HYP
pounded of the fame Abfcifs, and a given right Line call'd the Tranfverfe Axis , as another given right Line, call'd the Parameter of the Axis, is to the Tranfverfe Axis : Or, it is a Curve Line, wherein ay z =abx-\-bxx, that is, b:a-=y 2 'i ax-]-x z .
In the Hyperbola, a mean Proportional between the tranf verie Axis and Parameter, is call'd the Conjugate Axis : And if the Tranfverfe Axis AB (Tab. Conicks, Fig. 27.) be join'd dire&ly to the Axis A X, and be biifefted in C ; the Point C is call'd the Centre of the Hyperbola. See Axis and Centre.
If a right Line DE pafs thro' the Vertex A, (Fig. 20.) parallel to the Ordinates Mm, it is a 'Tangent to the Hyper- bola in A. See Tangent.
If a right Line DE be drawn thro' the Vertex A of an Hyperbola, parallel to the Ordinates Mm, and be equal to the Conjugate Axis, viz. the Parts DA and AE equal to the Semi-axis ; and right Lines CF and CG be drawn from the Centre C thro' D and E ; thofe Lines are call'd AJfymp- totes of the Hyperbola. See Assymptotes.
The Square of the right Line CI or A I, is call'd the T-'c-iOT of the Hyberbola. See Power.
'Properties of the Hyperbola.
In the Hyberbola, the Squares of the Semi-ordinates are to each other, as Recfangles of the Abfcifs into a cer- tain right Line compos'd of the Abfcifs and Tranfverie Axis. — Hence, as the Abfciffes x increale, the Rectangles ax--x z , and, consequently, the Squares of the Semi-ordi- nares y z , and therefore the Semi-ordinates themielves, in- creaie. The Hyperbola, therefore, continually recedes from its Axis. , -
2 . The Square of the Conjugate Axis, is to the Square of the Tranfverfe, as the Parameter, to theTranfverie Axis.
— And hence, fince b:a: : I'M' : AP.PB, the Square of the Conjugate Axis, is to the Square of the Tranfverie ; as the Square of the Semi-ordinate is to the Rectangle of the Abfcifs into a Line compos'd of the Abfcifs and Tranfverfe
5?'. To defiribe an Hyperbola, in a continu'd Motion; the Tranfverfe Axis, and the Diftance from the Vertex being given. In the two Foci F and/, (Fig. 28.J fix two Nails or Pins ■ and to one of them, in F, tye a Thread F M C, fatten- ing the other End C to the Ruler Cf, which exceeds the fame by the Tranfverfe Axis A B. The other End of' the Ruler being perforated, put it on the Pin/; and fixing a Style to the Thread, move the Ruler. Thus will the Style trace out an Hyperbola.
Again, with' the fame Data, any Number of Points in an Hyperbola ate cafily found, which may be connected into an Hyperbola. Thus, from the Focus/ with any Interval greater than A B, delcribe an Arch ; and making/£ = AB ; with the remaining Interval bin, from the Point F, draw another Arch interfeaing the former in ill : For, as fm — p m — ^ g . m i s a Foi nt in the Hyperbola. And fo of the
reft
i° If in an Hyperbola, the Semi-ordinate PM, Fig. 20. be p-odu'e'd till it meet the Afymptote in R ; the Difference of the Squares of PM and PR is equal to the Square of the Conjugate Semi- axis DA.— Hence, as the Semi-ordinate PM increafes the right Line PR decreaies, and conlequently MR • and therefore the Hyperbola itielf approaches nearer to the Afymptote ; but it can never ablolutely meet it be- caufe, as PR'-PM' = DA' ; it is impoflible PR'-PM'
— PM' Ifiould ever become =0.
,° In an Hyperbola, the Rectangle of MR and Mr, is equal to the Difference of the Squares PR' and PM'. -- And hence, the fame Reaangle is equal to the Square of the Conjugate Semi-axisDA; and confequently all Refl- anglesform'd in the lame Manner are equal.
«° If am be parallel to the Afymptote C F, the Rea- angle of qm into Cq is equal to the Power of the Hyper- Ma -And hence, 1°. If we make GI = AI — a, Lq — r an d qm=y, we (hall have a' = .iy<; which is the E- ouation expreffing the Nature of the Hyperbola within its Afymptotes. 2=. The Afymptotes therefore be.ng given in Politico, and the Side ot the Power of CI or A I ; if in one of the Afymptotes C G, you take any Number of Ab- fciffes - lb manv Semi-ordinates will be found, and by them any Number of' Points in an Hyperbola will be determin d by finding third Froportionals to the Abfcifs, and the Side ot the Power CI.- 3 . If *<= Abfci{fo be not computed from the Centre C, but from fome other Point L; andCL be fuppos'd=t; we (hall have Cq=b+x ; and confequent-
' y t^x^ Hyperbola, as the Tranfverfe Axis is to the Parameter; fo is the Aggregate of the Tranfverfe Semi- axis and Abfcifs, to the Subnormal : And as the Aggregate of the Tranfverfe Semi-axis and Abfcifs is to the Ablciis; fo is the Aggregate of the entire Tranfverfe Axis and Ab- fcifs to the Subtangent. See Subnormal and Suetan-
( 2 77 )
HYP
8°. If within the Afymptotes of an Hyperbola, from a ioint thereof m, (r,g. 25.) be drawn two right Lines Hm and 111K and other two LN and NO, parallel to the fame; Him, »K=Lh,NO. And the fame will hold, if you draw LNo parallel to the right Line thus drawn Hmk, viz. in this Cale likewife Hm.mk^L'N.'No. — Con- lequently, all Reaangles form'd after this Manner of right Lines drawn parallel either to the fame Line HA, or to two, H m and m K, are equal to each other.
