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 assign'd Point in the Ellipfis Periphery may (without finding the aforesaid Point P) be easily deduced from the following Property of Tangents drawn to a Circle, which is this. If to any Radius of a Circle, as CB, there be drawn a Tangent Line (as H K) 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= SCBK = 90°. In like Manner the two Angles, made between the Tangent and the two Lines drawn from the Foci of any Ellipsis to the point of Contaal, will always be equal, but not Right Angles, fave only at the two Ends of the Transverse Diameter. These being weli consider'd, and compar'd with what hath been said in Page 366, it must needs be easy to understand the following Way of drawing Tangents to any asign’d Point in the Ellipfis Periphery ; which is thus : Having by the transverse 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 asign'd Point of Contact, as fb and Fb (or f B. and FB) in the annex'd Figure. Next set off (viz. make) bd=b F(or BD =BF) and join the Points Fd (or ti FD) 'with a Right Line. Then, I say, if a Right Line be drawn through the Point of Contact H & (or B) parallel to d F, or D F, it will be the Tangent requir’d. For it is plain, that as the “f NHS SFNK when the Tangent is parallel to the Transverse Diameter, even so is the fbb== F Bk, (and 5f BH= FBK) and will be every where so, as the point of Contact b (or B) and its Tangent is carry'd about the Ellipfis Periphery with the Lines f 6 F (or f B F). ia · 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. ... From the Confideration of these Proportions, it will be caly 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 bised that Line (by Problem. 2. Page 287.) marking the point where the bisecting Line doth intersect or cross the Axis, as at E (or e) and with the Radius S E (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 a r) be the true La. tus Reflum required, For SA:BA::BA: A R, and Sa:ba :: ba: or, by Theor. 13. therefore AR= L. And ar=L, by the ift and 2d Steps above. This peculiar Property of the Parabola was first publish'd Anno 1684, by one Mr. Thomas Baker, Rector of Bishop Nympion in Devonshire, in a Treatise intituled, The Geometrical Key : Or, the Gate of Æquations unlock’d; wherein he hath shew'd the Geometrical Construction and Solution of all Cubick and Biquadratick Adfected Æquations 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. Seet. 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 Abscissa. Thus, suppose the Point at Fio be the Focus, Sthe Vertex, the Ordinate RFR = L the Latus Rectum, and bab any other Or. dinate. Then will SF = L. Demonttration. And 2 FR= L; for the Ordinate RFREL as above, 2 o 310 FR=OL={LXIL 1,= 3141SFXL=IOL 4 • I 5 SF= L, as by Definition 4. Sect. 4. Page 359. oba Again 6 =L, by the third Step in Page 375. . Q. E. D. Sect. 4. To describe, or draw a Parabola feveral Ways. Note, There are two or three Ways of drawing a Parabola inftrumentally at one Motion; but because those Instruments or Machines are not only too perplex'd for a Learner to manage, but also a litele 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. |