2. Every ellipse may be considered as the projection of a circle whose diameter equals the axis major of the ellipse. R Let DBD', RPR be two circular sections of the cone made by planes at right angles to the axis through C, N, the middle and any other point of AA', and cutting the plane of the curve in BCB', PNP' at right angles to AA'. Then NP = RN. NR and BC = DC. CD'. Also RN:DC: AN: AC and NR: CD' :: NA: A'C; .: RN, NR : DC. CD' :: AN. NA': AC", .. NP : BO :: AN.NA: AC?. Now if NQ be the ordinate of a circle drawn on AA' as diameter, we shall have NQ=AN. NA', and NP: BC? :: NQ: AC", or then if a circle, diameter AA', be inclined to the plane of the paper at such an angle that the maximum ratio of diminution of the planes be AC: BC, every ordinate of the circle will be diminished by projection in this ratio, and will therefore after projection equal the corresponding ordinate of the ellipse, and the circle whose diameter is AA' will project into an ellipse whose axes equal AA', BB', 3. We must now prove that in every case the diameter BB' of the ellipse at right angles to the plane of the paper is < AA' in this plane. BOP= DC. CD': we have to shew that DC. CD <AC?. Draw AE perpendicular to DD', and AF parallel to D'A' so that CF will = CD'. Then CA = CE + AE = DC. CF+EF + AE (Euclid 11. 6) = DC. CD + AF (or AD'), thus DC. CD' is always < C4%, and the ratio BC: AC one of less inequality. We may now trace some of the properties of the circle in the ellipse by means of the principles of projection already proved. 4. In the circle if two diameters are at right angles to one another each bisects all chords parallel to the other. But by Chap. I. § 4, the projection of a point which bisects a line bisects the projection of the line; therefore chords of the circle which are bisected by a diameter will project into chords of an ellipse bisected by the projection of the dia meter. Hence diameters of the circle at right angles to one another project into diameters of the ellipse which bisect each the chords parallel to the other. These diameters of the ellipse are said to be conjugate on account of the relation between them being mutual. Also the tangents to a circle at the extremities of a diameter are at right angles to the diameter and parallel to the chords bisected by it: and the parallelism of straight lines is not destroyed by projection; hence the tangents at the two extremities of a diameter of an ellipse are parallel to the chords bisected by it and to the conjugate diameter. Those angles at the centre contained by conjugate diame ters in which the axis major lies will in all cases be less than a right angle. The accompanying figure shews how diameters of the circle at right angles to one another project into conjugate diameters of the ellipse. 5. Supplemental Chords. The angle in a semicircle is a right angle in whatever point of the circumference the chords containing it meet : therefore when projected these chords are parallel to conjugate diameters of the ellipse : such chords when projected are called supplemental chords; they form two sides of a triangle the base of which is a diameter, and its vertex on the circumference of the ellipse. Hence supplemental chords are parallel to conjugate diameters. 6. If QV be a semi-chord of a circle at right angles to a diameter PCp, and CD the radius parallel to QV, then QV = PV. Vp; or since CP, CD are equal, QV : CD :: PV. Vp : CP. These ratios are compounded of the ratios of parallel lines, and are therefore unaltered by projection, therefore also in the ellipse QV : CD :: PV. Vp : CP?. DEFINITION. The half of a chord bisected by a diameter (as Q V by PCp) is called an ordinate to that diameter. 7. If QT, the tangent to the circle at Q, meet CP produced in T, CV.CT= CQ* or = CP?; or CP is a mean proportional between V and T. The ratios of these lines are not altered by projection; therefore also in the ellipse CV. OT= CP As a particular case of this proposition, if the tangent at P meets the axis major in T, and PN be the ordinate, CN. OT= CA. 8. The tangents to a circle at the extremities of any chord meet in the diameter (produced) which bisects the chord: and the tangent at the extremity of the diameter when terminated by the pair of tangents is bisected at its point of contact. These properties are not affected by projection, and therefore hold true in the ellipse. COR. Hence to draw tangents to an ellipse from any point T, join CT and let it cut the curve in P; draw the tangent at P by the latter part of the last Art. In CP take a point V such that CV is a third proportional to CT, CP. Through V draw an ordinate QVq parallel to the tangent at P: join TQ, Tq, then by the proposition of the last Art. these will be the tangents required. 9. The above propositions might equally bave been made to take the form, if Qu be the ordinate to CD, and the tangent at Q meet CD in t, Q * : CP* :: D% . vd : CD”, and Cv. Ct = CD. . As a particular case of the latter, if Pn be the ordinate from P to the axis minor, and the tangent at P meet the axis minor in t, Cn. Ct= CB2. Let the normal at P (the line through Pat right angles to the tangent) meet the axes and DCd the diameter conjugate to CP in G, g and F; and let the ordinates PN, Pn produced meet Dd in R, r. Then in the quadrilateral GFRN, the angles at F and N are right angles; and the quadrilateral may be inscribed in a circle. n P G N B |