SEPTEMBER 15, 1899. ] 
operation it represents.” This is a depar- 
ture from conventional definitions to rules 
founded upon the universal properties of 
that which is represented. 
In the paper first quoted, Gregory con- 
siders five classes of operations. He sup- 
poses + and — to be defined by the rules 
of signs; and he finds in arithmetic a pair 
of operations which come under it, namely, 
addition and subtraction ; and in geometry 
another pair, namely, turning through a 
circumference, and a semi-circumference re- 
spectively. But it is instructive to note 
that the difficulty referred to in the title of 
the later paper is none other than the view 
that + and — represent the operations of 
addition and subtraction; and he there 
shows that addition (including subtraction) 
is subject to a couple of very different 
laws, the commutative and the associative, 
though he does not use the latter term. It 
may be observed that the rule of signs ap- 
plies to x and = also; henceif + and — em- 
braced addition and subtraction, so would x 
and =. The truth of the matter is that in 
ascending from arithmetic to algebra, we re- 
place the coérdinate ideas of addition and sub- 
traction by the more general idea of sum and 
the subordinate functional idea of opposite. 
Similarly the coordinate ideas of multiplica- 
tion and division are replaced by the more 
general idea of a product and the subordinate 
functional idea of reciprocal. The symbols 
— and + then denote opposite and recipro- 
eal respectively, while the ideas of sum and 
product are not expressed by symbols, but 
are sufficiently indicated by the manner of 
writing of the several elements. This dif- 
ficulty appears to have upset his belief in 
the existence of classes of operations sub- 
ject to the same laws of combination, yet 
totally dissimilar in nature, and without 
any real analogy binding them together. 
According to Gregory, the second class 
of operations are index operations, subject 
to the two laws: 
SCIENCE. 
349 
fn(4) flo) = fn +n(a) and fnfa(@) = fnn()- 
The third class comprises the ordinary sym- 
bol of algebra, and the symbols d and 4 of 
the calculus; they are subject to the dis- 
tributive law 
f(a) + f(b) =fla + 4), 
and to the commutative law 
fF(@) =fF,(4)- 
The fourth class comprises the logarithmic 
operations subject to the law 
f(a) + f() = f(b). 
The fifth class are the sine and cosine func- 
tions, subject to the laws expressed by the 
fundamental theorem of plane trigonometry, 
namely, the connection between the sine 
and cosine of the sum of two angles and 
the sines and cosines of the component 
angles. 
Following as far as may be the chrono- 
logical order, we come next to Augustus De 
Morgan, distinguished for his contributions 
alike to logic and to mathematics. In his 
‘Formal Logic’ he takes a formal view of 
the nature of reasoning in general, and in 
his ‘ Trigonometry and Double Algebra’ he 
lays down an excessively formal foundation 
for algebra. Indeed, it may be said that 
he carries formalism to its logical issue; 
and, thereby, he renders a service, for its 
inadequacy then becomes the better evident. 
In the chapter of the book mentioned, 
which is headed, ‘On Symbolic Algebra,’ 
he thus expresses the view he had arrived 
at: ‘In abandoning the meanings of sym- 
bols, we also abandon those of the words 
which describe them. ‘Thus addition is to 
be, for the present, a sound void of sense. 
It is a mode of combination represented by 
+; when + receives its meaning, so also 
will the word addition. It is most impor- 
tant that the student should bear in mind 
that, with one exception, no word nor sign of 
arithmetic or algebra has one atom of 
meaning throughout this chapter, the ob- 
ject of which is symbols, and their laws of 
