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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 easily perform'd by Help of a Theorem ; as in the Ellipsis, Seet. 6. Chap. 2. DTS the Transverse Diameter.
in the Transverse cut by
the Tangent. Then, if y be given, x may be found by Dy tyy
[ from that in the Ellipses only in Signs. Vide Page 371.] Or, if z be given, then y may be found by this Theorcm : Theozem. VDD+23
nates ; as before in the Ellipfis, &c.
Dy L + yy L
Dy L --yy L-2y* L- DxL + xx L
O ab Per Figure 712: AB::24:ab, viz. PA: AB::Paal
7 in O’s 8 zz: 0 AB :: 22–2 zx+xx: a b Suppose 9x=0 and every where rejected (as in the Ellipfis.)
:: zz-22: a b
Dy L + yy L-2yL-DL 6,
DyLzz tyyLzz--2 DyL2-2yyLz
Dzz 12 reduced
13 Dz+zy=Dy + yy 13 Analogy 14D+y:y::D y:z, viz. CA:SA:: TA: AP 13++D+y 15 2=
Dytyy which is the first Theorem. 13 — zy (16|yy + Dy-zy = Dz
DD – 2Dz tzz
DD + 16 OC 17|yy+Dyzy+
4 17 18|y+:D—z=vPD+
which is the fe18 + 199=VDE
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. 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, consider 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 same Transverse Diameter common to both, &c. (see Selt. 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.
And the two Diagonals of the Right-angled Parallelogram, which circumscribes the Ellipsis (or is inscribid to the four Hyperbola’s) being continued, will be such Afymptotes to those Hyperbola's as are defined, Chap. 1. Sett. 5. Defin. 4.
Sect. 6. To draw the asymptotes of any Vyperbola, &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 n, infinitely continued (as in the following Figure) and they will be the Asymptotes 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.
Suppose the Semi-ordinates a b and A B to be rightly apply'd to the Axis T A; and produced both Ways to the Afymptotes, as at f g and FG; then will the a CS N, A Cag, and AC AG be alike.
Let d=CS=TC. And L= the Latus Rectum; as before.
dL=OSN. per Sect. 3.
dd L+2 del teel
= 0 ag
OCS: OSN::OČA: O AG 3,12-13
dd L + zdy L+yy L=Q AG
2dy + yy L 14 115
D AB 20
d L 13 – 15
18 AG - ABZBG per Fig. 17 x 1819) AG-OAB=BFXBG
1920|BFX BG=;d L 11,820 21105 = . And BG=
From the last Step 'tis evident, that the Afymptates are nearer the Hyperbola at G than at g, and consequently will continually approach to its Curve : For B F) dL (= B G is less than b) dLi=bg, because the Divisor B F is greater than the Divifor bf; and it must 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, 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 Afymptote.
Confe&taryFrom hence it follows, that every Right Line which pafles thro' the Center and falis within the Afymptotes, will cut the H.perbola ; and all such Lines are callă Diameters (as in the Ellips) because the Properties of the Hyperbola and Ellipfis are the same.
Note. Every Diameter, both in the Ellipfis, Parabola, and Hyperbola, hath its particular Latus Rectum and Ordinates ; which (should they be diftinctly handled, and the Effection of all such Lines as relate to them, as also the Nature and Properties of such Figures as may be inseribed and circumfcribed to all the Sections, with the various Habitudes or Proportions of one Hyperbela to another, &c.) would afford Matter sufficient to fill a large Volume. But thus much may fuflice by way of Introduction; I fhall therefore desist 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.
HE 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 callid Arithmetick of
Infinites; which I shall briefly deliver in the following Lemma's, and apply them to Practice in finding the fuperficial and folid Contents of Geometrical Figures farther on.
LEM MA I. If any Series of equal Numbers (representing Lines or other
Quantities) as, 1. j. 1. 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.
LE M M A II. If the Series of Numbers in Arithmetick Progression begin with a
Cypher, and the common Difference be I; 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 ;