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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 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 detinitions, 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 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 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,
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 conditions "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 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 30, 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 shew 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 that those 'solids are equul 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 other. Sitnson observes, that Euclid needed only 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
pect, 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 nottheir equal and 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 new kind of magnitude. The nature of the difficulty he had to contend with, will perhaps be the more readily admitted as an apolosy 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 another, and has treated of equal and similar solids, so as to exclude the symmetrical altogether, to which indeed he seems not 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 contrary 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 edition of his Euclid in 1756, the very same year that M. le Sage communicated 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 shewn 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 the 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, BA are equal to FB, BA, and the base EA is equal to the base FA ; wherefore the angle EBA is equal to the angle FBA, and the triangle EBA equal to the triangle FBA, 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
F them by three triangles, the common vertex of which is the point G, and their bases the straight lines AB, BC, CA, and by three other triangles 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 contained 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.” “Cor. From this it appears, that two unequal solid angles may
be contained by the same number of equal plane angles."
“ For the solid angle at B, which is contained by the four plane angles EBA, EBC, GBA, GBC is not equal to the solid angle at the same point B, which is contained by the four plane angles FBA, FBC, GBA; GBC ; for this last contains the other : And each of them is contained by four plane angles, which are equal to one another, each to each, or are the self-same, as has been proved : And indeed, there may be innumerable solid angles all unequal to one another, which are each of them contained by plane angles that are equal to one another, each to each: It is likewise manifest, that the before-mentioned solids are not similar, since their solid angles are not all equal."
RIGONOMETRY is defined in the text to be the application of
Number to express the relations of the sides and angles of triangles. It depends, therefore, on the 47th of the first of Euclid, and on the 7th of the first of the Supplement, the two propositions which do most immediately connect together the sciences of Arithmetic and Geometry.
The sine of an angle is defined above in the usual way, viz. the perpendicular drawn from one extremity of the arch, which measures the angle on the radius passing through the other; but in strictness the sine is not the perpendicular itself, but the ratio of that perpendicular to the radius, for it is this ratio which remains constant, while the angle continues the same, though the radius vary. It might be convenient, therefore, to define the sine to be the quotient which arises from dividing the perpendicular just described by the radius of th circle.
So also, if one of the sides of a right angled triangle about the right angle be divided by the other, the quotient is the tangent of the angle opposite to the first-mentioned side, &c. But though this is certainly the rigorous way of conceiving the sines, tangents. &c. of angles, which are in reality not magnitudes, but the ratios of magnitudes; yet as this idea is a little more abstract than the common one, and would also involve some change in the language of trigonometry, at the same time that it would in the end lead to nothing that is not attained by making the radius equal to unity, I have adhered to the common method, though I have thought it right to point out that which should in strictness be pursued.
A proposition is left out in the Plane Trigonometry, which the astronomers make use of in order, when two sides of a triangle, and the angle contained by them, are given, to find the angles at the base, without making use of the sum or difference of the sides, which, in some cases, when only the logarithms of the sides are given, cannot be conveniently found.