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DEFINITIONS.I A TRArcia line AB, andi a potiat with Fig. i. out it, arealven in position. On the plane of ABC, there is placed a rule DEF, havin iis fidem applied io AB, and iis other side EFon the fame fide of AB with the potnt C. Astring FG Ccis ahen equat in tengthao EF andone en of this stringaeing fixe in F, and theother in C, a partis it FG is by means os a i
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III. An theloint C is named the focus of the parabola. IV. A straight line perpendicular to the directriX, is a med a diameter and the oint Where a diameter meet the parabola, is named the verte of that diameter and the diameter Whicli passe through the focus, is a med thearis of the parabola; and the vertex of the axis is amed the principat verter.
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V. When a traight in terminate both waysi a parabola, is bisected by a diameter,it is satyto e ordinateis applied to that diameter or it is named, Simply, an ordinate to that diameter. VI A straight line quadruple of that segmentis a diameter hicli is intercepte be- tween iis vertex and the directrix, is amen the latus reclum, orthoestiriameἱCT, O that dia
VII. A straight line meeting a parabola onbi one potnt, and whicli, hen produced both ways, fatis ithout the parabola, is sal totoue the parabola in that potnt. PROPOSITIO I. THEoREΜ.
straight line drawn perpendiculario the directri from any oin of parabola, is e lunt to the traight line dram to the focus rom that
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Fig. i. et Gie a potnt in the parabola, and Ea strato line perpendicular to the directrix AB dra GC to the focus C, an leti beequa to the tength of that fide of the uter Whicli is o the fame fide of AB with the focus C: heresere EF is quai to the tengst of the string FGC: Aeaway the Common PartFG, and the remainde EG will e qua toste remainder GC. COROLLARY Hence that segment of the axis hicli is intercepte between the focus and the directrix, is bisected in the vertex of the axis Thus CB is bisected in II. PROP. II. THEOR.
I the distance of any oin sto the
focus os a parabola be equa to the perpendicular dras seo the Same oin to tho directriX, that pollit is in the parabola.
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Let there e a parabola the directrix o Fig. Lwhicli is AB, and the focus C; and et Dae n. 2. potnt, the distance of whicli frona the focus is the strat glit line DC; fronam dra D perpendicular to the directrix. . I DC ie qualto DE, the poliat reis in the parabola.
From the centre C, at the distanc CD, describe a Circle, meetin the axis in the poliat F leti e the vertex of the axis, and otii CE rhen, beCause any two sides of a trian gleare together greater 2O. I. Elements of Euclid)than the thir fide, CD, D are together greater than Ec much more, then, are theytogether I9. I. Elem. greater than B: ut CD is equat o DE, as also CH Cor. I. I. to HB oherefore CD, that is CF is greater than sin the parabola, there re, With respectuo iis vertex II, is illiin the circle GDE: of Consequence, it must meet the Circi Some-Where, sincerit maybe extended def. l. to a distanc seo the focus C rhich shali exceedan given distance No i meet the circle
in the oin D; for i this is no true, it
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must meet the circle in ome ther mini.
Let, then, the potnti, hic is o the sameside of the axis illi the oin D e thalother potnt then, having oine CL dra LM perpendicular to the directrix, and LN parallel to the fame and Iet L mee DE in Ν; and because thelointi is in the parabola, CL is l. 1. equat o LM and accordindito the hypothesis, Cois equat o DE; and, be-ing the etair of the circle, Lis equa to miliere re LM that is, NE, is qua to DE;whicli is impossible the parabola, theresere, meet not the circle in the potnti, o any where but in D: thereserem is a potnt in theparabola. PROP. III. THEOR.
s Cus Cet the parabola an astriaght line draKn from an potnt Within a parabola to the focus, istes than the perpendicula draWnfroni that oin to the directriY.
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Let there e a parabola, the directricos Fic. i. whichris AB, and the oint C the focus; anystraight line drawm through C meet that pa
First, is CB, a straight line drawn through the focus, be perpendicular to the directrix, thepoin H, bisectini cor. I. I. the segment, interceptedietween the focus and the directrix, is in the 2. I. parabola: ut is an otherstraight line P e drawn through the focus, bisec the angle BCP by the straight line M, and et M meet the directrix in M, and dra MN parallel to the axis C then, be-Caus the angies PCM, MN are together Ies stan two right angies for ach of them istes than ne right angie, the traight ines
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CP, MN meet ench ther; et the meet in theso in O then the angi OCM is qua tolli angle MO; foris ach of the two is qualto 29. 1. Elem. BCM; os consequence Mis 6. I. Elem. equa to in therei ore thepoliat O scin 2. l. the parabola.
To proceed o demonstrate themther part of
the proposition First, et there e an potiat Κ with in the parabola, that is, et itiei thesam fide Lit,ith the focus C, an Urais KLat right an gles to the directrix; dra likewi se
Και the focus minis es than L. I et C meet the parabola in , and let there bedra wn to the directrix the traight line M parallel o L, and et L bebo ined. Since, then, heloint O is in the parabola, Gis equalto MM; ut OMis I9. l. Elem. Ies than
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more, then, are the greater than QR. Cis, there re greater than R. COR Henc it is evident, ibat an potiat is within Ormittiout a Parabola, CCOrding as the distance of that potnt seo the focus is es orgreater than a perpendicula drawn from that fame potnt to the directrix. PROP. IV. THEOR.
A perpendicular to the directrix meet the parabola ni in onepotiat; and when produced doWn-wards, it alis illiin the parabola.