C 333 ] 



Every number which is not Prime, is a multi- 

 ple of fome Prime number, as Euclid hath demon- 

 flrated (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 dii- 

 tances ; by attending to which circumftance, all the 

 multiples, that is, all the Composite numbers, 

 may be eafily diftinguimed 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 

 (15) 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 ib on. Again,, 

 between 5 and its firft multiple (15) four numbers 

 intervene, which are not multiples of 5. And be- 

 tween 15 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 

 Ewo 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 toy 

 have been the Operation of the- Sieve, and. is as 

 follows* 



2. $hr 



