62 GREEK GEOMETRY. 



thing eminently unlikely, as he never published any part of 

 his restoration, and it was not found after his decease. In 

 1637, Fermat restored the ' Loci Plani ' of Apollonius, but in 

 a manner so little according to the ancient analysis, that we 

 cannot be said to approach by means of his labours the lost 

 book on this subject. In 1615, De la Hire, a lover and a 

 successful cultivator of the ancient method, published his 

 Conic Sections, but synthetically treated ; he added after- 

 wards other works on epicycloids and conchoids, treated on 

 the analytical plan. L'Hopital, at the end of the seventeenth 

 century, published an excellent treatise on Conies, but purely 

 algebraical. At the beginning of the eighteenth century, 

 Viviani and Grandi applied themselves to the ancient 

 geometry; and the former gave a conjectural restoration 

 (Divinatio) of Aristaeus's ' Loci Solidi,' the curves of the 

 second or Conic order. But all these attempts were exceed- 

 ingly unsuccessful, and the world was left in the dark, for the 

 most part, on the highly interesting subject of the Greek 

 Geometry. We shall presently see that both Fermat and 

 Halley, its most successful students, had made but an incon- 

 siderable progress in the most difficult branches. 



How entirely the academicians of France were either care- 

 less of those matters, or ignorant, or both, appears by the 

 ' Encyclopedic ;' the mathematical department of which was 

 under no less a geometrician than D'Alembert. The definition 

 there given of analysis, makes it synonymous with algebra : 

 and yet mention is made of the ancient writers on analysis, 

 and of the introduction to the seventh book of Pappus, with 

 only this remark, that those authors differ much from the 

 modern analysts. But the article ' Arithmetic ' (vol. i. p. 677), 

 demonstrates this ignorance completely ; and that Pappus's 

 celebrated introduction had been referred to by one who 

 never read it. We there find it said, that Plato is supposed to 

 have invented the ancient analysis; that Euclid, Apollonius, 

 and others, including Pappus himself, studied it, but that we 

 are quite ignorant of what it was: only that it is by some 

 conceived to have resembled our algebra, or else Archimedes 



