[ 335 ] 
of ii, by elevens, and expunge every eleventh 
number, if not expunged before. Thus all the 
multiples of 1 1 are expunged, which were not be- 
fore expunged among the multiples of 3,5, and 7. 
Continue thefe expunfdions, till the firft uncancelled 
number that appears, next to that whofe multi- 
ples have been laid expunged, is fuch, that its 
fquare is greater than the laid and greatefd 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 laid and greatefd 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 fhould drew 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 Eratofdhenes 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 likewife (hewed whether two or 
* The Comment of Joannes Grammaticus is extant in ma- 
nufcript in the Savilian Library at Oxford, to which 1 have 
frequent accefs, by the favour of the Reverend and Learned 
Mr, Hornfby, the Savilian PofelTor of Aftronomy. 
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