Ptolemy's theorem and the chords of the Almagest

The addition formulae have a very long history, one that predates the modern sine–cosine notation by more than a thousand years. Greek mathematicians did not use the sine of an angle, but the chord of the corresponding arc: in a fixed circle they associated with each arc of measure $\theta$ the chord $\text{crd}(\theta)$, the length of the segment subtending that arc. The link with the sine is simple: in a circle of radius $R$, $\text{crd}(\theta) = 2R\,\sin\!\frac{\theta}{2}.$

The earliest table of chords we know of is due to Hipparchus of Nicaea (2nd century BC), regarded as the founder of trigonometry. It was Claudius Ptolemy, however, in his great astronomical treatise the Almagest (2nd century AD), who produced an extraordinarily accurate version, with chords tabulated in half-degree steps (Boyer, Katz).

Ptolemy’s theorem

The key tool behind that computation is an elegant geometric result, known today as Ptolemy’s theorem: in a quadrilateral inscribed in a circle, the product of the diagonals equals the sum of the products of the opposite sides. If $ABCD$ is the cyclic quadrilateral, $AC\cdot BD = AB\cdot CD + BC\cdot DA.$

By applying this theorem to suitable quadrilaterals, with one side lying on the diameter, Ptolemy derived the relations between the chord of the sum and the chord of the difference of two arcs. Translated into modern language, those relations are exactly the addition and subtraction formulae we use today: for instance the construction that leads to $\sin(\alpha-\beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta$ is, in essence, a special case of Ptolemy’s theorem (Dunham, Katz).

From these identities, together with a refined estimate of the chord of $1°$ obtained by interpolation, Ptolemy was able to compile a table that remained unsurpassed in Europe until the Renaissance. It is a fine example of how a purely geometric theorem — a statement about inscribed quadrilaterals — became the computational engine of ancient astronomy.

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Topics: Trigonometric formulae
Concepts: Addition formulae
People: Claudius Ptolemy · Hipparchus of Nicaea