Properties of the Conic Sections; Deduced by a Compendious Method. Being a Work of the Late William Jones, Esq; F. R. S. Which He Formerly Communicated to Mr. John Robertson, Libr. R. S. Who Now Addresses It to the Reverend Nevil Maskelyne, F. R. S. Astronomer Royal

XXXVI. Properties of the Conic Sections; deduced by a compendious Method. Being a Work of the late William Jones, Esq; F. R. S. which he formerly communicated to Mr. John Robertson, Libr. R. S. who now addresses it to the Reverend Nevil Maskelyne, F. R. S. Astronomer Royal. SIR, YOU well know that the curves formed by the sections of a cone, and therefore called Conic Sections, have, from the earliest ages of geometry, engaged the attention of mathematicians, on account of their extensive utility in the solution of many problems, which were incapable of being constructed by any possible combination of right lines and circles, the magnitudes used in plane geometry. The properties of these curves are become far more interesting within the two last centuries, since they have been found to be similar to those which are described by the motions of the celestial bodies in the Solar system. Two different methods have been taken by the writers who have treated of their properties; the one, and the more antient, is to deduce them from the properties of the cone itself; the other is to consider the curves, as generated by the constant motion of two or more strait lines moving in a given plane, by certain laws. There are various methods of generating these curve lines in plano; one method will give some properties very easily; but others, with much trouble: while, by another mode of description, some properties may be readily derived, which, by the former, were not so easily come at: so that it appears there may be a manner of describing the curves similar to the Conic Sections, by the motion of lines on a plane, which, in general, shall produce the most essential properties, with the greatest facility. That excellent mathematician, the late William Jones, Esq; F. R. S. had drawn up some papers on the description of these curves, or lines of the second kind, very different from what he gave in his Synopsis Palmariorum Matheseos, published in the year 1706; or from that of any other writer on this subject. A copy of these papers he was pleased to let me take about the year 1740. He had not finished them as he intended; but, in their present state, they appear of too much consequence to be lost; as, it is much to be feared, his own copy, together with many other valuable papers, are; and therefore, I am desirous of preserving them in the Philosophical Trans- Transactions, in the manner I at first transcribed them; although, I am aware, they might have been put into a form more pleasing to the generality of readers: I have indeed annexed larger diagrams than what accompanied the author's copy, in order to render the lines more distinct, as all the relations are to be represented in a single figure, of each kind. Mr. Jones, having laid down a very simple method of describing these curves, seems to have been desirous of arriving at their properties in as expeditious a way as he could contrive; and therefore he has used the algebraic method, in general, of reducing his equations; and, on some occasions, has used the method of fluxions, to deduce some properties chiefly relating to the tangents; and, by a judicious use of these, he has very much abridged the steps which otherwise he must have taken, to have deduced the very great variety of relations he has obtained: these he intended to have arranged in tables, from whence an equation expressing the relation between any three or more lines of the Conic Sections, might be taken out as readily as a logarithm out of their tables; this he has only partly executed; but it may easily be continued by those who are desirous to have it done, and are sufficiently acquainted with what follows. From the House of the Royal Society, April 29, 1773. The description of Lines of the second kind. Let the right lines $AB$, $AQ$, be drawn on a plane, at any inclination the one with the other. See Plate XIV. Fig. 1, 2, 3. In $AD$, $AQ$, take $AA'$, $AM$, of any given magnitude, and draw $MN$ parallel to $AD$. On the points $A$, $A'$, let two rulers $AP$, $AP'$ revolve, and cut $MN$, $AQ$, in $N$ and $Q$, so that $AQ$ be everywhere equal to $MN$. Then shall the intersection $P$ of the rulers describe lines of the second kind, or curves of the first kind. Where the right-line $AA'$, is the first, or transverse diameter. The point $C$, bisecting the diameter $AA'$, is the center; The right-line $PD$, drawn parallel to $AQ$, is the ordinate to the diameter $AA'$. The part $AD$, or $CD$, of the diameter, is the abscissa, when reckoned to begin from $A$ to $C$, or from $C$ to $A$. The right line $BB'$ drawn from the center $C$ parallel to the ordinate $PD$, and terminated in the curve, is called the second, or conjugate diameter. Those diameters to which the ordinates are perpendicular, are called the axes. And $AM$ is the parameter to the diameter $AA'$. The Properties of Lines of the Second Kind. 1. Put \( AA = d = 2AC = 2t; BB = d = 2BC = 2c; AM = 2p; PD = y; CD = x; AD = a \). Then \( PD^2 = \frac{p}{t} \times ADA \). Or \( yy = \frac{p}{t} \times u \times d + u = 2pu + \frac{p}{t} uu = \frac{p}{t} \times \pm tt \mp xx = \pm pt \mp \frac{p}{t} x^2 \). For \( PD = \frac{AM \times AD}{MN} = \frac{AQ \times DA}{AA} \) (by sim. \( \triangle s \)). Th. \( PD^2 = \frac{AM}{AA} \times ADA \). 2. Consequently \( \frac{t}{p} yy = \pm tt \mp xx = du \mp uu = \pm dd \mp xx \). 3. Hence \( \frac{t}{p} yy = \mp xx = t \mp u \times u = xu \). 4. And \( pt = cc \), or \( 2pd = ss \); for when \( y = c = \frac{1}{2}x \), then \( x = 0 \). 5. Therefore \( \frac{PD^2}{ADA} = \frac{yy}{\pm tt \mp xx} = \frac{p}{t} \frac{2p}{d} \frac{2pd}{dd} \frac{pt}{tt} \frac{ss}{dd} \frac{cc}{AE} \). 6. The curve line whose property is \( yy + \frac{p}{t} uu - 2pu = 0 \), (where the abscissa begins at the curve), Or \( yy + \frac{co}{tt} xx - cc = 0 \), (where it begins at the center), is called an Ellipsis. This curve returns into itself. For when \( x = 0 \), then \( y = c \); and when \( y = 0 \), then \( x = t \). Which can happen but two ways. 7. The curve line whose property is \( yy - \frac{p}{t} uu - 2pu = 0 \), Or \( yy - \frac{cc}{tt} xx + cc = 0 \), is called an Hyperbola. This curve spreads out infinitely. For \( y \) increases as \( x \) increases; and has four legs tending contrary ways: for \( xx \), or \( yy \), may be produced as well from \(-x\), or \(-y\); as from \(+x\), or \(+y\). 8. If the point \( a \), is supposed to be at an infinite distance from \( A \), so that a ruler \( AP \) moves in a parallel position to \( AD \); then is \( yy = 2pu \), or \( yy - 2pu = 0 \), the property of the curve described, and is called a Parabola. This curve spreads out infinitely; for \( y \) increases as \( u \) increases. 9. Let 9. Let \( Aa, pp \), be any two first diameters; \( bb, qq \), their second diameters. PLATE XIV. Fig. 4. Draw the ordinates \( PD, QE \), to the diameter \( Aa \), and the ordinate \( Pd \), to the diameter \( bb \). Let \( PT \) be a tangent, and \( PM \) be perpendicular, to the curve, in \( P \); \( PT \) cutting \( Aa, bb \), produced, in \( T, t \); and \( PM \), in \( M, m \). Put the subtangent \( DT = s, dt = \sigma \). Let \( AT = r, PM = \tau, CE = w, CT = q = x + s = i + r \). Put \( s, s' \), sine and cosine of the angle \( MPD \), or angle \( PMB \). \( R \) = tabular radius. Then \( cdt = xs = t + u \times s = \frac{t}{p} yy = \frac{tt}{cc} yy = \pm tt + xx = u \times d + u = ADA \). For \( u \times t + u = \left( \frac{t}{p} yy \right) ayj \). Therefore \( \left( \frac{u}{j} = \frac{ay}{t + u} = \right) \frac{ay}{x} = \frac{s}{y} \) (by sim. \( \Delta s \)). And \( cd = y\sigma = \frac{p}{q} xx = \frac{cc}{tt} xx = \pm cc + yy = bdb \). For \( \left( \frac{u}{j} = \right) \frac{ay}{x} = \frac{x}{\sigma} \). 10. Hence \( AC^2 = tt = xx + s = xx + r = xq = DCT \). 11. And \( BC^2 = cc = y \times y + \sigma = y \times ct = dct \). 12. Consequently \( \frac{PD^2}{BD} = \frac{xx}{\pm cc + yy} = \frac{tt}{cc} = \frac{AC^2}{BC^2} \). 13. Also \( CE^2 = ww = \left( \frac{DT^2}{PD^2} \times \frac{QE^2}{PD^2} = \frac{DT^2}{ADa} \times \frac{AEa}{ADA} \right) ss \times \frac{tt - ww}{sx} = sx = \frac{tt}{cc} yy = ADA = CDT \). 14. Therefore \( AEa = \left( \frac{CE^2}{DT^2} \times ADA = \frac{ss}{xx} \times ss = \right) xx = CD^2 \). 15. And \( QE^2 = \left( \frac{PD^2}{ADa} \times AEa = \right) \frac{cc}{tt} xx = \frac{p}{t} xx \). In the general schemes. **Plate xv.** and Fig. 5. **Plate xiv.** 16. Let $Aa$, $Bb$, be the longest and shortest axes. Draw $c\varphi$ perpendicular to the tangent $PT$, cutting it in $\varphi$. Put $CP = T; CQ = c; C\varphi = \varphi$. Then $c\varphi = \varphi = \left(\frac{cT \times cE}{CQ} = \frac{cc}{yy} \times \frac{ty}{cc}\right) \frac{tc}{c} = \frac{AC \times CB}{CQ}$. Hence the parallelogram, under the two axes, is equal to the parallelogram under any two diameters. 17. Draw the tangents $AN$, $an$, to any vertices $A$, $a$, meeting any diameters $PP$, $QQ$, produced in $v$, $u$, and $v$, $u$, and the tangent $PT$ in $N$, $n$. Then $AU = \left(\frac{CA \times EQ}{CE} = \frac{tx}{p}\right)$ And $AV = \left(\frac{CA \times PD}{CD} = \frac{ty}{x}\right)$. 18. Also $CU = \left(\frac{AU \times CQ}{EQ} = \frac{cc}{y}\right)$. And $CV = \left(\frac{CP \times CA}{CD} = \frac{tT}{x}\right)$. 19. Hence $PV = (CV \times CP = ) T \times \frac{tccx}{x}$. And $PV = T \times \frac{t+x}{x}$. Also $QU = (CU \times CQ = ) c \times \frac{c+y}{y}$. And $QU = c \times \frac{c+y}{y}$. 20. When $Aa$ and $Bb$ are the longest and shortest axes; and when $y = p$, Then $xx = tt = \frac{t}{p} yy$ will become $tt = pt = tt = cc$, which call $ff$. And $CD = x$, will become $CF = cf = f$. The points $F$, $f$, are called the Foci. 21. Hence $AF = af = \pm t + f$; $Af = af = t + f$. 22. Also $CF^2 = CF^2 = ff = \pm tt + cc = \pm tt + pt$. And in the ellipsis, $\frac{AC^2}{BC^2 + CF^2} = \frac{BF^2}{BA^2}$. in the hyperbola, $CF^2 = \frac{AC^2 + BC^2}{BA^2}$. Hence, a circle described from $B$, with the distance $AC$ in the ellipsis, or from $C$, with the distance $AB$ in the hyperbola, will cut the axis $Aa$ in the focii $F$, $f$. 23. Draw 23. Draw \( FP, fP \), from the focii \( F, f \), to any point \( P \) of the curve; and draw the conjugate diameters \( PP, QQ \). Put \( PF = z; Pf = v; FC = t; QC = c; \frac{f}{t} = \phi; \frac{f}{c} = \gamma \). Then \[ PF^2 = vv = yy + xx + 2xf + ff = tt + 2fx + \frac{ff}{tt}xx = TT + ff + 2fx. \] \[ PF^2 = zz = yy + xx - 2xf + ff = tt - 2fx + \frac{ff}{tt}xx = TT + ff - 2fx. \] For \( yy = \left( \frac{cc}{tt} \times tt \right) \frac{ff}{tt} \times tt \). 24. \( Pf = v = \left( \sqrt{tt + 2fx + \frac{ff}{tt}xx} \right) t + \frac{f}{t}x = t + \phi x = \frac{tt + fx}{t}. \) \( PF = z = \left( \sqrt{tt - 2fx + \frac{ff}{tt}xx} \right) t - \frac{f}{t}x = t - \phi x = \frac{tt - fx}{t}. \) 25. \( PF \pm Pf = z \pm v = 2t = AA. \) 26. \( PF^2 + PF^2 = vv + zz = 2yy + 2xx + 2ff = 2tt + 2\phi\phi xx = 2TT + 2ff. \) 27. \( PF^2 - PF^2 = vv - zz = 4fx = v + z \times v - z = 2t \times v + z. \) 28. \( FPf = zv = z \times 2t \mp z = 2tx \mp zz = tt \cos \phi \phi xx = tt \cos xx \times \frac{cc}{tt}xx \) \[ = cc \cos \frac{ff}{cc} yy = \frac{tt}{cc} yy + \frac{cc}{tt} xx = CE^2 + EQ^2 \] \[ = CQ^2 = CC = T \times P. \] 29. Let \( m = t - z = v - t = \frac{fx}{t} = \phi x = \pm \frac{1}{2}v \mp \frac{1}{2}z \) \[ = \frac{fx}{\sqrt{cc + ff}} = \frac{x}{t} \sqrt{tt \mp cc} = \sqrt{tt \mp zz} \] \[ = \sqrt{tt \mp cc} = \sqrt{ff + cc - zz}. \] 30. Hence \( \dot{x} = -\dot{v} = \phi \dot{x} = -\frac{f}{t} \dot{x} = \frac{fy}{px} j. \) And \( \dot{v} = -\dot{x} = \phi \dot{x} = \frac{f}{t} \dot{x} = -\frac{fy}{px} j. \) 31. \( AC = t = \frac{1}{2}x + \frac{1}{2}v = \frac{cc}{p} = \sqrt{\frac{cc}{ff}} = \frac{fx}{m} = \frac{cx}{\sqrt{cc \cos yy}} \) \[ = \frac{c}{y} \sqrt{sx} = \sqrt{x \times x \cos s} = \frac{y^2 + \sqrt{y^4 + 4p^2x^2}}{2p} = \frac{yy}{2p} + \sqrt{\frac{y^4}{4p^2} + xx}. \] 32. \( CD^2 = PD^2 = xx = \frac{(tyy)^2}{ccs} = \frac{(tyy)^2}{ps} = \pm tt + \frac{t}{p} yy = \frac{tt}{cc} \times cc + yy \) \[ = \frac{ff + cc}{cc} \times cc + yy = tt + \frac{tt}{cc} yy = \frac{mm}{ff} = \frac{tt}{ff} mm \] \[ = \frac{cc + ff}{ff} mm = \frac{tt}{ff} \times tt + zz = \frac{tt}{ff} \times n + cc = \frac{tt}{q} \] \[ = \frac{tt}{cc} \times zz - \frac{tt}{cc} yy = \frac{tt}{p} \times zz - \frac{tt}{p} yy = \frac{tt}{ff} \times t - zz \] \[ = \frac{tt}{ff} \times v - i = \frac{tt}{ff} \times \frac{tt}{cc} \times ss = \frac{ss}{rr} \times rr. \] 33. And \( \dot{x} = -\frac{tyy}{px} = -\frac{ttyy}{ccx} = -\frac{tty}{qq} = -\frac{tz}{f} = \frac{tv}{f}. \) 34. \( PD^2 = CA^2 = yy = \frac{p}{t} sx = \frac{cc}{tt} sx = \frac{cc}{tt} \times \pm tt + xx = \pm cc + \frac{cc}{tt} xx \) \[ = \pm pt + \frac{p}{t} xx = \frac{2cc}{tt} u = \frac{cc}{tt} uu = \pm tt + \frac{ff}{tt} \times \pm tt + xx \] \[ = \frac{cc}{tt} \times zz - \frac{cc}{tt} xx = \frac{cc}{tt} \times cc - \frac{cc}{tt} xx = \frac{cc}{ff} \times zz - cc \] \[ = \frac{cc}{ff} nn = \frac{pt}{ff} nn \text{ (putting } nn = zz - cc) \] \[ = \frac{tt}{ff} \times cc = \frac{tt}{cc} \times \frac{ss}{rr} \times rr = \frac{ss}{rr} \times rr. \] 35. And \( \dot{y} = -\frac{px \dot{x}}{ty} = -\frac{ccx \dot{x}}{tty} = \frac{px \dot{x}}{ty} = -\frac{px \dot{v}}{fy}. \) 36. Also \( \frac{ff}{cc} yy = zz - cc = \frac{ff}{tt} \times tt + xx = ff + \frac{ff}{tt} xx \) \[ = nn = ff + mm = cc - cc. \] 37. \( \overline{PC}^2 = TT = xx + yy = xx + cc + \frac{cc}{tt}xx = tt - ff + \frac{ff}{tt}xx = cc + \phi \phi xx \) \[ = cc + tt \phi z v = cc + mm = cc + tt \phi cc \] \[ = tp + \phi \phi xx = 2cc + ff \phi z v = 2pt + ff \phi z v. \] 38. And \( \dot{\tau} = \frac{\phi \phi x \dot{x}}{\tau} = \frac{\phi \phi x \dot{x}}{\sqrt{tp + \phi \phi xx}} = \left( \frac{\phi \phi x}{\tau} \times \frac{ty \dot{y}}{px} \right) \frac{ff \dot{y}}{pt \tau} = \frac{ff \dot{y}}{cc \tau}. \) 39. Also \( \overline{CQ}^2 + \overline{CP}^2 = (cc + tt) \overline{CB}^2 + \overline{CA}^2. \) 40. \( cT = q = \frac{tt}{x} = \left( \frac{tt}{tm} \right) \frac{ft}{m} = \frac{ft}{t \phi x} = \frac{ft}{\sqrt{tt \phi xv}} \) \[ ct = \frac{cc}{y} = \frac{cf}{n} = \frac{cf}{\sqrt{xxv - cc}}. \] 41. And \( \dot{q} = -\frac{q \dot{x}}{x} = -\frac{qq \dot{x}}{tt} = \frac{qq \dot{y}}{ccx} = \frac{qq \dot{y}}{ptx} = -\frac{tt \dot{x}}{xx} \) (for \( tt = qx \)). 42. \( AT = (\pm cT \mp CA) \pm \frac{tt}{x} \mp t = \frac{t}{x} \times \pm t \mp x = \pm \frac{ft}{m} \mp t = \pm \frac{ft}{t \phi x} \mp t. \) \[ aT = (cT + ca) \frac{tt}{x} + t = \frac{t}{x} \times t + x = \frac{ft}{m} + t = \frac{ft}{t \phi x} + t. \] 43. \( fT = (\pm cT \mp CF = \pm \frac{ft}{m} \mp f = \pm \frac{ft}{m} \mp fm) = \frac{fx}{t \phi x} = \frac{fx}{m} = \frac{tx}{x}. \) \[ ft = (cT + cf = \frac{ft}{m} + f = \frac{ft}{m} + fm) = \frac{fw}{t \phi t} = \frac{fw}{m} = \frac{tw}{x}. \] On \( Aa \) describe a circle, draw the tangent \( TP' \), and draw \( CP' \). Continue \( DP \) to \( P' \), and at right angles to \( TP' \), draw \( FR' \), Then \( FR' = \left( \frac{fT}{ct} \times CP' = \frac{tx}{x} \times t \times \frac{x}{tt} = \right) z = FP. \) \[ fr' = \left( \frac{fT}{ct} \times CP' = \frac{tw}{x} \times t \times \frac{x}{tt} = \right) v = fp. \] 44. \( DT = s = \frac{t y y}{p x} = \frac{t y y}{c c x} = \pm t + \frac{x x}{x} = \frac{t n n}{f m} \). \( dt = \sigma = \frac{p x x}{t y} = \frac{c c x x}{t y} = \frac{c c + y y}{y} = \frac{c m m}{f n} \). 45. \( AN = \left( \frac{c t}{c t} \times AT = \right) \frac{c m}{n x} \times t - x = \frac{c}{n} \times f - m \). \( an = \left( \frac{c t}{c t} \times a T = \right) \frac{c m}{n x} \times t + x \). 46. \( VN = \left( \frac{p v \times c t}{p c} = \frac{t - x}{T} \times \frac{T}{x} \times \frac{f c}{n} = \right) \frac{f c}{n x} \times t - x = \frac{f c}{n m} \times f - m \) \( = \frac{f c}{\sqrt{z v - c c \times \sqrt{t t - z v}}} \times f - \sqrt{t t - z v} \) \( UN = c t = \frac{c c}{y} \). 47. \( AD = (\pm AC \mp CD = \pm t \mp \frac{t m}{f} = ) \frac{t}{f} \times \pm f \mp m \) \( a D = (AC + CD = t + \frac{t m}{f} = ) \frac{t}{f} \times f + m \). 48. Produce \( p F, p f, \) so meet the curve in \( \Pi, \#; \) Draw \( \Pi A \) perpendicular to \( A q \). Put \( F \Pi = x', f \pi = v', F \Delta = x' \). Now, \( FD = (\pm CF \mp CD = ) \pm f \mp x = \pm f \mp \frac{t m}{f} = \frac{t v - c c}{f} \) \( fD = f + \frac{t m}{f} = \frac{t v - c c}{f} \). Then \( F \Delta = x' = (c \Delta - c F = ) \frac{t m}{f} - f = \left( \frac{t}{f} \times t - x - f = \right) \frac{c c - t z}{f} \) \( f \Delta = f + \frac{t m}{f} = \frac{t v - c c}{f} \). 