The astronomer and geometer 1\Ianelaus of Alexandria, lived in the second century of the Christian era : he com , posed a treatise on Trigonometry, in six books ; and ano ther on Spherics, in three books, which is still extant. He appears also to have treated of the geometry of curve lines.
The astronomer Ptolemy must be reckoned among the geometers of the second century. His work on Optics, which however is now lost,* is supposed to have contained some beautiful applications of geometry.
There were several geometers who flourished in the pe riod of the three or four first centuries of the Christian era ; but the exact time of each is not certainly known ; as Sere nus of Antissensis, who wrote on cylinders and cones ; Hypsicles of .Alexandria, who wrote two books on regular solids ; Perseus Citticus, the inventor of certain lines called which were curves made by the section of a plane and a solid, formed by the revolution of an arc of a circle about a given axis. Philo of Thyaneus, who appears to have treated of certain curves, which were also considered by Menelaus, but whose nature is not now known. Tap pus also mentions Demetrius of Alexandria, as the author of a work which treated of curves, and hence it has been conjectured that the ancients had gone farther into this sub ject. than has been generally supposed.
We are now come to the period when learning began to decline, so that instead of brilliant discoveries, and origi nal treatises, we have only commentaries and annotations on the works of former times, a presage of the approach of ignorance and barbarism. Of this nature were the works of Pappus, and Thcon of Alexandria, two mathematicians who lived towards the end of the fourth century. The former of these, however, ranks in a higher class, on account of the genius displayed in his writings. Geometry is particu larly obliged to him for his Mathematical Collections, which originally consisted of eight books ; but of these, the first and half of the second are now lost. He seems to have in tended to collect, into one body, several scattered discove ries, and to illustrate and complete, in many places, the writings of the most celebrated mathematicians, in parti cular Apollonius, Archimedes, Euclid, and Theodosius ; for this purpose he has given a multitude of lemmas, and curious theorems, which they had supposed known ; and he has also described the di fferent attempts which had been made to resolve the more difficult problems, as the duplication of the cube, and the trisection of an angle. The preface to
his seventh book is inestimably precious, for it has preserv ed from oblivion many analytical works on geometry, of which we should otherwise have been entirely ignorant. The abridgment which he has given of these is all that re mains of the greater number ; yet it has served to give a continuity to the history of geometry, and to inspire mo dern mathematicians with a high opinion of the theories of the ancients. In fact, such of their geometrical writings as have descended to our times are merely elementary ; their more recondite works have either been entirely lost, or are only known by the account which Pappus has given of them. The books that remain of Pappus have suffered much from the injuries of time ; there are many inaccura cies, and some passages so mutilated as to be hardly intel ligible. The original Greek, except some extracts, has never been published. The only translation that has been given, which is by Commandinus, was published at Pcsara in 1588, and again, with little variation, in 1660, at Bologna. Commandinus appears to have had access to only one ma nuscript, which wanted the first two books, and throughout was very faulty. There are, however, several manuscripts of Pappus in the libraries ; the university of Oxford pos sesses two ; one has half of the second hook, and this part was published, with a Latin translation, in 1688, by Dr Wallis. It treats of arithmetic, so that probably the first two books treated of this subject. The university has al ready conferred a great favour on geometrical science, by the elegant editions it has given of Euclid, Apollonius, and Archimedes ; and it is to be wished that the obligation were increased by an edition of Pappus. Our limits will not ad mit of a detailed statement of the contents of this valuable work ; some account of it may be seen in Dr Hutton's Ma thematical Dictionary, and also in Dr Traill's Life of Sim son.