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be bisected in E. From E draw ED perpendicular ta the plane ABC, and from ), 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 B two triangles be applied to one another, 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, ihough the equal planes be applied to one another, because they are on the opposite sides of those planes.
It may be said, ther, on what ground do we conclude the pyramids to be equal ? The answer is, because their construction is entirely the same, and the conditions that determine the magnitude of the one identical with those that 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 that are precisely the sanje in the two pyramids, so that there is nothing that can determine one of them to be greater than another.
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 one another can be determined. Neither is this a case that occurs rarely; it is one, that in the comparison of magnitudes having three dimens sions, presents itself continually; for, though two plane figures that are equal and similar can always be made to coincide, get, with rea gard to solids that are equal and similar, if they have got a certain similarity in their position, there will be found just as many cases in which they cannot, as in wbich they can coincide. Even figures described on surfaces, if they are not plane surfaces, may be equal and similar without the possibility of coinciding. Thus, in the figure described on the surface of a sphere, called a spherical triangle, if we suppose it to be isosceles, 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 one another, and which, nevertheless, cannot be so laid on one another 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 ihis principle is bas also appeared very clearly in the course of these remarks; and it is indeed no other than the principle so cele. brated in the philosophy of Leibnitz, under the name of THE SUFFICIENT REASON. Forit was shown, that the pyramids DABE and DACE are concluded to be equal, because each of them is determined to be of a certain magoitude, rather than of any other, by conditions that are the same in both, so that there is no REASON for the one being greater than the other. This Axiom may be rendered general by saying, That things of which the magnitude is determined by conditions that 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 Axiom 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 pbilosophers. 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 seldom 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 salse conclusion, because he knew nothing of the rotation of the eartb 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 quan. tities, whether they be of one, of two, or of three dimensions.
Legendre in his Elements has made the same remark that has been just stated, that there are solids and other Geometric Magnitudes, wbich, 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 80 concise as the nature of a simple and elementary truth would seem
to require, and consequently not such as to render the axiom proposa ed above altogetber unnecessary.
But a circumstance for which I cannot very well account is, that Legendre, and after him Lacroix, ascribe to Simson the first mention of such solids as we are here considering. Now I must be permitted to say, that no remark to this purpose is to be found in any of the writings of Simson, which have come to my kpowledge. He has jadeed made an observation concerning the Geometry of Solids, which was both new and important, viz. that solids may have the condition which Euclid thought sufficient to determine their equality, and may nevertheless be unequal; whereas the observation made here is, that solids may be equal and similar, and may yet want the condition of be ing able to coincide with one another. These propositions are widely different; and how so accurate a writer as Legendre should have mis. taken the one for the other, is not easy to be explained. It must be observed, that he does not seem in the least aware of the observation which Simson kas really made. Perhaps having himself made the remark we now speak of, and on looking slightly into Simson, having found a limitation of the usual description of equal solids, he had without much inquiry, set it down as the same with his own notion; and so, with a great deal of candour, and some precipitation, he has ascribed to Simson a discovery which really belonged to himself. This at least seems to be the most probable solution of the difficulty.
I have entered into a fuller discussion of Legendre's mistake than I should otherwise have done, from having said in the first edition of these elements, in 1795, that I believed the non-coincidence of similar and equal solids in certain circumstances, was then made for the first time. This it is evident would have been a pretension as ridiculous as ill-founded, if the same observation had been made in a book like Simson's which in this country was in every body's hands, and which I had myself professedly studied with attention. As I have not seen any edition of Legendre's Elements earlier than that published in 1802, I am ignorant whether he or I was the first in making the remark here referred to. That circumstance is, however immaterial; for I am not interested about the originality of the remark, though very much interested to show that I had no intention of appropriating. to myself a discovery made by another.
