But we can imagine we hear it said that this principle, though some times employed in pure physics, is never introduced into mathematical reasoning except after direct demonstration, in order to confirm the mind of the learner by making him sec how difficult it would have been to inuigine the possibility of any contradiction being successfully maintained against the proposition just proved. 'We believe, indeed, that this principle is seldom employed, and always without necessity, so that we could wish its use were entirely abandoned. But we can show that a tacit appeal to it is sometimes made ; and this is the worst possible mode of employing it. If the principle be dangerous, and liable to be unaonndly used, it should be most carefully stated when it is used. Whenever we see a proposition assumed, not as an express postulate, but in a definition for instance, or as a self-evident truth, wo may trace the operation of this principle on our minds. For instance, take the proposition which is, if there be such a thing in any one pro position, a digest of all the methods of mathematics, namely, that if the same operations be performed on equal magnitudes, the resulting mag nitudes are equal. Try to imagine this not true, and want of sufficient reason interferes to prevent success. What can make a difference ? In this question the principle claims to be applied.
Now, first, in examining the definitions of Euclid, we find an asser tion of theorems which we can hardly :suppose that Euclid overlooked, though it is very possible that the impossibility of imagining other wise may have been his guide. For instance, the assertion of the equality of the two parts into which a diameter divides a circle, follow ing immediately upon the definition of a circle ; and the definition of equal solids as those which are contained by the mine number of plane figures equal each to each. These and such little matters have been, or may be, corrected ; but we will now point out a use of the principle which exists in our elementary works of the present day in an unacknowledged form.
In proving the celebrated proposition of Albert Girard relative to the dependence of the area of a spherical triangle upon the sum of its angles, it is assumed that two spherical triangles which have their sides and angles equal, each to each, are equal in area. Now it is easily shown (Srsistyrnv] that there may be two such triangles of which it is impossible to make one coincide with the other, nor is any process ever given for dividing each into parts, so that the parts of one may be capable of coinciding with those of the other. Let the angular points of one be placed upon the angular points of the other (which is always possible), and the triangles will not coincide ; in common language, they will bulge in different directions. When the triangles are so placed, and the common chords drawn, there is no difficulty in seeing that if ever a want of sufficient reason can be granted upon perception, it is for there being any inequality of the areas of the two triangles. And the equality of these areas is accordingly assumed: for instance, in the proposition above alluded to, a pair of unsymmetrically equal triangles always occurs, except when the given triangle is isosceles. Aud thus the appeal to this principle may be avoided ; for it is easy to make the given triangles into the sum or difference of isosceles triangles, in which each of one set is capable of being actually applied to one of the other.
Leaving the subject of pure mathematics, let us now consider the application of this principle in physics. We have observed [STATICS] that the line of separation between pure mathematics and the more exact parts of mathematical physics is very slight indeed : this means as to the clearness and fewness of the first principles, and the rigour of the demonstrations. If we cut the link which ties the sciences of
statics and dynamics to the properties of the matter which actually exists around us, we may go farther, and say that we have not only pure sciences, but pure sciences in which the principle of the want of sufficient reason is strictly applicable, because it is our own selves who have, by express hypothesis, excluded sufficient reason. In propo sitions of pure mathematics, we have seen that we cannot invent or deny for any hypothetical purpose ; is and must be, is not and cannot be, are synonymes, in all the truths which these sciences teach. But the properties of matter which are not also those of space, are not, in our conceptions, necessary : we can imagine them other than they are, without any contradiction of ideas.
We shall now proceed to consider the point mentioned in STATICS, namely, the character of the axioms of that science. Are they "self evidently true," and " not to be learnt from without, but from within ?" We will not here inquire whether the first must be the second, not being sufficiently clear as to what is meant by knowledge "from without " and knowledge "from within," to enter upon any such inves tigation. It will do for our purpose to take knowledge " from within " to be a phrase descriptive of such truths as that two straight lines can not inclose a space, and knowledge " from without " another phrase indicating such truths as are found, say iu the facts of political history or geography. Let its separate from the rest one axiom of statics, say " equal weights at the ends of equal arms of a horizontal straight lever balance one another." First, " equal weights " is a synonymic for equal and parallel pressures. We have no objection to placing the idea of pressure on the same footing as that of a straight line, for be the name we give the former conception what it may, it is probable 'that those powers of communication with the external world which are certainly necessary to the development, at least, of pressure, arc not less necessary to that of straightness. Nor are equal pressures difficult of definition ; let them be those which are interchangeable, so that either may be put in the place of the other. The rest of the terms of the axiom are geometrical, and to balance each other is to produce no motion,—motion, independently of producing causes, being, we think, as much an idea of geometry as any other. Let A and B be the two ends of a lever (a rigid bar without weight), and c its middle point, which is the pivot ; that is to say, the middle point cannot move, the only possible motion of the lever being revolution, in the plane of the pressures, about that middle point. On these hypotheses, we may certainly say that the axiom is self-evident, for want of sufficient reason, that is. of a possibility of sufficient reason for anything in con tradiction of it. We have, before the pressures are applied, no cause of motion, by hypothesis : we are to conceive a lever, which, if it move at all, does so by reason of the pressures. We have made these pressures equal, and applied them symmetrically : there is then, and can be, no reason why one should predominate, which does not hold as much of the other. In the very notion of equality of pressures there is interchangeability ; that is, each may be substituted for the other without alteration of effect. Suppose then the left-hand pressure to predominate : it will do so if the pressures be interchanged. But after the interchange the same reasons which made the left hand pre dominate, will make the right hand predominate : or both ends will move in the direction of the pressures, which is impossible.