[ 335 1 



of ii, by elevens, and expunge every eleventh 

 number, if not expunged before. Thus all the 

 multiples of r I are expunged, which were not be- 

 fore expunged among the multiples of 3, 5, and 7. 

 Continue thefe expunctions, till the firft uncancelled 

 number that appears, next to that whofe multi- 

 ples have been laft expunged, is fuch, that its 

 iquare is greater than the laft and greateft num- 

 ber to which the feries is extended.. The. 

 numbers which then remain uncancelled are all 

 the Prime- numbers, except the number 2, which 

 occur in the natural progreffion of number from 1 

 to the limit of the- feries. By the limit of the fe- 

 ries I mean the laft and greateft number to which 

 it is thought proper to extend it. 



Thus the prime numbers are found to any given- 

 limit., 



Nicomachus propofes to make fuch marks 

 over the Compofite numbers, as mould mew all 

 the divifors of each. From this circumftance*. 

 and from the repeated intimations both of Nico- 

 machus, and his commentator Joannes Gramma- 

 ticus *, one would be led to imagine, that the Sieve 

 of Eratofthenes was fomething more than its- name 

 imports, a method of lifting out the Prime num- 

 bers from the indifcriminate mafs of all numbers 

 Prime and Compofite, and that, in fome way or 

 other, it exhibited all the divifors of every Compo- 

 fite number, and like wife mewed whether two or 



* The Comment of Joannes Grammatieus is extant in ma- 

 nufcript in the Savilian Library at Oxford, to which I have 

 frequent accefs, by the favour of the Reverend and Learned 

 Mr, Hornfby, t^e Savilian Pofeflbr of Aftronomy. 



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