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. DTS the Tranfverfe Diameter. Let Lthe Latus Rectum. y=SA the Abfciffa. Andz-AP the Distance between the Then, if y be given, z may be found by this Theorem, {D [which differs from that in the Ellipfis only in Signs. Vide Page 371.] T Or, if z be given, then y may be found by this Theorem: Theozem. VDD+zz 4 Demonftration. Draw the Semi-ordinate b a, as in the Figure, and put x = aan infinite fmall Space between the two Semi-ordinates; as before in the Ellipfis, &c. Then ID: L:: Dy+yy:□ AB That is, 2 TS: L::TS+SAXSA: □ AB I Dy Lyy L 3 D QAB Dy+yy—2yx Again 4 D: L:: Dyyy-27 — Dxxx:□ ab 4 6 Per Figure 7: A B :: z—x: a b, viz. PA: AB:: Pa: ab 7 in 's 8 zz□ AB :: zz—2 zx+xx: a b Suppofe 90 and every where rejected (as in the Ellipfis) Dy L+yy L Dy L + y y L − 2 y L — DL D DyLzz+yyLzz-2 DyLz—2yy Lz Dzz Dz+zy=Dy+yy 13 13 Analogy 14D+y:9::D+y:z, viz. CA: SA :: TA: AP Dy+y which is the first Theorem. 13 − z y |16|3y+Ďy—zy = ¦ Dz DD-2 Dx zz DD +2 18 y + ÷ D — z = √ √ D D + zz 4 18 ±19|y=V D + % + 4 which is the fecond Theorem. Q. E. D. The Geometrical Effection of the firft of thefe Theorems is very eafy; for, by the 14th Step, 'tis evident that there are three Lines given to find a fourth proportional Line, [By Problem 3, Page 308.] Scholium. From the Comparisons, which have been all-along made in this Chapter, between the Hyperbola and the Ellipfis, 'twill be cafy (even for a Learner) to perceive the Co herence that is in (or between) thofe two Figures; but, for the better understanding of what is meant by the Center and Afymptotes of an Hyperbola, confider the annex'd Scheme, wherein it is evident (even by Inspection) that the oppofite Hyperbola's will always be alike, because they will always have the fame Transverse Diameter common to both, &c. (lee Sect. 1, of this Chap.) Alfo, that the middle Point, or common Center of the Ellipfis, is the common Center to all the four con. jugal Hyperbola's. And the two Diagonals of the Right-angled Parallelogram, which circumfcribes the Ellipfis (or is infcrib'd to the four Hyperbola's) being continued, will be fuch Afymptotes to those Hyperbala's as are defined, Chap. 1. Sect. 5. Defin. 4. Sect. Sect. 6. To draw the Afymptotes of any Hyperbola, &c. Having found the Latus Rectum (by Sect. 2.) and the Conjugate Diameter in n SN in its true Pofition, by Sect 3. Then, thro' the Center C of the Hyperbola, and the extream Points n N of its Conjugate Diameter, draw two Right Lines, as C N and C n, infinitely continued (as in the following Figure) and they will be the Afymptotes required. That is, they are two fuch Right Lines as, being infinitely extended, will continually incline to the Sides of the Hyperbola, but never touch them. Demonftration. Suppose the Semi-ordinates a b and AB to be rightly apply'd to the Axis TA; and produced both Ways to the Afymptotes, as at fg and FG; then will the ACS N, ▲ Cag, and a CAG be alike. Let d-CSTC. And L the Latus Rectum; as before. = Put the Abfciffe. Then Then Id: SN:: d+e: ag. viz. CS: SN:: Ca: ag Iin's 2 dd: SN:: dd +2de +ee: Dag 5 4 Sect. dL OSN. per 3. ddL+2de Lee L 2 d =0 ag Again 52d: L:: 2 de +ee: a b, per Sect. 2. : But { 6 67 8a 8+ ab = bf } per Fig. 8 x 9 10 8×9 fx Dag-ab= bf xbg 7, 10 11 bfx bg= d L Again 12 dd: SÑ::dd +2dy+yy:□AG That is, 3,12. 13 CS:SN:: □ČA:□ AG dd L+ 2dy L+yyLAG 2 d But 14 2d: L:: 2dy+yy:□ AB, per Sect. 2. 2dy Lyy L a N 2d d L 13-15 16 2 AGABBF1 Alfo {17 4G+ 4B = B F 17 X 18 19 .16 }per AG-DAB=BFXBG 19 20 BFX B G = 1 d L 2 11,820.21bg= . And BG= bf idL BF From the laft Step 'tis evident, that the Afymptotes are nearer the Hyperbola at G than at g, and confequently will continually approach to its Curve: For B F) dL (= BG is lefs than b dL (bg, because the Divifor B F is greater than the Divifor bf; and it must needs be fo where-ever the Ordinates are p oduc'd to the Afymptotes, from the Nature of the Triangles. Again; From the 7th and 16th Steps 'tis evident, that the Afymptotes can never really meet and be co-incident with the Curve of the Hyperbola, altho' both were infinitely extended, becaufed L will always be the Difference between the Square of any Semi-ordinate and the Square of that Semi-ordinate, when 'tis produc'd to the Asymptote. Confectary. From hence it follows, that every Right Line which paffes thro the Center and falls within the Afymptotes, will cut the Hyperbola; and all fuch Lines are call'd Diameters (as in the Ellipfis) because the Properties of the Hyperbola and Ellipfis are the fame. Note. Every Diameter, both in the Ellipfis, Parabola, and Hyperbola, hath its particular Latus Rectum and Ordinates; which (fhould they be diftinctly handled, and the Effection of all fuch Lines as relate to them, as also the Nature and Properties of fuch Figures as may be inferibed and circumfcribed to all the Sections, with the various Habitudes or Proportions of one Hyperbsla to another, &c.) would afford Matter fufficient to fill a large Volume. But thus much may fuffice by way of Introduction; I fhall therefore defift purfuing them any further, being fully fatisfied, that, if what I have already done be well understood, the reft muft needs be very eafy to any one that intends to proceed farther on that Subject. AN ΑΝ 397 INTRODUCTION T TO THE Mathematicks. PART V. HE Method of finding out any particular Quantity (viz. either any Line, Superficies, or Solid) by a regular Progreffion, or Series of Quantities continually approaching to it, which, being infinitely continued, would then become perfectly equal to it, is what is commonly call'd Arithmetick of Infinites; which I fhall briefly deliver in the following Lemma's, and apply them to Practice in finding the fuperficial and folid Contents of Geometrical Figures farther on. LEMMA I If any Series of equal Numbers (reprefenting Lines or other This is fo very plain, and easy to be underftood, that it needs no Example. If the Series of Numbers in Arithmetick Progreffion begin with a Cypher, and the common Difference be 1; as, O. 1. 2. 3. 4. &c. (reprefenting a Series of Lines or Roots beginning with a Point) if the laft Term be multiply'd into the Number of Terms, the Product will be double the Sum of all the Series. That is, putting L the laft Term, N= the Number of Terms, and S the Sum of all the Series: Then |