MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000

So far, we have seen Ptolemy find two Crd functions.  However, his circle is divided into 360 units, or parts.  Therefore, there are 358 more Crd functions to find.  Actually, there are 718 left to be computed (he also included half degrees in his table).  How could Ptolemy calculate all these remaining functions?

Ptolemy's Theorem

After setting up his circle (with a total of 360 parts) and his calculations of the first two Crd functions, Ptolemy continued by deriving what we now know as Ptolemy's Theorem.  Stated in simplest form, it states:  The sum of the products of the opposite sides of a quadrilateral is equal to the product of its diagonals. Now, Ptolemy inscribed his quadrilateral in a circle as shown at the right.  Then, using Euclid's Elements, he proved that ac + bd = ef.  A through proof can be found at www.cut-the-knot.com.  

After proving a general case of the theorem above, Ptolemy designed a special case of his theorem, with one of the sides of the quadrilateral the diameter of the circle.  Designing the quadrilateral in this manner gave Ptolemy the chance to calculate several other chords, such as 72º, 60º, 12º, 6º, 3º, 1 ½º, and ¾º.

To calculate the function Crd 12º, Ptolemy developed a theorem which enabled him to compute the chord of the difference between two given arcs and the corresponding chords.  Because he already calculated Crd 72º and Crd 60º, he calculated Crd 12º by computing the Crd 72º - Crd 60º.  Consider the figure at the left.  We know by the previous theorem AG X BD = (AB X GD) + (BG X AD).  Now,  suppose that AG X BD  is given to us and  AB X GD is given.  Then, by subtraction, BG X AD is also given.  Now suppose AG X BD  is the arc  of 72º and AB X GD is the arc of 60º.  Then, the arc BG X AD is, by subtraction, is the arc of 12º.  Since AD is the diameter of the circle, then the chord BG is given.  Thus, BG is chord in question, and we can calculate Crd 12º, which turns out to be 12;32,36.

Approximating Crd 1º

Now that Ptolemy calculated two functions close to 1º , he attempted to approximate Crd 1º.   In his book Almagest, Ptolemy completes some more computations, each one important in his approximation.  He knew it was less than Crd 1 ½º and Crd ¾º.  Finally, he approximates Crd 1º to be 1;2,50, and Crd ½º = 0;31,25.  With this calculation, the remaining functions can be computed and documented.  A complete chord table can be found in Almagest, and at the following site:  http://hypertextbook.com .

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