Cor. 1. Take each of the equal triangles CPT, CAK, from the common space CAOP; there remains triangle OAT = OKP. Cor. 2. Also take the equal triangles CPT, CAK, from the common triangle CPM; there remains triangle MPT = trapez. AKPM. PROP. XVIII The same being supposed, as in last proposition, then any straight lines, Q&, QE, drawn parallel to the two tangents shall cut off equal spaces. That is, triangle GQE = trapez. AKXG Draw the ordinate PM. The three similar triangles CAK, CMP, CGX, are to each other as But, .. CA2, CM2, CG2, KR P TEAT AKPM trap. AKXG :: CM2 - CA2: CG2 - CA, dividendo. QG :: CM. CA2: CG-CA3, PM2: .. trap. AKPM: trap. AKXG :: But, trian. MPT trian. GQE :: are similar. PM : QG' PM : QG', the triangles .. trap. AKPM: trian. MPT :: trap. AKXG: trian. GQE, Cor. 1. The three spaces AKXG, TPXG, GQE, are all equal. Cor. 2. From the equals, AKXG, EQG, take the equals AKRr, Eqra there remains, RrXG = rqQG. Cor. 3. From the equals RrXG, rqQG, take the common space rqvXG; there remains, triangle vQX = triangle vqR. Cor. 4. From the equals EQG, TPXG, take the common space EvXG; there remains, TPvE = triangle vQX. Cor. 5. If we take the particular case in which QG coincides with the minor axis, The triangle EQG becomes the triangle IBC, The figure AKXG becomes the triangle .. triangle IBC = triangle AKC HH 2 Cor. Hence, any diameter divides the hyperbola into two equal parts. PROP. XX. The square of the semi-ordinate to any diameter, is to the rectangle under the abscissæ, as the square of the semi-conjugate to the square of the semi-diumeter, That is, If Qq be an ordinate to any diameter CP, Qv2 Pv. vp :: CD. CP'. Let Qq meet the major axis in E; Draw QX, DW, perpendicular to the major axis, and meeting PC in X and W. Then, since the triangles CPT, CvE, are similar, trian. CPT: trian. CvE:: CP2: Cv2 or, trian, CPT: trap. TPvE :: CP2: Cv2 - CP2 Again, since the triangles CDW, vQX, are similar, But, And triangle CDW : triangle vQX :: CD2 : vQ*;` Cor. 1. The squares of the ordinates to any diameter, are to each other as the rectangles under their respective abscissæ. Cor. 2. The above proposition is merely an extension of the property already proved in Prop. 12, with regard to the relation between ordinates to the axis and their abscissæ, ON THE ASYMPTOTES OF THE HYPERBOLA. DEFINITION.-An Asymptote is a diameter which approaches nearer to meet the curve, the farther it is produced, but which, being produced ever so far, does never actually meet it. PROP. XXI. If tangents be drawn at the vertices of the axes, the diagonals of the rectangle so formed are asymptotes to the four curves. Now, as CM increases, the ratio of CM to CM3 — CA continually approaches to a ratio of equality; but CM3 - CA' can never become actually equal to CM3, however much CM may be increased. Hence, MP is always less than MQ, but approaches continually nearer to an equality with it. In the same manner it may be proved, that CQ is an asymptote to the conju gate hyperbola BP'. Cor. 1. The two asymptotes make equal angles with the axis major and with the axis minor. Cor. 2. The line AB joining the vertices of the conjugate axes is bisected by one asymptote and is parallel to the other. Cor. 3. All lines perpendicular to either axis and terminated by the asymptotes are bisected by the axis. PROP. XXII. All the parallelograms are equal, which are formed between the asymptotes and curve, by lines drawn parallel to the asymptotes. That is, the lines GE, EK, AP, AQ, being parallel to the asymptotes CH, Ch, then the parallelogram CGEK = parallelogram CPAQ. For, let A be the vertex of the curve, or extremity of the semi-transverse axis AC, perpendicular to which draw AL or Al, which will be equal to the semi-conjugate, by definition xix. Also, draw HEDeh parallel to Ll. Then, and by parallels, P CA: AL' :: CD'- CA: DE', D DE or rect. HE. Eh; therefore, by subtract. CA: AL' :: CA: DH'. consequently, the square AL = the rectangle HE. Eh. But, by similar trian. PA: AL :: GE : EH, and, by the same, QA : Al :: EK: Eh; therefore, by comp. PA. AQ: AL:: GE. EK: HE. FA; and, because AL = HE. Eh, therefore PA. AQ = GE. F.K. But the parallelograms CGEK, CPAQ, being equiangular, are as the reʊtangles GE. EK and PA. AQ. And therefore the parallelogram GK the parallelogram PQ. That is, all the inscribed parallelograms are equal to one another. Q.ED Corol. 