f 337 ] 



the method of the Sieve. And he could not mean, 

 to apply this method to a problem, to which ano- 

 ther was better adapted. 



Laftly, Eratofthenes could not mean, that the 

 method of the Sieve mould be applied to the find- 

 ing of all the poffible divifors of any Compoiite 

 number propofed, becaufe he could not be unac- 

 quainted with a more ready way of doing this, 

 founded upon two obvious Theorems, which could 

 not be unknown to him. 



The Theorems I mean are thefe. 



ift. If tws Prime numbers multiply each other, 

 the number produced hath no divifors but the two- 

 prime faBors. 



id. If a Prime number multiply a Compofde num- 

 ber, and likewife multiply all the divifors of that 

 compofite fever ally, the numbers produced by the mul- 

 tiplications of thefe divifors will be divifors of the 

 number produced by thefirfl multiplication: And the 

 number produced by the firft multiplication will have 

 no divifors, but the two faffors, the divifors of the 

 Compofte faffior, and the numbers made by the multi- 

 plication of thefe divifors by the Prime fa El or fever ally. 



The method of finding all the divifors of any 

 Compoiite number, delivered by Sir Ifaac New- 

 ton in the Arithmetica Univerfalis, and by Mr. 

 Maclaurin in his Treatife of Algebra, may be 

 deduced from thefe proportions, as every ma- 

 thematician will eafily perceive. This method 

 requires indeed that the lead prime divifor mould 

 be previoufly found ; and, if the leaf! prime di- 

 vifor ihould happen to be a large number, as it 

 is not affignable by any general method, the 



Vol. LX1J. X x inve. 



