SEPTEMBER 15, 1899. ] 
case do we get a digital number. He draws 
the line so as to exclude the fraction as a 
multiplier but not as a multiplicand ; accord- 
ing to his own principle it should be wholly 
excluded from arithmetical algebra. But 
arithmetic so restricted would be a very nar- 
row science, and the logical result would be 
to divide arithmetic itself into an arithmet- 
ical arithmetic and a symbolical arithmetic. 
He then describes what he means by 
‘symbolical algebra.’ ‘‘Symbolical algebra 
adopts the rules of arithmetical algebra but 
removes altogether their restrictions ; thus 
symbolical subtraction differs from the 
same operation in arithmetical algebra in 
being possible for all relations of value of 
the symbols or expressions employed. All 
the results of arithmetical algebra which 
are deduced by the application of its rules, 
and which are general in form, though par- 
ticular in value, are results likewise of sym- 
bolical algebra, where they are general in 
value as well as in form; thus the product 
of a” and a” which is a”t” when m and n 
are whole numbers, and therefore, general 
in form, though particular in value, will be 
their product likewise when m and n are 
general in value as well as in form; the 
series for (a + 6)” determined by the prin- 
ciples of arithmetical algebra when n is any 
whole number, if it be exhibited in a gen- 
eral form, without reference to a final term, 
may be shown upon the same principle to 
be the equivalent series for (a + 0)” when 
m is general both in form and value.” 
The principle here brought forward was 
named by Peacock the ‘principle of the 
permanence of equivalent forms ’; by means 
of it the transition is made from arithmet- 
ical algebra to symbolical, and at page 59 
of ‘Symbolical Algebra’ it is thus enun- 
ciated: ‘‘ Whatever algebraical forms are 
equivalent, when the symbols are general in 
form but specific in value, will be equiva- 
lent likewise when the symbols are general 
in value as well as in form.” 
SCIENCE. 
B47 
One asks naturally, ‘ What are the limits 
set to the generality of the symbol?’ Pea- 
cock’s answer is, ‘Whatsoever.’ In the 
theory of reasoning the great question is 
not, ‘ How do we pass from generals to par- 
ticulars ?’ but ‘How do we pass from par- 
ticulars to generals?’ The application of 
general principles is plain enough—the 
difficulty is in explaining how we arrive at 
the truth of the general principles. The 
logician, seeking for light on this question, 
is apt to turn to exact science, and es- 
pecially to algebra, the most perfect branch 
of exact science. Should he turn to Pea- 
cock, he finds that all that is offered him is 
this ‘ principle of the permanence of equiva- 
lent forms’; which, paraphrased, amounts 
to the following: We find certain theorems 
to be true when the symbol denotes in- 
teger number; let these theorems be true 
without restriction, and let us try to find 
the different interpretations which may be 
put on the symbol. Is not the following 
attitude more logical? We. find certain 
theorems to be true, when the symbol de- 
notes number ; how far and no further may 
the conception of number be generalized, 
yet these theorems remain true without any 
alteration of form?; and, should the con- 
ception of number be still further general- 
ized, what is the modified form which the 
theorems then assume? This-is the logical 
process of generalization, whereas Peacock’s 
process is ‘‘essentially arbitrary, though 
restricted with a specific view to its opera- 
tions and their results admitting of such 
interpretations, as may make its applica- 
tions most generally useful.’? (Report on 
Recent Progress in Analysis, p. 194.) 
The two processes may be illustrated by 
their application to the binomial theorem, 
proved to be true for a positive integer 
index. According to Peacock’s process, 
(a + b)” = a at na"—*b + a a"™— 7h? -- 
