TS, and it will affign the Point of Contact B in the Ellipfis Periphery, through which the Tangent must pass.

But the Practical Method of drawing Tangents to any affign'd Point in the Ellipfis Periphery may (without finding the aforefaid Point P) be eafily deduced from the following Property of Tangents drawn to a Circle, which is this.

If to any Radius of a Circle, as C B, there be drawn a Tangent Line (as HK) to touch the Radius at the Point B; the two Angles, which the Tangent makes with the Radius, will always be two Right Angles (16, 17, 18, 19 Euclid 3.) that is, HBC=CBK=90°.

C

B

In like Manner the two Angles, made between the Tangent and the two Lines drawn from the Foci of any Ellipfis to the Point of Contact, will always be equal, but not Right Angles, fave only at the two Ends of the Tranfverfe Diameter.

Thefe being well confider'd, and compar'd with what hath been faid in Page 366, it muft needs be eafy to understand the following Way of drawing Tangents to any affign'd Point in the Ellipfis Periphery; which is thus:

H

K

k

Having by the tranfverfe and conjugate Diameters found the two Foci f and F, by Sect. 3. from them draw two Right Lines to meet each other in the affign'd Point of Contact, as fb and Fb (or f B and FB) in the annex'd Figure. Next fet off (viz. make) b db F (or BD = BF) and join the Points Fd (or FD) with a Right Line.

T

f

D

B

H

Then, I fay, if a Right Line be drawn through the Point of Contact b (or B) parallel to d F, or DF, it will be the Tangent requir'd. For it is plain, that as theƒNH< FNK when the Tangent is parallel to the Tranfverfe Diameter, even fo is the <fbb =<F Bk, (and ƒB H = < FB K) and will be every where fo, as the Point of Contact b (or B) and its Tangent is carry'd about the Ellipfis Periphery with the Lines fb F (or f B F).

CHA P. III.

Concerning the Chief Properties of every Parabola. NOTE, in every Parabola, the intercepted Diameter, or that

Part of its Axis, which is between the Vertex and that Ordinate which limits its Length, as Sa or SA, &c. is call'dan Abfciffa. Sect. 1. The Plain or Figure of every Parabola is proportion'd by its Ordinates and Abfciffe, as in the following Theorem.

Theorem.

As any one Abfciffa: is to the Square of its Semi-ordinate :: fo is any other Abfciffa: to the Square of its Semi-or

dinate.

Let the following Figure HVG reprefent a Right Cone cut into two Parts by the Right Line SA, parallel to its Side VH. Then the Plain of that Section, viz. BbSb B will be a Parabola, by Sect. 4. Page 364, wherein let us fuppofe SA to be its Axis, and bab, BAB to be Ordinates rightly apply'd to that Axis. Again, imagine the Cone to be cut by the Right Line hg parallel to its Bafe HG. Then will hg be the Diameter of a Circle, by Sect. 2. Page 363. and A Sag like to A SAG.

Thefe Proportions being prov'd to be the common Property of every Parabola, all that is farther requir'd about that Section, or Figure, may from thence eafily be deduced.

Sect. 2. To find the Latus Kedum or Right Parameter of any Parabola.

The Latus Rectum of a Parabola hath the fame Ratio or Proportion to any Abfciffa, and its Semi-ordinate, as the Latus Rectum of any Ellipfis hath to its Tranfverfe and Conjugate Diameters, and may be found by this Theorem.

Theorem. { is that Semi-ordinate: to the Latus Rectum.
As any Abfciffa is in Proportion to its Semi-ordinate : :

Let L the Latus Rectum.

Then I Saba::ba: LUS where-ever the Points a, and And 2 SA: BA:: BA: L S Y A, are taken in the Axis.

I '.' 3

ba

=L: Or Sa XL=□ ba

Sa
Ba

Sa

ОВА

3 4 5

SA

=L: Or SAXL=□ BA

5 x 6 Sa X □ BA = SAX□ ba, which gives this Analogy Saba::SA: BA, the fame as at the 7th Step of the laft Process; therefore L (thus found) is the true Latus Rectum, by which all the Ordinates may be regulated and found, according to its Definition in Section 4, Page 364. For by the third Step Sa XL= ba, and by the 4th Step SAX LO BA. Confequently Sax Lba and ✔ SAX L = B A ; and fo for any other Ordinate.

