AN INTRODUCTION ΤΟ CONICK SECTIONS. SECTION I. OF THE ELLIPSIS. Definition 1. IF two pins be fixed at the points F, S; and a thread PSFP, put about them and knotted at P; then if the thread be drawn tight, and the point P and the thread be moved about the fixed centres F, S; the point P will describe the curve PDpBEAP, called an Ellipsis. See Fig. 1. F Def. 2. The points or centres F, S, are called the foci. Def. 3. The line A, B, drawn through the foci to the curve, is call .ed the transverse axis. Def. 4. The point C in the middle of the axis AB, is the centre. Def. 5. The line DE, (drawn through the centre C) perpendicu lar to the transverse AB, is called the conjugate axis. Def. 6. Any line TO, drawn through the centre C to the curve, is called a diameter. And the extremity T (or O) its vertex. Def. 7. If TO be a diameter, then the diameter GK, drawn parallel to the tangent at its vertex T, is called its conjugate. And the two diameters TO, GK, are said to be conjugates to one another. A R Fig. 2. M T LH Def. 8. The line LR (drawn through the focus F, perpendicular to the transverse axis AB,) is called the parameter or latus rectum. Def. 9. A line drawn from any point of the curve (as HI) perpendicular to the transverse axis, is called an ordinate to the transverse. And, in general, any line drawn from the curve to any diameter TO, parallel to its conjugate GK, (as HN,) is an ordinate to that principal diameter, TO. If it go quite through the figure, as Hh, it is called a double ordinate. Def. 10. 1 Def. 10. A right line meeting the ellipsis in one point T, but not cutting it, is called a tangent to it in that point, as TM. Def. 11. The part of the diameter between the vertex and the ordinate, is called the abscissa, TN, AI. And the vertex is the extremity of any diameter. PROPOSITION I. The sum of the lines FP, SP drawn from the foci, to any point of the curve, is equal to the transverse axis AB. See Fig. 1. For by construction, PF+PS-AF+AS AF+AF+FS=2AF+FS. And the same PF+PS=2BS+FS; therefore 2AF+FS=2BS+FS, and 2AF=2BS, or AF-BS. Whence PF+PS=2AF+FS=AF” +ES+FS=AB. COR. The two foci are equally distant from the vertices, and also from the centre: AF-BS; and FC-SC. For it is proved that AF BS; and since AC=CB (Def. 4.) therefore AC-AF-CB-BS, or FC=SC. PROP. II. A line, drawn from the end of the conjugate axis, to the focus, is equal to half the transverse; DF=CA. See Fig. 3. Draw DS to the other focus. Then the two right angled triangles CDF and CDS are similar and equal. For SCCF, the angles at C are right, and CD common: therefore SD= B DF; and since the sum SD+DF-the transverse (Prop. 1,) one of them DF = half the transverse CA. 2 Fig. 3. E COR. The distance of the foci is a mean proportional between the sum and difference of the transverse and conjugate axis, SF2= BA+DEXBA-DE: For CA2=DF2=DC2+CF2; and CF2= CA2-CD2-CA+CDXCA-CD; and 4CF2 or SF2=2CA+2CD x2CA-2CD. PROP. III. The rectangle of the focal distances, from either vertex, is equal to the square of the semiconjugate: AFXFB-DC2. See Fig. 3. For DC2-DF2-CF2= (Prop. 2.) CA2-CF2=CA+CF × CA-CF-BC+CFXCA-CF=BFXFA. PROP. IV. As the transverse axis to the conjugate, so the conjugate to the latus rectum of the transverse: AB : DE :: DE : LR. See Fig. 3. For SL+LF-BA=2CA (Prop.1.); and SL-2CA-LF, and, by squaring, SL2-4CA2-4CAxLF+LF2. And in the right angled triangle SLF,SL2=SF2+LF2; whence 4AC2-4CA×LF+LF, =SF2+LF2, and 4AC2-4CA×LF-SF2-4CF2, and 4AC2=4CA XLF+4CF2=4CAxLF+4DF2-4DC2, and 4AC2+4DC-4CAX LF+4DF2; but CA2-DF2 (Prop.2.); therefore 4DC2-4CAXLF= 2CAX2LF; that is, DE2=BA×LR. COR. 2. COR. 1. As the semitransverse is to the semiconjugate, so the semi- COR. 2. As the semitransverse, to the distance of the focus from For CF2 DF2— DC2-CA2—CD2=CA2—CAXLF. COR. 3. The rectangle BFA = half the transverse x half the latus SCHOLIUM. Since, as the transverse axis is to the conjugate, so the PROP. V. From any point M in the curve, drawing the lines MF, As the semitransverse CA : To the distance of the focus from the centre, CF :: To half the difference of the lines