49. From 49. From \( f \), \( f' \), draw \( FR \), \( fr \), perpendicular to the tangent \( PT \), and cutting it in \( R \), \( r \). \[ FR = \frac{FT \times c}{ct} = \frac{cz}{c} = \frac{cz}{\sqrt{zv}} = \frac{cz}{\sqrt{2tz - zz}} = c\sqrt{\frac{z}{v}} \] \[ fr = \lambda = \frac{ft \times c}{ct} = \frac{cv}{c} = \frac{cv}{\sqrt{zv}} = \frac{cv}{\sqrt{2tv - v}} = c\sqrt{\frac{v}{z}} \] \[ \dot{\lambda} = \frac{t \cdot \dot{z}}{v \sqrt{zv}}. \quad \text{For } \lambda^2 = \frac{c^2 z}{v}. \quad \text{Th. } 2\lambda \dot{\lambda} = \left( \frac{c^2 v \dot{z} - c^2 x \dot{\phi}}{vv} \right) \frac{2c^2 t \dot{z}}{v}. \] 50. \( TR = \left( \frac{AT \times RF}{AN} = \frac{tnz}{mc} \right) = \frac{tnz}{fcx} = \frac{tn}{fx} \sqrt{\frac{z}{v}} = \frac{tn}{m} \sqrt{\frac{z}{v}}. \) \[ TR = \frac{at \times rf}{an} = \frac{tn}{m} \sqrt{\frac{v}{z}}. \] 51. Draw \( PM \) perpendicular to the tangent \( PT \), meeting the axes \( Aa \), \( Eb \), in \( M \), \( m \). \[ DM = \left( \frac{PD \times PD}{TD} = \frac{ccm}{ft} \right) = \frac{cc}{ft} \sqrt{tt \pm zv} = \frac{cc}{tt} x = \frac{p}{t} x \] \[ dm = \left( \frac{pd \times pd}{dm} = \frac{tt}{fc} \right) = \frac{tt}{cc} \sqrt{zv - cc} = \frac{tt}{cc} y = \frac{t}{p} y. \] 52. \( CM = (cd \mp DM = x \mp \frac{cc}{tt} x = ) \frac{tt \mp cc}{tt} x = \frac{ff}{tt} x = \frac{f}{t} m \) \[ = \frac{f}{t} \sqrt{tt \mp zv} = \frac{f}{t} \times t \mp z = \frac{f}{t} \times \sqrt{tt \mp tt \pi w} \] \[ cm = (dm \mp cd = \frac{tt}{cc} y \mp y = ) \frac{tt \mp cc}{cc} y = \frac{ff}{cc} y = \frac{f}{c} n = \frac{f}{c} \sqrt{zv - cc} \] 53. \( FM = (cf \mp cm = f \mp \frac{fm}{t} = ) \frac{f}{t} \times t \mp m = \frac{fz}{t} = \pm f \mp \frac{ff}{tt} x. \) \[ fm = (fc + cm = f + \frac{fm}{t} = ) \frac{f}{t} \times t + m = \frac{fv}{t} = f + \frac{ff}{tt} x. \] 54. \( \overline{fm}^2 = f^m = (\overline{f}^2 + c^m) = ff + \frac{ffnn}{cc} = \frac{ffnv}{cc} = \frac{ff}{cc} \times cc. \) 55. \( AM = (AC \propto CM = ) t \propto \frac{fm}{t} = \frac{tt \propto fm}{t} = t \propto \frac{ff}{tt}. \) \( AM = (ac + cm = ) t + \frac{fm}{t} = \frac{tt + fm}{t} = t + \frac{ff}{tt}. \) 56. \( TM = (TF + FM = \frac{fx}{m} + \frac{fx}{t} = ) \frac{fxw}{tm} = \frac{xw}{x} = \frac{cc}{x}. \) \( tm = (ct + cm = \frac{fc}{n} + \frac{fn}{c} = ) \frac{fxw}{cn} = \frac{xw}{y} = \frac{cc}{y}. \) 57. \( PM = \pi = \left( \frac{FR \times TM}{FT} \right) \frac{c}{t} \sqrt{zv} = \frac{c}{t} c = \frac{p}{c} = \frac{c}{t} \sqrt{tt \propto \phi^2 x^2}. \) \( = \frac{c}{t} \sqrt{t + \phi x \times t - \phi x} = \frac{cc}{tt} \times \frac{R}{s} = \frac{cr}{ts} \sqrt{ce \mp yy}. \) \( = \frac{c}{tt} \sqrt{t^2 - ffxx} = \frac{c}{t} \sqrt{cc + yyy} = \frac{1}{t} \sqrt{c^2 + ffyy}. \) \( = \frac{c}{t} \sqrt{2tz \mp zz} = \sqrt{2pz \mp \frac{p}{t} az}. \) \( pm = \pi' = \left( \frac{PM \times PD}{DM} \right) \frac{t}{c} \sqrt{zv} = \frac{t}{c} c = \frac{c}{p} = \frac{t}{c} \sqrt{tt \propto \phi^2 x^2}. \) \( = \frac{t}{c} \sqrt{t + \phi x \times t - qx} = \frac{t}{c} \sqrt{t^2 \propto ffxx} = \frac{t}{c} \sqrt{cc + yyy}. \) \( = \frac{t}{cc} \sqrt{c^2 + ffyy} = \frac{t}{c} \sqrt{2tz \mp zz} = \sqrt{\frac{2tt}{p} z \mp \frac{t}{p} zz}. \) 58. \( mm = (pm \mp PM = \frac{t}{c} c \mp \frac{c}{t} c = ) \frac{tt \mp cc}{ct} c = \frac{ff}{ct} \sqrt{zv}. \) \( = \frac{ff}{ct} \sqrt{tt \propto \phi^2 x^2}. \) 59. \( PT = \tau = \left( \frac{PD \times PM}{DM} \right) \frac{n}{m} c = \sqrt{\frac{zw - cc}{tt - zv}} \times zv = \frac{tn}{fx} c. \) \( = \frac{ty}{cx} = \frac{ty}{cx} \sqrt{tt \propto \phi^2 x^2} = \sqrt{zv - cc \times zv}. \) \( pt = \tau' = \left( \frac{PT \times PD}{DT} \right) \frac{m}{n} c = \sqrt{\frac{tt - zv}{zv - cc}} \times vz = \frac{fx}{cz} c. \) \( = \frac{cx}{cz} = \frac{cx}{cz} \sqrt{tt - \phi^2 x^2}. \) 60. \( TN \) 60. \( Tn = \left( \frac{PT \times TA}{DT} \right) = \frac{n}{m} \times \frac{t}{m} \times f \times m \times f \times m = \frac{f}{mn} \times f \times m = \frac{c}{xy} \times t \times x \) \( Tn = \left( \frac{PT \times Ta}{DT} \right) = \frac{n}{m} \times \frac{t}{m} \times f + m \times f \times m = \frac{f}{mn} \times f + m = \frac{c}{xy} \times t + x \) 61. \( PN = \left( \frac{AD \times PT}{DT} \right) = \frac{t}{f} \times f \times m \times n \times f \times m = \frac{c}{n} \times f \times m = \frac{c}{xy} \times t \times x \) \( PN = \left( \frac{aD \times PT}{DT} \right) = \frac{t}{f} \times f + m \times n \times f \times m = \frac{c}{n} \times f + m = \frac{c}{xy} \times t + x \) 62. \( PR = \left( \frac{PT \times FM}{TM} \right) = \frac{n}{c} = n \sqrt{\frac{z}{v}} = \sqrt{\frac{z}{v}} \times zv - cc = \frac{f}{c} \times y \times \sqrt{\frac{z}{v}} \) \( PR = \left( \frac{PT \times f_{Ma}}{TM} \right) = \frac{n}{c} = n \sqrt{\frac{v}{z}} = \sqrt{\frac{v}{z}} \times zv - cc = \frac{f}{c} \times y \times \sqrt{\frac{v}{z}} \) 63. \( Tg = \left( \frac{PT \times TC}{TM} \right) = \frac{n}{c} = \frac{tt}{t-z} \sqrt{\frac{zv}{vz}} = \frac{f}{cc} \times \frac{yt^2}{xcc} \) 64. \( P \epsilon = (Tg + TP) = \frac{n}{c} = \frac{n}{m} \times \frac{t}{m} \times tt + cc = \frac{n}{mc} \times mm = \frac{n}{c} \) 65. \( Tt = \left( \frac{PT \times c}{PD} \right) = \frac{n}{c} = \frac{c}{n} \times \frac{f}{cn} = \frac{ff}{mn} = \frac{ff}{\sqrt{\frac{zv}{tt-zv \times zv-cc}}} \) \( Tt = \frac{f}{xy} = \frac{f}{xy} \sqrt{\frac{tt}{ff} \times \frac{zz}{tt}} \) 66. \( n = \left( \frac{PT \times Aa}{DT} \right) = \frac{2f}{c} = \frac{2c}{y} = \frac{2c}{y} \sqrt{\frac{zv}{zv-cc}} = \frac{2f}{\sqrt{\frac{zv}{zv-cc}}} \) From \( c \) and \( f \), draw \( cy \) and \( fe \), perpendicular to \( pf \) Put \( ps = b; \Sigma = co-\text{sine of the } \angle Afp; R = \text{tabular radius}. \) Then \( f \gamma = \left( \frac{cf \times fD}{pf} \right) = \frac{tw-cc}{f} = \frac{tw-cc}{w} = \left( \frac{\Sigma}{R} \times cf \right) \frac{\Sigma}{R} \) And \( pf = v = \frac{cc}{t-\frac{\Sigma}{R}f} = \frac{tt-ff}{t-\frac{\Sigma}{R}f} \) \[ r = b = \left( \frac{\pm xz + vv - 4ff}{2v} \right) = \frac{\pm tt - ff}{v} = \frac{\pm vt - vv}{v} = \frac{\pm zt - vz}{v} \] \[ = \pm \frac{zp}{v} \mp z = \pm \frac{x + v}{v} p \mp z. \] Hence \( p = \frac{\pm b \mp z}{x + v} \times v; \quad v = \frac{pz}{\mp b \pm z - p}; \quad z = \frac{p - b}{v - p} \times v \) Therefore \( p \) is less than, equal to, or greater than \( \pm b \mp z \), in the Ellipsis, Parabola, or Hyperbola. 67. Draw \( m \mu \), making the \( \angle Pm \mu = \angle PCM \), and cutting \( PC \) in \( \mu \). Then \( P \mu = \left( \frac{PM \times PM}{PG} \right) = \frac{cc}{tt} = \frac{zz}{tt} = \frac{tt - \phi \phi xx}{cc + \phi \phi xx} \) \[ = p = \frac{1}{2} \text{ parameter to } PC. \] 68. Let \( F \Phi \) be an ordinate to the axis \( AA \), at the focus \( F \), and \( FG \) a tangent to the curve in \( \Phi \), meeting \( AA, BC, AN, an \), in \( G, g, s, s \). Then \[ F \Phi = \sqrt{\frac{cc}{tt}} \times AFa = \sqrt{\frac{cc}{tt}} \times cc = \frac{cc}{tt} = \frac{pt}{t} = p \] \[ = \frac{1}{2} \text{ parameter to } AA. \] 69. \( CG = \frac{tt}{f} = \frac{ff \pm cc}{f} = \frac{tt}{\sqrt{tt - cc}} = \frac{tt}{\sqrt{tt - p}} \) 70. \( FG = (CG \circ CF = \frac{tt}{f} \circ f = \frac{tt \circ ff}{f} = \frac{cc}{f} = \frac{pt}{f}. \) 71. Draw \( PH \) parallel to \( AA \); and \( GH \) perpendicular to \( AA \), meeting \( PH, PT \), in \( H, b \); then \[ PH = DG = (CG \circ DC = \frac{tt}{f} \circ x = \frac{t}{f} z = \frac{t}{f} \times PF. \] 72. \( t \sigma = (ct \cdot cg = \frac{ft}{m} \cdot \frac{tt}{f} = \frac{t}{m} \times \sqrt{f \omega x}) = \frac{t}{f} \times \frac{\pm tx - cc}{f} = \frac{tt}{fx} \times \sqrt{f \omega x}. \) 73. \( gh = \left( \frac{PD \times TG}{DT} = \frac{cn}{f} \times \frac{t}{m} \times \sqrt{f \omega x} \times \frac{fm}{inn} = \frac{c}{n} \times \sqrt{f \omega x} = \right) \) \[ \frac{cc}{fy} \times \sqrt{f \omega x} = \frac{c}{\sqrt{xv - cc}} \times \frac{tx - cc}{f}. \] 74. \( pb = \left( \frac{TP \times DG}{DT} = \right) \frac{zc}{n} = \frac{cz}{fy} c = z \times \sqrt{\frac{zv}{xv - cc}} = \frac{cz}{fy} \sqrt{zv}. \) 75. \( rh = (pb - pr = ) \frac{zc}{n} - \frac{zn}{c} = \frac{zc}{nc} \times \frac{cc - nn}{c} = \) \[ \frac{zcc}{nc} = \frac{zcc}{\sqrt{zv - cc + xv}} = \frac{zcc}{fy c}. \] 76. \( fb = (\sqrt{fr^2 + rh^2} = ) \frac{cz}{n} = \frac{cz}{\sqrt{zv - cc}} = \frac{ccz}{fy} = \frac{ptx}{fy}. \) 77. Let \( d \Sigma \) be any Ordinate to the axe \( Aa \), cutting the curve in \( \Sigma \), and the focal tangent \( \Phi G \) in \( \sigma \); Then \( d \sigma = \left( \frac{F \Phi + DG}{FG} = \frac{cc}{t} \times \frac{tx}{f} \times \frac{f}{cc} = \right) z = FP = f \Sigma. \) 78. Therefore \( AS = AF; as = af; cg = ca; \) by lim. \( \Delta s. \) 79. \( DF^2 = (D \sigma^2 - D \Sigma^2 = D \sigma + D \Sigma \times D \sigma - D \Sigma = ) P \sigma \times \sigma \Sigma. \) 79. Let the tangents \( PN, pL \), to the opposite vertices \( P, p \), cut the tangents \( AN, an \), to the opposite vertices \( A, a \), in \( N, n, L, l. \) Then \( pn = pl; an = al; pn = pl; an = al. \) For the Trapezia's \( PCan, pCAL \), are similar and equal; And so are the Trapezia's \( PCAN, pCAL. \) Vol. LXIII. A a a 81. PF 81. \( PF \times MF = \left( \frac{f}{t} \times v \right) = PF \times MF = \frac{f}{t} \times FP f. \) 82. \( CEQ = (xy) = CDP = dCD = DM \times dm. \) 83. \( AN \times AD = \frac{ctn}{f} = an \times AD = AC \times PD = Dal = Dal. \) 84. \( RPR = nn = TP \). 