f 



INTRODUCTION. _ 27 



of everything known respecting these curves before his time. The last 

 four books contain his own discoveries. In the fifth book he treats of 

 the greatest and smallest lines which can be drawn from each point of 

 their circumference, and many other intricate questions, which required 

 the greatest sagacity and the most unremitting attention to investigate. 

 The sixth book is not very important nor difficult ; but the seventh con- 

 tains many very important problems, and points out the singular analogy 

 that exists between the properties of the various conic sections. The 

 eighth book has not come down to us. The fifth, sixth, and seventh 

 books, were discovered by Borelli, in Arabic, in the library of the grand- 

 duke of Tuscany. He got them translated, and published his translation, 

 with notes and illustrations, in the year 1661. Dr. Halley published an 

 t edition of Appolonius in 1710, and has supplied the eighth book from 

 the account given by Pappus of the nature of its contents. 



Archimedes was, beyond dispute, the greatest mathematician that an- 

 tiquity produced. He was born in Sicily, about the year 2S7 before the 

 Christian era, and is said to have been a relation of Hiero, king of Syr- 

 acuse. So ardent a cultivator was he of the mathematics, that he was 

 accustomed to spend whole days in the deepest investigations, and was 

 wont to neglect his food, and forget his ordinary meals, till his attention 

 was called to them by the care of his domestics. His studies were par- 

 ticularly directed to the measurement of curvilinear spaces ; and he in- 

 vented a most ingenious method of performing such measurement, well 

 known by the name of the " Method of Exhaustions." 



When it is required to measure the space bounded bv curve lines, 

 the length of a curve, or the solid bounded by curve surface. 1 he inves- 

 tigation does not fall within the range of elementary geoiT. 1 -y. Recti- 

 linear figures are compared on the same principle as superposition ; but 

 this principle can not be applied to curvilinear figures. It occurred to 

 Archimedes, that, by inscribing a rectilinear figure within, and another 

 without the figures, two limits would be obtained, the one greater and 

 the other smaller than the area required. It was evident that, by in- 

 creasing the number, and diminishing the sides of these figures, these 

 two limits were made continually to approach each other. Thus they 

 came nearer and nearer to the curve area which was intermediate be- 



