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Now, first, let K and C be supposed equal, then it is evident, that L and D are also equal; and therefore, since by construction a :c :: A:K, we have also a : 0 ::A:C; and, for the same reason, 6:d :: B: D, and these analogies form the first of the two conditions, of which one is affirmed above to be always essential to the truth of Torelli's proposition. Next, if K be greater than C, then since
A+K: A+C :: B+L:B+D, by division,
A: B::C:D. Wherefore, in this case the ratio of A to B is equal to that of C te D, and consequently, the ratio of a to b equal to that of c to d. The same may be shewn, if K is less than C; therefore in every case there are conditions necessary to the truth of Torelli's proposition, which he does not take into account, and which, as easily shewn, do not belong, to the magnitudes to which he applies it.
In consequence of this, the conclusion which he meant to establish respecting the circle, falls entirely to the ground, and with it the general inference aimed against the modern analysis.
It will not, I hope, be imagined, that I have taken notice of these circumstances with any design to lessen the reputation of the learned Italian, who has in so many respects deserved well of the mathematical sciences, or to detract from the value of a posthumous work, which by its elegance and correctness, does so much honour to the English editors. But I would warn the student against that narrow spirit which seeks to insinuate itself even into the abstractions of geometry, and would persuade us, that elegance, nay truth itself, are possessed exclusively by the ancient methods of demonstration. The high tone in which Torelli censures the modern mathematics, is imposing, as it is assumed by one who had studied the writings of Archimedes with uncommon diligence. His errors are on that account the more dangerous, and require to be the more carefully pointed out.
This enunciation is the same with that of the third of the Dimensio Circuli of Archimedes ;' but the demonstration is different. though it proceeds, like that of the Greek Geometer, by the continual bisection of the 6th part of the circumference.
The limits of the circumference are thus assigned ; and the method of bringing it about, notwithstanding many quantities are neglected in the arithmetical operations, that the errors shall in one case be all on the side of defect, and in another all on the side of excess, (in which I have followed Archimedes,) deserves particularly to be observed, as affording a good introduction to the general methods of approximation:
DEF. VIII, and PROP. XX.
Solid, angles, which are defined here in the same manner as in Euclid, are magnitudes of a very peculiar kind, and are particularly to be remarked for not admitting of that accurate comparison, one with another, which is 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 in 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 anomalý, 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.
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
А. 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
D cases the demonstration which we have given, and which is the same with Euclid's, will no longer apply. Indeed, it were easy to shew, that on
B bases of this kind, by multiplying
C the number of sides, solid angles may be formed, such 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 arches 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 arches may evidently exceed four right angles, and that too, by multiplying and extending the arches 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
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 all those who had read or explained Euclid before M. le Sage, appears to have been sensible of this mistake. (Mémoires de l'Acad. des Sciences 1756, llist. 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.
This very elegant demonstration is from Legendre, and is much elsier 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. This 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.
The problem contained in this proposition, of drawing a straight line perpendicular to two straight lines not in the same plane, tainly to be accounted elementary, although not given in any book of elementary geometry that I know of before that of Legendrė. 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.
DEF. H. 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 shewing 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.
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
A 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.
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 deterinined. Neither is ihis a case that occurs rarely; it is one, that in the comparison of magnitudes having three dimen. sions, presents itself continually ; 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 tigures de scribed on sorfaces, 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 cvident, 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 brated in the philosophy of Leibnitz, under the name of THE SUFFICENT PEASON. 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