OPERATION. This article is to be considered as a continuation of ALGEBRA, and as a development of the views of the nature of algebra there laid down. It cannot be read entire, except by students who have some acquaintance with the Differential Calculus, &c.
The great considerations on which the mathematics are founded have always, until lately, been stated as those of number and space ; so that arithmetic and geometry have been called the wings of this branch of exact science. This similitude, suggested by the twofold character of its objects of comparison, may be carried a step further ; for us wings will not enable a bird to fly without nerves and sinews, so the mere consideration of space and number will never make a mathema tician, without an organised method of using the ideas of both. We have already [MATHEMATICS] suggested that the science of operation must be a constituent part of mathematics ; but it has always been so mixed up with the sciences bearing names derived from number and measure, that until lately it has had neither separate name nor exist ence ; and even now, what has been done in it is only the mere begin ning of a system.
The use of symbols of operation not standing for magnitudes but for directions how to proceed with magnitudes, began with Leibnitz and Newton, before whose time all algebraical characters denoted ,simple numbers. The progress of the Differential Calculus forced the attention of mathematicians upon modes of denoting, not results of processes, but ways of proceeding. The generalisations arising out of the organisation which notation gave to processes led to the use of indefinite and arbitrary symbols of operation. [FUNCTION.] Finally, it was observed that the symbols of operation employed in many general theorems would give simple and well-known relations if their meaning as symbols of operation were forgotten, and they were con sidered as symbols of quantity. For example, if Ase (.4 be co(x + 1) —0x, A being a symbol, not of a quantity multiplying cb.r, but of an operation to be performed upon car ; and if 13 fix, VIskr, &e. denote the successive differential coefficients of sax, Taylor's theorem gives If A and D had stood for quantities (which they do not), the pre ceding equation might have been divided by eps, and the result would have been If such a result had been obtained by those mathematicians who first ventured on the use of a negative quantity, they would doubtless have given to operations a sort of existence as quantities, and would have felt no repugnance to say that the direction to change sae into se(x +1) — sex was equal to e raised to the power of a direction to differentiate sex, diminished by a unit. This might have beat their negative quan
tity (or arithmetical quantity less than nothing) in the complication of its absurdities, but not iu absolute impossibility. Let two persons be required, the one to take four yards out of three, and the other to subtract a unit from—not the differential coefficient of 0.e, but—the direction to take the differential coefficient of ipx, and it could hardly be said that the first had a more hopeless task than the second.
The modern mathematicians, with Lagrange at their head, had had too much experience of the nature of extensions to hazard any assump tion upon the properties of symbols of operation, when separated from the quantity to be operated upon. The first step made was the ob servation that certain theorems involving symbols of operation might be easily remembered by the resemblance of the formula to well-known expressions; in fact, by the coincidence of those fonnuhe with the expressions, on the supposition that the symbols of operation are changed in meaning, and become symbols of quantity. And if it be said that these mathematicians were saved from introducing a diffi. culty analogous to that of negative quantities by the want of resem blances already existing in common modes of speaking and common views of arithmetic, it may be answered that such was not the ease, but that it would have been easy, and was not without precedent, to consider arithmetic itself as a science of operations upon one single magnitude, the unit. If wo always express the unit by r, we may, if wo please, consider 2 not as 1+ r, but as the direction to perform upon the operation in 1+ I ; so that 2 being merely a direction what to do, 21 may represent the result of so doing : similarly 3 may be the direction to proceed as in I+ I+ t, and 31 its result. And 3 x 2 would be a direction to perform 3 upon the result of 2, or to take 2 t+ 2i + 2 t, or (t + t) + (1+ 0+ (I+ 1), or 6L If then we say 3 x (21)=61, we have an equation between magnitudes ; but if we throw away t, as we just now did ifox, we have 3 x 2=6, an expression of equivalence of operations. Now it might very reasonably have been asked whether these operations must be the only ones which will admit of being treated by themselves and viewed independently of the subjects of operation ; and a direct assumption of such modes of notation as that marked (A), even when A and n were considered independently, though it might not have been fully explicable, would have appeared to be nothing but a natural extension of views which had already been taken to a limited extent.