Power

But the greatest improvement in the operation of division, and one which contains the principle of a class of improvements, is one which Homer called the synthetic method, which amounts to deferring the actual steps of subtraction until they are wanted. If we were to proceed one step farther with the preceding division, —44 in the quotient would be followed by +77. This + 77, if we look at all its components from the beginning, arises from + 0-11— 0+ 88. In like manner —44 arises from + 0+4-0-48. Now arrange the process as follows :— 14— 3+ 2+ 0 + 0-11— 1 9: — 8+ 0+ 4 —11+24-44 +22+0+0+0 +1 —48 +88 4-11+24-441+77+13-45 Write the coefficients of the,dividend horizontally a, b, c, &e., and of the divisor vertically p, q, r, &c., taking care to change the sign of every term of the divisor except the first.

p'a+6+c+d+e+f+g+h +q + tiq ttr + us + ut + + let +.rt +yt +r + vg + + rs + tea+ xa+ys +s +mg + ter+ xr +yr + +xo +yq et+ v + +x+y1+ + x' Divide a by p, giving u, and then write vg, us, us, and ea in the suc cessive columns which follow that of v. Make up + 6+ ag, the second column, and divide by p, giving e : write vg,rr, vs, rt, in the successive columns which follow that of v. Make up c + yr + rg and divide by p, giving w : write wq, ter, we, set, in the columns which follow that of w, and so on. Then a + w + , &e. will give the coefficients of the quotient, and n' + v' + , &c., made fie., the columns which have not been used to find quotient terms, will give the coefficients of the remainder. For example, we want to find some terms of the quotient of divided by :— 11+0+0+0+ 1 —1 —1+3+0+ 0 + 0+ 0+0 +3 +1-3+12 —21+60 +0 —4+ 7 —20 1-1+4-7+201-41+60+0 Hence the quotient is x-1+ and the remainder is When the first coefficient is anything but unity, fractions are intro duced into the quotient. To avoid this, proceed as follows : Let a be the coefficient of the first term of the divisor. Multiply the successive coefficients of the dividend by 1, a, as, &c. : turn the first coefficient of the divisor into 1, and multiply the second, third, fourth, fic., by 1, a, as, &c. Proceed as above with the coefficients thus altered, and suppose that in the last line the quotient terms become u +r +se +,&c.,

and those for the remainder iil+v'-1-, &c. To find the true quotient coefficients write W — &c., a at a' • The Royal Society has been very unfortunate in Its decisions about papers involving improvement of calculation. Horner's paper on equations was nearly rejected, and was only saved by the earnest remonstrance of Mr. Davies Gilbert. Barrett's impravement of life contingency calculations was rejected; so was Weddle's remarkable paper (afterwards published by himself) on the solution of equations by successive factors. Itorner's paper on transformation was with drawn, not rejected. The report on it set forth that Mr. Horner had proved himself the most able and the most dexterous of all who had recently investigated the solution of equations ; but there was a doubt as to the methods of the paper being to much better than others, and possessing so much of novelty, as to fit them for the Transactions. The council were embarrassed : they could neither reject nor accept the paper; so they suggested that It should he withdrawn. it seems then that Mtor methods than any In use were not fit for the Phil. Trans., unless thu superiority was very decided. Now, first, this was a very unsound principle, and very unworthy of a 'dentine body ; secondly, Mr. liorner's methods do possess a very decided superiority, as all who have used them know, and for the true remainder coefficients write v' — + &e., ce. where a. is the last power used in the quotient terms, repeated, not the next one to it. Suppose, for example, we are to divide + by 2.ci + Here, since x descends and m ascends regularly, we throw out x and m, and the abridged dividend and divisor become 1+1+1+5 and 2+0+1+0-3 1 2 4 8 1 2 4 8 underneath which we have written the multipliers. Hence we begin with 1+2+4+40 and 1+0-2+0-24: 1:1+2+4+40 +0 +0-2+ 0 +24+48+48+864 +0-4+ 0+ 0+ 0 • +0 + 0 — 4—i +24 + 0 1+2+2+361+20-21+48+864 Hence the quotient is 1 1_,_ 2 m + 2 36 — x + ce and the remainder is 20 24 48 864 16 x 32 al + 64 1.3 + 123 x'One of the easiest modifications of this rule is tho division of ax.? + . . . by x —I) or x +p, as explained in FRACTIONS, DECOMPOSITION or.