tors. 



2 EXHIBITION OF A SERIES Oh PRIMES, &C. 



iuppolition the produa will not be greater than the next lefs 

 fquare, Teeing the factors are whole numbers. ""Now Tuppofe 

 / 2 — c to reprefent any number whatever, where/ 2 is the 

 next greater fquare, and c the excefs of that fquare above the 

 given number : then any two factors ~ of/ 3 — c may be ex- 

 preffed by/— p, and/+ p + 2 n, from whence the follow- 

 ing method of determining all the factors to two fucceflive 

 fquares, and their intermediate numbers, is eafily derived. 

 Method of de- In a perpendicular column at the left hand of your future 

 termining fac- vvor ]^ j^ fab. 1ft,) which exhibits the divifors of all the 

 numbers from 4-9 to 36, write down the arithmetic progreffion 

 1, 2, 3, &c. beginning at the top with unity, and ending 

 with/; let all thefe numbers fucceffively reprefent the divifor 

 / — p, and correfponding to each of them, draw a double 

 horizontal column : when/ — p zz 1, make c in the upper- 

 moil horizontal column ~2/ — 1, 3/— 2, 2/ — .3, &c. 

 to 0, becaufe 2/ — 1 is the difference of the two fquares in 

 all cafes ; and for /-f p -f 2 n, write/ — 1 l > / — l]* -f. l, 

 /■— l^ -f 2, &c. to/ 2 . In order to fill up the other co- 

 lumns, divide/' by the refpeclive value of/ — p, and place 

 the remainder, if there be any, in the upper part of the hori- 

 zontal column, perpendicularly under the fame value of c in 

 the higheft column ; but if there be no remainder, place a 

 cypher in its room : then increafe this remainder as often as 

 you can by the addition of/— p, not to make it exceed 

 2/ — 1, and place thefe, each directly under the fame num- 

 ber in the higheft column, and write c on the left hand. 



f 

 Under the firft remainder or cypher, place the quotient ;r ■ 



in whole numbers; from w hie k, fubrracl unity as oft as/ — p 

 was added to the remainder, and call thefe /-f- p -f. 2 n ; but 

 when/ — p zzf, put alone for the value of c : when all the 

 columns are filled up in this manner, the factors of any given 

 number are thus found. Find the value of c correfponding to 

 thai number in the higheft column ; then find how often the 

 fame number recurs, by cajiing your eye perpendicularly 

 down the columns, the factors will be the value of /-}- p -f 2 n, 

 directly under that value of c in each column ; and the corre- 

 fponding value of/—/;: confequcntly, thefe values of c 

 which are only found in the higheft column, indicate that the 



number 



