296 Professor Hamitton on Conjugate Functions, 
already formed and fixed, but rather as nascent, or in process of generation: and employs, as its pri- 
mary conception, the thought of a flowing point. And, generally, the revolution which NEwron* made in 
the higher parts of both pure and applied Algebra, was founded mainly on the notion of fluwion, which 
involves the notion of Time. 
Before the age of Newron, another great revolution, in Algebra as well as in Arithmetic, had been 
made by the invention of Logarithms ; and the “ Canon Mirificus” attests that Naprert deduced that in- 
vention, not (as it is commonly said) from the arithmetical properties of powers of numbers, but from 
the contemplation of a Continuous Progression ; in describing which, he speaks expresssly of Fluxions, 
Velocities and Times. 
In a more modern age, Lacranes, in the Philological spirit, sought to reduce the Theory of Fluxions 
to a system of operations upon symbols, analogous to the earliest symbolic operations of Algebra, and 
professed to reject the notion of time as foreign to such a system; yet admitted} that fluxions might be 
considered only as the velocities with which magnitudes vary, and that in so considering them, abstrac- 
tion might be made of every mechanical idea. And in one of his own most important researches in pure 
Algebra, (the investigation of limits between which the sum of any number of terms in TayLor’s Series 
is comprised,) Lackance|| employs the conception of continuous progression to show that a certain vari- 
able quantity may be made as small as can be desired. And not to dwell on the beautiful discoveries 
made by the same great mathematician, in the theory of singular primitives of equations, and in the al- 
gebraical dynamics of the heavens, through an extension of the conception of variability, (that is, in 
fact, of flowingness,) to quantities which had before been viewed as fixed or constant, it may suffice for 
the present to observe that LacranGE considered Algebra to be the Science of Functions§, and that it is 
not easy to conceive a clearer or juster idea of a Function in this Science, than by regarding its essence 
as consisting in a Law connecting Change with Change. But where Change and Progression are, there is 
Time. The notion of Time is, therefore, inductively found to be connected with existing Algebra. 
If. The argument for the conclusion that the notion of time may be unfolded into an independent 
Pure Science, or that a Science of Pure Time is possible, vests chiefly on the existence of certain priori 
* Considerando igitur quod quantitates zqualibus temporibus crescentes et crescendo genite, pro velocitate majori vel mi- 
nori qua crescunt ac generantur evadunt majores vel minores ; methodum quierebam determinandi quantitates ex velocitati- 
bus motuum yel incrementorum quibus generantur; et has motuum vel incrementorum yelocitates nominando Fluxiones, 
et quantitates genitas nominando Fluences, incidi paulatim annis 1665 et 1666 in Methodum Fluxionum qua hie usus sum in 
Quadratura Curvarum—Traetatus de Quad. Curv., Introd., published at the end of Sir I. Newton’s Opticks, London 1704. 
+ Logarithmus ergo cujusque sinus, est numerus quam proximé definiens lineam, que xqualiter crevit intered dum sinus 
totius linea proportionaliter in sinum illum decrevit, existente utroque motu synchrono, atque initio equiveloce. Baron 
Napier’s Mirifici Logarithmorum Canonis Descriptio, Def. 6, Edinburgh 1614.—Also in the explanation of Def. 1, the 
words flucu and fluat occur. 
¢ Calcul des Fonctions, Legon Premiere, page 2. Paris 1806. 
|| Done puisque V devient nul lorsque i devient nul, il est clair qu’ en faisant croitre i par degrés insensibles depuis 
zéro, la valeur de V croitra aussi insensiblement depuis zéro, soit en plus ou en moins, jusqu’ a un certain point, aprés 
quoi elle pourra diminuer.— Calcul des Fonctions, Legon Neuviéme, page 90. Paris 1806. An instance still more strong 
may be found in the First Note to Lagrange’s Equations Numeriques. Paris, 1808. 
§ On doit regarder l'algébre comme la science des fonetions.—Cale. des Fonct., Legon Premiere. 
© The word “ Algebra” is used throughout this whole paper, in the sense which is commonly but improperly given 
by modern mathematical writers to the name “ Analysis,” and not with that narrow signification to which the unphilosophi- 
eal use of the latter term (Analysis) has caused the former term (Algebra) to be too commonly confined. The author 
confesses that he has often deserved the censure which he has here so freely expressed. 
