POWER. In FacroniaLs, the manner in which the term power was introduced into arithmetic is seen. By definition, the fourth power of x means the product of four xes, or xxxxxxx; and the same of other powers. But it is far more symmetrical to begin from unity ; and to say that the fourth power of x is the meta of four multi. pliestiona by x, unity being understood as the commencement. Thus the auccemise powers of x, first, second, third, fie., are 1 x x, 1 x x x .r, larxrx x, dc.: denotes" by .r', rt, &c. And the term root is the inverse of power, as follows : If A be the mth power of sot is the lath root of A, denoted by VA. The peculiar algebraical character of the roots is explained in Hoot..
It is thus easily proved that when as and a are any two integers, that when as is greater than a, Also that and that whenever ist is divisible by a without remainder. These rules, if applied in defiance of the restrictions first mentioned, lead to such results as , &c., which are unintelligible so far as the definitions have yet been stated. Their proper interpretations [INTERPRETATION] are as follows : First, a" must be allowed to stand for unity, whatever x may be ; secondly, must be understood to be 1 -;-x; thirdly, x, en and n being positive integers, must stand for Vrm. When these new definitions are added, all the rules remain true, whether m and a be positive or negative, integral or fractional : and the system of algebraic powers is complete.
An algebraic expression is said to be arranged in powers of a letter, say r, when the powers of that letter which enter are made to enter in ascending or descending order of algebraic magnitude. Thus ars+ is not at present arranged at alL To arrange it in ascending or descending powers of x, wo must write it thus - + ax= ascending ; + descending.
But even yet it is incomplete for many algebraical purposes, having no written indication of the fact that the ascent or descent is interrupted. Completely written in ascending powers, it should be + + Ox' + ars + Written in this form, which may remind us of the use of a cipher in writing ordinary numbers, it is clear that we hardly read the expression less easily, and write it much more briefly, if we omit x and its powers altogether, and snake some distinctive mark, analogous to the decimal point, between the parts which belong to the positive and negative powers. Thus the above might be written
3+0-1+ 1 0+0+a+0-1, or 1 +0+a+04-0 I-1+0+3; the mark 1 being on that side of the adjacent + or which belongs to the positive powers. This mark however is not necessary in what follows.
The late Mr. Horner [INvoLuTioN, &e.] was the first who suggested the systematic rejection of the ascending or descending powers. An example of multiplication and division will sufficiently explain its use.
Suppose it required to multiply and + 4z-5 : 2 + 0 + 1-4 5 7 2 + 0-3 14+0+7-28-35 4+0 2+ 8+10 6+ 0 3+12+15 14-4+7-36-27+ 7+12+15 Accordingly the answer is 12x +15; and every stroke of the pen which the usual method contains, more than is in the preceding, is mere waste, and risk of error into the bargain. Now let it be proposed to divide + at by re + 2a.rs as : 4 3+ 2+ 0+ 0-11-1(1+2+0-1 4+ 8+ 0 4 4 11 + 24-44 11+ 2+ 4+ 0 11-22+ 0+11 24+ 4-11-11 , 24+48+ 0-24 44-11+l3 1 44-83+ 0+44 77 +18 45 Accordingly the quotient is liars + 44e, and the re mainder is Mr. Horner himself did not lice to print this suggestion, which, simple as it ia, seems never to have been made before him. The possessor of his papers, Mr. T. S. Davies, of Woolwich, published somo extracts from those papers in an appendix to a reprint of the paper on the solution of equations, which reprint appeared in the' Ladies Diary' for 1838 ; having previously introduced the simplification into the I 1 th edition of Hutton'a Conroe. Since that time a paper on ' Algebraical Transformation,' sent by Mr. Horner to the Royal Society, but not printed in the Philosophical Transactions,' has been pub lished in the first and second volume of the' Mathematician.' Details and example* are given in Mr. Davies" Solutions of Questions con tained in Hutton, Course,' 1840, and In the 12th edition of that course, 1841.