SEPTEMBER 15, 1899. ] 
fessedly scientific work, some of these fun- 
damental laws are treated as self-evident, 
others as requiring geometric proof, and 
others yet are merely enunciated. If in 
the case of multiplication the commuta- 
tive law requires proof, so does it also in 
the case of addition, for itis just as self- 
evident that 2x 3=3x 2 as that 243 
=3 +2. 
The manner in which Hankel passes from 
arithmetic and arithmetical algebra to gen- 
eral algebra is as follows: Algebra, being 
formal mathematics, can be founded on any 
system of independent rules ; but, in order 
that its results may be interpretable and 
that it may be capable of application, it is 
found convenient to choose the system of 
fundamental rules satisfied by common 
arithmetic ; in other words, the laws of in- 
teger arithmetic are made the laws of al- 
gebra. This he calls the ‘principle of the 
permanence of the formal laws,’ and enun- 
ciates as follows (p. 11): “If two expres- 
sions stated in terms of the general symbols 
of arithmetical algebra (arithmetica uni- 
ver'salis) are equal to one another they shall 
remain equal to one another when the 
symbols cease to denote simple magnitudes 
and the operations receive any other mean- 
ing.”? Peacock speaks of the permanence 
of equivalent forms; Hankel of the perma- 
nence of the formal laws. Peacock says, 
“Tet any general equivalence in arithmetical 
algebra be true also in universal algebra’; 
Hankel says, ‘‘ Let the fundamental laws of 
the former be made the fundamental laws 
of the latter.”” Hankel gives a more scien- 
tific form to what was meant by Peacock. 
However, Hankel labors under a logical 
difficulty from which Peacock was exempt, 
for he does not take the laws of arithmetical 
algebra without exception ; he rejects the 
commutative law for a product, in order 
that quaternions may be included among 
his complex numbers. But, it may be asked, 
why not reject the commutative law for ad- 
SCIENCE. 
309 
dition also? So far as arithmetical algebra 
is concerned they stand on the same basis. 
Tf, as has been shown, the sum of quater- 
nion indices is not commutative we are 
logically bound, on his principles, to reject 
the commutative rule for addition also. We 
are reduced to the alternative: the choice 
of the fundamental rules’is arbitrary, or 
else they must be founded on the properties 
of the subject analyzed. The permanence 
of the formal laws is nothing but hypothesis, 
and in the case of any generalization must 
be tested by real investigation. 
One of the clearest thinkers on mathe- 
matical subjects in recent times was Pro- 
fessor Clifford, who like several of the 
mathematical philosophers we have spoken 
of, was cut down in the midst of his scien- 
tifie activity. In his posthumous work en- 
titled ‘The Common Sense of the Exact 
Sciences’ there are chapters on number and 
quantity in which he explains his views of 
the fundamental principles of algebra. He 
starts out from the principle, which he at- 
tributes to Cayley and Sylvester, that the 
number of any set of things is the same 
in whatever order we count them, and 
deduces from it, by means of diagrams, 
the commutative and associative rules for 
positive integer number. Hesays that they 
amount to the following: ‘‘ If we can in- 
terchange any two consecutive things with- 
out altering the result then we may make 
any change whatever in the order without 
altering the result.”” It may be remarked 
that this shows that the commutative and 
associative properties are not independent, 
but that the former involves the latter. He 
next shows, by a diagram, that the distribu- 
tive rule is true for the two forms a(b + c) 
=ab+ace and (b-+-c)a=ba+ ca, but he 
does not consider the complete form of the 
rule (a+ 0) (e+ d)=ac+ ad + be + bd. 
As regards the impossible subtraction and 
division he says (p. 383): ‘ Every opera- 
tion in mathematics that we can invent 
