13 13÷1D+y|15|2 = Dy+yy Dy L+yy L− 2 x L — DL D Dy Lzz+yy Lzz-2 Dy Lz—2yy Lz Dzz Dz+zy=Dy+yy 13 Analogy 14D+y:y::D+y: z, viz. CA: SA ::TA: AP 13-zy 16 yy + Dy—zy = 1 Dz which is the first Theorem. 2 DD + 17 18y + D—1%= พ +ལལྤ 4 18 ± 19 y = √ D + xx: + ± x—;D {cond 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 caly (even for a Learner) to perceive the Coherence that is in (or between) those 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. (fee Sect. 1, of this Chap.) Alfo, that the middle Point, or common Center of the Ellipfis, is the common Center to all the four conjugal 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 HyperLola'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 A CSN, A Cag, and a CAG be alike. Let d CSTC. And L Put {SAS e=Sal the Latus Rectum; as before. the Abfcife. Then + y = CA. Then Id: SN:: d+e: ag. viz. CS: SN:: Ca: ag But 2, 3*.* 5 3 dL OSN. per Sect. 3. 4 ddL+2de Lee L 2 d =0 ag Again 52d: L:: 2 de +ee:□ a b, per Sect. 2. 6 4 6-7 But { 2 de Lee L d L 2 2 d Dab Dag-ab 8agabbf1 8 x 9 10 ag-ab=bfx b g 7, 10 11 bfx bg = d L 13 Alfo d L 15 16 = AG-AB 2 { | 17 | A G + AB = BF 17 X 18 19 16 19 20 BFX BGdL 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 less than bf) dL (bg, becaufe the Divifor B F is greater than the Divifor bf; and it muft needs be fo where-ever the Ordinates are poduc'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 Afymptote. 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) becaufe 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 alfo the Nature and Properties of fuch Figures as may be infcribed and circumfcribed to all the Sections, with the various Habitudes or Proportions of one Hyperbola 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 satisfied, that, if what I have already done be well understood, the rest muft needs be very easy 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 Quantities) as, 1. I. I. I. &c. or 2. 2. 2. 2. &c. or 3. 3. 3. 3. &c. if one of the Terms be multiply'd into the Number of Terms, the Product will be the Sum of all the Terms in the Series. This is fo very plain, and easy to be understood, 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. (representing 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 Then will NL = 2 S. Confequently, NLS. viz. one Half of fo many times the greatest Term as there are Number of Terms in the Series. And this will always be fo, how many Terms foever there are, by Confect. 1, Page 185. LEMMA III. If a Series of Squares whofe Sides or Roots are, in Arithmetick Progreffion, beginning with a Cypher, &c. (as in the last Lemma) be infinitely continued; the laft Term being multiply'd into the Number of Terms will be Triple to the Sum of all the Series, viz. NL L = 3S, or NLL=S. That is, the Sum of fuch a Series will be one Third of the laft or greatest Term, fo many times repeated as is the Number of Terms in the Series. So+1+4 +9_14 9+9+9+9 30 18 3·16+16+16+16 80 I I 3 -+ 18 3 9 I I 8 24 3 24 From thefe Inftances 'tis evident, that, as the Number of Terms in the Series does encrease, the Fraction or Excess above does I decrease, the faid Excefs always being 6N—6, which, if we fup pofe the Series to be infinitely continued, will then become infi nitely small, viz. in Effect nothing at all. Confequently, NLL may be taken for the true or perfect Sum of fuch an infinite Series of Squares. LEMMA IV. If a Series of Cubes whofe Roots are in Arithmetick Progreffion, beginning with a Cypher, &c. (as above) be infinitely continu'd, NLLL=S. the Sum of all the Series will be That is, one Fourth of the laft or greatest Term so many repeated as is the Number of Terms. times Inflances |