a 2. If the Transverse Diameter and Lutus Rectum are given, the Ellipfis is truly limited, because by them the Conjugate may be found, by Sect. 2. 3. Or if only the Transverse, and the Proportion it hath either to the Conjugate or Latus Rietum, be given, the Ellipsis is thereby limited. As for Initance; suppose the given Ratio between the Transverse and Conjugate to be, as a : tod: TSXd 4. If either the Transverse or Conjugate, and the Distance of, the Focus from the Conjugate be given, the Ellipfis is limited, because by them the Conjugate or Transverse may be found. These being premis'd, and the precedent Work a little consider'd, it must be easy to describe or delineate any Ellipfis in Plano, either Geometrically or Numerically. 1. To describe an Ellipsis Numerically, by Points. Suppose the Transverse Diameter TS=20, and the Conjugate Nn = 12, (either Inches, or any Nabd other equal Parts) and let them cross each other at Right Angles in their Middles, as in the Point C; then will TC-CS = 10, and C1?} S NC=Cn = 6 and it will be 20:12 :: 12 : 7, 2 = the Latus Rectum. nabd Again 20 : 7, 2. Or rather take their Ratio. T 2 : 1:0, 36 : : 10+1 X10 1:0 a. || 1. Thus 1:0, 36 : : 10 + 2 X 10 O b. ll 2. ICC-1X0,36=a.1. Hencev 99x0.39=5,97 &c.=a.I Viz. X100—4x0,36= 6.2. V 96-0,36=5,88 &c.=6.2 100~9x0,36= d.3. V 9120,36=5,72 &c.=d.3 If so many Semi-ordinates as may be thought convenient (the me the better) be found in this Manner, and every one of them be set off at Right Angles from its respective Point in the Transverse Diameter each Way, viz. from í to a, from 2 to be from 3 to d, &c. Then if a Curve Line be carefully drawn with an even Hand thro' those extream Points a, b, d, &c, it will be the Ellipfis's Periphery requir’d. 2, To B G 2. To describe an Ellipfis Geometrically by Points. Having the Transverse and Conjugate Diameters given, viz. T3 and Nn, placed at Right Angles in their Middles, as before : Then from either End of the Conjugate, viz. N (or n) set off half the Transverse Diameter to x. That N is, make N * = T C (continuing the Conjugate Nn when it is shorter than TC) Or, which is all one, С make CX=TC-NC. 'T Then 16/ E A take any Point in the Line C * ac Pleasure ; suppose it at G, and from 120 that Point at G set off the Distance n Cx to the Transverse (as at E) viz. make GE = C, and join the Points G E with a Right Line, produced so far beyond E as to make E B = NC. Confequently GB=TC. Then, I say, where-ever the Point G was taken between C and * the Point B will juft touch (or fall in the Ellipfis Periphery. Deinon&ration. Draw the Right Line B A perpendicular to TS, viz. let B A be a Semi-ordinate rightly apply'd to the tranfverfe Diameter T S; then A GCE and A BAE will be alike. Consequently 1 CE: AE:: EG: EB, by Theorem 13. 2CE+AE: AE:: EG+ėB: EB. See p. 192. But 3CE+AE=CA.EG+EB=TC. And EB=NG Therefore 4 CA: AE :: TC : NC 6, in O's 5 o CA: AE :: 0 TC: NC O CAX ONC O AE ОАЕ AB= O AE INC OTC -=ONC - OAB 8 X OTS 9 CANC=ONCX TC-AB x TC 9 + 10 NCXTC_OCAX ONC=AB OTG Io, Analogy OTC: NG :: TC -OCA:0 AB That is, l121 TCxCS: ONC:: TC+CA TC-CA: ( AB which is according to the common Properties of the Ellipsis : Therefore the Point B is truly found. Q. E, D. Hence 1, And Hence it follows, that if a convenient Number of such Lines as G E B be so drawn (as above directed) from the like Number of Points taken between C and *, &c. their extream Points (as at B) will be those Points by which (with an even Hand) the Ellipsis may be truly describ'd; as before. But, if this be well understood, it will be very easy to conceive how to describe an Ellipsis very readily, without drawing those Lines, by having a thin, streight, narrow Ruler just the Length of TC, made fomewhat sharp at both Ends, upon which, from one of its Ends, fet off the Length of NC. Then, if the Point upon the Ruler which represents E be gradually or easily moved along the Transverse TS, and at the fame Time the Point or End representing G be kept sliding close along the Conjugate Nn, 'cis evident from the Work above, that the End of the Ruler representing B will, by that Motion, allign the true Periphery of the Ellipsis requir’d; for by that Motion the streight Edge of the Ruler doth supply an infinite Number of the aforesaid Lines; as will