may then write Ptolemy's Theorem in the following trigonometric form: Applying certain conditions to the subtended angles and it is possible to derive a number of important corollaries using the above as our starting point. In what follows it is important to bear in mind that the sum of angles . Corollary 1. Pythagoras's theorem Let and . Then (since opposite angles of a cyclic quadrilateral are supplementary). Then: Corollary 2. The law of cosines Let . The rectangle of corollary 1 is now a symmetrical trapezium with equal diagonals and a pair of equal sides. The parallel sides differ in length by units where: It will be easier in this case to revert to the standard statement of Ptolemy's theorem: The cosine rule for triangle ABC. Corollary 3. Compound angle sine (+) Let Then Therefore, Formula for compound angle sine (+). Corollary 4. Compound angle sine (−) Let . Then . Hence, Formula for compound angle sine (−). This derivation corresponds to the Third Theorem as chronicled by Copernicus following Ptolemy in Almagest. In particular if the sides of a pentagon (subtending 36° at the circumference) and of a hexagon (subtending 30° at the circumference) are given, a chord subtending 6° may be calculated. This was a critical step in the ancient method of calculating tables of chords. Corollary 5. Compound angle cosine (+) This corollary is the core of the Fifth Theorem as chronicled by Copernicus following Ptolemy in Almagest. Let . Then . Hence Formula for compound angle cosine (+) Despite lacking the dexterity of our modern trigonometric notation, it should be clear from the above corollaries that in Ptolemy's theorem (or more simply the Second Theorem) the ancient world had at its disposal an extremely flexible and powerful trigonometric tool which enabled the cognoscenti of those times to draw up accurate tables of chords (corresponding to tables of sines) and to use these in their attempts to understand and map the cosmos as they saw it. Since tables of chords were drawn up by Hipparchus three centuries before Ptolemy, we must assume he knew of the 'Second Theorem' and its derivatives. Following the trail of ancient astronomers, history records the star catalogue of Timocharis of Alexandria. If, as seems likely, the compilation of such catalogues required an understanding of the 'Second Theorem' then the true origins of the latter disappear thereafter into the mists of antiquity