Ptolemy’s Contribution in Mathematics


Claudius Ptolemy mostly known as Ptolemy was a Greek philosopher, mathematician, and astronomer. There is no authentic information about his life. Analysing his name carefully, Claudius Ptolemy is a combination of Greek-Egyptian names. It can therefore be claimed that he belonged to a Greek family living in Egypt and was a citizen of Rome. As mentioned by Theodore Meliteniotes in around 1360 (one thousand years after Ptolemy lived) that Ptolemy was born in Hermiou cannot be treated as a true statement. Moreover, there is no evidence of Ptolemy moving ever outside Alexandria. He was born around 90 AD. Some scholars believe that he was best at mathematics, while others believe that he was very good at explaining ideas and theories. Some of them even claimed that he betrayed his fellow scientists by going against the ethics of his profession. He is mainly known for his significant contributions to the field of astronomy. Ptolemy used works of Theon of Smyrna who is believed to be a teacher of Ptolemy. His most important work is Amalgest that is a treatise in thirteen books, the only ancient surviving work on astronomy. Initially, its original Greek title was ‘The Mathematical Compilation,’ but this title was later replaced by another Greek title that means ‘The Greatest Compilation’. This was translated into Arabic as “al-majisti,” and lastly, the title Almagest was given to the work when it was translated from Arabic to Latin. This book constitutes the motion of the sun, moon, and planets explained by mathematical theory. His other works include ‘Optics’ in which he explained reflection, refraction, and colour. In the field of geography, he wrote a book, Geaoraphika, on how to draw maps. He also wrote an astrological treatise known by the Greek term Tetrabiblos, which means ‘four books’ because his work was completed in four parts. Ptolemy died in 168 AD at the age of 78. His epigram, which he is said to have written himself, read:

Well do I know that I am mortal, a creature of one day.
But if my mind follows the winding paths of the stars
Then my feet no longer rest on earth, but standing by
Zeus himself I take my fill of ambrosia, the divine dish.’

1. Foundation of Trigonometry

Ptolemy and Hipparchus are both credited with the foundation of trigonometry. For the use of astronomers, Hipparchus introduced this branch, but it was put to exposition by Ptolemy in such a way that it was followed for the next 1400 years. Almagest is purely a work of astronomy, but astronomy is dependent on trigonometry. So, Ptolemy, in his book ‘Chords in a circle’, in Amalgest, had mentioned how to create a table of chords. Ptolemy’s table of chords gives the lengths of chords of a circle of diameter 120 as a function of the number of degrees n in the corresponding arc of the circle, for n ranging from 1/2 to 180 by increments of 1/2. He divides the circumference into 360 equal parts and then bisects part. Further, he divides the diameter into 120 equal parts, and then, for the subdivisions of these, he employs the sexagesimal method, i.e., he divides each of the sixty parts of the radius into sixty equal parts, and each of these parts he further subdivides into sixty equal parts. In the Latin translation, these subdivisions become “partes minutae primae ” and ” partes minutae secundae,” whence our ” minutes “. It can not be said that the concept of sexagesimal divisions came from Ptelomy, other scholars might have used these divisions before him. Also, the table of chords was not suggested by Ptolemy. Hypatia’s father,’ Theon of Alexandria,’ in his remarks on Almagest, said,

Hipparchus had already given the concept of chords inscribed in a circle in twelve books, and that Menelaus had done the same in six books, but, every one must be astonished at the ease with which Ptolemy, by means of a few simple theorems, has found their values; hence it is inferred that the method of calculation in the Almagest is Ptolemy’s own’.

2. Ptolemy’s Sum and Difference Formula

After creating the table of chords, Ptolemy needed ways to compute the trigonometric function for sum and difference of angles. Chord of the table was his basic trigonometric function. When we translate his formulas to sines and cosines, we get the following identities.

sin(α+β)= sinα cosβ+cosα sinβ

cos(α+β)=cosα cosβ-sinα sinβ

sin(α-β)=sinα cosβ-cosα sinβ

cos(α-β)=cosα cosβ+sinα sinβ

To establish these formulas, he first proved a theorem, which is known as Ptolemy’s theorem. It states,

    For a cyclic quadrilateral (that is, a quadrilateral inscribed in a circle), the product of the diagonals equals the sum of the products of the opposite sides. If the cyclic quadrilateral is ABCD, then the Ptolemy’s theorem equation is AC BD= AB CD+AD BC.’


3. Plane Trigonometry

To study the concept of plane triangles, Greeks inscribed a triangle into a circle, so they must have known the theorem which states that

The sides of a triangle are proportional to the chords of the double arcs which measure the angles opposite to those sides”.

This theorem gives a complete solution in the case of a right-angled triangle. To get a complete solution, other triangles were converted into a right-angled triangle by drawing a perpendicular from one vertex on the opposite side. Ptolemy solved a triangle in which the three sides are given by finding the segments of a side made by the perpendicular on it from the opposite vertex. However, the eleventh chapter of Almagest is about spherical trigonometry, but some theorems and problems of plane trigonometry are also contained in it.

plane trigonometry

4. Spherical Trigonometry

The concept of spherical geometry is mentioned in chapter IX of Almagest. The chapter starts with the theorem,

The segments of any side of the triangle are in a ratio compounded of the ratios of the segments of the other two sides of the triangle”

In the view of Chasles, this theorem of spherical triangles was deduced by Hipparchus from plane triangles. He further added and gave credit to Euclid by saying that this theorem was mentioned in his book Porisms. Ptolemy considered only two cases of this theorem, and it is not mentioned in the general form in his book Almagest. Ptolemy also mentioned two lemmas, and with the help of these lemmas, he deduced from the ‘theorem of Menelaus’ for a plane triangle, the corresponding theorem for a spherical triangle, “If the sides of a spherical triangle are cut by an arc of a great circle, the chords of the doubles of the segments of any one side will be related to each other in a ratio compounded of the ratios of the chords of the doubles of the segments of the other two sides”, which is again not generalised. Theon generalised this theorem by adding two mare cases to it. With the help of this generalised theorem, four of Napier’s formulae for the solution of right-angled spherical triangles can be easily established. Ptolemy did not mention any of them but used to apply the theorem of Menelaus for spherics directly whenever required.

spherical trigonometry

5. Degrees of Obliquity of the Sphere

Ptolemy hear found the height of the pole. Using the same data he demonstrated how to find, at what places and time the shadow of the sun will be vertical, and how to calculate the ratios of gnomons (the projecting piece on a sundial that shows the time by the position of its shadow) to their equinoctial and solstitial shadows at noon and conversely. Ptolemy worked out in detail for the parallel of Rhodes. Theon appreciated Ptolemy for his selection of parallel for three reasons. Firstly, the height of the pole at Rhodes is a whole number, i.e., 36, whereas, at Alexandria it was 30° 58′; secondly, Hipparchus had made many observations at Rhodes; and thirdly, climate of Rhodes holds the mean place of the seven climates.

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