VOL. LXII.] PHILOSOPHICAL TRANSACTIONS. 315 



But before entering expressly on the subject, I must take the liberty to ani- 

 madvert on a certain table, which, among other pieces ascribed to Eratosthenes, 

 is printed at the end of the beautiful edition of Aratus published at Oxford in 

 the year 1672, and is adorned with the title of Koa-xuo* EfXTooQtvxi. It contains 

 all the odd numbers from 3 to 1 1 3 inclusive, distributed in little cells, all the 

 divisors of every composite number being placed over it, in its proper cell, and 

 the prime numbers are distinguished, so far as the table goes, by having no 

 divisors placed over them. It has probably been copied either from a Greek 

 comment on the Arithmetic of Nicomachus, preserved among the manuscripts 

 of Mr. Selden in the Bodleian library, in which, though the manuscript is now 

 so much decayed as to be in most places illegible, I find j)lain vestiges of such a 

 table,* which might be more perfect 100 years ago, when the Oxford Aratus was 

 published; or else, from another comment, translated from a Greek manuscript 

 into Latin, and published in that language, by Camerarius, in which a table of 

 the very same form occurs, extending from the number 3 to lOQ inclusive. It 

 may sufficiently skreen the editor of Aratus from censure, that he had these 

 authorities to publish this table as the Sieve of Eratosthenes; especially as they 

 are in some measure supported by passages of Nicomachus himself. But the 

 Sieve of Eratosthenes was quite another thing. 



The Oxford editor has annexed to his table, to explain the use of it, some 

 detached passages, which he has selected from the text of Nicomachus, and from 

 a comment on Nicomachus ascribed to Joannes Grammaticus. In these passages 

 the difference between prime and composite numbers is explained, in many words 

 indeed, but not with the greatest accuracy ; and it is proposed to frame a kind 

 of table of all the odd numbers, from 3 to any given limit, in which the com- 

 posite numbers should be distinguished by certain marks.-|- The primes would 

 consequently be characterised, as far as the table should be carried, by being un- 

 marked. But, on what principles, or by what rule, such a table is to be con- 

 structed, is not at all explained. It is obvious that, in order to mark the com- 

 posite numbers, it is necessary to know which are such. And, without some 

 rule to distinguish which numbers are prime, and which are composite, inde- 

 pendent of any table in which they shall be distinguished by marks, it is impos- 

 sible to judge whether the table be true as far as it goes, or to extend it, if requi- 

 site, to a further limit. Now it was the rule by which the prime numbers and 



* This manuscript seems to have contained the text of Nicomachus with Scholia in the margin. 

 But the table evidently belongs to the Scholia, not to the text. — Orig. 



f Nicomachus and Joannes Grammaticus propose that these marks should be such, as should not 

 only distinguish the composite numbers, but likewise serve to express all the divisors of every such 

 number. It will be shown, in a proper place, that this was no part of the original contrivance of the 

 Sieve. — Orig. 



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