The interest which mankind in general take in the mathematical sciences, is but little in comparison to that which is excited by works of poetry, oratory, history, and the like, and hence it has happened that the writings of the ancients on these subjects have had a far better chance than their mathematical theories, of descending to the present times. However much we may regret the cir cumstance, it will not therefore appear wonderful that we now know nothing more than the names of the early cultivators of the conic sections: and of these Arista:us deserves to be particularly mentioned. Pappus of Alex andria informs us, in his Alathematical Collections, that this geometer composed five books De Locis Solidis, and as many on conic sections, all which are now entirely lost. The celebrated geometer Euclid is supposed to have been a disciple of Aristxus, at any rate he must have been his very particular friend. We learn front Pappus, that Euclid composed a treatise in four books on conic sections, but that also has been lost.
Of several other geometers who appear to have cul tivated this theory, we shall only mention Conon the friend of Archimedes ; but he is better known as an astronomer than as a geometer.
The vtritings of Archimedes spew, that before his time, codsiderable progress had been made in the dis covery of the properties of the conic sections, as he refers to many of them incidentally, and speaks of others as commonly known. Although he did not compose a complete treatise explaining the whole theory, yet he added a new branch to it, viz. that which treats the areas of the sections, and the solids formed by their revo lution about an axis : he demonstrated in two different ways, that the area of a parabola is two thirds of mat of its circumscribing parallelogram ; and tnis, for many ages, was the only true quadratu•e of a curvilineal space that was known. He also sheaved what was the propor tion of elliptic areas to their circumscribing circles, and of solids formed by the revolution of the different sec tions to their circumscribing cylinders. His various dis coveries on this subject may be regarded as the sublime mathematics of that period.
Apollonius of Perga may be reckoned the next in rank to Archimedes among the ancient geometers. He lived at a period about forty years later, that is, about the middle of the second century before the Christian era. He studied in the Alexandrian school under the successors of Euclid, and, besides writing treatises on the more abstruse branches of the mathematics culti vated at that time, he enriched the science by a work on conic sections, possessing a high degree of merit. It consisted of eight books. The first four is supposed to comprehend all that was known on the subject before his time, and the remaining books are reckoned to have contained his own discoveries. Several geometers of antiquity wrote commentaries on this work. Among
the Greeks we find Pappus, who illustrated them by lemmas or preliminary propositions prefixed to each book. The learned Hypatia, the daughter of Theon, also wrote a commentary, which, however, is now lost, but another by Eutocius, on the first four books, is still extant. In later times, when the Arabians began to collect the fragments of knowledge that had escaped the wreck of the sciences in preceding ages of barbarism, the conics of Apollonius were one of the first works of which they undertook a translation. It was begun under the Caliph Almamon in the year 830 of the Christian era, and what had been prepared was revised and con tinued in the course of the same century by Thebit Ben-Cora. A new translation was made under the Caliph Abu-Calighiar, in 994 : This version afterwards fell into the hands of the Italian geometer Borelli, as we shall presently have occasion to state. The Persian geometer and astronomer Nassir-Eddin, wrote notes on this work in the middle of the 13th century, and Abdol melee of Scheeraz, another Persian, abridged it : all these versions in manuscript were at last found in Eu rope.
For a long time, however, only the first four books were known ; and these in the Greek tongue, are the only part of the original work that has descended to modern times. When, or by what accident, the remain der was lost, is unknown. It existed, however, in the days of Pappus, who lived in the fourth century ; for that geometer has, in his mathematical collections, given some account of each book, and of the lemmata employ ed in the demonstration of the propositions. Guided by these, mathematicians in modern times undertook the restoration of the books supposed to be lost ; and in par ticular Mau•olicus, a Sicilian geometer of the 17th cen tury, composed a work, containing what he conceived to be the substance of the fifth and sixth books, which was published by Borelli in 1654. Viviani, a disciple of Galileo, and one of the most skilful geometers in Italy, had also begun a similar labour ; and, while he proceeded slowly and in silence to prepare materials, the learned Golius returned from the East, bringing with him many Arabic manuscripts, among which were the first seven books of Apolionius' conics ; but although this discovery was communicated to mathematicians as early as the year 1 644. yet, as no translation appeared, the last four books were still regarded as lost. In the year 1658, Borelli discovered in the library of the Medici at Florence, an Arabic manuscript with an Italian title, stating it to consist of the eight books of Apollonius. This, by the liberality of the Duke of Tuscany, he was allowed to carry to Rome. and there, aided by Abraham Ecchellensis, a learned oriental scholar, lie undertook a translation of it into Latin.