ADDRESS. 19 
postulate. We consider an abstract (real and positive) magnitude, and 
regard it as susceptible of continuous variation, without in anywise 
concerning ourselves about the actual expression of the magnitude by a 
numerical fraction or otherwise. 
There is an interesting paper by Sir W. R. Hamilton, ‘Theory of 
Conjugate Functions, or Algebraical Couples: with a preliminary and 
elementary Essay on Algebra as the Science of Pure Time,’ 1833-35 
(Trans. R. I. Acad. t. 17), in which, as appears by the title, he purposes 
to show that algebra is the science of pure time. He states there, in the 
General Introductory Remarks, his conclusions : first, that the notion of 
time is connected with existing algebra; second, that this notion or 
intuition of time may be unfolded into an independent pure science; and, 
third, that the science of pure time thus unfolded is coextensive and 
identical with algebra, so far as algebra itself is a science; and to sustain 
his first conclusion he remarks that ‘ the history of algebraic science shows 
that the most remarkable discoveries in it have been made either expressly 
through the notion of time, or through the closely connected (and in some 
sort coincident) notion of continuous progression. It is the genius of 
algebra to consider what it reasons upon as flowing, as it was the genius 
of geometry to consider what it reasoned on as fived. .. . And generally 
the revolution which Newton made in the higher parts of both pure and 
applied algebra was founded mainly on the notion of fluxion, which 
involves the notion of time.’ Hamilton uses the term algebra in a very 
wide sense, but whatever else he includes under it, he includes all that 
in contradistinction to the Differential Calculus would be called algebra. 
Using the word in this restricted sense, I cannot myself recognise the 
connection of algebra with the notion of time: granting that the notion of 
continuous progression presents itself, and is of importance, I do not see 
that it is in anywise the fundamental notion of the science. And still less 
can | appreciate the manner in which the author connects with the notion 
of time his algebraical couple, or imaginary magnitude a + bi (a + b /—1, 
as written in the memoir). 
IT would go further: the notion of continuous variation is a very 
fundamental one, made a foundation in the Calculus of Fluxions (if not 
always so in the Differential Calculus) and presenting itself or implied 
_ throughout in mathematics: and it may be said that a change of any 
kind takes place only in time; it seems to me, however, that the changes 
which we consider in mathematics are for the most part considered 
quite irrespectively of time. ~ 
Tt appears to me that we do not have in Mathematics the notion of 
time until we bring it there: and that even in kinematics (the science 
of motion) we have very little to do with it; the motion is a hypo- 
thetical one; if the system be regarded as actually moving, the rate 
of motion is altogether undetermined and immaterial. The relative rates 
of motion of the different points of the system are nothing else than the 
ratios of purely geometrical quantities, the indefinitely short distances 
c2 