appear very plain and easy in Practice. Scholium. Now from hence was deduced the first Invention of that wellcontriv'd Instrument for drawing an Ellipfis by one Motion, commonly call'd the Elliptical Compafjes, being usually made of Brass, and compos’d of three Parts, two of which represent (or rather supply) the transverse and conjugate Diameters set together at Right Angles; and the third Part is a moveable Ruler, which performs the Office of the last-mention'd thin Ruler. But because the making of it is so well known to most Mathematical Instrumentmakers, especially to that accurate and ingenious Artist Mr. JOHN ROWLEY, Mathematical Instrument-maker, under St. Dunstan's Church in Fleet-street, London; who, for his great Skill in contriv. ing, framing, and graduating all kind of Mathematical Instruments, may, I believe, be justly calld one of the best Workmen of his Trade in Europe; I think it needless therefore to give a particular Description of that Instrument. Also from hence came that ingenious Invention of making Engins for turning all Sorts of elliptical or oval Work, as oval Boxes, Pico ture-Frames, &C. Seet. H N D m. n Sect. 5. Any Elipsis being given, to find its Transverse and Conjugate Diameters. Suppose the given Ellipsis to be TNS n (in the annexed Scheme) in which let it be required to find the transverse Diameter T'S and its Conjugate Nn. Draw within the Ellipfis any two Right Lines parallel to each other as Hh and M m, and bisect B thofe Lines, viz. find the Middle Point of each, as at K and P; T S then thro' those Points K and P C A draw a Right Line, as D A, and it will be a Diameter ; for it will divide the Ellipfis into two equal Parts, [See Defin. 1. Page 363.) consequently the Middle of D A 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 Ellipsis (as 'tis in a Circle) to cut or cross one ansther in the common Center or Middle of the Figure; as at C. Upon the Point C describe an Arch of any Circle that will cut the Ellipsis's Periphery in two Points, as at B and b; then join those Points Bb with a Right Line, and it will be an Ordinate, through whose Middle (as at a) and the common Center C, the transverse Diameter TS; must pass. For BS=Sb, and Ba is at Right Angles with IS; therefore the Line B b is an Ordinate rightly apply'd to TS; the transverse Diameter. And if thro’tbe 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 Langent, or Right Line that may touch the Ellipsis's Periphery in any align’d Point. The Drawing of Tangents to or from any asignd Point in the Ellipfis's Periphery, admits of three Cafes. Case 1. If it be requir’d to draw a Tangent that may touch the Ellipfis in either of the extream Points of its transverse Diameter, as at I or S, it is plain the Tangent must be drawn parallel to the conjugate Diameter Nn; as H K in the following Figure is suppos'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 N or n, 'tis as evident that it must be drawn parallel to the Transverse Diameter T S, as K M. Consequently, if that Tangent and the Transverse were both infinitely continu'd, they н Bb would never meet. Case 3. But if it be requir'd CI Аа P. to draw a Tangent that may touch the Ellipfis in any other K M N Point, as at B, &c. Then, if the Tangent and the Transverse Diameter be both continu'd, they will meet in some point, as at P; and those two Points (viz. B and P) do so mutually depend upon each other, that one of them must be assign’d in order to find the other, that so the Tangent may by them be truly drawn. Let D=TS, y=A TS, y= AS, and % = A P. Then, if y be given, z may be found by this Or, if z be given, y may be found by DD+zz this Theorem 4 Theorem { po It Demonftration. Draw the Semi-ordinate b a, as in the Figure ; then will A BAP and a ba P be alike. Put * = Aa the Distance between the two Semi-ordinates (viz. between B A and b a) which we suppose infinitely small. Then 112: 2-*::BA: ba, by Theorem 13. But 2 Dayxy: D-3+xxy--*:: OBA:Oba That is, 3 Dy-yy: Dymyy+2bx-Dx-xx:: 0 BA: oba i in O's 4 zx : 22—22*+xx :: OBA: o ba Suppose 5*= 0, that so x may be every where rejected. 3, Then 6 Dy—yy: Dy—yy+2y-D:: BA: ba 42 And 7 zz: zz--22:: 0 BA: o ba 6, 71 8 Dy-yy: Dymyy+2y-D::%%: zz-22. 9/2yzz Dzz = 2yyz - 2 Dyz 9 22 10 yx - Dz=y— Dy Ili Dm- y2 = Dy — 99 10 |