Dy —ÿÿĮ Į which is the 1ft Theorem, and gives II 12% = D- the following Analogy. 2 Ꭰ Analogy 13 D-y: y:: D-y: z. Viz. CA: SA: TA: AP 10-yz 14 yy - Dy-yz - Dz = 1 C15yy-Dy—yz+÷DD—{Dz+izz=1DD+1zz 14 2 15 ແ 16 D That is, 17y= { D + { % ± √ DD+zz which is the 2d Theor. Q. E. D. The Geometrical Performance of these two Theorems is very eafy, as by the following Figure. 1. Suppofe the Point B in the Ellipfis Periphery were given, and it were requir'd to find the Point P, &c. Make TC Radius, and upon the common Center C defcribe the Semi-circle Td S, and join the Points Cand d with a Right Line; then bifect that Line (by Prob. 2, Pag. 287) and mark the Point where the bifecting Line would cross the Transverse, as at e. Upon that Pointe, with the Radius Ce (or Cd) defcribe another Semi-circle, producing the Tranfverfe Diameter to its Periphery, and it will affign the Point P. For if DTS, y = AS, z=AP, as before. Therefore the Point P is truly found. Confequently, if a Right Line be drawn through thofe Points B and P, it will be the Tangent requir'd, according to the firft Theorem. 2. The Converfe of this is as easy, to wit, if the Point P be given, thence to find the Point B in the Ellipfis Periphery. Thus, circumfcribe half the Ellipfis with the Semi-circle TdS, as before; and bifect the Distance between the Points C and P, as at e, viz. Let Cee P. Then making Ce Radius, upon the Point c, defcribe the Semi-circle CdP; and from the Point where the two Semi-circles interfect or cross each other, as at d, draw the Right Line d A perpendicular to the Tranfverfe TS, 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 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, HBC CBK = 90°. 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: 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 d=b F (or BD BF) and join the Points Fd (or FD) with a Right Line. Then, I fay, if a Right Line be drawn through the Point of Contact b (or B) parallel to d F, or D F, it will be the Tangent requir'd. T H f D B H K F 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 ƒBH=< FBK) 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 fBF). 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'd an Abfciffa. Sect. 1. The Plain or Figure of every Parabola is proportion'd by its Ordinates and Abfciffe, as in the following Theorem. As any one Abfciffa: is to the Square of its Semi-ordinate Theorem.fo is any other Abfciffa: to the Square of its Semi-ar dinate. Let the following Figure HVG reprefent a Right Cone cut into two Parts by the Right Line S A, parallel to its Side V H. Then the Plain of that Section, viz. Bb Sb B will be a Parabola, by Sect. 4. Page 364, wherein let us fuppofe SA to be its Axis, and ba b, 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 A Sag like to ASAG. 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. { As any Abfciffa: is in Proportion to its Semi-ordinate:: fo is that Semi-ordinate: to the Latus Rectum. Let L the Latus Rectum. Then I Saba:: ba: Ls where-ever the Points a, and And 2 SA: BA::BA: LA, are taken in the Axis. 5 X 6 Sa X OBA SAXO ba, which gives this Analogy 7 Saba: SA: BA, the fame as at the 7th Step of the laft Procefs; 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 S AXL=0 BA. Confequently Sax Lba and ✔ SAX L=BA; and fo for any other Ordinate. Or if the Ordinates are given, to find their Abfciffe; then it will be, L: baba: Sa, and L: BA:: BA: SA, &c. Ba Confequently, = $a, and = SA, &c. From the Confideration of these Proportions, it will be easy to conceive how to find the Latus Re&tum Geometrically, thus: Join the vertical Point S of the Axis, and either extream Peint of any Ordinate, as B (or b) with a Right Line, viz. SB (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. B R E 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 eafy to deduce and demonftrate this following Theorem. As the Latus Rectum : Is to the Sum of any two SemiTheorem.ordinates: fo is the Difference of thofe 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 D = AB+ab, DB AB-ab, and b D=SA-Sa. Then it will be L:ED::DB: b D, according to the Theorem. Which gives 3XL 4 SA - Sax L=□ BA- But 5BA-Oba=BA+baxBA-ba the following 45,6 6SA- Sa XL=BA+bax BA ba Analogy. 6, Analogy 7 L: BA+ba :: BA—ba: SA-S a Or 8 L: ED::DB:bD |