Another observation on the subject of those solids, which with Legendre we shall call Symmetrical, has occurred to me, which I did not at first think of, viz. that Euclid himself certainly had these solids in view when he formed his definition (as he very improperly calls it) of equal and similar solids. He says ihat those solids are equal and similar, which are contained under
the same number of equal and simifar planes. But this is not true, as Dr. Simson has shown in a passage just about to be quoted, because two solids may easily be assigned,
bounded by the same number of equal and similar planes, which are obviously unequal, the one being eontained within the other. Simson observes, that Euclid needed oply to have added, that the equal and similar planes must be similarly situated, to have made his description exact. Now, it is true, that this addition would have made it exact in one respect, but would have rendered it imperfect in another; for though all the solids having the conditions here enumerated are equal and similar, many others are equal and similar which have not those conditions, that is, though bounded by the same equal number of similar planes, those planes are not similarly situated. The symmetrical solids have not their equaland similar planes similarly situated but in an order and position directly contrary. Euclid, it is probable, was aware of this, and by seeking to render the description of equal and similar solids so general, as to comprehend solids of both kinds, has stript it of an essential condition, so that solids obviously unequal are included in it, and has also been led into a very illogical proceeding, that of defining the equality of solids, instead of proving it, as if he had been at liberty to fix a new idea to the word equal every time that he applied it to a pew kind of magnitude. The nature of the difficulty he had to contend with, will perhaps be the more readily admitted as an apolo. gy for this error, when it is considered that Simson; who had studied the matter so carefully, as to set Euclid right in one particular, was himself wrong in auother, and has treated of equal and similar solids, 80 as to exclude the symmetrical altogether, to which indeed he seeins bot to have at all adverted.
I must, therefore, again repeat, that I do not think that this matter can be treated in a way quite simple and elementary, and at the same time general, without introducing the principle of the sufficient reason as stated above. It may then be demonstrated, that similar and equal solids are those contained by the same number of equal and similar planes, either with similar or contrary situations. If the word contra: ry is properly understood, this description seems to be quite general.
Simson's remark, that solids may be unequal, though contained by the same number of equal and similar planes, extends also to solid angles which may be unequal, though contained by the same number of equal plane angles. These remarks he published in the first edi: tion of his Euclid ip 1756, the very same year that M. le Sage commů. nicated to the Academy of Sciences the observation on the subject of solid angles, mentioned in a former note; and, it is singular, that these two geometers, without any communication with one another, should almost at the same time have made two discoveries very nearly connected, yet neither of them comprehending the whole truth, so that each is imperfect without the other.
Dr. Simson has shown the truth of his remark; by the following reasoning.
“Let there be any plane rectilineal figure as the triangle ABC, and from a point D within it, draw the straight line DE at right angles to
the plane ABC; in DE take DE, DF equal to one another, upon the opposite sides of the plane, and let G be any point in EF; join DA, DB, DC; EA, EB, EC; FA, FB, FC; GA, GB, GC: Because tho straight line EDF is at right angles to the plane ABC, it makes right angles with DA, DB, DC, which it meets in that plane; and in the triangles EDB, FDB, ED and DB are equal to FD and DB, each to each, and they contain right angles; therefore the base EB is equal to the base FB; in the same manner EA is equal to FA, and EC to FC: And in the triangles EBA, FBA, EB, BÀ are equal to FB, BA, and the base E A is equal to the base FÁ; wherefore the angle EBA is equal to the angle FBĀ,and the triangle E B Aequal to the triangleFBA, and the other angles equal to the other angles; therefore these triangles are similar: In the same manner the triangle EBC is similar to the triangle FBC, and the triangle EAC to FAC; therefore there are two solid figures, each of which is contained by six triangles, one of
them by three triangles, the common vertex of which is the point ; and their bases the straight lines AB,BC, CA, and by three other tri. angles the common vertex of which is the point E, and their bases the same lines AB, BC, CA. The other solid is contained by the same three triangles, the common vertex of which is G, and their bases AB, BC, CA; and by three other triangles, of which the common vertex is the point F, and their bases the same straight lines AB, BC, CA: Now, the three triangles GAB, GBC, GCA are common to both solids, and the three others EAB, EBC, ECA, of the first solid have been shown to be equal and similar to the three others FAB, FBC,FCA of the other solid, each to each; therefore, these two solids are eontained by the same number of equal and similar planes: But that they are not equal is manifest, because the first of them is contained in the other: Therefore it is not universally true, that solids are equal which are contained by the same number of equal and similar planes."