85. \( RT r = \left( \frac{nn}{mm} \times tt = \frac{tt - xx}{xx} \times tt = \right) ATa. \) 86. \( RT f = \left( \frac{tnz}{mc} \times \frac{f}{m} = \frac{ft}{m} \times \frac{n}{m} = \right) CTP = r TF. \) 87. \( NT n = \left( \frac{ffcc}{mmnn} \times \frac{ff}{mm} = \right) \frac{ffcc}{mm} = PT f. \) 88. \( PN \times CB = \left( \frac{t}{ox} = \frac{fc}{tn} c = \right) AN \times CQ. \) 89. \( CAV = \left( t \times \frac{ty}{x} = \frac{tt}{x} \times y = \right) CT \times PD. \) 90. \( PR b = \left( \frac{cczz}{cc} = \right) FR^2. \) Hence \( fh \) is perpendicular to \( FP. \) 91. \( FM f = \left( \frac{fz}{t} \times \frac{fv}{t} = \right) \frac{ff}{tt} \times zv = \frac{ff}{tt} \times cc = \frac{ff}{tt} \times CQ^2. \) 92. \( AMa = \left( \frac{tt}{tt} \times \frac{ffmm}{tt} = \right) cc \times \frac{ff}{tt} \times zv \) \( = CB^2 + \frac{ff}{tt} \times CQ^2 = CB^2 + FMf. \) 93. \[ \frac{PN^2}{TN^2} = \frac{xx}{tt} = \frac{CD^2}{AC^2} = \frac{AN^2}{VN^2} = \frac{CA^2}{CT^2} = \frac{ADA}{AT^2} = \frac{AD^2}{RT^2} = \frac{aD^2}{aT^2} = \frac{CDT}{AT^2}. \] 94. \[ \frac{FT}{CT} = \frac{z}{t} = \frac{FP}{CA} = \frac{FR}{C^2}. \] Sim. \( \triangle TFR, TCG \). And \[ \frac{AD}{DT} = \frac{x}{t+x} = \frac{CD}{AD} = \frac{AC}{aT}. \] 95. \[ \frac{fT}{CT} = \frac{w}{t} = \frac{fP}{AC} = \frac{fR}{C^2}. \] And \[ \frac{AT}{DT} = \frac{t}{t+x} = \frac{AC}{AD} = \frac{CT}{aT}. \] 96. \[ \frac{PM}{PM^2} = \frac{cc}{tt} = \frac{p}{t} = \frac{BC^2}{AC^2}. \] And \[ \frac{pm}{fm} = \frac{f}{f} = \frac{AC}{CF}. \] 97. \[ \frac{FMf}{PM^2} = \frac{ff}{cc} = \frac{t+c\times t-c}{cc} = \left( \frac{tt-t^p}{tp} \right) \frac{t-p}{p} = \frac{CF^2}{GB^2}. \] 98. Let \( PM \), the perpendicular to the tangent \( PT \), cut the axis \( Aa \) in \( M \); and \( fP \), produced, cut \( FR \) in \( \phi \). Then will \( PM \) bisect the angle \( FPf \). For \( PF \times MF = Pf \times MF \). 99. And the angle \( FPT \) is equal to the angle \( fPt \). For \( \angle TPF + \angle FPM = \angle IPf + \angle fPM \). 100. Therefore \( PT \) will bisect the angle \( FP\phi \). For \( \angle \phi PT = (\angle fPt = ) \angle FPT \). 101. Consequently \( P\phi = Pf \). And \( R\phi = RF \). 102. A circumference of a circle described from \( t \), with the radius \( TN \), will cut the axis \( Aa \) in the focii \( F, f \). For \( TN \times TN = TF \times Tf \). 103. A circumference of a circle described from \( c \), with the radius \( ca \), will cut the tangent \( pt \) in \( r \), \( r' \). Whence the perpendiculars \( rf \), \( rf' \), to that tangent, will cut the axis \( aa \) in the focii \( f \), \( f' \). For \( tr \times tr = ta \times ta \). 104. A circumference described from \( b \), with \( ac \), in the Ellipsis, or from \( c \), with \( ab \) in the Hyperbola, will cut the axis \( aa \), in the focii \( f \), \( f' \). For, in the Ellipsis, \( tt = cc + ff \), or \( ac^2 = (bc^2 + cf^2) = bf^2 \). And in the Hyperbola, \( ff = tt + cc \), or \( cf^2 = (ac^2 + bc^2) = ab^2 \). 105. Let \( cq \) produced, cut \( pf \), \( pf' \), in \( z \), \( x \); draw \( mz \), \( mx \), and \( mz \), \( mx \), parallel to them, cutting \( pf \), \( pf' \) in \( z \), \( x \). Then \( px = cr = ca = pz = t \). 106. Hence \( \angle pzx = \angle pzr \); and \( \angle mxz = \angle mzx \). For \( pm \) is perpendicular to \( zx \). Consequently, the angles \( pzm \), \( pxm \); \( pzr \), \( pxr \) are equal. 107. And the triangles \( pzr \), \( pxm \), are similar and equal: And so are the triangles \( pzr \), \( pxm \). Consequently, the trapezias \( pzmx \), \( pzmx \), are similar. 108. Let \( cr \), \( cr' \), cut \( pf \), \( pf' \), in \( k \), \( k' \). Then \( ck = \left( \frac{cf \times pf}{ff} \right) \frac{1}{2} pf = pk = kr \). And \( ck' = \left( \frac{cf' \times pf'}{ff} \right) \frac{1}{2} pf' = pk' = kr' \). 109. Also \( px = pz = \left( \frac{pm}{pm} \times px = \frac{cc}{t} \times \frac{c}{tc} \times t = \right) \frac{cc}{t} = \left( \frac{pt}{t} = \right) p \). 