common in the other subjects of geometrical investigation. It cannot, for example, be said of one solid angle, that it is the half, or the double of another solid angle; nor did any geometer ever think of proposing the problem of bisecting a given solid angle. In a word, no multiple or sub-multiple of such an angle can be taken, and we have no way of expounding, even to the simplest cases, the ratio which one of them bears to another. In this respect, therefore, a solid angle differs from every other magnitude that is the subject of mathematical reasoning, all of which have this common property, that multiples and sub-multiples of them may be found. It is not our business here to inquire into the reason of this anomaly, but it is plain, that on account of it, our knowledge of the nature and the properties of such angles can never be very far extended, and that our reasonings concerning them must be chiefly confined to the relations of the plane angles, by which they are contained. One of the most remarkable of those relations is that which is demonstrated in the 21st of this Book, and which is, that all the plane angles which contain any solid angle must together be less than four right angles. This proposition is the 21st of the 11th of Euclid. A This proposition, however, is subject to a restriction in certain cases, which, I believe, was first observed by M. le Sage of Geneva, in a communication to the Academy of Sciences of Paris in 1756. When the section of the pyramid formed by the planes that contain the solid angle is a figure that has none of its angles exterior, such as a triangle, a parallelogram, &c. the truth of the proposition just enunciated cannot be questioned. But, when the aforesaid section is a figure like that which is annexed, viz. ABCD, having some angles such as BDC, exterior, or, as they are sometimes called, re-entering angles, the proposition is not necessarily true; and it is plain, that in such cases the demonstration which we have given, and which is the same with Euclid's, will no longer apply. Indeed, it were easy to show, that on bases of this kind, by multiplying the number of sides, solid angles may be formed, such A B D that the plane angles which contain them shall exceed four right angles by any quantity assigned. An illustration of this from the properties of the sphere is perhaps the simplest of all others. Suppose that on the surface of a hemisphere there is described a figure bounded by any number of arcs of great circles making angles with one another, on opposite sides alternately, the plane angles at the centre of the sphere that stand on these arcs may evidently exceed four right angles, and that too, by multiplying and extending the arcs in any assigned ratio. Now, these plane angles contain a solid angle at the centre of the sphere, according to the definition of a solid angle. We are to understand the proposition in the text, therefore, to be true only of those solid angles in which the inclination of the plane angles are all the same way, or all directed toward the interior of the figure. To distinguish this class of solid angles from that to which the proposition does not apply, it is perhaps best to make use of this criterion, that they are such, that when any two points whatsoever are taken in the planes that contain the solid angle, the straight line, joining those points, falls wholly within the solid angle or thus, they are such, that a straight line cannot meet the planes which contain them in more than two points. It is thus, too, that I would distinguish a plane figure that has none of its angles exterior, by saying, that it is a rectilineal figure, such that a straight line cannot meet the boundary of it in more than two points. We, therefore, distinguish solid angles into two species: one in which the bounding planes can be intersected by a straight line only in two points; and another where the bounding planes may be intersected by a straight line in more than two points: to the first of these the proposition in the text applies, to the second it does not. Whether Euclid meant entirely to exclude the consideration of figures of the latter kind, in all that he has said of solids, and of solid angles, it is not now easy to determine: it is certain, that his definitions involve no such exclusion; and as the introduction of any limitation would considerably embarrass these definitions, and render them difficult to be understood by a beginner, I have left it out, reserving to this place a fuller explanation of the difficulty. I cannot conclude this note without remarking, with the historian of the Academy, that it is extremely singular, that, not one of ali those who had read or explained Euclid before M. le Sage, appears to have been sensible of this mistake. (Memoires de l'Acad. des Sciences, 1756, Hist. p. 77.) A circumstance that renders this still more singular is, that another mistake of Euclid on the same subject, and perhaps of all other geometers, escaped M. le Sage also, and was first discovered by Dr. Simson, as will presently appear. PROP. IV. This very elegant demonstration is from Legendre, and is much easier than that of Euclid. The demonstration given here of the 6th is also greatly simpler than that of Euclid. It has even an advantage that does not belong to Legendre's, that of requiring no particular construction or determination of any one of the lines, but reasoning from properties common to every part of them. The simplification, when it can be introduced, which, however, does not appear to be always possible, is, perhaps, the greatest improvement that can be made on an elementary demonstration. PROP. XIX. The problem contained in this proposition, of drawing a straight line perpendicular to two straight lines not in the same plane, is certainly to be accounted elementary, although not given in any book of elementary geometry that I know of before that of Legendre. The solution given here is more simple than his, or than any other that I have yet met with it also leads more easily, if it be required, to a trigonometrical computation. BOOK III. DEF. II. and PROP. I. THESE relate to similar and equal solids, a subject on which mistakes have prevailed not unlike to that which has just been mentioned. The equality of solids, it is natural to expect, must be proved like the equality of plane figures, by showing that they may be made to coincide, or to occupy the same space. But, though it be true that all solids which can be shewn to coincide are equal and similar, yet it does not hold conversely, that all solids which are equal and similar can be made to coincide. Though this assertion may appear somewhat paradoxical, yet the proof of it is extremely simple. D Let ABC be an isosceles triangle, of which the equal sides are AB and AC; from A draw AE perpendicular to the base BC, and BC will be bisected 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 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 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 shewn 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 opposite sides of those planes. B E 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 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 same 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 dimensions, presents itself conti nually; for, though two plane figures that are equal and similar can always be made to coincide, yet, with regard to solids that 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 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 this principle is has also appeared very clearly in the course of these remarks; and it is indeed no other than the principle so celebrated in the philosophy of Leibnitz, under the name of THE SUFFICIENT REASON. For it was shewn, 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 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 Ele ments. 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 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 ques tions, 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 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 that has been just stated, that there are solids and other Geometrical 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 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 knowledge. He has indeed 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 quality, and may nevertheless be unequal; whereas the observation made here is, that solids may be equal and similar, and inay yet want the condition of being able to coincide with one another. These propositions are widely different; and how so accurate a writer as Legendre should have mistaken 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 has 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 1 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 intenton 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 that those solids are equal and similar, which are contained under the same number of equal and similar planes. But this is not true, as Dr. Simson has shewn 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 contained within the |