408 NOTION OF EQUAL SURFACES. same product may be obtained from an endless number of factors, provided that the one is increased, and the other diminished, in due proportion. Thus, if any surface or area whatever is expressed by the product of two definite factors, a and b, that is, by a b; and if we state the proportion ; m:a=bin, then, to every possible value of m, however great or however small, there will be a corresponding value of n, which will make the product m n= the product a b; and if we have the value of m given, the corresponding value of n can always be found by a simple application of the rule of three ; for in all a b cases n = m If supra-position as a test of equality will not apply to surfaces, of which our notion as definite magnitudes is always the result of a single multiplication only, much less will it apply to solids, of which our notion is that of the product of two multiplications of three factors, standing to each other in the relations of length, breadth, and thickness. In the case of ratios it is still less applicable ; because the ratios are not in themselves quantities at all: they are merely the results of the comparison of quantities, and therefore they do not occupy space in any sense of the word. Geometry would, however, be very imperfect and very useless, if it did not embrace the doctrine of ratios; and it really does appear that the tying down of the primary notion of equality in the student to this very limited fact of coincidence, is the chief reason why so much difficulty is always felt with the fifth book of Euclid's Elements, though that book is in reality the simplest of the whole. When we come to the general expression of quantities alge SUPRA-POSITION LIMITED. 409 braically, which is by far the most valuable portion of the mathematical sciences; or to the arithmetical expression of them, which is the only means whereby the principles can be reduced to practice; the notion of occupying the same space can have no application or meaning whatever; because neither the algebraical expressions, nor the figures of arithmetic, have the slightest reference to the occupation of any space, large or small; and therefore we have to judge of their equality or inequality upon very different principles. The one guiding principle in these is, that equal data equally dealt with invariably lead to the same results; and this ground of equality is perfectly satisfactory, and possesses the advantage of being applicable to all cases, whether they be of a mathematical nature or not. Thus, it brings geometry, as well as every other branch of mathematics, to the very same standard upon which we found our judgments in all the conduct of life ; and consequently, instead of making geometry stand apart, as if it were unconnected with our ordinary modes of thinking and acting, it brings it home to the mind as part of that general education which we derive from observation and experience, without in the least affecting that rigid accuracy which is the valuable part of mathematical study. It is impossible to extend this primary geometrical test of equality beyond the subjects of lines and angles, or to bring it in any way to bear upon our common modes of judgment; and surely, therefore, the wise plan is, to endeavour to bring these modes of judgment to bear upon geometry as well as other matters, in order that our modes of thinking may be the same upon every subject. By this means we are left perfectly untrammelled, to direct our attention to the peculiarities of the subject itself; and if there were nothing more than this to be gained, it would be well worthy of all the labour that it costs. 410 GENERAL TEST OF EQUALITY. We have alluded to this subject more than once in the previous part of this volume; but as we are now to proceed more directly to subjects which are geometrical, we have felt it necessary to dwell upon it at some length, in order that the reader may have it fresh in his memory, and thus be prepared for availing himself of any and every advantage which it may afford. We purposely treated of the doctrines of ratios, and of powers, and roots, at least in their most elementary forms, at an earlier stage than is usually done in books of geometry. We did SO, in order that we might carry our doctrine of proportion along with us as an element; and we feel convinced, that any one who chooses to look into the fifth book of Euclid's Elements, where he will find that, with the exception of one of the axioms, and a single proposition, which is very nearly self-evident, the four preceding books of the elements are not even once alluded to in the fifth one, which, in fact, stands alone, and has really nothing to do with the properties of figures ; as the ideas in the axiom, and the proposition to which allusion is made, are both strictly arithmetical in the particular case, and algebraical as taken generally Having given these explanations—which, however, are intended more for the guidance of the reader, than as an apology for departing from the established order of succession in the elements of geometry—we shall proceed to another section. SECTION XVI. COMPARISONS AND RELATIONS OF PLANE FIGURES, THEIR BOUND ING LINES, THE ANGLES DE BY THOSE LINES, LINES INTERSECTING THEM, AND THEIR SURFACES OR AREAS. In order that we may proceed with the requisite ease and expedition, in dealing with the subjects enumerated in the title of this section, it will be necessary that we carry along with us some preliminary notions, in addition to what have been treated of in the former sections. This is the more necessary, on account of the general view which we wish to take of the doctrine of equality, which doctrine also involves in it the opposite doctrine of inequality, in as much as where the one ends, the other begins as a matter of course. With regard to straight lines, to circles, and to angles as determined by circular measures, we believe that we can hardly make the matter plainer than we have already attempted to do; or indeed than it must appear to every one who understands the definition of a line, a circle, or an angle, when either of them is presented singly to the mind. When, however, we come to compare figures, or the sides or angles of figures, the quantities which we have to compare are results, and not simple data given singly; and consequently, in them the comparison of the results may involve, and very often does involve, the comparison of the means by which these results are arrived at. Indeed, it is this compound view of the matter in those cases, which gives them the greater part of their value; and therefore, if we do not familiarize ourselves with the result, so as to be able to analyze all the steps, and compare them, each with each, in the two parts of our general comparison, we never can be certain or satisfied that we are right; and thus, while we have the semblance of mathematical demonstration, we have nothing in reality but simple belief. This is another formidable obstacle in the way of the student of elementary mathematics ; and it is one, the removal of which is of far more importance than any apparent progress in detached theorems and problems, which could possibly be made by one before whom this obstacle were always presenting itself. GENERAL MAXIM. The same data which are sufficient for enabling us to construct any figure, are also sufficient for establishing the perfect equality in every respect of two figures of the same species ; that is to say, if the data are all exactly the same in the case of the one figure as in that of the other. This is a simple and general principle, not affected by any contingency; and, therefore, like all such principles, the converse of it is true ; that is to say, the same data which suffice for establishing the perfect equality in every respect of two figures, are quite sufficient for enabling us to construct those figures. The truth of this maxim, viewed both directly and conversely, is so clear and simple, that it cannot be made more so by any attempted demonstration; and yet it is one which, though seldom stated and generally overlooked, is very useful in shortening and simplifying many reasonings in mathematical science. When we consider it, we can readily see that it is of very general application, not to mathematical subjects only, but to all subjects in which any thing has to be done. If for instance; one states a proposition in writing, makes a drawing, or performs any other operation, in which a result is arrived at by the |