128 SIMSON. 



case of the 'Loci Plani' of Apollonius. Euclid's four 

 books on conic sections are also lost ; but of Apol- 

 lonius's eight books on the same subject, the most 

 important of the whole series, the ' Elements' excepted, 

 four were preserved, and three more were discovered 

 in the seventeenth century. His Inclinations, his 

 Tactions or Tangencies, his sections of Space and of 

 Ratio, and his Determinate section, however curious, 

 are of less importance ; all of them are lost. 



For many years Commandini's publication of the 

 * Collections' and his commentary did not lead to any 

 attempt at restoring the lost works from the general 

 account given by Pappus. Albert Girard, in 1634, 

 informs us in a note to an edition to Stevinus, that he 

 had restored Euclid's 'Porisms,' a thing eminently 

 unlikely, as he never published any part of his resto- 

 ration, 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 afterwards 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 cen- 

 tury, 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 exceedingly 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 inconsiderable 

 progress in the most difficult branches. 



