68 GEEEK GEOMETRY. 



contains are sufficiently explicit, as those of the Loci Plani 

 and the Determinate Section. Accordingly, former geo- 

 metricians 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 given purely algebraic solutions and demon- 

 strations. Fermat, one of the greatest mathematicians of the 

 seventeenth century, had proceeded more according to the 

 geometrical rules of the ancients ; but he had 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 TO-KOQ avaXvoptvov 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 de- 

 monstrations were all synthetical. His treatise, though 

 written as early as 1629, was only published among his col- 

 lected works in 1670. Schooten's was published among his 

 ' Exercitationes Mathematics ' 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 proposition are 

 thoroughly investigated. Many new truths of great import- 

 ance are added to those which had been unfolded by the 

 Greek philosopher. The whole is given with the perfect 

 precision and the pure elegance of the ancient analysis ; and 

 the universal assent of the scientific world has even confessed 

 that there is every reason to consider the restored work as 

 greatly superior to the lost original. 



The history of this excellent treatise shows in a striking 

 manner the cautious and modest nature of its author. He 

 had completed it in 1738 ; but, unsatisfied with it, he kept it 

 by him for eight years. He could not bring himself to think 

 that he had given the " ipsissimre propositiones of Apollonius 

 in the very order and spirit of the original work." Pie was 



