It is a curious circumstance, that the symbol which now represents equality, was first used to denote sub traction, in which sense it was employed by Albert Girarde, and that a w ord, signifying equality, was al ways used instead of a sign, until the time of Harriot, who at the same time proposed the signs and G. to denote " greater than" and " less than." Oughtred, in his " Clavis," appears first to have em ployed the x for multiplication, he sometimes also joins the letters without any intervening sign for the same purpose.
Such was the origin of the more common signs, which we now employ. An example of the different modes of expressing the same equation will exhibit this more clearly.
Since the time of Harriot, few alterations have been made in tbe mode of denoting the more simple opera tions of analysis, although the science itself has received great improvements. The first step which necessitated the invention of new methods of denoting operations, was the fluxional or differential calculus. Newton and Leibnitz each gave a notation to this new branch of analysis, and their respective followers, enteting with more zeal than judgment into the unfortunate contro versy respecting the claim of priority of invention, per tinaciously adhered to the notation of their masters, without inquiring into their relative merits, and probably without the knowledge requisite to form a correct judg ment on the question, had they turned their attention to the inquiry. In fact, the state of the science at the pe riod at which these modes of calculation wcre given to the world, could not admit of the formation of a correct estimate of their relative value, and the few reasons which could then be adduced were, on the whole, more favourable to the fluxional notation. It requires, even at the present time, an extensive acquaintance with the subsequent discoveries of the successors of Newton and Leibnitz, and an enlarged view of the present state of mathematical science, duly to appreciate the reasons which concur in rendering the notation of Leibnitz de cidedly the best of any which has hitherto becn proposed for accomplishing the same object.
As the proper adaptation of notation to the object of in quiry has contributed greatly to the progress of analysis, AVC propose offering a few general rules for the conside ration of those who may have occasion to express new relations, or who may desire to abbreviate those already in use ; and we shall compare several of the notations already received with these principles.
The great object of all notation is to convey to the mind, as speedily as possible, a complete idea of opera tions which are to be, or have been, executed ; and since every thing is to be exhibited to the eye, the more com pact and condensed the symbols are, the more readily will they be caught, as it were, at a glance.
The first and most obvious rule is, that all notation should be as simple as the nature of the operations to be indicated will admit. The reason of this is sufficiently evident, and it scarcely admits of an exception. It must, however, be remarked, that It is, in :nany cases, abso lutely impossible to express thc complicated operations required in the highest departments of analysis by for mulx that can be called simple. Still, however, they may be simple with reference to the multiplied relations they express.
The next rule we shall propose is, that we must ad here to cne notation for one thing. It is particularly un philosophical, and completely contrary to the whole spirit of symbolic reasoning, to employ the same signs for the representation of different operations, yet in stances of the infringement of this rule arc occasionally met with ; and as several examples are to be found in the works of an author of the very highest reputation, it is more particularly necessary to mark this deviation from the propriety of analytical language.
The notation employed by Lagrange to explain the difietential calculus will be considered on another oc casion. It is curious to observe, that his zeal for intro ducing the system of accentuation compelled him to Obandon a most valuable contrivance, which he had for merly employed for designating the variations of quan tities which harmonized with the notation of Leibnitz, whilst that which he substituted for it, (the systern of dots,) besides the inconveniences which it possessed in its original signification, had the additional disadvantage of having been employed in a different sense by all the followers of Newton. In the first edition of the Theorie des Fonctions, this notation was not employed, but in the Leccns sur le C'alcul des Fonetions, published in 1806, the fluxional notation is used for explaining the calculus of variations. The same author has also assumcd the cluadrant as the unity of angular measure, and represent ed sin.-2- : thus : sin. ',which, unless previously defined, would indicate the sine of an arc equal to the radius.