4 12 reduced 13 Dz+zy = Dy + yy 13 Analogy 14 D+y: y:: D+y: z, viz. CA: SA::TA : AP Dytyy which is the first Theorem. 2 Dty 13-zy 16 yy+Dy-zy =D z DD-2Dz + zz 4 17 yy+Dy-zy+ 4 DD +27 4 which is the second Theorem. Q. E. D. The Geometrical Effection of the first of these Theorems is very easy; 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. T From the Comparisons, which have been all-along made in this Chapter, between the Hyperbola and the Ellipsis, 'twill be easy (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, consider the annex'd Scheme, wherein it is evident (even by Inspection) that the opposite 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.) Also, that the middle Point, or common Center of the Ellipsis, is the common Center to all the four con. jugal Hyperbola's. S And the two Diagonals of the Right-angled Parallelogram, which circumfcribes the Ellipsis (or is infcrib'd to the four Hyperbola's) being continued, will be such Afymptotes to those Hyperbala's as are defined, Chap. 1. Sect. 5. Defin. 4. Sect. Sect. 6. To draw the Asymptotes of any Hyperbola, &c. Having found the Latus Rectum (by Sect. 2.) and the Conjugate Diameter in n S N in its true Position, 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 CN and Cn, infinitely continued (as in the following Figure) and they will be the Asymptotes 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. Demonstration. Suppose the Semi-ordinates a b and AB to be rightly apply'd to the Axis TA; and produced both Ways to the Asymptotes, as at fg and FG; then will the CSN, A Cag, and AC AG be alike. Let d = CS=TC. And L = the Latus Rectum; as before. Put se Sa {} =SA} the Abfciffe. Then d+e = Ca. Then Id:SN::d+e:ag. viz. CS:SN::Ca:ag I in's 2dd:SN::dd+2de+ee: ag But 3 dL = S N. per Sect. 3. 2,3... 4ddL+2 de L+ e eL 5 4 2 d Again 52d: L:: 2 de But { 62 de LeeL dL =0ag ee: ab, per Sect. 2. 2 d 67=ag-ab 8×9 10 2 8 ag+ab=bf 9ag-ab=bg}per Fig. □ag-ab=bfxbg 7,10 11 bfx bg = dL Again 12 dd:SN::dd+2dy+yy: AG f But 14 2d: L:: 2dy + yy: AB, per Sect. 2. Eee 2 13-15 17 AG+AB=BF) per Fig. 17 × 18 19□AG-AB=BFXBG 16 19 20 BFX BG=dL 11,20 21 bg= dL dL And BG = 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 BF)d L (= B G is less than bf) dL(=bg, because the Divifor B F is greater than the Divifor bf; and it must needs be so 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, becaused 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 passes thro the Center and falls within the Asymptotes, 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 Ellipsis, Parabola, and Hyperbola, hath its particular Latus Rectum and Ordinates; which (should they be distinctly handled, and the Effection of all fuch Lines as relate to them, as also the Nature and Properties of fuch Figures as may be infcribed and circumscribed 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 shall therefore defift pursuing them any further, being fully fatisfied, that, if what I have already done be well understood, the rest must needs be very easy to any one that intends to proceed farther on that Subject. AN 1 397 AN INTRODUCTION T 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 shall briefly deliver in the following Lemma's, and apply them to Practice in finding the superficial and solid Contents of Geometrical Figures farther on. LEMMA Ι. or If any Series of equal Numbers (representing Lines or other This is so very plain, and easy to be understood, that it needs no Example. If the Series of Numbers in Arithmetick Progression begin with a Cypher, and the common Difference be 1; as, 0. 1. 2. 3. 4. &c. (representing a Series of Lines or Roots beginning with a Point) if the last 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 last Term, N= the Number of Terms, and S = the Sum of all the Series : Then Consequently, NL = S. Then will NL = 28. viz. one Half of so many times the greatest Term as there are Number of Terms in the Series. Thus +1+2+3+4 4+4+4+4+4 10 = the Sum of the Series = NL. 20 NL. = And this will always be so, how many Terms soever there are, by Confect. 1, Page 185. LEMMA III. If a Series of Squares whose Sides or Roots are, in Arithmetick Progreffion, beginning with a Cypher, &c. (as in the last Lemma) be infinitely continued; the last Term being multiply'd into the Number of Terms will be Triple to the Sum of all the Series, viz. NLL = 3S, or NLL = S. That is, the Sum of such a Series will be one Third of the laft or greatest Term, so many times repeated as is the Number of Terms in the Series. From these Instances 'tis evident, that, as the Number of Terms in the Series does encrease, the Fraction or Excess above does I decrease, the faid Excess always being 6-6; which, if we fup pofe the Series to be infinitely continued, will then become infinitely small, viz. in Effect nothing at all. Confequently, NLL may be taken for the true or perfect Sum of such an infinite Series of Squares. If a Series of Cubes whose Roots are in Arithmetick Progreffion, beginning with a Cypher, &c. (as above) be infinitely continu'd, the Sum of all the Series will be + NLLL=S. That is, one Fourth of the last or greatest Term so many times repeated as is the Number of Terms. Inflances |