Proof of the rule for adding the square of half the coefficient of x in the quantity x2±px, in order to complete the square. Let a the quantity to be added, •. x2± px+a is a complete square. .. since four times the product of the extremes = the square of the mean, we have, 4 a x2= p2x2 or 4a= p2 and a= XXV. To prove that the sums of the reciprocals of the th powers of the odd and even numbers are to each other .. the sum of the odd: the sum of the even:: + 4.n 6n 1 1 I + 115 1 + &c. =S. + + &c. = 1 1 + + &c. = S = 2n In every scale of notation, whose radix is (r), the sum of all the digits, expressing any number, when divided by r−1, will leave the same remainder as the number itself when divided by r-1. 1 N=a(-1)+b(p?−1 − 1) + &c. +p (r22 −1)+q (r− 1) +a+b+c+ &c. +p+q+w. Now each of the quantities ( ” — 1) (p”—1 — 1) (r2 — 1) and (r-1) is divisible by (r-1); therefore no remainder can arise from the first line; hence it is evident that must r N leave the same remainder as (a+b+c+d+ &c.) divided by r-1, and if there be no remainder in the former case, there will be none in the latter. Hence if any number in the common scale be divisible by 9, without remainder, the sum of its digits will be so likewise; and if there be any remainder, it will be the same in both cases. XXVIII. Proof of multiplication by casting out the nines. Let (a) the multiplicand contain (m) nines with remainder a', and (b) the multiplier contain (n) nines with remainder b'. Then, a=9m+a', and b 9n+V'. .. ab=81mn+9 na'+9mb'+a'b', =9(9mn+na+mb')+a'b' ; .. ab divided by 9 leaves the same remainder as a' divided by 9; but (Prop. XXVII.) the remainder of a÷÷9 is the same as that of the digits of a÷9; and the same is true of b, and the same is true of the product ab. Hence the rule: Cast out the nines of the two factors,' &c. COR. The same will manifestly be true of 3, it being a divisor of 9. XXIX. Every number (N) is divisible by 8, if 4a+2a,2+a, be divisible by 8; a, a, a,, being the three first digits of the number, reckoning from the place of units. N=a.10"+a2-1•10′′~1+an_2•10′′-2 + &c...+a2.102+a1.10+ao. =an (10”—2′′)+a„_1(10"-1 — 2′′-1) + &C...+a2 (102 −22)+a,(10—2) +a".2" + a. 2′′-1 + &C... + a2.22 + a1.2+a ̧• Now, every term of the first line is divisible by (10-2), or 8, as also every term in the second line as far as a.29; ... when .. when Nis divided by 8, it must leave the same remainder as (a,.22+a1.2+a。)÷by 8, or as 4a2+2α1+α ̧• COR. 1. If a, be an even number, 4a, is divisible by 8, and in that case N÷8 leaves the same remainder as (2a1+a)÷8. COR. 2. If a, be divisible by 4, 2a, will be divisible by 8; and in this case, N÷8 leaves the same remainder as ao÷8. XXX. In any scale of notation, whose radix is (r), the difference of the remainders of the sum of the 1st, 3d, 5th, &c. digits divided by (r+1), and the sum of the 2nd, 4th, 6th, &c. digits divided by r+1, is equal to the remainder of the whole number divided by r+1. N=ar n―1+b(r n−1 + 1 ) + c{ v n−2 −1)+,&c. +p(r2 — 1)+q(r+1) Now (Prop. V.) when n is even, rn-1 is divisible by r+1, and so also are 1„-2 — 1, &c. . . . and r2-1. Also, in this case n— 1 is odd. .. r n−1+1, &c. and r+1, are all divisible by r+1. .. N+r+1 leaves the same remainder as (a+b+p+w÷r+1−b+, &c. . . . +q)÷by r+1. Again, if n be odd, N=a(r"+1)+b(r. n=1 — 1 ) +c(r n−2 + 1)&c...+p(r2—1)+q(r+1) |