Supplement Let ABC be an ifofceles triangle, of which the equal fides are AB and AC; from A draw AE perpendicular to the bafe BC, and BC will be bifected in E. From E draw ED 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 difputed, cannot be fo applied to one another as to coincide. For, though the triangles ABE, ACE are equal, BE being equal to CE, EA common A A B E to both, and the angles AEB, AEC equal, because they are right angles, yet if these two triangles be applied to one another, fo as to coincide, the folid DACE will, nevertheless, as is evident, fall without the folid DABE, for the two folids will be on the oppofite fides of the plane ABE. In the fame way, though all the planes of the pyramid DABE may easily be fhewn to be equal to those of the pyramid DACE, each to each; yet will the pyramids themselves never coincide, though the equal planes be applied to one another, because they are on the oppofite fides of thofe planes. It may be faid, then, on what ground do we conclude the pyramids to be equal? The answer is, because their conftruction is entirely the fame, 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 pofition of ED, in respect of the plane ABE; three circumstances that are precisely the fame in the two pyramids, fo that there is nothing that can determine one of them to be greater than another. This reafoning appears perfectly conclufive and fatisfactory, and it seems also very certain, that there is no other principle equally fimple, on which the relation of the folids 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 dimenfions, prefents itself continu ally; ally; for, though two plane figures that are equal and fimilar Book III. can always be made to coincide, yet, with regard to folids that are equal and fimilar, if they have not a certain fimilarity in their pofition, there will be found just as many cafes in which they cannot, as in which they can coincide. Even figures defcribed on furfaces, if they are not plane furfaces, may be equal and fimilar without the poffibility of coinciding. Thus, in the figure defcribed on the furface of a sphere, called a spherical triangle, if we suppose it to be isofceles, and a perpendicular to be drawn from the vertex on the bafe, it will not be doubted, that it is thus divided into two right angled spherical triangles equal and fimilar to one another, and which, nevertheless, cannot be fo laid on one another as to agree. The fame holds in innumerable other inftances, 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 remarks; and it is indeed no other than the principle fo celebrated in the philofophy of Leibnitz, under the name of THE SUFFICIENT REASON. For it was fhewn, that the pyramids DABE and DACE are concluded to be equal, because each of them is determined to be of a certain magnitude, 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 faying, That things of which the magnitude is determined by conditions that are exactly the fame, are equal to one another; or, it might be expreffed thus: Two magnitudes A and B are equal, when there is no reason that A fhould exceed B, rather than that B should exceed A. Either of these will ferve 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, fuch 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 fomewhat of a metaphyfical kind, and belongs to the doctrine of the fufficient reafon, which is looked on with, a fufpicious eye by fome philofophers. But this is no folid objection; for fuch reafoning may be applied with the greatest fafety to Ff thofe Supplement thofe objects with the nature of which we are perfectly acquainted, and of which we have complete definitions, as in pure mathematics. In phyfical queftions, the fame principle cannot be applied with equal fafety, because in fuch cases we have feldom a complete definition of the thing we reafon about, or one that includes all its properties. Thus, when Archimedes proved the spherical figure of the earth, by reafoning on a principle of this fort, he was led to a falfe conclufion, because he knew nothing of the rotation of the earth on its axis, which places the particles of that body, though at equal diftances from the centre, in circumstances very different from one another. But, concerning thofe 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 fufficient reafon, which at the fame time furnishes us with the only fingle Axiom, by help of which we can compare together geometrical quantities, whether they be of one, of two, or of three dimenfions. Legendre in his Elements has made the fame remark that has been juft ftated, that there are folids and other Geometric Magnitudes, which, though fimilar and equal, cannot be brought to coincide with one another, and he has distinguished them by the name of Symmetrical Magnitudes. He has alfo given a very fatisfactory and ingenious demonftration of the equality of certain folids of that fort, though not fo concife as the nature of a fimple and elementary truth would feem to require, and confequently not fuch as to render the axiom propofed above altogether unneceffary. But a circumftance for which I cannot very well account is, that Legendre, and after him Lacroix, afcribe to Simfon the first mention of fuch folids as we are here confidering. Now, I must be