42 HISTORY OF SCIENCE. 



latter, if alloyed with silver or bronze, would displace more water than 

 the lump of gold. When this idea flashed upon the bather's mind, he 

 was so overjoyed at his discovery that he leaped from the bath, and 

 ran home, unclad as he was, crying ''Eureka! Eureka! I have found 

 it out ! I have found it out ! " Thus led to make experiments on the 

 weight of bodies in air and in water, Archimedes soon arrived at the 

 general fact which is still called in our text-books the principle of Ar- 

 chimedes, and is thus expressed : " Every body immersed in a liquid 

 loses of its weight a portion equal to the weight of the liquid it dis- 

 places/' The true explanation, however, of this apparent loss of weight 

 in a solid body immersed in a liquid was reserved for modern science. 



In geometry Archimedes determined by calculation the perimeters 

 of inscribed and circumscribed polygons of ninety-six sides, and that 

 the ratio of the circumference of the circle to the diameter lay between 

 3^ and 3^. Treatises by him on the spiral called by his name, on 

 spheroids and conoids, and on the sphere and the cylinder, have come 

 down to us, and these contain some beautiful and interesting theorems. 



He devised methods of calculating the surfaces and solid contents 

 of these forms and of given portions of them. One celebrated de- 

 monstration proves the singularly simple relation which exists between 

 a sphere and a cylinder exactly enclosing"trTe~2priere, namely, that the 

 surface of the sphere has two-thirds the extent of the surface of the 

 cylinder (including in the latter its two bases), and that the solid con- 

 tents of the sphere are also two-thirds of those of the cylinder. Archi- 

 medes showed also that the curved surfaces of the cylinder and sphere 

 comprised between any two planes perpendicular to the axis of the 

 cylinder are equal. The discovery of these relations was so gratifying 

 to Archimedes, that he requested that after his death his tomb should 

 have a cylinder and a sphere figured upon it. The method of exhaus- 

 tions received at the hands of Archimedes a new and beautiful appli- 

 cation in his method of determining the area of a parabola. He in- 

 scribes a triangle in the parabola, then in each of the two remaining 

 segments another triangle, after that another triangle in each of the 

 four, eight, sixteen, etc., segments successively remaining. The sum 

 of these triangular areas continually approaches that of the parabola. 

 He shows that if the area of first inscribed triangle be called i, that 

 of the next two triangles will be i, that of the next four T T , and so on. 

 Therefore the sum of the series is i +-f yV+^4 -f- etc - It mav surprise 

 n reader unacquainted with mathematics, that the addition of any 

 number whatever of successive terms will not increase the sum of this 

 series beyond a certain limit. Continual addition only brings the sum 

 nearer to the limit, which is i^-. This investigation, therefore, involved 

 the notion of a new kind of limit, namely, one to which a numerical 

 quantity might continually approach. We see, therefore, that the area 

 bounded by the parabolic curve must be precisely equal to i^ times 

 the area of the inscribed triangle. The mere enumeration of all the 



