9

CONIC SECTIONS.

Prop. XXXIX. Pros. Fig. til.

Two right lines being given, one of which is bisected by theother at right angles ; to describe a parabola, in which the rightline bisected shall be an ordinate, and the other line the axis.

Let AC, B/> be the two given lines, one of wllich B h, whichis perpendicular to AC, is bisected in C. Find a third propor-tional to AC, CB; and produce CA to D, so that Al) may be afourth part of that third proportional ; take AS = AID, and drawDX perpendicular to DC. Let a ruler, the sides of which HE,LL, are perpendicular to each other, be placed in the plane CDX,so that the side EL may be applied to DX ; and take a stringfqnal in length to the side HE, one extremity of which must liefixed at 11, and the other at S ; and let part of the siring be ap-plied by means of a pin P, to the side of the ruler 11E; andwhilst the side EL moves along DX, let the string be stretchedby the pin, and constantly applied to^HE. Then, because thewhole length of the string UPS is equal to HE, the part SP willalways be equal to PE ; therefore the point P will describe a pa-rabola, by Prop. 12, of which AC is the axis, S the focus, andDX the "di rectrix ; and BC b will be an ordinate, because it isperpendicular to the axis, and CB is a mean proportional betweenthe absciss AC and -4AS, or the latus rectum.

Prop. XL. Prob. Fig. 40, 41.

To draw a tangent to a conic section from any given point with-out it, which is not the centre of the hyperbola.

if the given point H be in the directrix; draw IIS to the focuswhich is nearest to the directrix ; draw SP perpendicular to SlI,meeting the curve in P, and join IIP, which will touch the conicsection in P, Cor. 1, Prop. 20.

If the. given point be in any other situation, as at 1.; join LS,and draw LX perpendicular to the directrix. Take LD to LXin the constant ratio, Prop. 11, and from the centre L, at the dis-tance LD, describe a circle D\l</. From S draw SQ a tangentto the circle, meeting the directrix in II. Join 1.(4, and drawSP parallel to it, or perpendicular to S.1L Join IIL, and produceit to meet SP in P, winch is in the conic section, and the line IIPtouches the curve at P. For the triangles I1QL, IISP, are simi-lar, as also LIIX, PIIE, therefore SP:TH : : QL : LII, and PII:PE: : LII: LX ; therefore SP: PE : : QL: LX, that is, in theconstant ratio ; therefore P is a point in the curve, and becausePSH is a right angle, PII "is a tanjynt. Cor. 1. Prop. 20.

Cor. Because two lines SQ, S(/, may be drawn from the pointS to touch the circle; two tangents LP, L p, may be drawn fromL to the conic section.

Prop. XLL

If a circle touches a conic section, and cuts off from the diameter, which passes through the point of contact, a segment equal bits parameter, the conic sedtion is of the same curvature with thcircle at the point of contact.

First let a tangent DM be drawn to any point D in the parabola, (fig. 62,) draw also the diameter DF, and the perpeudiculaDL : through any point (I in the curve, near to D, let the circVDQO be described to touch DM in D, and meet DF in P, jobl’Q, DQ, and draw QN parallel to MD, meeting DF in XThen because the angle DPQ = MDQ = DQN, the triangleDN Q, PQD, having a common angle at D are equiangularhence PD : DQ.: : DQ : DN, andPD X DN = DQ 3 : also PD 3PQ 3 : : DQ 3 : QN 3 , therefore PD 3 : PQ 3 : : PDx DN : Px DNwhere P = parameter of DF. Now, it is evident, that the nearethe point Q is to the point D, the nearer will the cireumfereucjof the circle be to a coincidence with the curve at that pointand therefore, as no portion of these curves, however small, caibe the same, the circumference of the circle will have approacheithe nearest possible to a coincidence with the curve at D, wheithe point Q falls upon it; in which ease, the last analogy becomePD‘ 3 : PD 3 : : PD : P, therefore PD = P, the parameter of DFtherefore the proposition in the case of the parabola is manifest.

Next let DM he a tangent at any point m the ellipse or hyperbola, (fig. 63,) DF, EG, conjugate diameters, and DUO a perpendicular to the two parallels DM, EG.

I hrough any point Q in the curve, near to the point D, let thcircle DQO be described, to touch DM in D, and meet DF ii

I. Let PQ, QD, be joined, and QN drawn parallel to DM, h

VOL, It,—Ko, 50.

meet DF in N. The triangles DNQ, PQD, being similar, DN{DQ :: DQ : DP, hence DN : DP : : DN 3 : DQ 3 ,

Or DN : DP : : QN 3 : PQ 3 .