9°. If a righr Line Bk, be drawn in any Manner between the Afymptotes of an Hyperbola ; the Segments HE and m K intercepted on each Side between the Hyperbola and Afymp- totes ate equal. — And hence, if Em=o i the right Line Hfc is a Tangent to the Hyperbola : Confequently, the Tan- gent FD intercepted between the Afymptotes, is biffcaed in the Point of Contaa V. Laltlv, the Reaangle of the Segments Hz» and mk, parallel 'ro the Tangent FD, is equal to the Square of haft rhe Tangenr D V.
10°. The Square of the Semi-ordinate in an Hyperbola, is to the Reaangle of the Abfcifs, and Aggregate of the Iranlvtrle Diameter A B, (Fig. 50.) and Abfcifs AP, as the Square ot the Conjugate Semi-diameter AD, is to the Square of the Tranfverfe Semi diameter C A. — Hence, if youfuppofe AP.v, and 2 r' = A B=a, you will have v" r x
ax-j-x' ; confequently y % = (c'ajc -)-!;'»') : { aa
^~-^- + 4 *■ . Make 4c' .-«=£,; then will y 1 =bx +
bx':a. So that the fame Equation defines the Nature of the Hyperbola in refpea of irs Diameter, as expreffes it in refpea of its Axis ; an.l the Parameter is a third Propor- tional to the Conjugate Diameters DE and AB.
n°. If from the Vertex A, and any Point of a Parabola N, you draw AF and TN parallel to the Afymptote C tv ; the Reaangle of TN inio TC, will be equal to the Rcflangie of F A into F C. — Hence, i f T C = x, T N = j> ; the Equation expreifing the Nature of a Hyperbola within Afymptotes, in refpea of its Diameter, will be xy=ab.
ii°. An Afymptote being taken for a Diameter; divid- ed into equal Parts, and thro' all the Divifions, which form fb many Abfciffes continually increafing equally, Ordinates to the Curve being drawn, parallel to the other Afymptote ; the Abfciffes will repreient an infinite Series of natural Numbers ; and the correfponding Hyperbolic or Afymptotic Spaces, will reprefent the Series of Logarithms of the fame Numbers. See Logarithm and Logarithmic Curve.
Hence, different Hyperbola's will furnifh different Series of Logarithms to the fame Series of natural Numbers ; fo that to determine any particular Series of Logarithms, choice muft be made of fbme particular Hyperbola. — Now, the molt fimple of all the Hyperbola's is the Equilateral one ; i. e. whole Afymptotes make a right Angle between them. This, M. de Lagni alledges in Favour of the Binary Arith- metic, as being the Refult of iiich Equilateral Hyperbola. See 'Binary Arithmetic.
For the Locus of an Hyperbola. See Locus.
For the Quadrature of an Hyperbola. See Quadra- ture.
Equilateral Hyperbola, is that wherein the Conjugate Axis A B (Fig. 20.) and D E are equal.
'Properties of the Equilateral Hyperbola.
Since the Parameter is a third Proportional to the Conju- gate Axis, it is alio equal to the Axis.
Wherefore, if in rhe Equation y z =^bx-\-bx z : a, you fiippofe b=.a ; the Equation jy'=tf.v-[-A will expreis the Natute of the Equilateral Hyperbola.
And hence, the Squares of the Ordinates y* and z z , are ro each other a.sax-\~x z and avj-v~ : That is, as the Reaangles of the Abfciffes into righr Lines compos'd of the Abfcifs and Parameter.
If you fuppofe CP = *-, CA = r, and AP=# — r, and PB— r-j-A:. Confequently v'~tf' — x z .
And fince A E = C A ; the'Angle A C E will be half right ; and confequently the Angle of the Afymptotes F C G a right Angle.
Infinite Hyperbola's, or Hyperbola's of the higher Kinds, are thofe defindby the Equation ay™-j-"=bx' a _ (^-4-3.*)". See Hyperboloides.
Hence, in infinite Hyperbola's ay a --\ av m -\- % =bx" {a+xy-.bz" (a+z)": That is, jv"+" : v'-j-"=x- {a-\-xY:Z"{a + zy.
As the Hyperbola of the firfl Kind or Order has two Afymptotes,"that of the fecond Kind or Order has three, that of the third, four, (£c. See Asymptote, Curve,
In refpea of thefe, the Hyperbola of the firfl Kind, is call'd the Apollonian or Conical Hyperbola.
Apollonian Hyperbola is the common Hyperbola, or the Hyperbola of the firfl Kind : Thus call'd in Contra- diflinition ro the Hyperbola's of the higher Kinds.
M * Z z z HYPERBOLE
