6 
Science, like theory, is a systematical statement 
of rules and propositions ; art, like practice, is 
the application of any of these rules or proposi- 
tions. But if we reflect that the theoretical 
rule will in many cases be moulded so as to 
meet the practical object, the reaction of prac- 
tice, or art, on theory, or science, is at o ice 
suggested, and with it the difficulty of elimi- 
nating, in all cases at least, a practical ele- 
ment from theory. I have made these obser- 
vations with a view, not to open, but rather to 
close any discussion of the distinction between 
the sciences and the arts. The arrogant levity 
which should despise, or affect to despise, the 
latter, I do not expect to find among my hearers. 
But if there be any scientific purists who think 
that it is in itself desirable to make the chasm 
between the sciences and the arts as wide as 
possible, I would by an illustration show how 
difficult is the task of dissevering them. Take 
the case of arithmetic. In this case there 
are ho doubt propositions which, for 
their enunciation or conception, need no 
material or mechanical aid whatever 
beyond clothing them in words. Such a pro- 
position is this, that “ the sum of two numbers 
multiplied into their difference is the difference 
of their squires.” But on the other side there 
are numerous arithmetical questions which no 
one would dream of performing, except by or- 
ganised processes, and with the aid of the cur- 
rent notation. Now this notation being arbi- 
trary, and tne digit 5 for instance, capable of 
r presenting any other number, and on any 
other scale of notation, we see that an arbitrary 
or artificial character, a character which, in a 
certain sense, would justify us in calling arith- 
metic an art, tinctures that science. If we 
apply the term mechanical in a general sense, 
and to denote any process that is not purely 
menial, and if, moreover, we proscribe any- 
thing mechanical, whether in the subject mat- 
ter, the end in view, or the processes of deduc- 
tion by which that end is attained, we shall 
leave little to which the term science can be 
strictly applied. Pure geometry indeed might 
be placed in the category of sciences, for the 
diagrams of geometry are not, like the symbols 
of algebra, arbitral y symbols, but, at least, in 
plane geometry, direct r presentations of the 
conceptions which they are intended to suggest 
to the mind. I shall now, however, use the 
term science in that sense with which I may 
presume my heavers to be familiar, merely re- 
minding them that, while they view with dis- 
trust attempts to divorce the very name of art 
from science, they should lend no ready ear to 
those who think that art can flourish where 
science wither ; , and that, under the epithet of 
“ verbal truths,” her teachings can be 
placed in an unfavorable contrast with 
the deliverances of practical men, or 
remitted to a despised obscurity. That 
“ one and one make t-wo” may look like a verbal 
truth ; but he would be rash who would 
venture to say that the proposition that “ three 
times three added to four times four, is equal 
to five times five,” is so. It may not be un- 
likely that the man who would make such an 
assertion would be led into a hasty and false 
generalization of the proposition in question. 
The sciences may be conveniently divided into 
three divisions. 1. The mathematical sciences ; 
2. the naturalsciences; 3. the graphical sciences. 
In the case of the mathematical sciences, so far 
as they are mathematical, their truths are 
capable of being verified without any appeal to 
experience; although, of course, we may 
appeal to experience, if we will. Thus, in 
order to find the sum of the three angles of a 
triangle, we may measure and add them, unless 
we are satisfied with the apodictic proof that 
their sum is two right angles. The mathematical 
sciences are either (1) pure or (2) mixed , in which 
latter case the science is also a physical or a 
graphical one, or both. The pure mathematical 
sciences have for their subject matter pure 
conceptions of the mind, such as number, — 
conceptions which have their seat in that por- 
tion of the mind, which there is authority for 
calling the understanding. The pure mathe- 
matical sciences are arithmetic, logic, algebra, 
the differential and integral calculus, the 
calculus ot finite differences, the cal- 
culus of functions, &c. These may be called 
the logical sciences. There is another branch 
of pure mathematical science, in which the 
conceptions which form its subject are not — like 
number — pyre conceptions of the under- 
standing, but, on the contrary , are represen- 
tations of something exterior to, and inde- 
pendent of, the mind, and which, partaking of 
the qualities both of conceptions and of per- 
ceptions, may conveniently and briefly be called 
perconceptions. The perconceptions are two 
in number — namely, space and time. The 
former is the subject matter of geometry ; the 
latter, of Sir W. Rowan Hamilton’s (of Dublin) 
science of pure time. These two sciences may 
be called the intellectual sciences, inasmuch as 
there is authority for calling that faculty 
which is the source of the ideas of space and 
time the intellect. By a combination of the 
logical and intellectual sciences, we arrive at 
the sciences of analytical geometry, analytical 
trigonometry, the calculus of quaternions, &c. 
These may be called the logico-intellectual 
sciences. The logical, intellectual, and 
logico-intellectual sciences comprise the 
pure mathematical sciences. I find that 
the topics presented to myself, and 
the limits within which such an address as this 
must necessarily be contained, prevent me 
from filling up the sketch, which, when I 
commenced, I thought I might have rendered 
more perfect. I shall therefore content my- 
self with such remarks as I have time to 
make, and reserve until a future communica- 
tion to the society the residue of the scheme. 
I may say, however, that by the term graphi 
