The Spirit of Mathematical Analysis; And Its Relation to a Logical System (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1843 Excerpt: ...number. (C) If a is actual or imaginary but of the form p + q.i, all the m values of r/a are actual or imaginary but of the same form P+Q.i. (D) For these general roots the following laws hold: (I.) G/-)-= a; (II.) (O--; (1) V(ab) = ya.yb; (2) y(a: b) = ya: $b; (3) VWa) = ""Ja; in such a manner, namely, that exactly as many and exactly the same values stand on both sides of these equations, as by (sect. 3) was required of a correct equationt. In the same sense the equation (4) #(-")=w-r, where n is considered as a positive or negative whole number or zero, is not correct, because the expression on the left has m values, while that on the left may have less J, and will only have as many values when m and n have no common divisor. The unskilful application of this equation (No. 4.) therefore is to be looked upon as one of the sources of the paradoxes of calculation. (E) We may not put (p + q) X/a for p. a + q. %/a, nor (z/a)2 for Z/a./a, nor from ifa = a and $a = /3 conclude that a--(3, unless we have previously ascertained that the same symbol %a wherever it occurs in the same expression always represents one and the same of its m values, --a certainty which, in perfectly general investigations, we are, owing to the nature of the subject, usually unable to obtain. In the same way we may not substitute a for v/(am) because the latter root denotes one of m different values which has no need to be a exactly, but which is certainly contained in a. Z/l. Generally, therefore, treat such a general ma root as what it is, viz. a symbol which represents, wherever it occurs, one of m different forms, or all the m forms together, but leaves indeterminate which one is intended, --and you will never fall into contradictions, whereas the many-mean...