102 A SHORT HISTORY OF SCIENCE 



about Syracuse, nor even that in Sicily, but also that on the whole 

 mainland, inhabited and uninhabited. There are others again 

 who do not indeed assume this number to be infinite, but so great 

 that no number is ever named which exceeds this. ... I will 

 attempt to show however by geometrical proofs which you will 

 accept that among the numbers which I have named . . . some 

 not only exceed the number of a sand-heap of the size of the earth, 

 but also of that of a pile of the size of the universe." He assumes 

 that 10,000 grains of sand would make the size of a poppy-seed, 

 that the diameter of a poppy-seed is not less than one-fortieth of a 

 finger-breadth, that the diameter of the earth is less than a million 

 stadia, that the diameter of the universe is less than 10,000 di- 

 ameters of the earth. To express the vast number which results 

 .from these assumptions 10 63 in our notation he employs 

 an ingenious system of units of higher order comparable with the 

 modern use of exponents, an immense advance on current arith- 

 metical symbolism. 



MECHANICS OF ARCHIMEDES. In mechanics Archimedes is 

 a pioneer, giving the first mathematical proofs known. In two 

 books on Equiponderance of Planes or Centres of Plane Gravities, 

 he deals with the problem of determining the centres of gravity 

 of a variety of plane figures, including the parabolic segment. 

 A treatise on levers and perhaps on machines in general has been 

 lost, as also a work on the construction of a celestial sphere. A 

 sphere of the stars and an orrery constructed by him were long 

 preserved at Rome. He describes an original apparatus for deter- 

 mining the angular diameter of the sun, discussing its degree of 

 accuracy. 



The lever and the wedge had been practically known from 

 remote antiquity, and Aristotle had discussed the practice of 

 dishonest tradesmen shifting the fulcrum of scales towards the 

 pan in which the weights lay, but no previous attempt at exact 

 mathematical treatment is known. 



Archimedes assumes as evident at the outset : 



(1) Magnitudes of equal weight acting at equal distances from 

 their point of support are in equilibrium ; 



