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Part IV. 1. If any Ordinate and its Abfciffa are given, there may by them be found as many Ordinates as you please to affign or take Points in the Parabola's Axis; (by Sect. 4. Page 380) and the Curve of the Parabola may be drawn by the extream Points of thofe Ordinates, as the Ellipfis was Page 373.

2. If the Latus Rectum, and either any Ordinate, or its Abscissæ, are given, then any affign'd Number of Ordinates may by them be found (by Sect. 2. Page 381.) either Numerically or Geometrically, &c.

3. If only the Distance of the Focus from the Vertex of the Parabola be given, any affign'd Number of Ordinates may be found by it. For SFL the Latus Rectum, and LFR as in

the laft Section; and it will be, as SF: is to □ FR:: fo is any other Abfciffa, viz. (Sa, or SA, &c.) to the Square of its Semi-ordinate (viz.ba, or □BA) according to the common Property of the Parabola.

Altho' any of thefe Ways of finding the Ordinates are easy enough, yet that Way which may be deduced from the following Propofition will be found much more easy and ready in Prac

tice.

Propofition.

The Sum of any Abfciffa and focal Distance from the Vertex, will be equal to the Distance from the Focus to the extream Point of the Ordinate, which cuts off that Abfciffa.

For Inftance, fuppofe S to be the Vertex of any Parabola, the Point F to be its Focus, and AB any Semi-ordinate rightly apply'd to its Axis SA: Then I fay, where-ever the Point A is taken in the Axis, it will be SA + SF FB. Confequently, if Sf=S F, it will be ƒ A= FB.

First

Demonstration.

SFL by the 7th Step, Sect. 3.

S

Ergo 2fA FA+L by Conftruction above.

2.2

3ƒA=QFA+FAXL+ ÷ L L

Again 4 SAFA+L by the Suppofition and Figure. 4XL 5 SAXL=FAXL+LL, but SAX L=□ AB Ergo 6 AB=FAXL+ LL

3-6 70A-AB-FA,confe. fA

But 8

FA+□AB

FA÷QAB=QFB, by Theorem 11.

Ergo fa=FB 90

2

9 ww 10 f AFB

Q. E. D.

This Propofition being well understood, 'twill be very eafily apply'd to Practice, fuppofing the Focal Distance given, or any other Data by which it may be found. Thus draw any Right Line to

represent the Parabola's Axis, and from its vertical Point, as at S, fet off the Focal Distance both upwards and downwards, viz. make Sf SF, the Distance of the given Focus from the Vertex; as in the Scheme: Then by the Propofition 'tis evident, that, if never fo many Lines be drawn Ordinately at Right Angles to the Axis, the true Distance between the Point fout of the Parabola, and any of thofe Lines (or Ordinates) being meafur'd or fet off from the Focus F to the fame Line or Ordinate, 'twill affign the true Point in that Line through which the Curve muft pafs; that is, it will fhew the true Limits or Length of that Ordinate; as at B in the laft Scheme.

Proceeding on in the very fame Manner from Ordinate to Ordinate, you may with great Expedition and Exactness find as many Ordinates (or rather their Points only, like B) as may be thought convenient, which, being all join'd together with an even Hand, will form the Parabola requir'd.

N. B. The more Ordinates (or their Points) there are found, and the nearer they are to one another, the easier and exacter may the Curve of the Parabola be drawn. The fame is to be obferv'd when other Curve is requir'd to be drawn by Points.

any

Sect. 5. To draw a Tangent to any given Point in the Curve of a Parabola.

Tangents are very eafily drawn to the Curve of any Parabola;

For, fuppofing S to be its Vertex, B the Point of Contact (viz. the Point where the Tangent muft touch the Curve) and P the Point where the Tangent will interfect (or meet with) the Parabola's Axis produced: Then if from the Point of Contact B there be drawn the Semi-ordinate BA at

bB

a A

P

Right Angles to the Axis SA, wherefoever the Point A falls in the Axis, 'twill be SPS A.

Demonftration.

