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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 C N and C ne infinitely continued (as in the following Figure) and they will be the Afymptotes required. That is, they are two such'Right Lines as, being infinitely extended, will continually incline to the Sides of the Hyperbola, but never touch them.
From the last Step 'tis evident, that the Afymptotes are nearer the Hyperbola at G than at g, and consequently will continually approach to its Curve : For B F) idL(= B G is less than bf)
dL=hg, because the Divifor B F is greater than the Divifor bf; and it must needs be so 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, because įd L will always be the Difference between the Square of any Semi-ordinate and the Square of that Semi-ordinate, when ʼtis produc'd to the Afymptote.
Conseetary. From hence it follows, that every Right Line which passes thro' the Center and falls within the Afymptotes, will cut the Hyperbola ; and all such Lines are callà Diameters (as in the Ellipfis becaufe the Properties of the Hyperbola and Ellipsis are the same.
Note. Every Diameter, both in the Ellipfis, Parabola, and Hyperbola, hath its particular Latus Rectum and Ordinates ; which (Thould they be distinctly handled, and the Effection of all such Lines as relate to them, as also the Nature and Properties of fuch Figures as may be inscribed 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 defist pursuing them any further, being fully satisfied, 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.
T H E Method of finding out any particular Quantity (viz.
either any Line, Superficies, or Solid) by a regular
Progression, or Series of Quantities continually approaching to it, which, being infinitely continued, would then become perfectly equal to it, is what is commonly calld 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.
L E MM A I. If any Series of equal Numbers (representing Lines or other
Quantities) as, I. 1. I. 1. & 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 so very plain, and easy to be understood, that it needs no Example.
LEMMA II. 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 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 :