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Every number which is not Prime, is a multi- 
ple of fome Prime number, as Euclid hath demon- 
ftrated (Element. 7. prop. 33.) Therefore the 
foregoing feries confifts of the Prime numbers, and 
of multiples of the Primes. And the multiples, of 
every number in the feries, follow at regular dif 
tances ; by attending to which circnmftance, all the 
multiples, that is, all the Compolite numbers, 
may be eafily diftinguifhed and exterminated. 
I fay, the multiples of all numbers, in the fore- 
going feries, follow at regular diftances. 
For between. 3 and its firft multiple in the feries 
(9) two numbers intervene, which are not multi- 
ples of 3. Between 9 and the next multiple of 3 
( 1 5) two numbers likewife intervene, which are 
not multiples of 3. Again between 15 and the 
next multiple of 3 (21) two numbers intervene, 
which are not multiples of 3 ; and lb on. Again, 
between 5 and its firft multiple (15) four numbers 
intervene, which are not multiples of 5. And be- 
tween 1 5 and the next multiple of 5 (25) four 
numbers intervene which are not multiples of 5 ; 
and fo on. In like manner, between every pair of 
the multiples of 7, as they ftand in their natu- 
ral order in the feries, 6 numbers intervene which, 
are not multiples of 7. Univerfally, between every 
two multiples of any number «, as they ftand in 
their natural order in the feries, n — 1 numbers in- 
tervene, which are not multiples of n. 
Hence may be derived an Operation for extermi- 
nating the Compofite numbers, which I take to 
have been the Operation of the Sieve, and is as 
follows. 
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