28 INTRODUCTION. 



tween them. He observed, by thus increasing the number of sides for a ? 

 great number of times successively, that he approached a certain as- 

 signable rectilinear area, and could come nearer to it than any difference 

 how small soever. It was evident that this rectilinear area was the 

 real size of the curvilinear area to be measured. It was in this way that 

 he found that two thirds the rectangle under the abscissa and crdinate 

 of a parabola, is equal to the area contained by the abscissa and ordi- 

 nate, and that part of the circumference of the parabola lying between 

 them. In the same way he obtained an approximate measure of the 

 area of the circle, demonstrating that if the radius be unity, the circum- 

 ference is less than 3}, and greater than 3}f. His two books on the 

 sphere and cylinder were conducted by a similar method of reasoning. 

 He measures the surface and solidity of these bodies, and terminates his 

 treatise by demonstrating that the sphere (both in surface and solidity) is 

 two thirds of the circumscribed cylinder. 



In the same spirit his " Treatise on Conoids and Spheroids" was 

 conducted. These names he gave to solids formed by the revolutions 

 of the conic sections round their axis. We pass over his researches on 

 the " Spiral el Archimedes," as it is usually called, though in reality dis- 

 covered by Conon, one of his friends ; but must notice the treatise enti- 

 tled " Psany nites," or " Arenarius." Some persons had affirmed that 

 no number, however great, was sufficient to express the number of 

 grains of sand situated on the seashore. This induced Archimedes to 

 write his treatise, in which he demonstrated that the fiftieth term of a 

 ducuple increasing progression is more than sufficient to express all the 

 grains of sand contained in a sphere, having for its diameter the distance 

 bf> \ween the earth and the sun, and totally filled with grains rf sand. 

 7 :ie treatise is short, but abstruse, in consequence of its imperfect method 

 o expressing numbers employed by the Greeks. Were our figures 

 ? bstituted for the Greeks letters, the reasoning would be sufficiently 

 .< nple and clear. 



Archimedes did not confine himself to pure mathematics : he turned 

 his attention likewise to mechanics, and may in some measure be con- 

 sidered as the founder of that important branch of physical science. He $ 

 first laid down the true principles of statics and hydrostatics. The fqr- / 





