astronomical observations at Alexandria between 127 and 151 AD, and for that reason he is also often referred to as Ptolemy of Alexandria.

Ptolemy was an astronomer of some note. As a prerequisite to his work on astronomy and the accurate measurements he needed to make, Ptolemy

studied trigonometry, and his greatest work was a thirteen book treatise on trigonometry named the

frequently known by the Arabic name

written by Aristarchus and others which was referred to as ‘the lesser’.

Ptolemy’s Almagest contains very accurate trigonometric tables. Central to the compilation of these tables was the theorem that has become known

as Ptolemy’s Theorem (see below).

Ptolemy also made a contribution to the way we measure angles. The division of the circumference of a circle into 360 equal parts seems to have

been in use since the time of Hipparchus around 430 BC, and this idea translated into dividing a complete turn into 360 degrees. However, it was

Ptolemy who subdivided the degree into sixty equal parts, named ‘minutes’, and the minute into sixty equal parts named ‘seconds’.

Ptolemy’s calculations also led him to an approximation for pi which he gave as 3.1416 in modern notation.

ABCD is a cyclic quadrilateral drawn inside the circle centre O.

Ptolemy gave the following theorem named Ptolemy’s Theorem:

If ABCD is a cyclic quadrilateral, then the product of the lengths of the diagonals is equal to the sum of the products of pairs of opposite sides.

i.e.

Ptolemy gave the following theorem named Ptolemy’s Theorem:

If ABCD is a cyclic quadrilateral, then the product of the lengths of the diagonals is equal to the sum of the products of pairs of opposite sides.

i.e.

The proof of this theorem uses similar triangles.

It also requires knowledge of another well-known theorem variously called*The Angles subtended by the same arc theorem* or *The Angles in the *

same segment theorem which states that:

If CD is an arc of a circle and A and B are any other two points on the circumference of the circle on the same side of chord CD, then Angle DAC =

Angle DBC

It also requires knowledge of another well-known theorem variously called

same segment theorem

If CD is an arc of a circle and A and B are any other two points on the circumference of the circle on the same side of chord CD, then Angle DAC =

Angle DBC

Equal angles are shown on the above diagram as follows:

Angle ABE = AngleDBC = Angle DAC (one arc)

Angle BAC = Angle BDC (two arcs)

Angle ACB = Angle ADB (three arcs)

Now consider two pairs of similar triangles:

Angle ABE = AngleDBC = Angle DAC (one arc)

Angle BAC = Angle BDC (two arcs)

Angle ACB = Angle ADB (three arcs)

Now consider two pairs of similar triangles: