribed in a circle (geogebra applet number 1), then AB*CD+BC*DA=AC*BD; that is to say that the sum of the products of the opposite sides of a cyclic(inscribed) quadrilateral is equal to the product of the diagonals.
Below is a geogebra applet that you can manipulate to be able to see that this relationship holds true in all cases! I invite you to try and prove this yourself using this applet as a way to guide you through!

This hyperlink will lead you to an actual proof done by Ken Ward: Proof of Ptolemy's Theorem
Now, Ptolemy was armed with formulas and a table of chords of half an arc, and having a good value of chord 1/2°, he could go on to build up his table! He began with the chords x+y and x-y assuming that chords of x and y were known. He was able to discover the chord of 36 °. Now this lead him to be able construct the chord of 1 1/2 °. If x>y, then chd x/chd y (chd means chord) Taking
x= 1 1/2° and y=1°, we have chd 1°. 2/3 chd 1 1/2 ° > 2/3 * 1;34,14,41.1,2,49,47. (minutes, seconds respectively).

Similarly, taking x= 1° and y= 3/4 ° we can show that chd 1°,1,2,49,55. This was significant because they were able to get more accurate estimations of the distances of the Sun and Moon. Thus they were able to have better navigation. For more application of these table of chords go to the application section of this website!

Calculating the Radius of the Moon's Epicycle.

Amazingly enough Ptolemy and other astronomers following him were able to better make calculations. Ptolemy, who followed the teachings of Hipparchus, believed that there was not a perfect circular orbit in the planetary system. He claimed that the orbital motions were more elliptical. There was a system of cycles, epicycles, and eccentrics.
The geogebra applet below is a depiction of what Ptolemy predicted the planetary orbit to be around the earth. If you click the trace P box it will turn a trace on. If you click the start Animation button then it will begin to trace out the orbit of a planet in Ptolemy's geocentric Universe.

Ptolemy could calculate the radius of the moon's epicycle's from data obtained by observing three eclipses. Below is an example of one calculation made by Ptolemy presented in the book Hipparchus' Table of Chords : Ptolemy using eclipses observed by the Babylonians in the first and second years of the reign of Marduk-apal-iddina, about 720 B.C. The time intervals between the eclipses, reduced to mean solar time, were 354 days, 2 hours, 34 minutes from the first to the second, and 176 days, 20 hours, 12 minutes from the second to the third. From the anomalistic period Ptolemy calculated how far round the epicycle the moon traveled in these two intervals. If the positions of the moon on the epicycle at the times of the eclipses are PI, P2, and P3, respectively, then, measured clockwise
arc P1*P2=306° *25', arc P2*P3=150° *26' (1)
From the times of the eclipses, converted to Alexandria time, Ptolemy found the longitudes of the sun and hence of the moon. From these he found (see Figure A4.1, in which T denotes the earth)
angle P2*T*P1=3° *24', angle P2* T* P3= 0° *37'. (2)

(2) (1) and (2) are the numerical data for the calculation.
Ptolemy several times used the table of chords to find the proportions of a right-angled triangle. This is how it is done. Let ABC he a triangle with a right angle at B (see Figure A4.2). Suppose that the angle ACB is ~x and we want to find ABIAC. If 0 is the midpoint of AC, then A 0 B = x. If we look up x in the table of chords and find that chd x = y, this means that AB = Y on a scale in which AO = 60. Thus
AB/AC = y/120.

This is the reason for such items as ~ x 6° 48' or ~ x 1° 14' in various steps of the calculation.

Theorem of Menelaus

To my left is a figure with a brief explanation as to how Menelaus went about proving the identity we know today and use frequently in trigonometry. Now onto the theorem of Menelaus.
Here is a proof of Menelaus' Theorem by Alex Bogomolny: Proof of Menelaus Theorem
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