Draw the Semi-ordinate b a (as in the Figure) then will the BAP and Aba P be alike. Let y AS the Abfciffa,

Ddd

and

and z = SP; put = A a the Distance between the two Semiordinates, which we fuppofe to be infinitely near each other, as in the Ellipfis, Page 377.

Then I, Or

Theorem 13.

Iy+z: BA::y+z+x:ba, per 2y+2y+2+x:: BA:ba. See Page 192. Again 3y BA :: y+x: □ ba, per Theorem Page 380, 3, Or 4yy+x::□ BA:□ ba

2in's 5

4 5 6

Syy + zy z + zz:yy+2yz+2yx+zx+ 2xx + xx::OBA: ba

{y: y+x :: y y +2yz+zz:yy+2yz+ {2yx+xx+2xx + xx

Sxy + 2y z + yx+xx+2xx+

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y

( y y + 2 y z + 2 y x + x z + 2 x x + xx.

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Suppofe 9o and rejected, as in the Ellipfis, Page 377.

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Concerning the chief Properties of the Hyperbola.

NOTE, any Part of the Axis of an Hyperbola, which is intercepted between its Vertex and any Ordinate (viz. any intercepted Diameter) is call'd an Abfciffa; as in the Parabola.

Sect. I.

The Plain of every Hyperbola is proportion'd by this general Theorem.

As the Sum of the Tranfverfe and any Abfciffa multiply'd into that Abscissa is to the Square of its Se

Theorem. mi-ordinate: fo is the Sum of the Transverse and any other Abfciffa multiply'd into that Abfciffa: to the Square of its Semi-ordinate.

That

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Let the following Figure HVG reprefent a Right Cone cut into two Parts by the Right Line SA; then will the Plain of that Section be an Hyperbola (by Sect. 5. Chap. 1.) in which let SA be its Axis, or intercepted Diameter, bab and BAB Ordinates rightly apply'd (as before in the Parabola) and TS its Tranfverfe Diameter. Again, if the Cone is fuppos'd to be cut by hg, parallel to its Bafe HG, it will also be the Diameter of a Circle, &c., as in the Ellipfis and Parabola. Then will the A Sg a and ASGA be alike, alfo the A Tah and ▲ TAH will be alike; therefore it

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Т

And

72

Sax Tax

AB=

Ε

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H

A

which give the following

8, Anal. 9 Sa X Ta:□ab:: SAXTA: □ AB, &c.

B

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B

T

A

B

Part IV. Thefe Proportions are the common Property of every Hyperbola, and do only differ from thofe of the Ellipfis in the Signs + and as plainly appears in the following Proportions. That is, if we fuppofe TS the Tranfverfe Diameter common to both Sections (viz. both the Ellipfis and Hyperbola) as in the annexed Scheme: then in the Ellipfis it will be TS-Sax Sa: □ ab:: TS-SA XSA: AB as by Sect. 1. Chap. z. and in the Hyperbola it is TS + Sa × Saab::TS+SA× SA: □ A B, as above. Therefore all, that is farther requir'd in the Hyperbola, may (in a manner) be found as in the Ellip

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P

A

b

B

fis, due Regard being had to changing of the Sines.

Sca. 2. To find the Latus Rectum, or Right Parameter, of any Hyperbola.

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From the laft Proportion take either of the Antecedents and its Confequent, viz, either Tax Sa: ab. Or TAX SA: □ AB, to them bring in the Tranfverfe TS for a third Term, and by those three find a fourth Proportional (as in the Ellipfis) and that will be the Latus Rectum.

TaxSa

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OabXTS

= the Latus

Then

2,

Rectum, which call L (as in the Parabola.)

2TS: L:: Tax Sa:☐ ab.

But 3 Tax Sa□ab:: TAX SA:
34TS: L:: TAX SA: □ AB, &c.

AB, therefore

Confequently L is the true Latus Rectum, or right Parameter, by which all the Ordinates may be found, according to its Definition in Chap. 1. And becaufe TS+Sa Ta, let it be TS+

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=

OabXTS
TSXSa+Sa

□ abxTS

TSX Sa

L and in the

=LR=L.

Sa

Se&t.

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