314 PHILOSOPHICAL TRANSACTIONS. [anNO 1772. 



-XJT. -^n Essay on the Periodical Appearing and Disappearing of certain Birds 

 .. at different Times of the Year. By the Hon. Daines Barrington, Fice Pres. 

 R. S. p. 265. 



" See this paper, with some additions, in Mr. B.'s collected works. 



■t» 



XXJL KOIKINON EPATO20ENOT2; ; or, the Sieve of Eratosthenes. Bein^ 

 an Account of his Method of Ending all the Prime Numbers.* By the Rev 

 S. Horsley, F.R.S. p. 327. 



A prime number is such a one, as has no integral divisor but unity. A 

 number which has any other integral divisor, is composite. Two or more 

 numbers, which have no common integral divisor, besides unity, are said to be 

 prime with respect to each other. Two or more numbers, which have any 

 common integral divisor besides unity, are said to be composite with respect to 

 each other. 



To determine, whether several numbers proposed be prime or composite with 

 respect to each other, is an easy problem. The solution of it is given by Euclid, 

 in the first 3 propositions of the 7th book of the Elements, and is to be found 

 in many common treatises of arithmetic and algebra. But to determine, con- 

 cerning any number proposed, whether it be absolutely prime or composite, is a 

 problem of much greater difficulty. It seems indeed incapable of a direct solu- 

 tion, by any general method: because the successive formation of the prime 

 numbers does not seem reducible to any general law. And for the same reason, 

 no direct method has hitherto been hit on, for constructing a table of all the 

 prime numbers to any given limit. Eratosthenes, whose skill in every branch 

 of the philosophy and literature of his times, rendered his name so famous 

 among the sages of the Alexandrian school, was the inventor of an indirect 

 method, by which such a table might be constructed, and carried to a great 

 length, in a short time, and with little labour. This extraordinary and useful 

 invention is at present little, if at all, known, being described only by two 

 writers, who are seldom read, and by them but obscurely; by Nicomachus 

 Gerasinus, a shallow writer of the 3d or 4th century, who seems to have been 

 led into mathematical speculations, not so much by any genius for them, as by 

 a fondness for the mysteries of the Pythagorean and Platonic philosophy; and 

 by Boethius, whose treatise on numbers is but an abridgment of the wretched 

 performance of Nicomachus.-|~ I flatter myself therefore, that a succinct ac- 

 count of it will not be unacceptable to this learned society. 



* Other tracts on the subject of prime and composite mimbers, may be seen, collected with other 

 pieces, with this learned prelate's edition of Euclid's Data, printed at Oxford in 1803. 



+ There are more pieces than one of thi» Nicomachus extant. That which I refer to is intitled 

 Eiraywyi A^/*i)tixi). — Orig. 



