Sect. 5. Any Ellipfis being given, to find its Transverse and Conjugate Diameters. Suppofe the given Ellipfis to be TNSn (in the annexed Scheme) in which let it be required to find the tranfverfe Diameter TS and its Conjugate Nn. Draw within the Ellipfis any two Right Lines parallel to each other, as Hb and Mm, and bifect those Lines, viz. find the Middle HN B Point of each, as at K and P; then thro' thofe Points K and P draw a Right Line, as DA, and it will be a Diameter; for it will divide the Ellipfis into two equal T n M S A Parts, [See Defin. 1. Page 363.] confequently the Middle of DẠ will be the true Middle or common Center of the Ellipfis, as at C. For 'tis the Nature or Property of all Diameters, howsoever they are drawn in any Ellipfis (as tis in à Circle) to cut or cross one another in the common Center or Middle of the Figure; as at C. = Upon the Point C defcribe an Arch of any Circle that will cut the Ellipfis's Periphery in two Points, as at B and b'; then join thofe Points Bb with a Right Line, and it will be an Ordinate, through whose Middle (as at a) and the common Center C, the tranfverfe Diameter TS muft pafs. For BS Sb, and Ba is at Right Angles with TS; therefore the Line Bb is an Ordinate rightly apply'd to TS, the tranfverfe Diameter. And if thro' the Point C there be drawn the Right Line Nn parallel to Bb, it will become the Conjugate; as was required. Sect. 6. To draw a Tangent, or Right Line that may touch the Ellipfis's Periphery in any affign'd Point. The Drawing of Tangents to or from any affign'd Point in the Ellipfis's Periphery, admits of three Cafes." Cafe 1. If it be requir'd to draw a Tangent that may touch the Ellipfis in either of the extream Points of its tranfverfe Diameter, as at Tor 8, it is plain the Tangent must be drawn parallel to the conjugate Diameter Nn; as HK in the following Figure is fuppos'd to be. Cafe Cafe 2. Or, if the Tangent must be drawn to touch the Ellipfis in either of the extream Points of its Conjugate Diameter, as at Nor n, 'tis as evident that it must be drawn parallel to the Tranfverfe Diameter TS, as K M. Confequently, if that Tangent and the Tranfverfe were both infinitely continu'd, they would never meet. Cafe 3. But if it be requir'd to draw a Tangent that may touch the Ellipfis in any other Point, as at B, &c. Then, if H BL the Tangent and the Tranfverfe Diameter be both continu'd, they will meet in fome Point, as at P; and those two Points (viz. B and P) do fo mutually depend upon each other, that one of them must be affign'd in order to find the other, that fo the Tangent may by_them be truly drawn. Let DTS, y = AS, and z = AP. Then, if y be given, z may be found by this Theorem { Dy-yy D this Theorem) Draw the Semi-ordinate b a, as in the Figure; then will ▲ BAP and Aba P be alike. Put x = Aa the Distance between the two Semi-ordinates (viz. between B A and ba) which we suppose indefinitely small. Then z: x-x:: BA: ba, by Theorem 13. But 2 D-yxy: D−y+xxy-x::BA: ba That is, 3 Dy-yy: Dy—yy+2yx-Dx-xx :: □ BA: □ ba Iin's 4 zz: zz−2zx+xx :: □ BA: □ ba [ being infinitely less than x may be every But 5 where rejected. 4, And 3,Then 6 Dy-yy: Dy-yy÷2yx-Dx:: □ BA: □ ba 922 10% Dxyy — Dy 10 +11 Dxyz Dy-yy = II 12 z = 2 } Dy-yy which is the 1ft Theorem, and gives 144 D-y the following Analogy. Analogy 13 D-y: y:: Dy: z. Viz, CA: SA:: TA: AP 10-yz 14 yy Dyyz - Dz 14C 15yy-Dy—yz+÷DD+÷Dx+¦zz={DD+1/zz 22 15 216y-D — 1 x = √ DD-zz That is, 17y D + ÷ ≈ ± √ ÷ DD+zz which is the 2d Theor. Q. E. D. The Geometrical Performance of these two Theorems is very casy, as appears 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 C and 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 Tranfverfe, as at e. Upon that Point e, 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, Then 1 Dyxy dA = D-yxz□ dA For 3. TA: dA:: dA: SA And CA: dA:: dA: AP But 15 CAD-y, &c. SDz―yz=Dy-yy 1, 2Pas at the 11th Step before. N a 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 Converse 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 Td S, 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 d, where the two Semi-circles interfect or cross each other, draw the Right Line d A perpendicular to the Tranfverfe T'S, TS, and it will affign the Point of Contact B in the Ellipfis Pe- But the Practical Method of drawing Tangents to any affign'd 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, LHBCL CBK = 90°. C H 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. These being well 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: H K k Having by the transverse and conjugate Diameters found the Foci f and F, by Sect. 3. draw from thofe Points two Right Lines to meet each other in the affign'd Point of Contact, as fb and Fb (or fB and FB) in the annex'd Figure. Set off (viz. make) b d=b F (or BD =BF) and join the Points F, d (or F, D) with a Right Line. T f H Then, 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. For it is plain, that as the f NH =LFNK when the Tangent is parallel to the Tranfverfe Diameter, even fo is the Lfbb LFBk, (and Lƒ BH-L 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). = СНАР. III. Concerning the Chief Properties of the Parabola. NOT OTE, in every common 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 Plane 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-ordiTheozem. nate: fo is any other Abfciffa: to the Square of its Semi-ordinate. That is, if we suppose the annex'd Figure to be a Parabola, wherein S a, and SA, are Abfciffe, and bab, BAB, Ordinates rightly apply'd, it will BAS wherefoever the Points a, And fo for any other Abfciffa, &c, Demonstration. Let the following Figure HVG reprefent a Right Cone cut into two Parts by the Right Line SA, parallel to its Side V H. Then the Plane 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 bab, BAB 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 A SAG. Therefore I {Baag:: Saag:: SA: AG 1.2 Sax AG = SĄ × ag 2xha 3 By Axiom 3. HA-ha, because SA But 4 is parallel to VH And 5 ba = agx ha P. 363. 3, 4, 56 { Sax □ BA=SAx-ba By Axiom 5. 6, Analogy 7 Vide Page 194. Q. E. D. V S h B Thefe Here, and in what follows concerning the Properties of the Parabola, the Ordinate s always fuppofed rightly applied to the Axis of the Figure, |