Or if the Ordinates are given, to find their Abfciffe ; then it will be, L: ba::ba: Sa, and L: BA:: BA: SA, &c.

Confequently ba— Sa, and ba,

Ова
I

=SA, &c.

From the Confideration of these Proportions, it will be cafy to conceive how to find the Latus Rectum Geometrically, thus :

Join the vertical Point S of the Axis, and either extream Point of any Ordinate, as B (or b) with a Right Line, viz. S B (or Sb) and bifect that Line (by Problem. 2.

Page 287.) marking the Point where the bifecting Line doth interfect or crofs the Axis, as at E (or e) and with the Radius SE (or Se) upon the Point E (or e) defcribe a Circle; (as in the annex'd Figure) then will the Distance between the Ordinate and that Point where the Circle's Periphery cuts the Axis, viz. AR (or a r) be the true Latus Rectum required.

R

S

E

For SA: BA:: BA: AR, and Sa: ba :: ba: or, by Theor. 13. therefore ARL. And ar=L, by the 1ft and 2d Steps above.

Confectary.

From thefe Proportions of finding the Latus Rectum, it will be eafy to deduce and demonftrate this following Theorem.

As the Latus Rectum: Is to the Sum of any two Semi Theorem.ordinates: fo is the Difference of those two Semi-ordi nates to the Difference of their Abfciffa.

Suppofe any Right Line drawn within the Parabola, as b D, parallel to its Axis SA; then will that Line (viz. b D) be a Diameter (by Def. 5. Pag. 365) which will make E DAB+ab, DB=AB-ab, and b D=SA-Sa. Then it will be L:ED::DB: b D, according to the Theorem.

3 X L

But

BA-ba

E

4S A—Sax L=□ BA-ba

Which gives the following

5 BA-ba=BA+bax BA-ba 4,5,6 6SA— Sa × L=BA+baXBA—ba Analogy. 6, Analogy 7L: BA+ba:: BA-ba: SA-Sa Or 8L: ED::DB:bD

This peculiar Property of the Parabola was first publish'd Anno 1684, by one Mr. Thomas Baker, Rector of Bishop Nympton in Devonshire, in a Treatife intituled, The Geometrical Key: Or, the Gate of Equations unlock'd; wherein he hath fhew'd the Geometrical Construction and Solution of all Cubick and Biquadratick Adfected Equations by one general Method, which he calls a Central Rule, deduced from this peculiar Property of the Parabola.

Sect. 3. To find the Focus of any Parabola.

The Focus of every Parabola is that Point in its Axis through which the Latus Rectum doth pafs. (See Definition 3. Sect. 4. Page 359.) Therefore its Distance from the Vertex of the Parabola may be eafily found, either by the Latus Rectum itself, or by any other Ordinate, and its Abfciffe.

Thus, fuppofe the Point at F to be the Focus, S the Vertex, the Ordinate RFR L the Latus

Rectum, and bab any other Or

dinate. Then will SF

S

R

L.

F

b

a

Firft

Demonstration.

SFX L□ FR. by Sect. 2. Page 375. And 2 FRL; for the Ordinate R FRL as above. 202 FROL = { LX L 1,3 4S FXL=OL

3

4 L 5SFL, as by Definition 4. Sect. 4. Page 359.

Oba

Sa

=L, by the third Step in Page 375.

Again 6

Oba Confeq. 7 4 S

L, &c. as above..

Q. E. D.

Sect. 4. To defcribe, or draw a Parabola feveral Ways.

Note, There are two or three Ways of drawing a Parabola inftrumentally at one Motion; but because thofe Inftruments or Machines are not only too perplex'd for a Learner to manage, but also a little fubject to Error, I have therefore chofen to fhew how that Figure may be (the beft) drawn by a convenient Number of Points, viz. Ordinates found, either Numerically or Geometrically, according to the Data; which, if the Work of the three laft Sections be well confider'd, muft needs be very eafy.