base BC, and BC will be bisected in E. From E draw ED Book III. perpendicular to the plane ABC, and from D, any point in it,

draw DA, DB, DC to the three

angles of the triangle ABC. The pyramid DABC is divided ' into two pyramids DABE, DACE, which, though their equality will not be disputed, cannot be so applied to one another as to coincide. For, though the triangles ABE, ACE are equal, BE being equal to CE, EA common to both, and the angles AEB, AEC equal, because they are right angles, yet if these

two triangles be applied to each other, so as to coincide, the solid DACE will, nevertheless, as is evident, fall without the solid DABE, for the two solids will be on the opposite sides of the plane ABE. In the same way, though all the planes of the pyramid DABE may easily be shown to be equal to those of the pyramid DACE, each to each, yet will the pyramids themselves never coincide, because, if the equal planes be applied to one another, they are on the opposite sides of those planes.

It may be said, then, on what ground do we conclude the pyramids to be equal? The answer is, because their construction is entirely the same, and the conditions which determine the magnitude of one identical with those which determine the magnitude of the other. For the magnitude of the pyramid DABE is determined by the magnitude of the triangle ABE, the length of the line ED, and the position of ED in respect of the plane ABE; three circumstances which are precisely the same in the two pyramids, so that there is nothing which can determine one of them to be greater than the other.

This reasoning appears perfectly conclusive and satisfactory; and it seems also very certain that there is no other principle equally simple, on which the relation of the solids DABE, DACE to each other can be determined. Nor is this a case that occurs rarely; it is one which, in the comparison of magnitudes having three dimensions, presents itself continually; for, though two plane figures which are equal and similar can always be made to coincide, yet, with regard

Suppl.

to solids which are equal and similar, if they have not a certain
'similarity in their position, there will be found just as many
cases in which they cannot, as in which they can coincide.
Even figures described on surfaces, if they are not plane sur-
faces, may be equal and similar without the possibility of co-
inciding. Thus, in the figure described on the surface of a
sphere, called a spherical triangle, if we suppose it to be isos-
celes, and a perpendicular to be drawn from the vertex on
the base, it will not be doubted that it is thus divided into
two right angled spherical triangles equal and similar to each
other, and which, nevertheless, cannot be so laid on each other
as to agree.
The same holds in innumerable other instances,
and therefore it is evident that a principle, more general and
fundamental than that of the equality of coinciding figures,
ought to be introduced into geometry. What this principle
is has also appeared very clearly in the course of these re-
marks; and it is indeed no other than the principle so cele-
brated in the philosophy of Leibnitz, under the name of
THE SUFFICIENT REASON. For it was shown that the pyra-
mids DABE and DACE are concluded to be equal, because
each of them is determined to be of a certain magnitude, ra-
ther than of any other, by conditions which are the same in
both, so that there is no REASON for one being greater than
the other. This axiom may be rendered general by saying,
That things of which the magnitude is determined by condi-
tions which are exactly the same are equal to one another;
or it might be expressed thus: Two magnitudes A and B
are equal, when there is no reason that A should exceed B,
rather than that B should exceed A. Either of these will
serve as the fundamental principle for comparing geometrical
magnitudes of every kind; they will apply in those cases
where the coincidence of magnitudes with one another has
no place; and they will apply with great readiness to the
cases in which a coincidence may take place, such as in the
4th, the 8th, or the 26th of the First Book of the Elements.

The only objection to this axióm is, that it is somewhat of a metaphysical kind, and belongs to the doctrine of the sufficient reason, which is looked on with a suspicious eye by some philosophers. But this is no solid objection; for such reasoning may be applied with the greatest safety to those objects with the nature of which we are perfectly acquainted, and of which we have complete definitions, as in pure mathematics. In physical questions the same principle cannot be

applied with equal safety, because in such cases we have sel- Book III. dom a complete definition of the thing we reason about, or one that includes all its properties. Thus, when Archimedes proved the spherical figure of the earth, by reasoning on a principle of this sort, he was led to a false conclusion, because he knew nothing of the rotation of the earth on its axis, which places the particles of that body, though at equal distances from the centre, in circumstances very different from one another. But concerning those things that are the creatures of the mind altogether, like the objects of mathematical investigation, there can be no danger of being misled by the principle of the sufficient reason, which at the same time furnishes us with the only single axiom, by help of which we can compare together geometrical quantities, whether they be of one, of two, or of three dimensions.

Legendre in his Elements has made the same remark which has been just stated, that there are solids and other geometric magnitudes which, though similar and equal, cannot be brought to coincide with one another, and he has distinguished them by the name of symmetrical magnitudes. He has also given a very satisfactory and ingenious demonstration of the equality of certain solids of that sort, though not so concise as the nature of a simple and elementary truth would seem to require, and consequently not such as to render the axiom proposed above altogether unnecessary.

THE END.