VOL. LXII.] PHILOSOPHICAL TRANSACTIONS. 3\Q 



though Nicomachus hath not attended to it. Eratosthenes therefore could not 

 intend the construction of such a table. 



In the next place, such a table not being had, Eratosthenes could not but per- 

 ceive, that the determining whether two or more numbers be prime or composite 

 with respect to one another, is in all cases to be done more easily, by the direct 

 method given by Euclid, than by the method of the sieve. And he could not 

 mean to apply this method to a problem to which another was better adapted. 



Lastly, Eratosthenes could not mean, that the method of the sieve 

 should be applied to the finding of all the possible divisors of any composite 

 number proposed, because he could not be unacquainted with a more ready 

 way of doing this, founded on two obvious theorems, which could not be 

 unknown to him. The theorems I mean are these: 1. If two prime numbers 

 multiply each other, the number produced hath no divisors but the two prime 

 factors. 2. If a prime number multiply a composite number, and likewise 

 multiply all the divisors of that composite severally, the numbers produced by 

 the multiplication of these divisors, will be divisors of the number produced by 

 the first multiplication: And the number produced by the first multiplication will 

 have no divisors, but the two factors, the divisors of the composite factor, and 

 the numbers made by the multiplication of these divisors by the prime factor 

 severally. 



The method of finding all the divisors of any composite number, delivered by 

 Sir Isaac Newton in the Arithmetica Universalis, and by Mr. Maclaurin in his 

 Treatise of Algebra, may be deduced from these propositions, as every mathema- 

 tician will easily perceive. This method requires indeed that the least prime 

 divisor should be previously found; and if the least prime divisor should happen 

 to be a large number, as it is not assignable by any general method, the investi- 

 gation of it by repeated tentations may be very tedious. A table therefore of 

 the odd numbers,* in which the composite numbers should each have its least 

 prime divisor written over it, would be very useful. But Nicomachus's project 

 of framing a table in which each composite number should have all its divisors 

 written over it, is ridiculous and absurd, on account of the insuperable difficulties 

 which would attend the execution of it. 



The extracts from Nicomachus, and from Boethius, are omitted, as unneces- 

 sary to be retained here. 



XXIII. On the Effects of Elder, in Preserving Growing Plants from Insects 

 and Flies. By Mr. Christopher Qullet. p. 348. 



This paper relates to the effects of elder; 1st. In preserving cabbage plants 



• A table of the odd numbers would be sufficient : for the number 2 is the least pril»e divisor of 

 every even number ; and it is easy, even in the largest numbers, to try whether they are divisible by 2. 

 In our method of notation, this may always be known, by observing the last figure in the expression 

 of the number proposed. — Orig. 