1. Because the rectangle GEK or CGE is constant, therefore GE is reciprocally as CG, or CG: CP:: PA: GE. And hence the asymptote continually approaches towards the curve, but never meets it; for GE decreases continually as CG increases; and it is always of some magnitude, except when CG is supposed to be infinitely great, for then GE is infinitely small or nothing. So that the asymptote CG may be considered as a tangent to the curve at a point infinitely distant from C. Corol. 2. If the abscissas CD, CE, CG, &c., taken on the one asymptote, be in geometrical progression increasing; then shall the ordinates DH, EI, GK, &c., parallel to the other asymptote, be a decreasing geometrical progression, having the same ratio. For, all the rectangles CDH, CEI, CGK, &c., being equal, the ordinates DH, EI, GK, &c., are reciprocally as the abscissas CD, CE, CG, &c., which are geometricals. And the reciprocals of geometricals are also geometricals, and in the same ratio, but decreasing, or in converse order. PROP. XXIII. The three following spaces, between the asymptotes and the curve, are equal ; namely, the sector or trilinear space contained by an arc of the curve and two radii, or lines drawn from its extremities to the centre; and each of the two quadrilaterals, contained by the said arc, and two lines drawn from its extremities parallel to one asymptote, and the intercepted part of the other asymptote. That is, The sector CAE = PAEG = QAEK, all standing on the same arc AE. For, as has been already shown, CPAQ= CGEK; So shall the paral. PI = the paral. IK; To each add the trilineal IAE, Then is the quadril. PAEG = QAEK. G P gain, from the quadrilateral CAEK, take the equal triangle CAQ, CEK, and there remains the sector CAE = QAEK. Therefore, CAE = QAEK = PAEG. Q.ED. APPLICATION OF ALGEBRA то GEOMETRY. WHEN it is proposed to resolve a geometrical problem algebraically, or by algebra, it is proper, in the first place, to draw a figure that shall represent the several parts or conditions of the problem, and to suppose that figure to be the true one. Then, having considered attentively the nature of the problem, the figure is next to be prepared for a solution, if necessary, by producing or drawing such lines in it as appear most conducive to that end. This done, the usual symbols or letters, for known and unknown quantities, are employed to denote the several parts of the figure, both the known and unknown parts, or as many of them as necessary, as also such unknown line or lines as may be easiest found, whether required or not. Then proceed to the operation, by observing the relations that the several parts of the figure have to each other; from which, and the proper theorems in the foregoing elements of geometry, make out as many equations independent of each other, as there are unknown quantities employed in them: the resolution of which equations, in the same manner as in arithmetical problems, will determine the unknown quantities, and resolve the problem proposed. As no general rule can be given for drawing the lines, and selecting the fittest quantities to substitute for, so as always to bring out the most simple conclusions, because different problems require different modes of solution; the best way to gain experience, is to try the solution of the same problem in different ways, and then apply that which succeeds best, to other cases of the same kind when they afterwards occur. The following particular directions, however, may be of some use. 1st, In preparing the figure, by drawing lines, let them be either parallel or perpendicular to other lines in the figure, or so as to form similar triangles. And if an angle be given, it will be proper to let the perpendicular be opposite to that angle, and to fall from one end of a given line, if possible. 2d, In selecting the quantities proper to substitute for, those are to be chosen, whether required or not, which lie nearest the known or given parts of the figure, and by means of which the next adjacent parts may be expressed by addition and subtraction only, without using surds. 3d, When two lines or quantities are alike related to other parts of the figure or problem, the way is, not to make use of either of them separately, but to substitute for their sum, or difference, or rectangle, or the sum of their alternate quotients, or for some line or lines in the figure, to which they have both the same relation. 44h. When the area, or the perimeter, of a figure, is given, or such parts of it as have only a remote relation to the parts required; it is sometimes of use to assume another figure similar to the proposed one, having one side equal |