110. The Trapezias \( FGBR, PDFR, frtc \), are similar, and consequently their corresponding parts are proportional. That is, \[ \begin{align*} \frac{FG}{PD} &= \frac{GB}{DF} = \frac{BR}{FR} = \frac{RF}{RP}, \\ \frac{fr}{rt} &= \frac{rt}{tc} = \frac{tc}{cf}. \end{align*} \] For the triangles \( RFB, RPF, \) and \( FBG, FPD, \) are similar. 111. The Trapezias \( CEPD, tCMP \) are similar, and consequently their corresponding parts are proportional. That is, \[ \frac{CE}{tC} = \frac{EP}{CM} = \frac{PD}{PM} = \frac{DC}{Pt}. \] 112. And \( cr, cr, \) are parallel to \( Pf, PF, \) and equal to \( ca. \) For \( rtf = ctp = rtf. \) 113. Let \( \Sigma' \) = sine of the \( \angle IPf, \) or \( TPF; \) \( R = \) tabular radius. Then \[ \frac{R}{\Sigma'} = \left( \frac{Pf}{fr} \right) = \frac{c}{c} = \sqrt{\frac{xv}{c}}. \] Put \( \Sigma = \) sine of the \( \angle PCQ, \) made by any diameter and its ordinate. Then \[ \frac{TT}{cc} = \frac{1}{2} ff \pm \sqrt{\frac{1}{4} f^2 + \frac{r^2}{\Sigma^2}}; \quad \frac{tt}{cc} = \frac{1}{2} FF \pm \sqrt{\frac{1}{4} F^2 + \frac{T^2}{c^2} \Sigma^2}. \] 114. Let the parallels \( PFP, cbr, pfw \) be drawn, cutting the curve in \( \Pi, \beta, \pi; \) and ordinarily applied to some diameter (\( 2\tau \)), whose parameter is \( 2\pi, \) and semi-conjugate \( c\beta = x, \) to which \( pw \) is ordinarily applied at \( \delta. \) Then \[ c\delta = \left( \frac{PP + pf}{2} \right) = \frac{1}{2} z + \frac{1}{2} z' = \frac{1}{2} \Pi = \left( \frac{c\beta}{cr} \right) = \frac{c\beta^2}{AC} = \frac{xx}{t} = \frac{\pi}{t}. \] 115. \( c\beta^2 = \left( \frac{1}{2} AC \times \Pi = \right) \frac{1}{2} t \times z + z' = \left( \frac{AC^2}{APa} \right) \frac{tt}{cc} = \frac{1}{2} zz' = \frac{1}{2} zz'. \] 116. \( \frac{PP}{\Pi} = \frac{z}{z} = \frac{RD}{FD} = \frac{tz - cc}{cc - tz} = \frac{z - p}{p - z}. \) 117. \( PFP = \frac{pzz}{2z - p} = \frac{1}{2} p \times z + z' = \frac{1}{2} p \times PFP. \) 118. \( \Pi = z + z' = \frac{2zz}{zz - p} = \frac{2\pi}{t} = \frac{\pi}{t} \times 2\pi. \) \[ \overline{CB}^2 = \alpha = p_i = \pm tt \mp ff = \pm i \mp f \times t + f = AFa = VAU = tc d = CFG \] \[ = PM \times Cq = FR \times fr = AN \times an = al \times AL = AC \times F \Phi = tt \times \frac{PM}{PM} \] \[ = tt \times \frac{DM}{DC} = tt \times \frac{dC}{dm} = tt \times \frac{PM^2}{CQ^2} = tt \times \frac{PD^2}{AD^2} = tt \times \frac{PD^2}{TDC} = ff \times \frac{PD}{Cm} \] \[ = \frac{CQ \times Cq}{AC} = \frac{AN \times CQ}{PN} = AMa - FMf = FPf - RPr = ff \times \frac{t}{i-p} \] \[ = TT + CC - tt = \frac{tt}{xx} \times zv + xx - tt = \frac{1}{2} zv \pm \sqrt{\frac{1}{4} z^2 v^2 ff y}. \] \[ \overline{CQ}^2 = CC = P \times T = MPm = NPn = TPT = FPf = CPn = LpL \] \[ = PN \times PL = CD \times TM = PD \times tm = \frac{tt}{cc} \times PM = \frac{cc}{tt} \times PM = \frac{cc}{ff} \times fm \] \[ = \frac{tt}{ff} \times FMf = \frac{ct}{ff} \times Mm = \frac{ACB}{Cq} = PM^2 + FMf = BC^2 + RPr \] \[ = tt + cc - TT = tt - \frac{ff}{tt} \times x = \frac{tt}{cc} \times yy + \frac{cc}{tt} \times xx = \frac{dc}{dt} \times PT^2. \] \[ \overline{CF}^2 = ff = tt - cc = tt - tp = \frac{cc}{p} \times t - p = \frac{tt}{xx} \times mm = Pq \times Tt \] \[ = \frac{tt}{cc} \times FMf = \frac{cc}{cc} \times fm = \frac{mm}{cc} \times FTf = \frac{mm}{cc} \times NTn. \] The semi-parameter \( (p) \) to the greater axis \( (Aa) \) is equal to \[ F \Phi = PZ = \frac{AFa}{AC} = \frac{VAU}{AC} = \frac{CFG}{AC} = \frac{tc d}{AC} = \frac{PF \Pi}{AC} = \frac{FM \times Cq}{AC} \] \[ = \frac{FR \times fr}{AC} = \frac{AN \times an}{AC} = \frac{AC \times PM}{PM} = \frac{AC \times DM}{DC} = \frac{AC \times dc}{dm} = \frac{BC \times PM}{CQ} \] \[ = \frac{BC \times CQ}{Pm} = \frac{BC^2}{AC} = \frac{cc}{tt} \times tyy = \pm tt \mp ff = \frac{cc}{xx} \times y \times t \] \[ = \frac{cc + TT - tt}{t} = \frac{cc + xx - tt}{xx} \times t = \frac{2t - z}{tx} \times FR = \frac{2t - w}{tv} \times fr^2. \] The semi-parameter \( (P) \) to any diameter \( (PP) \) is equal to \[ Pq = \frac{CQ^2}{PC} = \frac{CC}{T} = \frac{MPm}{PC} = \frac{NPn}{PC} = \frac{TPT}{PC} = \frac{FPf}{PC} = \frac{LpL}{PC} = \frac{PN \times PL}{PC} \] \[ = \frac{CD \times TM}{PC} = \frac{PD \times tm}{PC} = \frac{PT \times PN}{PV} = \frac{tt}{cc} \times \frac{PM^2}{PC} = \frac{cc}{tt} \times \frac{PM^2}{PC} \] \[ = \frac{cc}{ff} \times \frac{fm^2}{PC} = \frac{tt}{ff} \times \frac{PMf}{PC} = \frac{tt + cc - TT}{T}. \] [See Tab. XIV, XV.] XXXVII. An