TS, and it will assign 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 Ellipsis Periphery may (without finding the aforesaid Point P) be easily deduced from the following Property of Tangents drawn to a Circle, which is this.

B

If to any Radius of a Circle, as CB, 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,

C

HBC= <CBK = 90°.

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 Transverse Diameter.

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

Having by the transverse and conjugate Diameters found the two Foci f and F, by Sett. 3. from them draw two Right Lines to

meet each other in the assign'd Point

h

of Contact, as fb and Fb (orf B

and FB) in the annex'd Figure. Next

set off (viz. make) bd=bF (or BD

= BF) and join the Points Fd (or T

S

FD) with a Right Line.

Then, I say, if a Right Line be drawn through the Point of Contact b (or B) parallel to dF, or DF,

it will be the Tangent requir'd.

For it is plain, that as the <fNH=<FNK when the Tangent is parallel to the Transverse Diameter, even so is the <fbb = <FBk, (and <fBH = < FBK) and will be every where so, as the Point of Contact b (or B) and its Tangent is carry'd about the Ellipsis Periphery with the Lines fb F (or

f B F).

CHAP. III.

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

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

Theorem.

{

As

any one Abscissa: is to the Square of its Semi-ordinate :: so is any other Abscissa. to the Square of its Semi-or

dinate.

Let the following Figure HVG represent a Right Cone cut into two Parts by the Right Line SA, parallel to its Side V H. Then the Plain of that Section, viz. BbSb B will be a Parabola, by Sect. 4. Page 364, wherein let us suppose S A 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 bg parallel to its Bafe HG. Then will bg be the Diameter of a Circle, by Sect. 2. Page 363. and △ Sag like to ASAG.

These 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 easily be deduced.

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

The Latus Rectum of a Parabola hath the same Ratio or Proportion to any Abscissa, 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. {

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

Let L the Latus Rectum.

Then I Saba:: ba: Ls where-ever the Points a, and 2 SA: BA::BA:L}{ A, are taken in the Axis.

And

□ Ba=L: Or SAXL=□ BA

ba

=L: Or Sa XL= □ ba

Sa

Sa

BA Sa
ba

5X6 Sax □ BA=SAX ba, which gives this Analogy 7 Sa: ba::SA: BA, the fame as at the 7th Step of the last 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 SAXL= BA. Consequently ✓ SaXL=ba and √ SAXL=BA; and fo for any other Ordinate.

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

From the Confideration of these Proportions, it will be easy 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 bisect that Line (by Problem. 2.

Page 287.) marking the Point where

S

the bisecting Line doth interfect or cross the Axis, as at E (or e) and with

de

the Radius SE (or Se) upon the Point E (or e) describe a Cirele; (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 ar) be the true La

b

tus Rectum required.

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

Confectary.

From these Proportions of finding the Latus Rectum, it will be easy to deduce and demonstrate this following Theorem.

As the Latus Rectum : Is to the Sum of any two Semi. Theorem. ordinates :: : so is the Difference of those two Semi-ordinates: to the Difference of their Abscissa.

Suppose 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 D = AB + ab, DB=AB-ab, and bD=SA–Sa. Then it will be

L:ED: ;DB: b D, according to the Theorem.

3XL 4SA - Sa×L=BA- ba

But

Which gives

5BA-ba=BA+ba×BA—ba the following

4,5,6 6 SA- Sa x L=BA+ba×BA balAnalogy.

6, Analogy 7L: BA+ba::BA-ba: SA-Sa

Or8L: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 Treatise intituled, The Geometrical Key: Or, the Gate of Equations unlock'd; wherein he hath shew'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 pass. (See Definition 3. Sect. 4. Page 359.) Therefore its Distance from the Vertex of the Parabola may be easily found, either by the Latus Rectum itself, or by any other Ordinate, and its Abscisse.

Thus, suppose 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 = L.

S

F

First 1S FXL = FR. by Sect. 2. Page 375.

And

20

2FR=L; for the Ordinate RFR = L as above,

30FR=1□L=LXL

1,= 3 4 SFXL=4OL

4-L5SF=L, as by Definition 4. Sect. 4. Page 359.

Again 6 = L, by the third Step in Page 375.

ba

Sa

ba

4 Sa

Confeq. 7 =L, &c. as above.

E. D.

Sect. 4. To describe, or draw a Parabola several Ways.

Note, There are two or three Ways of drawing a Parabola instrumentally at one Motion; but because those Inftruments or Machines are not only too perplex'd for a Learner to manage, but also a little subject to Error, I have therefore chosen to shew how that Figure may be (the best) 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, must needs be very easy.