and on Algebra as the Science of Pure Time. 295 
forms of logic may build up from them a symmetrical system of expressions, and a practical art may be 
learned of rightly applying useful rules which seem to depend upon them. 
So useful are those rules, so symmetrical those expressions, and yet so unsatisfactory those principles 
from which they are supposed to be derived, that a growing tendency may be perceived to the rejection 
of that view which regarded Algebra as a SciENcE, in some sense analogous to Geometry, and to the 
adoption of one or other of those two different views, which regard Algebra as an Art, or as a Language : 
as a System of Rules, or else as a System of Expressions, but not as a System of Z’ruths, or Results having 
any other validity than what they may derive from their practical usefulness, or their logical or philological 
coherence. Opinions thus are tending to substitute for the Theoretical question—“Is a Theorem of 
Algebra true?” the Practical question,—‘* Can it be applied as an Instrument, to do or to discover some- 
thing else, in some research which is not Algebraical?” or else the Philological question,— Does its 
expression harmonise, according to the Laws of Language, with other Algebraical expressions ?” 
Yet a natural regret might be felt, if such were the destiny of Algebra; if a study, which is conti- 
nually engaging mathematicians more and more, and has almost superseded the Study of Geometrical 
Science, were found at last to be not, in any strict and proper sense, the Study of a Science at all: and 
if, in thus exchanging the ancient for the modern Mathesis, there were a gain only of Skill or Ele- 
gance, at the expense of Contemplation and Intuition. Indulgence, therefore, may be hoped for, by any 
one who would inquire, whether existing Algebra, in the state to which it has been already unfolded by 
the masters of its rules and of its language, offers indeed no rudiment which may encourage a hope of 
developing a Science of Algebra: a Science properly so called ; strict, pure, and independent; deduced 
by valid reasonings from its own intuitive principles ; and thus not less an object of priori contempla- 
tion than Geometry, nor less distinct, in its own essence, from the Rules which it may teach or use, and 
from the Signs by which it may express its meaning. 
The Author of this paper has been led to the belief, that the Intuition of Time is such a rudiment. 
This belief involves the three following as components: First, that the notion of Time is connected 
with existing Algebra ; Second, that this notion orintuition of Time may be unfolded into an independent 
Pure Science ; and Third, that the Science of Pure Time, thus unfolded, is co-extensive and identical with 
Algebra, so far as Algebra itself is a Science. The first component judgment is the result of an induction; 
the second of a deduction; the third is a joint result of the deductive and inductive processes. 
I. The argument for the conclusion that the notion of Time is connected with existing Algebra, is an 
induction of the following kind. The History of Algebraic Science shows that the most remarkable disco- 
veriesin it have been made, either expressly through the medium of that 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 on as flowing, as it was the genius of Geometry to consider what 
it reasoned on as fixed. Evcxip* defined a tangent to a circle, APoLLoniust conceived a tangent to an 
ellipse, as an indefinite straight line which had only one point in common with the curve ; they looked 
upon the line and curve not as nascent or growing, but as already constructed and existing in space ; they 
studied them as formed and fixed, they compared the one with the other, and the proved exclusion of any 
second common point was to them the essential property, the constitutive character of the tangent. 
The Newtonian Method of Tangents rests on another principle; it regards the curve and line not as 
* EdSia xdxrov iparreaSar Aéyerau, Aris awroutyn Tov xixAou xat ExCadrouiyn ow reaver Toy xixdore-— EUCLID, Book III. Def. 2. 
Oxford Edition, 1703. 
ft Edy ty xdvou rom amd ris xopupis ris romms ayS7 eieia wapx rerayutvus xzrnyucyny Exrds weosira Ths rosstis.—txros Spa 
megsiray diomwep tQanreras thi To4Hi.—APPOLLONIUS, Book 1. Prop. 17. Oxford Edition, 1710. 
