T B B Thefe Proportions are the common Property of every Hyperbola, and do only differ from thofe of the Ellipfis in the Signs + and; as plainly appears in the folThat is, if we lowing Proportions. fuppofe 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 XSA: AB as by Sect. 1. Chap. 2. and in the Hyperbola it is TS + Se X Saab::TS+SAX 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 Ellip fis, duc Regard being had to changing of the Sines. F Sect. 2. To find the Latus Kedum, or Right Parameter, of any Hyperbola. From the laft Proportion take either of the Antecedents and its Confequent, viz, either Tax Sa: ab. Or TAX 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. Then 2, ab::TS: OabXTS = the Latus Rectum, which call L (as in the Parabola.) 2TS: L:: T But 3 ax Sa: ab. Tax Sa: ab:: TAX SA: 3 4TS: L:: TAXSA: □ A B, &c. AB, therefore Confequently L is the true Latus Rectum, or right Parameter, by which all the Ordinates may be found, according to its DefiniAnd because TS+Sa Ta, let it be 7S+ tion in Chap. 1. □abXTS TSXSa+Sa abxTS Sa inftead of Ta, then it will be = L and in the Ellipfis it would be =LR=L. TSX Sa-Sa Sect. 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 país (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. Demonstration. 1 Suppofe 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 shall be hereafter fhew'd.) Again; let C S d. and S Fa = Then 12d: L:: 2d + axa: LL That is, 2 TS: L::TS+SFXFS: □ FR I 3d L2da + aa 3+ dd 4 dd + 1 d L = dd + 2da+aa 4 ww2 ຫນ 5 ad+dl=d+a= FC B Or 5,—d 6√dd¦dL—d=a=SF C S R F 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 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 Hyperbola draw the Right Linen SN at Right Angles to Cx; then join the Points CN with a Right Line, and 'twill be CN=d+a=FC. T Now here is not only found the Distance of the Hyperbola's Focus, either from its Center C, or Vertex S, but here is alfo found that 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: Nn :: Nn: LR. But TS d, Nn=SN, 플 144 and LR L. Therefore d: SN::SN: L. As at the = 2d Step above. Σ What Ufe 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, so 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 laft Scheme) then the Point fis 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 CFz, and S A=x, let C S, or TC=d, as before; then will ƒ A=d+x+z, and FA=d+x- Z. Again, let FBh, and fB=b, then 2db-b, by the Propofition. From thefe fubftituted Letters it follows, Å That I dd+2dx +2dz + xx + 2xx+zz = □ ƒ A But Per 4th of baft} 3 dd+id L=da +2da+aa= □ FC=zz. Again 6 2d: L ; : 2d +×××: □AB, by common Properties, dd+2dx+2dz+xx+2xx+zz+ ZZ- -dd 5, 6 7 2d: : : 2dx + xx : □AB I+89 2+810 =FA+QAB—b 2dxzxzxxx-2d3x-ddxx dd dd+2dx-2dx+xx¬2xx+x+ dd 9+d11d++2d3z+2ddzx+ddzz+2dzzx+zzxx=ddhh 10 X dd 12 d+2d1z-2ddzx+ddzz+2dzzx+zzxx=ddbb II w2 13 dd+dz+zx=dh 12 w2 14 dd-dz-zx=db 13÷d15d+x+27=h 14d 16 d-z But because I would leave no Room for the Learner to doubt Zx about changing the Equation, d — z———= b into that of ढ़ - db, it may be convenient to illuftrate the whole Procefs in Numbers, whereby (I prefume) 'twill plainly appear that b-b=TS. In order to that, let the Tranfverfe TS=2d=12, then d=6 fuppofe the Abfciffa SA=x=4, and the Semi-ordinate AB=3 Firft ITS+SAXSA:□AB:: TS: L, per Set 2. 12+4×4 64: 9: 12: 1,6875= L √dd+dL=d+aCF, per Sect. 3. √3645,06256,408 = CF = z I, viz. Again 3 3, viz. 14 Then 5d+x+x=6+4+ 6,408 = 16,408 = ƒ A And 6d+x-%=6+4—6,408 3,592 = FA = 5827269,2224 = FA But 9 12,9024OFA 9 AB, for AB 3 by Suppofition. 7910278,2224ƒA + □ A B = ƒB 21,9024 FA+Q AB = □ FB 16,68 fB 8 +911 10 w2 12 II w2 13 4,68 FB = 12-13 14 12,00ƒB-FB-TS. Which was to be prov'd. If this Propofition be truly understood, it must needs be easy 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 prece dent Rules) thus: Draw any freight Line at Pleafure, and on it fet off the Length of the given Tranfverfe T S, and from its extream Points or Limits, viz. TS, fet off Tf SF, the Distance of the given Foaus (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 Tranfverfe TS, making their Difference a fecond Radius, with which, upon the Point F within the Section, describe 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 last 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 Ordi nates, as by Section 1, &c. but I think none fo eafy and expediti ous 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. Sect. |