1 



) -2G INTRODUCTION. 



j foundation of the celebrated Alexandrian school. He collected all the I 

 elementary facts known in mathematics before his time, and arranged j 

 them in such an admirable order beginning with a few simple axioms, 

 and deducing from them his demonstrations, every subsequent demon- 

 stration depending on and rigidly deduced from those that immediately 

 precede it that no subsequent writer has been able to produce any- > 

 thing superior or even equal. His "Elements" still continue to be j 

 taught in our schools, and could not be dispensed with, unless we were ) 

 to give up somewhat of that rigor which has been always so much ad- ( 

 J mired in the Greek geometricians. Perhaps, however, we carry this j 

 admiration a little too far. The geometrical axioms might be somewhat ) 

 enlarged, without drawing too much upon the faith of beginners. And < 

 were the method followed, considerable progress might be made in ) 

 mathematics without encountering some of those difficult demonstrations ) 

 that are apt to damp the ardor of beginners. 



The elements of Euclid consist of thirteen books. In the first four 

 he treats of the properties of lines, parallel lines, angles, triangles, and 

 circles. The fifth and sixth treat of proportions and ratios. The sev- 

 enth, eighth, ninth, and tenth, treat of numbers. The eleventh and 

 twelfth treat of solids ; and the thirteenth of solids : also of certain pre- 

 liminary propositions about cutting lines in extreme and mean ratio. It 

 is the first four books of Euclid chiefly that are studied by modern ge- 

 ometricians. The rest have been, in a great measure, superseded by 

 more modern improvements. 



Appolonius was born at Perga in Pamphylia, about the middle of the 

 second century before the Christian era. Like Euclid, he repaired to ( 

 Alexandria, and acquired his mathematical knowledge from the succes- s 

 sors of that geometrician. The writings of Appolonius were numerous , 

 and profound ; but it is upon his " Treatise on the Conic Sections," in \ 

 eight books, that his celebrity as a mathematician chiefly depends. 



The conic sections, which, after the circle, are the most important of / 

 all curves, were discovered by the mathematicians of the Platonic school ; ( 

 though who the discoverer was is not known. A considerable number j 

 of the properties of these curves were gradually developed by the Greek ! 

 geometricians. And the first four books of Appolonius are a collection < 



