CONIC SECTIONS. 



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as commonly known. Although he did not compose a 

 complete treatise explaining the whole theory, yet lie 

 added a new branch to it, viz. that which treats of the 

 areas of the sections, and the solids formed by their re- 

 volution about an axis : he demonstrated in two diffe- 

 rent ways, that the area of a parabola is tMjp thirds of 

 that of its circumscribing parallelogram ; and this, for 

 many ages, was the only true quadrature of a curvili- 

 neal space that was known. He also shewed what wr.s 

 the proportion of elliptic areas to their circumscribing 

 circles, and of solids formed by the revolution oY'the 

 different sections to their circumscribing cylinders. His 

 various discoveries on this subject may be regarded as 

 the sublime mathematics of that period, 

 i 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 geometer., 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 barba- 

 rism, the conies 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 re- 

 vised and continued in the course of the same century 

 by Thebit Ben-Cora. A new translation was made 

 under the Caliph Abu-Cabghiar, 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 Abdolmelec of Scheeraz, another Persian, 

 abridged it : all these versions in manuscript were at 

 last found in Europe. 



For a long time, however, only the first four books 

 were kno>vn ; 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 re- 

 mainder tvas lost, is unknown. It existed, hoAvever, 

 in the days of Pappus, who lived in the fourth centu- 

 ry ; for that geometer has, in his mathematical collec- 

 tions, given some account of each book, and of the 

 lemmata employed in the demonstration of the propo- 

 sitions. Guided by these, mathematicians in modern 

 times undertook the restoration of the books supposed 

 to be lost ; and in particular Maurolicus, a Sicilian geo- 

 meter of the 17th century, composed a work, contain- 

 ing 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 simi- 

 lar Labour ; and, while he proceeded slowly and in si- 

 lence to prepare materials, the learned Golius returned 



from the East, bringing with him marry Arabic man;?- 1 

 scripts, among which were the first seven books of 

 Apollonius' conies; but although this discovery wtfs """""V" 

 communicated to mathematicians as early as the year 

 1644, yet, as no translation appeared, the last four 

 books were still regarded as lost. In the year 165°-, 

 Borelli discovered in the library of the Medici at Flo- 

 rence, an Arabic manuscript with an Italian title, sta- 

 ting it to consist of the eight books of Apollbnius. 

 This, by the liberality of the Duke of Tuscany, he was 

 allowed to carry to Rome, and there, aided by Abra- 

 ham Ecchellensis, a learned oriental scholar, he under- 

 took a translation of it into Latin. 



Meanwhile, Viviani was advised by his friends not 

 to lose the fruit of his investigations, and accordingly, 

 without being made acquainted with the contents of 

 the books that had been found, he proceeded and pub- 

 lished the result of his labours, in 1659- The trans- 

 lation, made by Borelli, accompanied by learned notes, 

 was published in 1661 ; and it is remarkable, that in 

 the Arabic manuscript he had found, as well as in that 

 of Golius, and in the abridged version of Abdolmelec 

 which Ravius had brought from the East, and pub- 

 lished in 1 669, the eighth book is entirely wanting, so 

 that it is now, in all probability, lost for ever. Dr 

 Halley, however, attempted to restore it from the hints 

 afforded by Pappus, and published the fruit of his re- 

 searches along with the other seven books,, and two 

 books on the sections of cylinders and cones, written 

 by Serenus, a geometer who lived in some of the early 

 centuries of the Christian era. It is commonly sup- 

 posed, that the restoration is so excellent, as to leave 

 but little reason to regret the loss of the original. 



The conies of Apollonius procured him the appelia- . 

 tion of the Great Geometer: a character to which he 

 appears to have been justly entitled, whether we con- 

 sider the difficulty of the subjects on which he wrote, 

 or the subtlety of his investigations and the skill and 

 success with which he has conducted them. Among 

 the improvements which he introduced into the mode 

 of treating the subject, there is one particularly worthy 

 of remark, because it is one of many instances in the 

 history of science, of the slow progress of the human 

 mind in passing from particular to more general truths. 

 Before his time, the different curves were defined, by 

 supposing right cones to be cut by planes perpendicu- 

 lar to their sides. By this method, three different 

 cones were required to produce the three sections : a 

 right angled cone for the parabola ; an acute angled 

 cone for the ellipse ; and an obtuse angled cone for the 

 hyperbola : But Apollonius shewed how all the three 

 sections might be formed by any one cone, whether 

 right or oblique. 



In the early ages of science, the conic sections were 

 cultivated merely as a geometrical theory, that might 

 afford an agreeable subject of contemplation to the 

 mind, b\ without a prospect of its ever being appli- 

 cable to the explanation of the phenomena of nature. 

 The discoveries of modern tiir^s. however, have great- 

 ly extended its utility, and rendered it by far the most 

 interesting speculation in pure geometry. Galileo shew- 

 ed, that the path of a body projected obliquely is a pa- 

 rabola, and Kepler discovered that the planetary or- 

 bits are ellipses ; these facts alone were sufficient to en- 

 hance greatly the value of the theory of the conic seCr 

 tions ; but the numerous discoveries of Newton that 

 followed went much farther, and incorporated it with 

 those of astronomy and the other branches of natural 

 philosophy. 



