and SP; put x = Aa the Distance between the two Semiordinates, which we fuppofe to be infinitely near each other, as in the Elipfis, Page 377. Then I y +z:BA::y+z+x: ba, per Theorem 13. 1, Or 2y+:y+x+x:: BA: ba. See Page 192. 3y: BA :: y+x:□ba, per Theorem Page 380. 4y: y+x::BA: Ag in 3, Or ba Syy + 2 y z +zzyy+2yzx+2yx+zz + 2in's 522x + xx:: □ BA: ba 4,5 6 67 That is, 8 +2yz+zz:yy Syy+xyy + 2 y z + zzyy+2yx+ 2yx+2x+2xx + xx โ ZZ =yxxx, confequently =1+x But 9x being infinitely less than x, ZZ Then 10 =y, confequently zzyy 102112 = y, that is, SP=SA may be rejected, as Q. E. D. CHA P. IV. Concerning the chief Properties of the Hyperbola. NOTE, any Part of the Axis of an Hyperbola, which is intercepted between its Vertex and any Ordinate (viz. any intercepted Diameter) is call'd an Abjciffa; as in the Parabola. Sect. I. The Plane of every Hyperbola is proportion'd by this general Theorem. As the Sum of the Tranfverfe and any Alfifa multiply'd into that Abfciffa is to the Square of its Se Theorem. mi-ordinate: : fo is the Sum of the Transverse and any other Abfciffa multiply'd into that Abscissa : to the Square of its Semi-ordinate. That Let the following Figure HVG represent a Right Cone cut into = two Parts by the Right Line SA; then will the Plane of that Section be an Hyperbola (by Sect. 5. Chap. 1.) in which let SA be its Axis, or intercepted Diameter, bab and BAB Ordinates rightly apply'd (as before in the Parabola) and TS its Transverse Diameter. Again, if the Cone is fuppos'd to be cut by bg, parallel to its Bafe HG, it will also be the Diameter of a Circle, &c. as in the Ellipfis and Parabola. Then will the A Sge and ASGA be alike, alfo the A Tab and ATAH wil be alike; therefore it T Thefe Proportions are the common Property of every Hyperbola, and do only differ from thefe of the Ellipfis in the Signs + T and; as plainly appears in the following Proportions. That is, if we suppose TS the Tranfverfe Diameter common to both Sections (viz. both the Ellipfis and Hyperbola) as in the annexed Scheme: then in the Ellipfis it will be TS Sax Sa: ab::TS SA × SA: □ AB, as by Sect. 1. Chap 2. and in the Hyperbola it is TS + Sa × Sa: ab :: TS + SA × SA: 0 A B, as above. Therefore all, that is farther requir'd in the Hyperbola, may (in a manner) be found as in the Ellipfis, due Regard being had to changing of the Sines. b Sect. 2. To find the Latus Rectum, or Right Parameter, of any Hyperbola. From the laft Proportion take either of the Antecedents and its Confequent, viz. either Tax Sa: ab. Or TA x SA: □ AB, to them bring in the Tranfverfe TS for a third Term, and by those three find a fourth Proportional (as in the Ellipfis) and that will be the Latus Rectum. Thus Tax Sa: ab::TS: abxTS the Latus Tax Sa Rectum, which call L (as in the Parabola.) Then 2 TS: L:: Tax Saab. AB, therefore Confequently L is the true Latus Rectum, or right Parameter, by which all the Ordinates may be found, according to its Definition in Chap. I. And because TS+Sa Ta, let it be TS+ Sa inftead of Ta, then it will be Ellipfis it would be = ☐abxTS TSxSa+□ Sa □ abx TS TSX Sa-Sa = L and in the =LR=L. Seat. Sect. 3. To find the Focus of any Hyperbola. The Focus being that Point in the Hyperbola's Axis through which the Latus Rectum muft pafs (as in the Ellipfis and Parabola) it may be found by this Theorem. To the Rectangle made of half the Tranfverfe into half the Latus Rectum, add the Square of half the Theozem. Tranfverfe; the Square Root of that Sum will be the Diftance of the Focus from the Centre of the Hyperbola. Demonstration. 1 Suppose the Point at F, in the annex'd Scheme, to be the Focus fought; then will FR L. Let TC= CS be half the Tranfverfe; then is the Point C call'd the Center of the Hyperbola (for a Reason that fhall be hereafter fhew'd.) Again; let CS d. and SF = a In the Ellipfis, 'tis, 2d: L:: 2d-axa: LL. that is,dl= 2da — a a, &c. The Geometrical Effection of the laft Theorem is very eafily perform'd, thus: make Sx L, viz. half the Latus Rectum; and let CSd, as above. Upon Cx (as a Diameter) describe a Circle, and at S the Vertex of the Hy perbola draw the Right Line nS N at Right Angles to Cx; then join the Points C, N with a Right Line, and 'twill be CN=d+a=FC. For ICS: SN:: SN: Sx, per Fig. That is 2d: SN::SN: L 23dLOSN But 4dd+SN=□ CN 3 5 us +16√dd+dL=CN=d+a,&c. B Now Now here is not only found the Distance of the Hyperbola's Focus, either from its Center C, or Vertex S, but here is also found the Right Line ufually call'd its Conjugate Diameter, viz. the Line n SN, which bears the fame Proportion to the Tranfverfe and Latus Rectum of the Hyperbola, as the Conjugate Diameter of the Ellipfis doth to its Tranfverfe and Latus Rectum. For in the Ellipfis TS: Nn:: Nn: L R. per Sect. 2. Pag. 363. Confequently TS: NnNn: LR. But TS=d, Nn=SN, and LR L. Therefore d: SN:: SN: L. As at the 2d Step above. 1 2 What Use the aforefaid Line n SN is of, in relation to the Hyperbola, will appear farther on. Sect. 4. To defcribe an Hyperbola in Plano. In order to the easy defcribing of an Hyperbola in Plano, it will be convenient to premife the following Propofition, which differs from that of the Ellipfis in Sect. 3. Chap. 2. only in the Signs. If from the Foci of any Hyperbola there be drawn two Right Lines, fo as to meet each other in any Propolition. Point of the Hyperbola's Curve, the Difference of thofe Lines (in the Ellipfis 'tis their Sum) will be equal to the Tranfverfe Diameter. That is, if F be the Focus, and it be made Cf = C F (as in the last Scheme) then the Point ƒ is faid to be a Focus out of the Section (or rather of the oppofite Section) and it will be ƒ B→ FBTS. Demonftration. Suppose ƒ C, or CFz, and SA= x, let CS, or TC = d, as before; then will fAd+x+%, and FA = d -\- x —z. Again, let FB = b, and ƒ B = b, then 2db-b, by the Propofition. From thefe fubftituted Letters it follows, That rdd+2dx + 2dz + xx + 22x + zz = ƒA But Per 4th } of last ¤ ƒA+□ AB=¤ƒ B, and ¤ ƒ A+ □ AB= □ FB dd + 1⁄2 d L = da + 2da + aa = □ FC=zz. |