therefore

p(x-x)=MG2- CE2;

or, since x'x CD, and MN is bisected in G,

P.CD=MD.DN.

PROPOSITION XXXVI.

134. If from any point in a parabola a tangent and a chord be drawn; the segments of any diameter intercepted between the tangent, the curve line, and the chord, will be to one another in the same proportion as the segments of the chord intercepted between the diameter and the curve.

(M

Let PM be the tangent, PQ the chord, and MK part of any diameter intersecting the chord in K, the tangent in м, and the curve in N: then

MN NK:: PK: KQ.

Let TR be part of a diameter bisecting PQ in R, cutting the curve in v and the tangent in T, and draw NE parallel to PQ; thus both RQ and NE are ordinates to TR.

ני

Put p for the parameter of the diameter TR, z for VE, and x' for v R, or (Art. 104.) Tv: then (Art. 72.)

√px = EN or RK, and √pr' = PR or RQ;

therefore, adding and subtracting,

√p (√x+ √x') = PK, and √p ( √ x' — √x) = KQ. Hence

PK: KQ :: √x+ √x' : √x' — √x.

But (Art. 64. a.)

PT2 PM2: TV: MN:

again, by similarity of triangles and Euc. 22. vI.

while NK (ER) = x-x

= (√x+ √x) (√x' — √x):

therefore MN: NK:: √x+ √x′: √x −√x, which, as above, is also the ratio of PK to KQ: thus, finally,

MN NK:: PK: KQ.

PROPOSITION XXXVII.

135. If to any diameter of a parabola an ordinate be drawn, also from the vertex of the diameter, and from the intersection of the ordinate with the curve, ordinates be drawn to any other diameter, the rectangle contained by the parameter of the latter diameter and the abscissa of the former will be equal to the square of the difference between the two last ordinates.

Let PK, AE be parts of two diameters, and KM an ordinate to the former at any point K in it; let also MB, PC be ordinates to AE; then p being the parameter of AE,

p.PK = (PC-MB)2.

Produce MK to N, draw the lines N NH, KD parallel to PC, and MQ

A

M

B

C

P

D

K

H

parallel to AD, cutting KD in G; then (Art. 72.)

E

p.AB = MB2, p. ACPC2, and p. AH = NH2.

Subtracting the first of these equations from the second,

But MN being bisected in K, we have 2BD=BH, also
NH+MB=2KD or 2PC,

and

NH-MB (=NQ or 2 KG) = 2 (PC-MB);

therefore, substituting in (6) and dividing by 2, p.BD 2PC (PC-MB).

I

(a)

(b)

From this last equation subtracting (a) we have

30

or

p.CD (PC-MB) (PC-MB),

p.PK (PC-MB)2.

PROPOSITION XXXVIII.

136. Having a focus, with the lengths and positions of three radii vectores, and the angles contained between them; to find the axis of a conic section passing through the extremities of the radii.

and, since the three first terms are known, the point T is determined.

In-like manner, from the proportion

FP"-FP: FP" :: P'T'-P'T': P'T',

the point T is determined: and thus the position of the directrix is found.

From the points P, P', P" let fall PQ, P'Q', P"Q" perpendicularly on the directrix; and, through F, draw an indefinite line also perpendicular to the directrix, cutting it in D: this will be the direction of the transverse axis; and imagine E and c to be the extremities of that axis. Then, since (Art. 88.) FP: PQ::FE: ED, and FP: PQ :: FC: CD,

and

FP+PQ FP:: FD: FE,

PQ-FP FP:: FD: FC:

thus, both FE and FC are found; and, consequently, the major axis is obtained.

If FP is less than PQ the curve is an ellipse, if equal a parabola, and if greater an hyperbola.

PROPOSITION XXXIX.

137. To trisect an angle, or a circular arc, by means of an hyperbola.

Let FDA be a given circular arc which is to be trisected: draw the chord FA; and, having bisected it in c, draw an indefinite line cz perpendicular to FA. Make CV equal to FC; then if, with F as a focus, v one extremity of the transverse axis, and cz as a directrix, there be described (Art. 89. Scholium) an hyperbolic curve VPQ, it will cut the

D

P

S

circular arc in P; and the arc FP, will be one third of the given arc FDA.

For (Art. 88.) having joined F, P, and drawn PN perpendicular to CZ,

=

FV VCFP: PN;

But, since vc FC, FV = 2vc; therefore the straight line FP2PN.

Now, if PN be produced to cut the arc FDA in P,' PP' will be bisected in N, and the arc PDP', or FDA, in D: thus the arcs FP, PP', and P'A will be equal to one another, or the arc FDA is trisected in P.

If s represent the centre of the hyperbola, we shall have (Art. 88.)

SF SV SV: SC;

whence SF-sv:sv:: sv-sc: SC, or FV SV:: VC: Sc. But Fv=2vc; therefore sv=2sc and vc=sc:

Hence SV=FV and 2sv=SF, or twice the semi-transverse axis is equal to the excentricity.

SECTION III.

ANALYTICAL GEOMETRY OF THREE DIMENSIONS.

CHAPTER I.

THE EQUATIONS OF LINES AND PLANES IN SPACE.

138. THE position of a point in space is determined when its distances from any three given points are known; but, in order that the varying positions of the point may be expressed algebraically, it is found convenient to imagine the positions of three lines concurring in one point to be given, and to refer the point in space to these lines or to planes conceived to pass through them. The three lines are designated Axes of the Co-ordinates, and the planes passing through them are called Co-ordinate Planes: these may form any angles, right or oblique, with one another, and may be considered as three faces of a parallelepiped, about one of its solid angles; while the co-ordinate axes may be considered as three concurring edges of the figure. The position of a line and of a plane in space is also indicated by the distances of points in such line or plane from the three co-ordinate planes or axes.

When the situation of one of the co-ordinate planes can be assigned at pleasure, it is convenient to imagine that it coincides with the plane of the paper, the two other planes then making with the paper, and with each other, right or oblique angles. Thus, o being the common intersection of the three axes Ox, OY, oz or the origin of the co-ordinates, ox, OY produced if necessary towards x' and y' may be in the plane of the paper, and the planes zox, ZOY inclined to it at any angles, either rising above it, or, if necessary, produced so

m

Y

Z

X/

n