3l6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1772. 



the composite might be distinguished, not a table constructed we know not how, 

 that was the invention of Eratosthenes, to which from its use, as well as from 

 the nature of the operation, which proceeds, as will be shown, by a gradual ex- 

 termination of the composite numbers from the arithmetical series 3, 5, 7, 9, 1 1, 

 &c. infinitely continued, its author gave the name of the Sieve. I have thought 

 it necessary to premise these remarks, to remove a prejudice, which I apprehend 

 many may have conceivai, as this beautiful and valuable edition of Aratus is in 

 every one's hands, that this ill-contrived table, the useless work of some monk in 

 a barbarous age, was the whole of the invention of the great Eratosthenes, and 

 in justice to myself, that I might not be suspected of attempting to reap another's 

 harvest. 



I now proceed, to give a true account of this excellent invention ; which, for 

 its usefulness, as well as for its simplicity, I cannot but consider as one of the 

 most precious remnants of ancient arithmetic. I shall venture to represent it 

 according to my own ideas, not obliging myself to conform, in every particular, 

 to the account of Nicomachus, which I am persuaded is in many circumstances 

 erroneous. In stating the principles on which the operation of the sieve was 

 founded, he hath added observations on certain relations of the odd numbers to 

 one another, which are certainly his own, because they are of no importance in 

 themselves, and are quite foreign to the purpose. Every thing of this kind I 

 omit: and having stated what I take to have been the genuine theory of Eratos- 

 thenes's method, cleared from the adulterations of Nicomachus, I deduce from 

 it an operation of great simplicity, which solves the problem in question with 

 wonderful ease, and which, because it is the most simple that the theory seems 

 to afford, I scruple not to adopt as the original operation of the sieve, though 

 nothing like it is to be found in Nicomachus; though, on the contrary, 

 Nicomachus, and all his commentators, would suggest an operation very different 

 from it, and far more laborious. For the satisfaction of the curious and the 

 learned, I have annexed a copy of so much of Nicomachus's treatise, as relates 

 to this subject, with such corrections of the text, as it stands in the edition 

 of Wichelius, printed at Paris ann. 1538, as the sense hath suggested to me, 

 or I have thought proper to adopt, on the authority of a manuscript preserved 

 among those of Archbishop Laud, in the Bodleian library; which, in this part, 

 I have carefully collated. By comparing this with the account which I subjoin, 

 every one will be able to judge how far I have done justice to the invention I have 

 undertaken to explain. 



Problem. — To find all the Prime Numbers. 

 . The number 2 is a prime number; but, except 2, no even number is prime, 

 because every even number, except 2, is divisible by 2, and is therefore com- 

 posite. Hence it follows, that all the prime numbers, except the number 2, 



