# Mathematical Monthly Volume 2

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1860 edition. Excerpt: ... ri-j-1. Now let there be any number abcdefghih, and let it be separated into periods of three figures. For the sake of convenience, let each period be denoted by a single letter; then a, bed, efg, hi A = Impq. The number Imp q may be put under the form lmpq = 109/+ 106?w + l&p + q, = (In--1) /-f (7ri + 1) m + In"--l)p + q, = 7 (n-f ri-f ri')--I-f m--p + q. Now the part 7 (n-f-ri-f-ri') is divisible by 7, consequently, if the part--l--m--p-j-q is divisible by 7 the number Impq is also. But this latter part is equal to the difference of the sums of the odd and even periods into which the given number was separated. By the same method the proposition may be proved true for the" numbers 11 and 13.--H. Willey, Esq., New Bedford, Mass. 3. Note on the Ox Question.--Would not the following come more properly under the definition of an analytical solution? u Jf a oxen in m weeks eat b acres of grass, and c oxen in n weeks eat d acres, how many oxen will eat e acres in p weeks, the grass being supposed to grow uniformly?" Let x--the required number, a the grass at first on an acre, and 3 the growth of one acre in one week; then b (a-j-m /3) = grass on b acres at end of m weeks, d(a--nfi) = d " n e(a--pj3) = 'e'" p" The quantity of grass consumed will be proportional to both the number of oxen and the time together; hence (1) b (a--w/3): d(a--n8):: ma: nc, (2) b (a-f-m3):, e (a-j-:: ma: px. m gives 3--(ma"--ncb)a (O) crivesar--T '('+Pfi i) gives p--mn (ic _ ad), gives x-----j---. (m--p nee /n--p mae m--n/ p d m--nr pb Any one of the quantities may be assumed as unknown, and its value obtained from this last equation.--Samuel Schooler, Principal of Edge Hill School, ...