TRANSCENDENTAL, a mathematical term of description, the meaning of which is not very uniform. When any particular formula is incapable of being expressed by any particular range of algebraical symbols, it is, with respect to those symbols, transcendental—that is, it transcends or climber beyond the power of those symbols. The word was perhaps first used by Leibnitz (' Leipzig Acta,' I6S6), who says, "pLacet hoc loco, ut magic profutura dicamus, foracm aperire tranaccn dcntium quarttifatum, cur iiindrum quitdam problemata neque sint plans, neque solids, neque sursolida, aut ullius certi gradus, sod omnem tequationem algebraicam transeemiants" Here, then, is the first moaning of the word; a transcendental problem is one the equation of which is Infinitely high, or contains an infinite series of powers of an unknown quantity, so that its highest degree transcends every degree.
To form an idea of what is now most commonly meant by transcen dental, it will be desirable to recapitulate the steps by which algebra has arrived at its present state of expression,—or, rather, mathematical analysis, as those would say who do not like to call the differential calculus by the name of algebra.
And first we have the state which preceded the time of Vieta, in which formulie were mostly described in words, and the adoption of arbitrary symbols of quantity was only of casual occurrence.
Next, we have the introduction of arbitrary symbols of quantity by Vida, but not to the extent of using arbitrary number, of multipli cations, or algebraical exponents. Hero what we now call as was transcendental; Vieta could have described 00 by a cubo-cubutn, or by a quadrataquadrato-eubum, but as bad neither name nor symboL Thirdly, we have the stage which began with Ilarriot and Descartes, and which brought ordinary algebra into substantially its present form. During these periods, however, geometry and arithmetic, without help from algebra, had brought into use sines, cosines, fie, and logarithms, which were then properly transcendental. The words which described a particular mode of drawing lines in a circle, or the result of many interpositions of geometrical means between two given number., did not place those lines or means among the objects of algebra, and gave no clue to any algebraical properties.
Fourthly, we have the short but interesting period in which, before the formal invention of fiuxions or the differential calculus, infinite series began to be employed, and the transcendental. last alluded to ceased to be absolutely incapable of expression. This was the state in which Leibnitz found the science when he first proposed to distinguish between algebraical and transcendental problems.
Fifthly, we have the period succeeding the invention of the diffe rential calculus, in which the areas and lengths, &c., of curves could be expressed, whether they could be reduced into older language or not, by the new signs for fluents or integrals.
Sixthly, we have an alteration which it might have been supposed ,should have come long before, namely, the expression of the old tran scendental. as recognised functions, and the writing of them accordingly, as log x, sin x, cos x, &c. Strange as it may appear, this was never done till the time of Euler. And it is only in our own day that the system has been completed by the recognition of the number whose logarithm is x, the angle whose eine is x, &c., as functions of .r, and the adoption of the appropriate symbols x, x, &c.
Seventhly, a most important addition has been coming into use in the present century,—namely, the employment of definite integrals as modes of expression, not merely of functions of the variable of integra tion, but of other quantities which only enter as constants, or which, if they vary, vary independently of the variable used in integration. So powerful is this mode of expression, that it may almost be sus pected to be final; and the word transcendental is rapidly acquiring a new meaning. We predict that it will settle into the following : a transcendental result will be one which is incapable of expression except by a definite integral, or by an infinite series which cannot be otherwise expressed than by a definite integraL In the meanwhile there are two senses in which the word is used. The first is that just explained ; the second has reference to the old distinction of algebraical and transcendental. A function of x is alge braical when it is finite in form, and x is never seen, nor any function of it, in an exponent, nor under the symbols of a sine, cosine, fie., or a logarithm. No operation then enters with x unless it be one of the four great operations of arithmetic, or else involution or evolution with a definite exponent. Thus, in this sense of the word, log x and sin x are both transcendental,. But in the modern sense in which transcen dental is not opposed to algebraical, but to that which is expressible by ordinary means, log x and sin x are not transcendental, being among the most common of the present modes of expression, and being, in fact, connected with algebra in a way which, had it been understood when these symbols were first used, would probably have always saved them from the distinctive term.