permitted to fay, that no remark to this purpole is to be found in any of the writings of Simfon, which have come to my knowledge. He has indeed made an obfervation concerning the Geometry of Solids, which was both new and important, viz. that folids may have the conditions fuppofed by Euclid fufficient to determine their equality, and may nevertheless be unequal; whereas the observation made here is, that folids may be equal and fimilar, and may yet want the condition of being able to coincide with one anothefe propofitions are widely different; and how fo able a mathematician as Legendre fhould have mistaken the ther. one one for the other, is not easy to be explained. It must be ob- Book III. ferved, that he does not seem in the least aware of the obfervation which Simfon has really made. Perhaps having himfelf made the remark we now fpeak of, and on looking flightly into Simfon, having found a limitation of the ufual defcription of equal folids, he had, without much inquiry, set it down as the fame with his own notion; and fo, with a great deal of candour, and fome precipitation, he has afcribed to Simfon a discovery which really belonged to himself. This at least seems to be the most probable folution of the difficulty. I have entered into a fuller difcuffion of Legendre's miftake 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 fimilar and equal folids in certain circumstances, was then made for the first time. This it is evident would have been a pretenfion as ridiculous as ill-founded, if the fame observation had been made in a Book like Simfon's, which in this country was in every body's hands, and which I had myself profeffedly studied with attention. As I have not seen any but the fourth edition of Legendre's Elements, published in 1802, I am ignorant whether he or I was the first in making the remark here referred to. That circumftance is, however, immaterial; for I am not interested about the originality of the remark, though very much interefted to fhew that I had no intention of appropriating to myself a discovery made by another. Another obfervation on the fubject of thofe folids, 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 folids in view when he formed his definition (as he very improperly calls it) of equal and fimilar folids. He fays that thofe folids are equal and fimilar, which are contained under the fame number of equal and fimilar planes. But this is not true, as Dr Simfon has fhewn in a passage just about to be quoted, because two folids may eafily be affigned, bounded by the fame number of equal and fimilar planes, which are obviously unequal, the one being contained within the other. Simfon obferves, that Euclid needed only to have added, that the equal and fimilar planes must be fimilarly fituated, to have made his defcription exact. Now, it is true, that this addition would have made it exact in one re fpect, Supplement fpect, but would have rendered it imperfect in another; for though all the folids having the conditions here enumerated are equal and fimilar, many others are equal and fimilar which have not thofe conditions, that is, though bounded by the fame equal number of fimilar planes, thofe planes are not fimilarly fituated. The fymmetrical folids have not their equal and fimilar planes fimilarly fituated, but in an order and pofition directly contrary. Euclid, it is probable, was aware of this, and by seeking to render the description of equal and fimilar folids fo general, as to comprehend folids of both kinds, has fript it of an effential condition, (so that solids obviously unequal are included in it), and has alfo been led into a very illogical proceeding, that of defining the equality of folids, inftead 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 new kind of magnitude. The nature of the difficulty he had to contend with, will perhaps be the more readily admitted as an apology for this error, when it is confidered that Simson, who had ftudied the matter fo carefully, as to set Euclid right in one particular, was himself wrong in another, and has treated of equal and fimilar folids, fo as to exclude the fymmetrical altogether, to which indeed he seems not to have at all adverted. I muft, therefore, again repeat, that I do not think that this matter can be treated in a way quite fimple and elementary, and at the fame time general, without introducing the principle of the fufficient reafon as ftated above. It may then be demonstrated that fimilar and equal folids are those contained by the fame number of equal and fimilar planes, either with fimilar or contrary fituations. If the word contrary is properly understood, this defcription feems to be quite general. Simfon's remark, that folids may be unequal, though contained by the fame number of equal and fimilar planes, extends alfo to folid angles which may be unequal, though contained by the fame number of equal plane angles. These remarks he published in the first edition of his Euclid in 1756, the very fame year that M. le Sage communicated to the Academy of Sciences the obfervation on the fubject of folid angles, mentioned in a former note; and it is fingular, that thefe two geometers, without any communication with one another, fhould almost at the fame time have made two dif coveries |