Perseus is known as the discoverer and investigator of the spiric sections, i. e. certain sections of the σπειρα, one variety of which is the tore. The spire is generated by the revolution of a circle about a straight line in its plane, which straight line may either be external to the circle (in which case the figure produced is the tore), or may cut or touch the circle.
Zenodorus was the author of a treatise on Isometric figures, the problem in which was to compare the content of different figures, plane or solid, having equal contours or surfaces respectively.
Hypsicles (second half of second century B. C.) wrote what became known as ‘Book XIV’ of the Elements containing supplementary propositions on the regular solids (partly drawn from Aristaeus and Apollonius); he seems also to have written on polygonal numbers. A mediocre astronomical work (Αναφορικος) attributed to him is the first Greek book in which we find the division of the zodiac circle into 360 parts or degrees.
Posidonius the Stoic (about 135-51 B. C.) wrote on geography and astronomy under the titles On the Ocean and περι μετεωρων. He made a new but faulty calculation of the circumference of the earth (240,000 stades). Per contra, in a separate tract on the size of the sun (in refutation of the Epicurean view that it is as big as it looks), he made assumptions (partly guesswork) which give for the diameter of the sun a figure of 3,000,000 stades (39-1/4 times the diameter of the earth), a result much nearer the truth than those obtained by Aristarchus, Hipparchus, and Ptolemy. In elementary geometry Posidonius gave certain definitions (notably of parallels, based on the idea of equidistance).
Geminus of Rhodes, a pupil of Posidonius, wrote (about 70 B. C.) an encyclopaedic work on the classification and content of mathematics, including the history of each subject, from which Proclus and others have preserved notable extracts. An-Nairīzī (an Arabian commentator on Euclid) reproduces an attempt by one ‘Aganis’, who appears to be Geminus, to prove the parallel-postulate.
But from this time onwards the study of higher geometry (except sphaeric) seems to have languished, until that admirable mathematician, Pappus, arose (towards the end of the third century A. D.) to revive interest in the subject. From the way in which, in his great Collection, Pappus thinks it necessary to describe in detail the contents of the classical works belonging to the ‘Treasury of Analysis’ we gather that by his time many of them had been lost or forgotten, and that he aimed at nothing less than re-establishing geometry at its former level. No one could have been better qualified for the task. Presumably such interest as Pappus was able to arouse soon flickered out; but his Collection remains, after the original works of the great mathematicians, the most comprehensive and valuable of all our sources, being a handbook or guide to Greek geometry and covering practically the whole field. Among the original things in Pappus’s Collection is an enunciation which amounts to an anticipation of what is known as Guldin’s Theorem.
It remains to speak of three subjects, trigonometry (represented by Hipparchus, Menelaus, and Ptolemy), mensuration (in Heron of Alexandria), and algebra (Diophantus).
Although, in a sense, the beginnings of trigonometry go back to Archimedes (Measurement of a Circle), Hipparchus was the first person who can be proved to have used trigonometry systematically. Hipparchus, the greatest astronomer of antiquity, whose observations were made between 161 and 126 B. C., discovered the precession of the equinoxes, calculated the mean lunar month at 29 days, 12 hours, 44 minutes, 2½ seconds (which differs by less than a second from the present accepted figure!), made more correct estimates of the sizes and distances of the sun and moon, introduced great improvements in the instruments used for observations, and compiled a catalogue of some 850 stars; he seems to have been the first to state the position of these stars in terms of latitude and longitude (in relation to the ecliptic). He wrote a treatise in twelve Books on Chords in a Circle, equivalent to a table of trigonometrical sines. For calculating arcs in astronomy from other arcs given by means of tables he used propositions in spherical trigonometry.
The Sphaerica of Theodosius of Bithynia (written, say, 20 B. C.) contains no trigonometry. It is otherwise with the Sphaerica of Menelaus (fl. A. D. 100) extant in Arabic; Book I of this work contains propositions about spherical triangles corresponding to the main propositions of Euclid about plane triangles (e.g. congruence theorems and the proposition that in a spherical triangle the three angles are together greater than two right angles), while Book III contains genuine spherical trigonometry, consisting of ‘Menelaus’s Theorem’ with reference to the sphere and deductions therefrom.
Ptolemy’s great work, the Syntaxis, written about A. D. 150 and originally called Μαθηματικη συνταξις, came to be known as Μεγαλη συνταξις; the Arabs made up from the superlative μεγιστος the word al-Majisti which became Almagest.