348. 
is to be made a theorem in symbolical 
algebra, whether the series be finite or in- 
finite, and ail that remains is to find the 
different ways in which it may be inter- 
preted. The process of generalization pro- 
ceeds by steps. For instance, it asks : Will 
the series retain the same form when n is 
generalized so as to include any rational 
fraction? This is one of the questions 
which Newton proposed to himself, and 
settled in the affirmative; and it is re- 
corded that he verified the truth of his 
conclusion by squaring the series for 
(1—2°)3. Peacock’s principle does not 
distinguish divergent from convergent 
series; it is nothing but hypothesis, and 
any result suggested by it must stand the 
test of independent investigation. 
An important advance in the philosophy 
of the fundamental principles ofalgebra was 
made by D. F. Gregory, a younger member 
of the Cambridge school of mathematicians. 
Descended from a Scottish family, already 
famous in the annals of science, he early 
gave promise of adding additional luster to 
the name ; this he accomplished in a brief 
life of thirty-one years. In 1838 he read a 
paper before the Royal Society of Edin- 
burgh ‘On the Real Nature of Symbolical 
Algebra,’ in which he says: ‘ The light in 
which I would consider symbolical algebra 
is, that it is the science which treats of the 
combination of operations defined not by 
their nature, that is, by what they are or 
what they do, but by the laws of combina- 
tion to which they are subject. And as 
many different kinds of operations may be 
included,in a elass defined in the manner 
I have mentioned, whatever can be proved 
of the class generally, is necessarily true of 
all the operations included under it. This, 
it may be remarked, does not arise from any 
analogy existing in the nature of the opera- 
tions which may be totally dissimilar, but 
merely from the fact that they are all sub- 
ject to the same laws of combination. It is 
SCIENCE. 
(N.S. Vou. X. No. 246. 
true that these laws have been in many 
cases suggested (as Mr. Peacock has aptly 
termed it) by the laws of the known opera- 
tions of number; but the step which is 
taken from arithmetical to symbolical 
algebra is, that leaving out of view the 
nature of the operations which the symbols 
we se represent, we suppose the existence 
of classes of unknown operations subject to 
the same laws. We are thus able to prove 
certain relations between the different 
classes of operations, which, when ex- 
pressed between the symbols are called 
algebraical theorems. And if we can show 
that any operations in any science are sub- 
ject to the same laws of combination as 
these classes, the theorems are true of these 
as included in the general case; provided 
always that the resulting combinations are 
all possible in the particular operation 
under consideration.”’ 
It will be observed that he places algebra 
ou a formal basis; for its symbols are de- 
fined, not to represent real operations, but 
by laws of combination arbitrarily chosen. 
In a subsequent paper, however, entitled 
‘ On a Difficulty in the Theory of Algebra,’ 
he practically gave up the formal view, and 
appears inclined to adopt the realist view 
instead. He says: ‘‘In previous papers 
on the theory of algebra I have maintained 
the doctrine that a symbol is defined alge- 
braically when its laws of combination are 
given; and that a symbol represents a given 
operation when the laws of combination of 
the latter are the same as those of the 
former. This, or a similar theory of the 
nature of algebra seems to be generally en- 
tertained by those who have turned their 
attention to the subject; but without in 
any degree leaning on it, we may say that 
symbols are actually subject to certain laws 
of combination, though we do not suppose 
them to be so defined ; and that a symbol 
representing any operation must be subject 
to the same laws of combination as the 