But DF : P :: FN X ND : QN 3 , Prop. 32. Cor. I.

Therefore DF x DN : P X DP: FN x DN : PQA

and DF: I*'N : : P x DP : PQ*; this analogy, when Q coin-cides, with D, becomes DF: DF: : P X DP: DP J , in which caseP = PD, where P = parameter of DF as before.

Cor. If, from any point D in an ellipse or hyperbola, a diame-ter DF be drawn, "and a perpendicular Dll to its conjugate EG,the radius of curvature at the point D is a third proportional to theperpendicular Dll and the semi-conjugate diameter EC.

For since Dll: DC : : DP : DO,and DC : EC : : EG : P or DP ;

therefore DII: EC : : EG : DO :: EC : DII, the radius ofcurvature.

Prop. XLII. Fig. 64.

If any ordinate and absciss of a parabola be completed into aparallelogram ; the area of the parabola, included between the oi-dinute and the curve, is to the parallelogram as 2 to 3.

Let AN be the absciss, and PQ the ordinate ; let the parallelo-gram PQ.CB be completed, and let AN be divided into indefi-nitely small equal parts, of which ND is one ; through D drawId 1 parallel to PQ, cutting the parabola in 1*' and G, and throughF draw KF parallel to N A ; take K1I = KP, and draw RL pa-rallel to KE. By Prop. 25, 1IF x HG: PN 3 : : IIP: NA, butbecause I)N is indefinitely small, PQ or 2PN may be taken forI1G ; and PK = 11F, also NA = PR, therefore21’K x PX 3 : PN:: HP : PB, and 2PK, or PR : PN :: HP ; PB; now the parallelo-grams RB, PD, are equiangular ; therefore they are equal, andthe parallelogram PD : KB : : 2 : 1 ; and the sum of all the paral-lelograms in AI’N is to the sum of all those in APB in the sameratio of 2 to 1 ; but the sum of all the parallelograms in A PN ap-proaches indefinitely near to the curvdineal area AFPN, whentheir breadths are continually diminished ; and in like manner thesum of all the parallelograms in APB approaches to the curvilineararea AFPB ; therefore area AFPN : area AFPB : : 2 : 1, and thearea PAQ is to the parallelogram PBCQ as 2 to 3.

Prop. XLII I. Fig. 65, <36:

If two ellipses or two hyperbolas have a common axis, and anordinate be drawn through the same point in the axis to each ofthe curves ; the areas included between llie common absciss, theordinates, and the two curves, also the whole areas of the ellipseswill be to each other as the conjugate axes.

Let AP, AQ, be two ellipses or two hyperbolas; take any ab-sciss AN, which is not greater than half the axis of the ellipse,and draw the ordinates N P, NQ. Let the absciss AN be dividedinto any number of equal parts AE, EF, FG, GN, &e.; drawthe ordinates ER1, FSK, GTL, and complete the parallelogramsAR, Al, ES, EK, &c. also draw If, K k, LI parallel to AN.Then it is evident that the difference between the circumscribedparallelograms AJ, EK, FL, GQ, and the inscribed parallelogramsE i, F k, G l is equal to GQ ; and if parallelograms be inscribedin tin; same manner in the figure APN, the difference beta ecuthese and tiie circumscribed parallelograms would he equal to GP,therefore (he difference between each series of parallelograms, andthe areas AQN, APN will be less than the parallelograms GQ,

GP, and because'GP : GQ: : NP : NQ, and each parallelogramin the figure APN is to the corresponding parallelogram in thefigure AQN in the same ratio, the sum of all those in APN is tothe sum of all those in AQN as NP is to NQ, which is the sameratio with that of the conjugate axes. Conceive the breadths ofthe parallelograms to be now diminished, and their number in-creased ad infinitum, and the parallelograms APN, AQN will ireultimately equal to the areas APN, AQN, for the parallelograms

GQ, GP will now vanish, therefore the areas APN, AQN are toeach oilier as their conjugate axes ; and if the sections be ellipses,their whole areas are to each other in the same ratio.

Cor. 1. If a circle be described about an ellipse, the area of thecircle is to be the area of the ellipse as the transverse axis is to.the conjugate.

Cor. 2. The area of an ellipse is equal to that'of a circle whosediameter is a mean proportional between the two axes.

Cor. 3. The areas ot two ellipses ate to each other as the rec-tangles under their axes.

D CONIFER'S,