308 Professor Hamitton on Conjugate Functions, 
the later of those two. And, therefore, reciprocally, if in any case of two given 
dates € and b,. we can prove that no moment B" whatever, of all that can be imagined 
as uniserial with two given distinct moments a and s, falls thus between the momenta 
c and p, we shall then have a sufficient proof that those two moments c and D are 
identical, or, in other words, that the two dates c and p represent only one common 
moment of time, (c=p,) and not two different moments, however little asunder. 
And even in those cases in which we have not yet succeeded. in discovering 
a rigorous proof of this sort, identifying a sought moment with a known one, or dis- 
tinguishing the former from the latter, the conception of continued analogies offers 
always a method of research, and of nomenclature, for investigating and. expressing, 
or, at least, conceiving as investigated and expressed, with any proposed degree of 
approximation if not with perfect accuracy, the situation of the sought moment in 
the general progvession of time, by its relation to-a known equidistant series of 
moments sufficiently close. ‘This might, perhaps, be a proper place, in a complete 
treatise on the Science of Pure Time, to introduce a regular system of integer ordi- 
nals, such as the words first, second, third, &c., with the written marks 1, 2, 3, &c., 
which answer both to them and to the cardinal or quotitative numbers, one, two, 
three, &c.; but it is permitted and required, by the plan of the present essay, that we 
should treat these spoken and written names of the integer ordinals and cardinals, 
together with the elementary laws of their combinations, as already known and - 
familiar, It is the more admissible in point of method to suppose this previous 
acquaintance with the chief properties of integer numbers, as set forth in elementary 
arithmetic, because these properties, although belonging to the Science of Pure Time; 
as involving the conception of succession, may all be deduced fromthe unfolding of 
that mere conception of succession, (among things or thoughts as counted,) without 
requiring any notion of measurable intervals, equal or unequal, between successive 
moments of time. Arithmetic, or the science of counting, is, therefore, a part, indeed, 
of the Science of Pure Time, but a part so simple and familiar that it may be pre- 
sumed to have been previously and separately studied, to some extent, by any one 
who is entering on Algebra. 
On steps in the progression of time ; their application (direct or inverse) ta moments, 
so as to generate other moments ; and their combination with other steps, in the 
way of composition or decomposition. 
9. The foregoing remarks may have sufficiently shewn the importance, in the 
general study of pure time, of the conception of a continued analogy or equidistant 
