318' PHILOSOPHICAL TRANSACTIONS. [aNNO 1772. 



and expunge every 7th number, if not expunged before. Thus all the multiples 

 of 7 are expunged, which were not before expunged among the multiples of 3 or 

 5. The next uncancelled number which is now to be found in the series, after 7, 

 is 1 1 . Expunge the square of 1 1 , Count all the terms of the series, which 

 follow the square of 11, by elevens, and expunge every 1 1 th number, if not 

 expunged before. Thus all the multiples of 1 1 are expunged, which were not 

 before expunged among the multiples of 3, 5, and 7. Continue these expunc- 

 tions, till the first uncancelled number that appears, next to that whose multiples 

 have been last expunged, is such, that its square is greater than the last and 

 greatest number to which the series is extended. The numbers which then 

 remain uncancelled are all the prime numbers, except the number 2, which occur 

 in the natural progression of numbers from 1 to the limit of the series. By the 

 limit of the series I mean the last and greatest number to which it is thought 

 proper to extend it. Thus the prime numbers are found to any given limit. 



Nicomachus proposes to make such marks over the composite numbers, as 

 should show all the divisors of each. From this circumstance, and from the 

 repeated intimations, both of Nicomachus, and his commentator Joannes 

 Granimaticus,* one would be led to imagine, that the sieve of Eratosthenes 

 was something more than its name imports, a method of sifting out the prime 

 numbers from the indiscriminate mass of all numbers prime and composite, and 

 that, in some way or other, it exhibited all the divisors of every composite 

 number, and likewise showed whether two or more composite numbers were 

 prime or composite with respect to each other. I have many reasons to think 

 that this was not the case. I shall as briefly as possible point out some of the 

 chief, for the matter is not so important, as to justify my troubling the society 

 with a minute detail of them. First then, in the natural series of odd numbers, 

 3, 5, 7, &c. every number is a divisor of some succeeding number. Therefore 

 if we are to have marks for all the different divisors of every composite number, 

 we must have a different mark for every odd number. Therefore we must have 

 as many marks, or systems of marks, as numbers; and I do not see that it 

 would be possible to find any more compendious marks, than the common 

 numeral characters. This being the case, it would be impracticable to carry 

 such a table as Nicomachus proposes, and his commentators have sketched, to a 

 sufficient length to be of use, on account of the multiplicity of the divisors of 

 many numbers, and the confusions which this circumstance would create.-|~ It 

 is hardly to be supposed, that Eratosthenes could overlook this obvious difficulty, 



* The comment of Joannes Grammaticus is extant in manuscript in the Savilian library at Oxford, 

 to which 1 have frequent access, by the favour of the Rev. and learned Mr. Hornsby, the Savilian 

 professor of astronomy. — Orig. 



f The number 3i65 hath no less than 22 different divisors. — Orig. 



