+ Sin. x Sin. x', and Sin. (x x') for Sin. x Cos. x' Cos. x Sin. x', it then becomes, .{ Cos. (x. .r') + Sin. (s —x') V-1 I.
a quantity which has the form A + B ?—I as before. The function (a + 6 becomes, in like manner, rn (Cos. x + Sin. x v-1)n = in (Cos. n x + Sill. 72 X ?-1) which is still a quantity of the same form.
By employing the same substitution it is easy to prove, that the very general imaginary function (a + + n?-1 is still of the form A + B In applying imaginary expressions to analytical investi gations, it becomes an important question, what is the na ture of the evidence it affords for the truth of the results ? It must be confessed that this part of their theory is in volved in some degree of obscurity. John Bernoulli and Maclaurin allege, that when imaginary expressions are put to denote real quantities, the imaginary characters involv ed in the different terms of such expressions do then com pensate or destroy each other. To this it has been object ed, that an imaginary character being no more than a mark of impossibility, such a compensation is altogether unintel ligible; for to suppose that one impossibility can remove or destroy another, would be to bring impossibility under the predicament of quantity, and to make it the subject of arithmetical computation.
Professor Mayfair, of the university of Edinburgh, has advanced a different theory in the London Philosonhica! Transactions for 1778. He there observes, that impos sible expressions occur only when circular arcs or hyper bolic areas are the subject of investigation, and that corres ponding to every imaginary formula, there is a real for mula perfectly analogous; one of these is the analytic ex pression of some property of the circle, and the other the expression of a like property in the hyperbola. Thus, sup posing that in either curve the semiaxis = I, let c denote any abscissa, reckoned from the centre, s the ordinate, and x the double of the area contained by the semiaxis, a semi diameter drawn to the top of the ordinate, and the intercept ed hyperbolic arc ; also let e denote the number of which Nap. log. = I : In the circle .a: v-1) +e I ( I -x I ) s = - 2 — e —e and in the hyperbola (as will readily appear from FLUX IONS, § 150, Ex. 5.)
c — x — 5 (e ' + e )' e —e These expressions are perfectly analogous in their form, so that if in the first set V I be put instead instead of the imaginary symbol A/ it will be immediately trans formed into the second. By the first set of formula, the whole theory of the Arithmetic of Sines may be inves tigated, and by the second, a correspondin theory re lating to the co-ordinates of an hyperbola, many of the corresponding properties of the two curves will be iden tical, and some will differ only in the signs of the terms. In both, the results will be alike free from the ima ginary sign, although in the one case the steps by which they have been found arc unintelligible, and in the other they arc perfectly significant. This agreement ol two me thods so very different in the discovery of truth, the inge nious writer attributes to the analogy that takes place be tween the subjects of investigation, which is so close, that every property of the one may, with certain restrictions, be transferred to the other. Hence it happens, that the ope rations performed with imaginary characters,. although destitute of meaning themselves, are yet notes of reference to othets which are significant : They point out indirectly a method of demonstrating a certain property of the hy perbola, and then leave us to conclude from analogy that the same property belongs also to the circle. All that we are assured of by the imaginary investigation is, that its con clusions may, with all the strictness of mathematical rea soning, be proved of the hyperbola ; but if from thence we would tialsfer that conclusion to the circle, it must be in consequence of the principle which has been just now mentioned. The investigation, therefore, in every case re solves itself into an argument from analogy, and after the strictest examination, will be found to have no other claim to the evidence of demonstration. A proposition that is proved of the hyperbola only, and afterwards concluded to hold true of the code, merely from the affinity of the curves, will have precisely the same degree of certainty as when a proof is made out by imaginary symbols.