142 SIMSOX. 



by which his steps in this path must be guided. The 

 introduction to the Seventh book of Pappus contained 

 the whole that had reached our times to let us know 

 the contents of the lost works. Some of the sum- 

 maries which that valuable discourse contains are suffi- 

 ciently explicit, as those of the Loci Plani and the 

 Determinate Section. Accordingly, former geometri- 

 cians had succeeded in restoring the Loci Plani, or 

 those propositions which treat of loci to the circle and 

 rectilinear figures. They had, indeed, proceeded in 

 a very unsatisfactory manner. Schooten, a Dutch 

 mathematician of great industry and no taste, had 



flven purely algebraic solutions and demonstrations. 

 ermat, one of the greatest mathematicians of the 

 seventeenth century, had proceeded more acccrdin 

 to the geometrical rules of the ancients; but he h 

 kept to general solutions, and neither he nor Schooten 

 had given the different cases, according as the data in 

 each proposition were varied ; so that their works 

 were nearly useless in the solution of problems, the 

 great purpose of Apollonius, as of all the authors 

 of the TOTTOC avaXwojUfvou the thirty-three ancient 

 books. As for the analysis, it was given by neither, 

 unless, indeed, Schooten's algebra is to be so termed. 

 Fermat's demonstrations were all synthetical. His 

 treatise, though written as early as 1629, was only pub- 

 lished among his collected works in 1670. Schooten's 

 was published among his ' Exercitationes Mathema- 

 ticae' in 1657. Of the field thus left open, Dr. 

 Simson took possession, and he most successfully 

 cultivated every corner of it. Nothing is left without 

 the most full discussion; all the cases of each proposi- 

 tion are thoroughly investigated. Many new truths 

 of great importance are added to those which had 

 been unfolded by the Greek philosopher. The whole 

 is given with the perfect precision and the pure ele- 

 gance of the ancient analysis; and the universal assent 

 of the scientific world has even confessed that there is 



