A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . L>raw the normal PG and produce it to meet CD in F;then since CD is parallel to the tangent at P, .-. PPis at right angles to CD, . CD = AC. BC, (Prop. XXII. Cor.) and PF. PG = B C2 = BC . B C, (Prop XXIII.) .-. CD: PG:: AC : BC. (1) Again, SP: SG :: CA : CS, (Prop. XL) SP: SG :: CA : 5P. SP: SG. SG:: CA : GS2, .. SP. &P: SP. SP- S£ . S# :: CM2 : C^l2 - <7S2. But SP. SP- SG . SG = PG2, (Euclid
A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . L>raw the normal PG and produce it to meet CD in F;then since CD is parallel to the tangent at P, .-. PPis at right angles to CD, . CD = AC. BC, (Prop. XXII. Cor.) and PF. PG = B C2 = BC . B C, (Prop XXIII.) .-. CD: PG:: AC : BC. (1) Again, SP: SG :: CA : CS, (Prop. XL) SP: SG :: CA : 5P. SP: SG. SG:: CA : GS2, .. SP. &P: SP. SP- S£ . S# :: CM2 : C^l2 - <7S2. But SP. SP- SG . SG = PG2, (Euclid, VI. iVop. B) .-. SP. SP:PG2 :: CM2 :PC2. But from (1) CP2 : P672 :: CA2 : BC .-. SP. SP= CD2. This proposition may also be very easily deduced fromProp. XV. CONIC SECTIONS. 57 Prop. XXV. 42. The area of the ellipse is to the area of the auxiliarycircle as B G to A G. Let PN and PN be two ordinates of the ellipse neartogether. Produce NP, NP, to meet the auxiliary circle in Qand Q. Draw Pm, Qn, perpendicular to QN. Then the parallelogram PN : the parallelogram QN :: PN: QJSf, : AC. And the same will be true for all the parallelograms thatcan be similarly described in the ellipse and auxiliary circle. Hence the sum of all the parallelograms inscribed in theellipse is to the sum of all the parallelograms inscribed in thecircle as B G to A C. And this holds however the number of parallelograms beincreased. But when the number of parallelograms is increased, andthe breadth of each diminished indefinitely, the sum of theparallelograms inscribed in the ellipse will be equal to thearea of the ellipse, and the sum of those inscribed in thecircle to the area of the circle. Hence the area of the ellipse : the area of the circle :: BC : AG. 58 CONIC SECTIONS. 43. Def. If with a point 0 on the normal at P as centre,and OP as radius, a circle be described t
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
