Eratosthenes flourished in the Alexandrian school, about the time of Archimedes : his extensive acquirements in all branches of knowledge induced the third Ptolemy to make him his librarian. As a geometer, he might rank with Aris tens, Euclid, and Apollonius. His construction of the du plication of the cube, has come down to us in Eutocius' Commentary on Archimedes ; and we find it recorded in Pappus, that he wrote two books on a branch bf the geo metrical analysis, which were entitled De Loris ad medic tales ; these appear to have been conic sections. There is an arithmetical invention attributed to him, by which the prime numbers may be determined. Its nature has been described in our article ARITHMETIC. It may be presumed that Eratosthenes composed many works ; one is said to have been on the conic sections, and others on astronomy, but these are now completely lost.
About the time that Archimedes finished his career, another geometer of the highest order appeared. This was Apollonius of Pcrga, a town in Pamphylia. He was horn towards the middle of the third century, before the Chris tian cra, and he flourished principally under Ptolemy Phi lopater, or towards the end of that century. He studied in the Alexandrian school under the successors of Euclid ; and so highly esteemed were his discoveries, that he ac quired the name of the Great Geometer. It is mortifying to reflect, that sometimes consummate abilities are alloyed with gicat moral defects: Apollonius had a mind of the highest order, yet he was vain, jealous of,merit in others, and always disposed to detraction. He was, however, one of the most inventive and profound writers that has treated of the mathematics, and it was in a great measure from his works that the true spirit of the ancient geometry was to be learnt. In the introduction to our article Come SECTIONS, we have had occasion to speak of his treatise on that sub ject ; which contributed principally to his celebrity. The most material of his other works were the following trea tises: 1. On the Section of a Ratio; 2. On the Section of a Space; 3. On Determinate Section ; 4. On Tangencies ; 5. On Inclinations; 6. On Plane Loci : The nature and con tents of each of these has been particularly described in our article on ANALYSIS. We have understood that Peyrard, the learned French editor of the works of Euclid and Archi medes, had it in contemplation to give French translations of the writings of Apollonius, as well as the other ancient geometers, as far as they have been preserved ; but we fear that the state of France is not likely to be soon favour able to the execution of his views.
The names of several mathematicians of antiquity, con temporary with Archimedes and Apollonius, have come down to us. Apollonius has addressed the three first books of his conics to Eudemus of Pergamus, and speaks of him as a good judge in these matters, but he being dead before the fourth book was finished, Apollonius addressed it to At talus. He says, in his first address to Eudemus, that Nau crates had instigated him to study the conics ; and in that which precedes the second book, he requests Eudemus to communicate it to Philonides of Ephesus.
It appears that there was a geometer named Trasideus, who corresponded with Conon of Samos on the conic sec tions, and another Nicoteles the Cyrenean, who animadvert ed on some mistakes committed by Conon. Here, then, are five or six geometers besides Apollonius, who all cultiva ted the theory of conics. The regret which Archimedes expressed for the loss of Conon, gives us reason to think highly of him ; but this is almost the only ground upon which we can form an idea of his skill as a geometer.
Dositheus was also a friend of Archimedes, who address ed to him several of his works. It is probable that Nico medes, the inventor of the conchoid, lived about the period at which we are now arrived. This curve, and the appli cation he made of it to the finding of two mean proportion als, are the only vestiges that now remain of his labours.
As we descend towards the commencement of the Chris tian era, we find a numerous list of mathematicians, most of whom are chiefly known as cultivators of astronomy, and some as writers on geometry. In this number were Ge minus of Rhodes, who composed a work called Enarra tiones Geometricce, which consisted of six books ; Philo, who gave a solution of the problem of two mean proportionals ; Possidonius, who was a geometer, an astronomer, a mecha nician, and a geographer. Dionysiodorus, who resolved a difficult problem of Archimedes, namely, to divide a hemi sphere in a given ratio by a plane parallel to its base ; and Theodosius, the author of an excellent treatise on Spherics, in three books, which has been preserved, and which con stitutes a part of the precious remains of the ancient geo metry.