the result of the application must be the same in every possible case, and also the same whether applied to the one side of the line or to the other.

We may therefore assume as a postulate, that "if any figure is already applied, or can be supposed to be applied to any straight line in any manner, figure exactly equal in every respect may be applied in the same manner to any other straight line equal to the first, whithersoever that line may be situated."

And from the same principle we may conclude with equal certainty, that, "if a figure is applied to one side of a straight line, a figure equal in every respect may be applied to the opposite side of the same."

If figures of this last description are not regular, that is, if all their sides, and all the sides and angles of each, are not equal, they are symmetrical magnitudes, that is, magnitudes of “equal measures," but reversed in their position with regard to each other; thus, if a diameter is drawn across a circle, the two semicircles, into which it divides the circle, are exactly equal, but they are symmetrical, as their circular sides are turned in opposite directions; also, if one triangle is constructed upon the one side of a line, and a triangle equal in all respects is constructed on the opposite side of the same, or of another line, such triangles are symmetrical: as, for instance, the triangles B and c are symmetrical with the triangle A, B being constructed on the opposite

side of the same line with a, and c on the opposite side of a different line; but all these three triangles are in every respect equal to each other.

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SYMMETRICAL MAGNITUDES.

The body of a perfectly formed animal, if we imagine it to be divided exactly on a mesial plane, that is, a plane extending the whole length of the animal, and passing exactly through the middle of the upper and under parts, affords a very good instance of what is meant by symmetrical magnitudes, as applied to solids. The two portions into which the animal is thus supposed to be divided, are exactly equal to each other in every respect; and yet we cannot, even if we were actually to divide the animal, place them so as to have them in the same position at one view, and thus judge in detail of the perfect equality of all their individual parts; for if we placed one of them in the natural position of the entire animal, we could not show the external surface of the other one without turning it either end for end, or upside down in respect of the first. Many cases of symmetrical magnitudes occur in investigations which are purely geometrical; and therefore, if we are not aware of them beforehand, we are apt to feel less certain in our reasonings respecting them, than we are respecting magnitudes which present themselves to us in the same position with each other.

The admission of the postulate which we have mentioned, or rather of that necessary inference from the third postulate usually given in the elements, completely obviates this difficulty; and, when carefully considered, there is an axiom deducible from it, or rather arising necessarily and obviously out of it, which enables us to get the better of many difficulties. The axiom to which we allude, when stated in its most general terms, is as follows:

If we know with certainty all the circumstances upon which any two conclusions or results depend; and if we farther know that those in the one case are exactly the same as those in the other, each to each, in the same order; then the two results or conclusions, whether they be truths which are established, or

figures which are constructed, or quantities which are found, or, in fact, any results whatsoever, must be exactly the same.

It is true that this axiom has not the apparent simplicity of the usual axioms in elementary geometry, but the fact is that it embodies them all, and a good deal more, some of which has to be demonstrated, and some is taken for granted in the course of the elements.

All mathematical reasoning is by comparison of quantities of the same kind; and the conclusion arrived at in every single step of such comparisons, is the equality or the inequality of the quantities compared; for if we seek to find the difference or the ratio, the determination of this requires a second step, and that step is arithmetical in any one particular case—an instance of subtraction, if we seek the difference, and an instance of division, if we seek the ratio. Therefore, it becomes necessary that our original notion of equality should be such as to embrace all possible cases.

Now, the fundamental axiom usually given in elementary geometry, though the eighth in order, and not the first, is in these words;-" Magnitudes which coincide with one another, that is, which fill exactly the same space, are equal to one another."

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This axiom, by the introduction of the word magnitudes," not only limits the case to geometrical equality, but it actually does not reach nearly to the whole of that. In strict language, no magnitude but a solid can be said to "fill space," for a line occupies no space, neither does a surface, unless when we regard it with relation to a solid. A line may, no doubt, be called a magnitude, but then it is a magnitude of one dimension only; and a surface is but of two dimensions, while it is just as impossible to imagine the existence of space without three dimensions,

as it is to take up in one's hand a piece of board which has no thickness.

Farther, an angle is not a magnitude which fills space, and indeed it is not, strictly speaking, a magnitude of any kind; because, without the imagined existence of two lines which have an inclination toward each other, there cannot be even an imagined angle. In like manner, no ratio can be regarded as a magnitude capable of filling space; and therefore it is that the doctrine of the equality of ratios, as expressed, and clearly and beautifully expressed, in the fifth definition of V., Euc. El., is so difficult to every student, and proves an insuperable barrier to so many.

Now, the doctrines of lines, of angles, of surfaces, and of ratios, are quite as essential in geometry, as the doctrines of those magnitudes which can fill space, and of which the equality can be established by its being shown that they fill the same space. They are even more essential, because they are the elements by the relations of which to each other the form and extent of any magnitude which can fill space are determined; and therefore, either the nature of those lines, angles, and surfaces which are compared in the earlier parts of the elements, are not understood, or the truth of the axiom we are considering is tacitly assumed, without being stated, which is certainly a very ungeometrical method of proceeding.

But there is another advantage in taking the doctrine of equality at once in its most general form, which is of much more importance to us than anything in mere geometry, important as that branch of science is. This very axiom is our general, we may say, universal rule in all our reasonings, and all our actions; in every department of science, or subject of knowledge, be it what it may, and in every action of our lives,

if we act as rational beings, that is, if we have an object in view, and seek to accomplish that object by the most simple and most certain means. We proceed upon what is usually called the judgment of experience; that is, we observe personally, or we are informed upon testimony which we have no reason to doubt, that formerly certain data, or things, or circumstances known to us, applied or acting in a particular manner, to a particular extent, in the former cases, produced or led to a certain definite result; and upon the faith of the axiom (or maxim, as we call it in matters of real life), that, "in like circumstances a like result must take place," we pursue our artificial plan with confidence of success, and therefore with pleasure; or, if the result be one which must be brought about by natural causes, with or without our assistance, we wait that result with the patience of wisdom, and do not spoil by attempted hurrying, that which, in the nature of things, we cannot hasten.

If, in our acting upon this maxim, we could obtain a perfect knowledge of all the circumstances, that is, of all the data, and all the means of dealing with this data, our expected results would all be physical or moral certainties; and though, even in physical matters, we cannot exactly accomplish this, we can always do it the more nearly, the more completely that all the circumstances are known to us. Thus, for instance, because the data are few, well understood, and, generally speaking, reducible to mathematical laws, we can notwithstanding the many variations in the motion of the moon, tell what shall be the apparent distance of that luminary from any fixed star, at any time, long before that time arrives; or we can, in the case of an eclipse of either of the great luminaries, predict the moment of its commencement and termination, and the portion of the luminary which shall be eclipsed, for almost any number of