Sect. 3. To find the Focus of any Hyperbola. The Focus being that Point in the Hyperbola's Axis through which the Latus Rectum must pass (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 Theorem. Tranfverfe; the Square Root of that Sum will be the Distance of the Focus from the Centre of the Hyperbola. Demonftration. Suppofe the Point at F, in the annex'd Scheme, to be the Focus fought; then will FR L. Let TC= 1 = CS be half the Tranfverfe; then is the Point Then 12d: L: 2d+axa: ÷ LL That is, 2 TS: L::TS+SFXFS: I 3d L=2da + aa 2 1 S FR R F 3+ dd 4 dd + 1 d L = dd +2da+aa 2 4 w2 Or In the Ellipfis 'tis, 2d: L:: 2d-axa: LL. that is, id L The Geometrical Effection of the laft Theorem is very easily 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 n S N at Right Angles to Cx; then join the Points CN with a Right Line, and 'twill be CN=d+a=FC. 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 that Right Line ufually call'd its Conjugate Diameter, viz. the Linen SN, which bears the fame Proportion to the Transverse 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: Nn :: ¦ Nn: ¦ L R. But TS d, Nn SN, LR. 1⁄2 and LR L. Therefore d; SN::SN: L. As at the 2d Step above. 1 2 What Ufe the aforefaid Line n S N is of, in Relation to the Hyperbola, will appear farther on. Sect. 4. To describe an Hyperbola in Plano. In order to the easy describing 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 Propofition. 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 CfCF (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 FB TS. Demonftration. Suppofe fC, or CF=z, and SA=x, let C S, or TC=d, as before; then will ƒA=d+x+%, and F A= d +x — Zi Again, let FB5, and ƒ B=b, then 2db—b, by the Propofition. From thefe fubftituted Letters it follows, That1dd2dx +2dx + xx + 2xx+zz = □ ƒÂ And 2 dd2dx· -2dz + xx But Per 4th of last ·22x+zz=OFA □ƒA+□AB=☐ƒB, and □ƒ1⁄4+□ AB=■FB 3 dd+d L = da +2da+aa = □ FC=zz. Again 6 2d: L:: 2d +×××:□ AB, by common Properties, ZZ- -dd 5, 6 7 2d: 9+d11d++2d3z+2ddzx+ddzz+2dzzx+zzxx=ddhh 10 X dd 12 d+-2d3z—2ddzx+ddzz+2dxzx+zzxx=ddbb ! I w2 13|dd+dz+zx=dh (Altho' the Equation at the 16th 2 12 w2 14 dd-dz-zx=db Step be in itself impoffible, becaufe z is greater than d (by the 4th Step) yet from thence it will be eafy to conclude, that the Difference between d and z + 2x d will give the true Value of b; as in the 17th Step. But because I would leave no Room for the Learner to doubt about changing the Equation, d — % — ZX b into that of d z+2➡d = b, it may be convenient to illustrate the whole d Process in Numbers, whereby (I prefume) 'twill plainly appear that b-bTS. In order to that, let the Tranfverfe TS=2d=12, then d=6 suppose the Abfciffa SA=x=4, and the Semi-ordinate AB=3 Firft ITS+SAXS A:AB:: TS: L, per Sect 2. 1, viz. 2 12 + 4X4 64: 9:: 12: 1,6875=L Again √dd+dL=d+aCF, per Sect. 3. 3, viz. | 436+5,0625 = 6,408=CF=z Then 5d+x+x=6+4+ 6,408 = 16,408 = ƒ 4 And 6d+x-6+4—6,408 3,592 = FA 5 02 62 3 by Suppofition. But 9 9 AB, for AB +910 278,2224 = ƒA□ A В = □ƒB +911 21,9024 FA+□ AB=OFB 16,68 = ƒ B w2 10 12 1 1 ພ 13 12-13 14 4,68 FB 12,00 =ƒB—FB=TS. Which was to be prov'd. If this Propofition be truly understood, it must needs be eafy to conceive how to defcribe the Curve of any Hyperbola very readily by Points, when the Tranfverfe Diameter and the Focus are given (or any other Data by which they may be found, as in the precedent Rules) thus: Draw any ftreight Line at Pleasure, and on it fet off the Length of the given Tranfverfe T S, and from its extream Points or Limits, viz. TS, set off Tf=8F, the Distance of the given Focus (viz. the Point f without, and F within the Section, as before); that done, upon the Point f (as a Center) with any affum'd Radius greater than TS, defcribe an Arch of a Circle; then from that Radius take the Transverse TS, making their Difference a fecond Radius, with which, upon the Point F within the Section, defcribe another Arch to cut or cross the first Arch, as at B; then will that Point B be in the Curve of the Hyperbola, by the laft Propofition. And therefore 'tis plain, that, proceeding on in this Manner, you may find as many Points (like B) as may be thought convenient (the more there are, and nearer they are together, the better) which being all join'd together with an even Hand (as in the Parabola) will form the Hyperbola requir❜d. There are feveral other Ways of delineating an Hyperbola in Plano: One Way is, by finding a competent Number of Ordinates, as by Section 1, &c. but I think none so easy and expeditious as this mechanical Way: I fhall therefore, for Brevity's Sake, pafs over the reft, and leave them to the Learner's Practice, as being eafily deduced from what hath been already faid. Sea, Sect. 5. To draw a Tangent to any given Point in the Curve of an Hyperbola. The drawing of a Tangent, that will touch any given Point in the Curve of an Hyperbola, may be eafily perform'd by Help of a Theorem ; as in the Ellipfis, Sect. 6. Chap. 2. D=TS the Tranfverfe Diameter. Let L the Latus Rectum. y=SA the Abfciffa. the Distance between the Sordinate and Ordinate and that Point Andz AP in the Tranfverfe cut by the Tangent. Then, if y be given, z may be found by this Theorem, { Dytyy D+Y =z [which differs from that in the Ellipfis only in Signs. Vide Page 371.] T P Or, if z be given, then y may be found by this Theorem : Draw the Semi-ordinate ba, as in the Figure, and an infinite fmall Space between the two Semi-ordi put x = Aa{anties, is before in the Ellipfis, &c. as Again 4D: L:: Dyyy-2yx - Dx+xx: □ ab That is, 5 TS: L::TS+ Saxsa: 4.6 ab Dy L-yy L-2yx L-DxLxx L D = 0 ab Per Figure 7: A B :: z—x: a b, viz. PA: AB:: Pa: ab in's 8 zz: □ AB :: zz- -2 zx+xx: a b Suppofe 90 and every where rejected (as in the Ellipfis) |