MF,MS, or For, make SD = CA, then SM = CA+DM, B MS-MF. 2 Fig. 4. N M D S E tion from the former, SM-FM2-4CA×DM=4CF×CP, and CFX therefore SM-FM-2DM; therefore CFXCP=CAX 2 COR. 1. If FS be the foci, MP an ordinate; then it is CA: CF For CFXCP=C A×DM, and DM=SM—CA=CA—FM. COR. 2. If F, S, be the foci, MP an ordinate; then the difference For SM2-FM-SM+FMXSM-FM-2CAXSM-FM-4CFXCP, SCHOLIUM. If PM fall on the other side of F, as pm, then pF-Cp PROP. VI. If an ordinate MP be drawn to the transverse axis; it 3...M As As the square of the transverse, BA2 : To the square of the conjugate, NE2 :: So the rectangle of the segments of the transverse BPA : : For make SD-CA, then DM is half the difference of SM and MF; therefore by Prop. 5. CA: CF :: CP: DM, and CA: CA+ CF or BF:: CP: CP+DM, and CA: CP :: BF: CP+DM, and CA: CA+CP or BP :: BF: BF+CP+DM. But BF-BC+CF=SD +CF; and BF+CP+DM=SD+CF+CP+DM=SM+CS+CP=SM+SP; whence CA BP :: BF : SM+SP. Again, since CA: CF :: CP : DM; then CA: (CA-CF) AF :: CP: CP-DM; and CA: CP ::AF: CP-DM. And CA: (CA-CP) PA :: AF : AF_CP+ DM. But AF=CA_CF=SD-SC; therefore AF-CP+DM=SD -SC-CP+DM-SM--SP; therefore CA: PA :: AF : SM-SP, and we had before, CA: BP :: BF: SM+SP; then multiplying these proportions together, we have CA2: BPXPA :: BFXFA : SM-SP". But (Prop. 3.) BFXFA-CN2; and SM2-SP2-PM2; therefore CA: BPA :: CN2: PM2, or alternately, CA2 : CN2 :: BPA : PM2, or BA (4CA): NE (4CN) :: BPA: PM2. Cor. 1. CA: CN :: BFA : PM'. COR. 2. As the transverse BA: to its latus rectum :: So the recangle BPA to square of the ordinate PM'. COR. 3. The rectangles of the segments of the transverse are as the squares of the ordinates. For every rectangle is to the square of its ordinate, in the given ratio of CA to CN1, or of BA to the latus rectum. COR. 4. As the square of the semitransverse CA': Rectangle of the focal distances from vertex BFA :: To square of the ordinate PM'. SECTION II. OF THE PARABOLA. Definition 1. If one end of a thread, equal in length to CH, be fixed at the point F, and the other end fixed at H, the end of the square DCH. And if the side CD of the square be moved along the right line BD, and always coincide with it, then if the string FGH be always kept tight, and close to the side GH of the square, the point or pin G (where it leaves the square) will describe a curve MRALGK called a Para bola. See Fig. 5. Def. 2. Def. 2. The fixed point F is called the focus. Def. 3. The right line BD is called the directrix. Def. 4. If the line BN be drawn through the focus F, perpendicular to BD; then AN is called the axis of the parabola, and A the vertex. Def. 5. A line drawn through the focus F, perpendicular to the axis, as LR, is called the parameter or latus reðum.. Def. 6. Any line drawn within the curve, parallel to the axis, as GH, is called a diameter. And the point G, where it cuts the curve, is the vertex. Def. 7. A right line drawn from any diameter to the curve, and parallel to the tangent at the vertex, as PM, is called an ordinate. If it go quite through the curve, it is called a double ordinate. See Fig. 6. Fig. 6. M Def. 8. The part of the diameter between the vertex and ordinate, as GP, is called the abscissa. Def. 9. A right line, meeting the curve in one point G, but not cutting it, is called a tangent in that point. Fig. 7. D B PROPOSITION I. If BD be the directrix, G any point in the curve, the line GD drawn to the directrix, parallel to the axis, is equal to the line GF drawn from the same point G to the focus; GD=GF. See Fig. 7. For HG+GF-length of the string = HD; take away GH from both, and then G.D=GF. COR. 1. The distances of the focus, and of the directrix from the AB=AF. For when D is at B, G will be at A ; vertex are equal. consequently AB=AF. COR. 2. If GP be an ordinate to the axis; then AP+AF=FG, For AP+AF=BP=GD. COR. 3. FG-FP= half the latus rectum. PROP. II. The distance of the focus from the vertex is the latus rectum : AF-LR=4LF. See Fig. 5. For when the pin G comes to L, then LF-FB (Prop. 1. Cor. 1.) =2FA, and AFFL. For the same reason FA=FR, therefore FA=+LR, SCHOLIUM, |