fore a cm+2 am, and, confequently, by Reduction, ma+cxa-a, Q. E. I,

PROPOSITION XIX.

The fame Things being ftill fuppofed; as the Double of the lower Ordinate added to the exter

Mllage, Audio-Visual Serm:c::m:d;

the

ichigan.

Museum, and Greenfielquently cm+ dmtments and free loan t 56 frames, black and

"

Ohomas A. Edison) 38 f

OB: LB::

fore a cm+2 am, and, confequently, by Re

duction, m✔a+cxa-a, QE. I.

Hence from a Point B, without the Curve, and not in the Axe produced, may be drawn a Tangent. For if from the given Point B, the right Line BO be drawn perpendicular to the Axe, meeting the Curve in O; let b G be taken equal to 2 BIOI, about which as a Diameter, let a Circle be described; in that Diameter, take bi equal to BI, and from the Point i, erect the Perpendicular ik, meeting the Periphery in k; from k as a Center, with the Interval BI, describe ncm cutting i k in C; laftly, in IO, take IL equal to ci, and draw LF parallel to the Axe, and the Point F will be determined; through which if a right Line be drawn from the given Point B, terminating in the Axe produced, it will be a Tangent to the Curve.

PROPOSITION XXI.

If FP touch the Curve in F, and from any two Points M, S in that Tangent, the right Lines BM, SD be drawn parallel to the Axe, meeting the Or

dinate in B and D; then will MO: SR:: FB: FD'.

Let MOb, FB=c, SR=d, and FD=

=9,

alfo GV=VT=x; then (by Prop. 11.) :b:: FT: FM (by fimilar Triangles) y2: c2;

2

2

24

and :d :: FT *: FS:: (by fimilar Triangles) y :q2; therefore, by Equality, b:dc:q', or MO: SR::FB': FD. Q. E.D.

PROPOSITION XXII.

If from any Point in the Tangent, a right Line be drawn parallel to the Axe, meeting an Ordinate; the Rectangle of the Parameter of the Axe, into the external Part of that Line, will be equal to the Square of the Segment of the Ordinate, intercepted between that Line and the Point of Contact; that is, px MO=FB, or px RS=FD1.

DEMON

DEMONSTRATION.

2

By the last Propofition, *=y' = (by Prop. 1.) px; also 2* =y•=px; therefore pb = c2 and dp=q2; that is, px MOFB, and pxRS -FD'. Q.E. D.

PROPOSITION XXIII.

If FP touch the Parabola in F, and if from any Point S, in the Tangent, a right Line SD be drawn parallel to the Axe, cutting another right Line FC, drawn from the Point of Contact any how within the Curve; then the Curve will cut the first Line, in the fame Proportion, as the first Line cuts the 'fecond; that is, SR: RD: FD: DC.

DEMONSTRATION.

Draw PC parallel to SD, and let CP=r, RS=c, FS=d, RD=?, PS=m, FD=g, and DC=b; then crd:d+m: (by fimilar Triangles) g':g+b2, alfo (by fimilar Triangles) r:g+b c+p:g; therefore cg+2cgb+c b2

2

g

cg+ch+pg+pb, and confequently cgb

g

+cb2 = pε2 + pgh, or c h x g + b = p 8