in mathematics, denotes a certain way of performing investigations and resolutions, which occur on many occasions, particularly in mechanical philosophy. Thus we say, the antecedental calculus, the algebraical calculus, the arithmetical calculus, the differential calculus, the exponential calculus, the fluxional calculus, and the integral calculus. Of by much the greater part of these calculi some account has been given in the Encyclopedia Britannica; but there is one of them, of which no notice has been taken in that work.

It is,

The Antecedental Calculus, a geometrical method of reasoning, without any consideration of motion or velocity, applicable to every purpose to which the much celebrated doctrine of fluxions of the illustrious Newton has been, or can be, applied. This method was invented by James Glenie, Esq.; "in which (he says) every expression is truly and strictly geometrical, is founded on principles frequently made use of by the ancient geometers, principles admitted into the very first elements of geometry, and repeatedly used by Euclid himself. As it is a branch of general geometrical proportion, or universal comparison, and is derived from an examination of the antecedents of ratios, having given consequents and a given standard of comparison in the various degrees of augmentation and diminution they undergo by composition and decomposition, I have called it the antecedental calculus. As it is purely geometrical, and perfectly scientific, I have, since it first occurred to me in 1779, always made use of it instead of the fluxionary and differential calculi, which are merely arithmetical. Its principles are totally unconnected with the ideas of motion and time, which, briefly speaking, are foreign to pure geometry and abstract science, though, in mixed mathematics and natural philosophy, they are equally applicable to every investigation, involving the consideration of either with the two numerical methods just mentioned. And as many such investigations require compositions and decompositions of ratios, extending greatly beyond the triplicate and subtriplicate, this calculus in all of them furnishes every expression in a strictly geometrical form. The standards of comparison in it may be any magnitudes whatever, and are of course indefinite and innumerable; and the consequents of the ratios, compounded or decomposed, may be either equal or unequal, homogeneous or heterogeneous. In the fluxionary and differential methods, on the other hand, 1, or unit, is not only the standard of comparison, but also the consequent of every ratio compounded or decomposed."

This method is deduced immediately from Mr Glenie's Treatise on the Doctrine of Universal Comparison or General Proportion: And as the limits of the present work will not allow us to enter upon this subject, we therefore refer our readers to the two above mentioned treatises, and to the fourth volume of the Transactions of the Royal Society of Edinburgh.

We confess, however, that we do not expect such great advantage from the employment of this calculus as the very acute and ingenious author seems to promise from it. The mathematical world is truly indebted to him for the clear and discriminating view that he has taken of the doctrine of universal comparison, and we believe it to be perfectly accurate, and in some respects new. Notwithstanding the continual occupation of mathematicians with ratios and analogies, their particular objects commonly refracted their manner of conceiving ratio to some present modification of it. Hence it seems to have happened that their conceptions of it as a magnitude have not been uniform. But Mr Glenie, by avoiding every peculiarity, has at once attributed to it all the measurable affections of magnitude, addition or subtraction, multiplication or division, and ratio or proportion. He is perhaps the first who has roundly considered ratio or proportion as an affection of ratio; and it is chiefly by the employment of this undoubted affection of ratio that he has rendered the geometrical analysis so comprehensive.

But when we view this antecedental calculus, not as a method of expressing mathematical science, but as an art, as a calculus in short, and consider the means which it must employ, and the notation which must be used, we become less languid in our hopes of advantage from it. The notation cannot (we think) be more simple than that of the fluxionary method, justly called arithmetical; and if we insist on carrying clear conceptions along with us, we imagine that the arithmetical exposition of our symbols will generally be the simpler of the two. The science of the antecedental calculus seems to consist in the attainable perception of all the simple ratios, whether of magnitudes, or ratios, or both, which concur to the formation of a compound and complicated ratio. Now this is equally, and more easily, attainable in the fluxionary or other arithmetical method, when the consequent is a simple magnitude. When it is not, the same process is farther necessary in both methods, for getting rid of its complication.

We apprehend that it is a mistake that the geometrical trical method is more abstracted than the fluxionary, because the latter superadds to the notion of extension the notions of time and motion. These notions were introduced by the illustrious inventor for the demonstration, but never occupy the thoughts in the use of his propositions. These are geometrical truths, no matter how demonstrated; and when duly considered, involve nothing that is omitted in the antecedent calculus.

We even presume to say, that the complication of thought, in the contemplation of the ratios of ratios, is greater than what will generally arise from the additional elements, time and motion.

We do not find that any of our most active mathematicians have availed themselves of the advantages of this calculus, nor do we know any specimen that has been exhibited of its eminent advantages in mathematical difficulties. Should it prove more fertile in geometrical expressions of highly compounded or complicated quantities or relations, we should think it a mighty acquisition; being fully convinced that these afford to the memory or imagination an object (we may call it a sensible picture) which it can contemplate and remember with incomparably greater clearness and steadiness than any algebraical formula. We need only appeal to the geometrical expressions of many fluents, which are to be seen in Newton's lunar theory, in the physical tracts of Dr Matthew Stewart, and others who have shown a partiality for this method.

It would be very presumptuous, however, for us to say, that the accurate geometer and metaphysician may not derive great advantages from profiting by the very ingenious and recondite speculations of Mr Glenie, in his doctrine of universal comparison.