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...
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