rOL. LXn.J PHILOSOPHICAL TRANSACTIONS. 3l7 



are included in the series of the odd numbers, in their natural order, infinitely 

 extended; that is, in the series 3, 5, 7, 9. ^h 13, 15, 17, IQ, 21, 23, 25, 27, 

 29, 31, 33, 35, 37, SQ, 41, 43, 45, 47, 49, 51, &C. 



Every number which is not prime, is a multiple of some prime number, 

 as Euclid hath demonstrated (element 7, prop. 33). Therefore the foregoing 

 series consists of the prime numbers, and of multiples of the primes. And the 

 multiples, of every number in the series, follow at regular distances; by attend- 

 ing to which circumstance, all the multiples, that is, all the composite numbers, 

 may be easily distinguished and exterminated. 



I say, the multiples of all numbers, in the foregoing series, follow at regular 

 distances. For between 3 and its first multiple in the series, 9, two numbers 

 intervene, which are not multiples of 3. Between Q and the next multiple of 3, 

 (15), two numbers likewise 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 so on. Again, between 5 and its first multiple (15) 

 four numbers intervene, which are not multiples of 3. And between 15 and 

 the next multiple of 5 (25) four numbers intervene, which are not multiples of 

 5; and so on. In like manner, between every pair of the multiples of 7, as 

 tliey stand in their natural order in the series, 6 numbers intervene, which 

 are not multiples of 7- Universally, between every two multiples of any 

 number n, as they stand in their natural order in the series n — 1 numbers 

 intervene, which are not multiples of n. 



Hence may be derived an operation for exterminating the composite numbers, 

 which I take to have been the operation of the sieve, and is as follows. 



The Operation of the Sieve. 



Count all the terms of the series following the number 3, by threes, and 

 ^expunge every 3d number. Thus all the multiples of 3 are expunged. The 

 first uncancelled number that appears in the series, after 3, is 5. Expunge the 

 square of 5. Count all the terms of the series, which follow the square of 5, 

 by fives, and expunge every 5th number, if- not expunged before. Thus all the 

 multiples of 5 are expunged, which were not at first expunged, among the 

 multiples of 3. The next uncancelled number to 5 is 7. Expunge the square 

 of 7. Count all the terms of the series following the square of 7, by sevens, 



3. 5. 7. 9- II. 13. 15. 17. 19- 'il. 23. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 

 45.47. 49. 5i. 53. 55. 57.59.61. 63.65. 67.69. 71.73, 75. 77.79.81.83.85. 

 SV. 89. 91.93. 95.97. 99- 101. 103, 105. 107. 109. 111. 113. 115. 117. 119. 

 121. 123. 125. 127. 129, 131. 133. 135. 137. 139. l^V. 143.. 145. 147. I49. 

 15i. 153. 155. 157. 



