MATHEMATICS. 317 



The conoids and spheroids are solids described by the revolution of conoids and 

 a conic section about its axis. These he considers, as also the sections s P heroids - 

 which are made in them by planes, the solid content of the parabolic 

 conoid, &c. This subject appears to have given him more trouble 

 than the rest, for he informs his correspondent that he long kept back 

 the proofs of his theorems on it, because he found some difficulty and 

 j doubt; "at least," he says, "going over them more carefully, I 

 | satisfied my scruples." 



Besides these works which are addressed to Dositheus, we have his 

 measurement of the circle ; in which he determines the circumference 

 to be between 3 and 3f times its diameter. The method which he 

 uses might easily be extended to greater accuracy by the assistance of 

 a proper system of arithmetic. 



The Greek arithmetic is the subject of his ' Psammites, or Number- Numbering 

 ing of the Sand,' of which he thus explains the purpose to Gelo, the the sand< 

 son of his king Hiero, and associated with him in the throne : " There 

 are persons, king Gelo, who think that the grains of the sand are infi- 

 finite in number ; I mean not merely the sands about Syracuse and 

 the rest of Sicily, but those of the whole earth, inhabited and uninha- 

 bited. Others think that they are not infinite, but that no number 

 can be expressed which shall exceed this multitude. Now, I shall 

 attempt to show by geometrical proofs, which you will be able to 

 follow, that among the numbers which I have expressed and pub- 

 lished in my books to Zeuxippus, there are some which exceed, not 

 only the multitude of the sands which would fill the earth, but of 

 those which would fill the universe. You understand that by the 

 universe is meant, by most astronomers, the sphere of which the centre 

 is the earth, and the radius the distance of the sun from the earth." 

 He then proceeds to some reasonings to establish that this distance is 

 less than 10,000 of the earth's radii ; l and to show that if we conceive 

 a globe of this magnitude to be formed of grains of sand, the fortieth 

 of an inch in diameter, their number may be reckoned. With our 

 present mode of notation, there is no difficulty in increasing numbers 

 to any magnitude whatever. But the Greek system, less perfect than 

 the Arabic, though much superior to the numeration of other countries, 

 required some contrivance to carry it to the requisite extent. The 

 Greek geometer answered this purpose by dividing the figures into 

 periods, the unit in each period being a myriad myriad, or ten million 

 times the unit in the preceding. The Greeks could thus go on with 

 their numbers as far as they might choose, though still their method 

 i did not afford them the same facilities which we derive from ours, in 

 | arithmetical operations. 



Of the astronomical labours of Archimedes, none have reached our Astronomy. 



1 It is, in fact, about 24,000 of the earth's radii ; but this difference does not 

 1 effect the reasonings of Archimedes. He founded his calculations on the supposition 

 i made by Aristarchus of Samos, that the sun's diameter was not greater than thirty 

 i times the earth's. 



