356 
land, of the University of Edinburgh. It 
was his custom to teach the elements of 
quaternions to the students of his senior 
class, and I remember how all went well till 
he came to multiplication, where the part 
played by a vector as a multiplier was 
likened, in some mysterious manner, to the 
action of acorkscrew. In the introductory 
chapter of the ‘Introduction to Quater- 
nions’ he remaks as follows on the process 
by which algebra is generalized : ‘ It is only 
by standing loose for a time to logical ac- 
curacy that extensions in the abstract sci- 
ences—extensions at any rate which stretch 
from one science to another—are effected.” 
And further on: ‘‘ We trust, then, it begins 
to be seen that sciences are extended by the 
removal of barriers, of limitations, of condi- 
tions on which sometimes their very ex- 
istence appears to depend. Fractional 
arithmetic was an impossibility so long as 
multiplication was regarded as abbreviated 
addition; the moment an extended idea 
was entertained, ever so illogically, that 
moment fractional arithmetic started into 
existence. Algebra, except aS mere sym- 
bolized arithmetic, was an impossibility so 
long as the thought of subtraction was 
chained to the requirement of something 
adequate to subtract from. The moment 
Diophantus gave it a separate existence— 
boldly and logically as it happened—by 
exhibiting the law of minus in the forefront 
as the primary definition of his science, that 
moment algebra in its highest form became 
a possibility, and indeed the foundation 
stone was no sooner laid than a goodly 
building arose on it.”’ 
It seems to me that no greater paradox 
could be enunciated than to say that higher 
principles in exact science are reached by 
standing loose for a time to logical accuracy. 
How long a time does that which is illogical 
take to become logical? The true process 
is generalization, not illogical extension. 
No doubt, the generalized principle may at 
SCIENCE. 
[N. S. Vou. X. No. 246. 
first be merely an hypothesis, and in that. 
form it may be applied so that it may be 
verified by its results; but this is not 
standing loose to logical accuracy. 
The same author gives the following ac- 
count of how Hamilton extended algebra to 
space: ‘‘He had done a_ considerable 
amount of good work, obstructed as he was, 
when, about the year 1843, he perceived 
clearly the obstruction to his progress in the 
shape of an old law which, prior to that 
time, had appeared like a law of common 
sense. The law in question is known as 
the commutative law of multiplication. 
Presented in its simplest form it is nothing 
more than this: ‘five times three is the 
same as three times five’; more generally 
it appears’ under the form of ab = ba what- 
ever aand 6 may represent. When it came 
distinctly to the mind of Hamilton that this 
law is not a necessity with the extended 
signification of multiplication he saw his 
way clear and gave up the law. The bar- 
rier being removed, he entered on the new 
science as a warrior enters a besieged city 
through a practicable breech.”? This ac- 
count is, of course, inadequate, for Grass- 
mann jumped over the same barrier in the 
shape of an ‘ old law,’ yet he was unable to 
deal with angles in space. There is no oc- 
casion to speak disrespectfully of the law of 
commutation ; it has its own place; Hamil- 
ton did not cast it aside as an obstruction ; 
he modified it for a product of factors hay- 
ing a real order, and the modified form 
amounts to the law of association. 
We shall now go back to another inde- 
pendent source of the development of the 
principles of algebra—Hermann Grass- 
mann. Like his contemporary, Hamilton, 
he was remarkable alike for attainments in 
mathematics and philosophy, and, besides, 
he made important contributions to philol- 
ogy. No doubt specialists are necessary, 
but the investigation of the fundamental 
principles of a science requires one who is 
