other. Simson 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 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 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 apology 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 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 the 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 selfsame, 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." TRIGONOMETRY 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 arc, 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 the 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. THEOREM. If, as the greater of any two sides of a triangle to the less, so the radius to the tangent of a certain angle; then will the radius be to the tangent of the difference between that angle and half a right angle, as the tangent of half the sum of the angles, at the base of the triangle to the tangent of half their difference. Let ABC be a triangle, the sides of which are BC and CA, and the base AB, and let BC be greater than CA. Let DC be drawn at right angles to BC, and equal to AC; join BD, and because (Prop. 1.) in the right angled triangle BCD, BC: CD:: R : tan CBD, CBD is the angle of which the tangent is to the radius as CD to BC, that is, as CA to BC, or as the least of the two sides of the triangle to the greatest. D C A B But BC+CD: BC-CD :: tan (CDB+CBD): tan (CDB-CBD) (Prop. 5.); and also, BC+CA: BC-CA:: tan (CAB+CBA): tan (CAB-CBA). Therefore, since CD=CA, tan (CDB+CBD): tan (CDB-CBD):: tan (CAB+CBA): tan (CAB-CBA). But because the angles CDB+CBD=90°, tan (CDB+CBD): tan (CDB-CBD): R: tan (450-CBD), (2 Cor. Prop. 3.); therefore, R tan (45°—CBD) :: tan 1⁄2 (CAB+CBA): tan (CAB-CBA); and CBD was already shewn to be such an angle that BC CA:: R: tan CBD. COR. If BC, CA, and the angle C are given to find the angles A and B ; find an angle E such, that BC: CA :: R: tan E; then R: tan (45°—E) :: tan (A+B): tan (A-B). Thus (A-B) is found, and (A+B) being given, A and B are each of them known. Lem. 2. In reading the elements of Plane Trigonometry, it may be of use to observe, that the first five propositions contain all the rules absolutely necessary for solving the different cases of plane triangles. The learner, when he studies Trigonometry for the first time, may satisfy himself with these propositions, but should by no means neglect the others in a subsequent perusal. PROP. VII. and VIII. I have changed the demonstration which I gave of these propositions in the first edition, for two others considerably simpler and more concise, given me by Mr. JARDINE, teacher of the Mathematics in Edinburgh, formerly one of my pupils, to whose ingenuity and skill I am very glad to bear this public testimony. SPHERICAL TRIGONOMETRY. PROP. V. THE angles at the base of an isosceles spherical triangle are symmetrical magnitudes, not admitting of being laid on one another, nor of coinciding, notwithstanding their equality. It might be considered as a sufficient proof that they are equal, to observe that they are each determined to be of a certain magnitude rather than any other, by conditions which are precisely the same, so that there is no reason why one of them should be greater than another. For the sake of those to whom this reasoning may not prove satisfactory, the demonstration in the text is given, which is strictly geometrical. For the demonstrations of the two propositions that are given in the end of the Appendix to the Spherical Trigonometry, see Elementa Sphæricorum, Theor. 66. apud Wolfii Opera Math. tom. iii.; Trigonometrie par Cagnoli, § 463; Trigonometrie Spherique par Mauduit, § 165. FINIS. |