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PROPOSITION VIII. To find the perspective of a vertical line of a given length, having given the perspective of its base and the horizontal line in which the seat of its base is situated. · Let P be the perspective of its base, and AC the line in which its seat is.
In the horizontal line HR take any point V; draw VP, and produce it to cut AC in A ; draw the vertical line AB equal to the given line ; join V, B, and from P draw PN parallel to AB, and PN will be the
auto required perspective.
For, join the point of sight S and V, and draw AP' parallel to SV, and AP will therefore
Caja lie in a horizontal plane containing the base of the given line, and it will also lie in the same plane with SV and AV. Join SP, and produce it to cut AP' in P', which is the base of the given line, since it must lie in the line SP, and also in the plane P'AC. From P' draw a vertical line P'N' to meet SN produced in N', Then P'N':PN = SP' :SP = VA:VP=AB:PN; and therefore AB = P'N', and P'N' is the given line, and therefore PN is its perspective.
DEFINITIONS. 1. If a straight line and a point be given in position, the locus of a point which is equally distant from them, is a curve called a parabola. · 2. The given line is named the directrix, and the given point the focus.
3. The vertex of the parabola is the middle of the perpen
dicular, which falls upon the directrix from the focus : and the axis, or principal diameter, is that part of the perpendicular produced indefinitely which falls within the curve. !
4. Any straight line, drawn from a point in the curve, parallel to the axis, and in the same direction, is called a diameter, and the point in the curve its vertex.
45. An ordinate to a diameter is a straight line, terminated on both sides by the curve, and bisected by that diameter: the part of the diameter which it cuts off, is called an absciss.
6. The parameter of a diameter is four times the distance of its vertex from the directrix.
7. A tangent is a straight line, which meets the curve in one point only, and every where else falls without it. I
8. A subtangent is that part of a diameter which is inters cepted by a tangent, and an ordinate to that diameter, from the point of contact. .
1 PROPOSITION I. The distance of a point from the focus is greater than its distance from the directrix, if the point be without the parabola, but less if it be within.
Let GYN be a parabola, whose directrix is AB,, vertet V, and focus F; and let P AQ D O I ALBI be a point without the curve, that is, on the same side of
T ub the curve with the directrix : then, if PF be joined, and PQ be drawn perpendicular to AB, PF will be greater than PQ.
For, as PF necessarily cuts the curve, let G be the point of section, GD perpendicular to AB, and P, D, joined. Then, because ĜF = GD (I. Def. 1), PF = PG + GD. Hence PF is greater than PD (Pl. Ge. I. 20), and consequently still greater than PQ (Pl. Ge. I. 19).** '
Again, let O be a point within the curve. The perpendicular OD, upon AB, necessarily cuts the curve ; let G be the point of section, and let GF, FO, be joined. Then OD is equal to the sum of OG, GF, and therefore greater than OF (Pl. Ge. 1. 20), uds added to the realite ws
Cor. 1.-A point is without, in, or within, the curve, ac
cording as its distance from the focus is greater, equal,
or less, than its distance from the directrix. Cor. 2.- perpendicular to the axis, at its vertex, is a
tangent to the parabola. For HI = V0 = VF, and VF <FH, therefore HIZ FH; and hence (Cor. 1) H is without the curve. In a similar manner, every other point of VH may be shown to be without the curve; therefore VH is a tangent.
PROPOSITION II. Every straight line perpendicular to the directrix meets the parabola, and every diameter falls wholly within it.
Let DG be perpendicular to the directrix, at any point of it, then shall DG meet the curve (figure to next proposition).
For, DF being joined, let the angle DFG be made equal to GDF, and let FG meet DG (Pl. Ge. I. 29, Cor.), which is parallel to FC, in G. The triangle DGF, having the angles at D and F equal, will also have the sides GD, GF, equal ; and therefore G is a point in the curve (I. 1, Cor.1).
Again, the diameter GE falls wholly within the curve (figure to last proposition).
For, if any point o be assumed in it, it is evident that OD is greater than OF. Cor.Hence, the two legs of the curve continually diverge from the axis.
PROPOSITION III. A straight line bisecting the angle formed by two lines drawn from the same point in the curve, the one to the focus, and the other perpendicular to the directrix, is a tangent to the parabola in that point.
The straight line GE, bi- A secting the angle DGF, is a tangent to the parabola in G.
For, let H be any other point in GE, from which let there be drawn HF and HD, also HA perpendicular toy l l 1,830 AB. Then, because GE bi
sects the vertical angle of the isosceles triangle GDF, it will also bisect the base DF at right angles (Pl. Ge. I. 4). Hence the triangles HED, HEF, are equal in every respect. Thus, HF is equal to HD, and therefore greater than HA. Consequently, the point H is without the curve (I.1, Cor. 1). COR. 1.—Hence the method of drawing a tangent from
any point in the curve. COR. 2.-If a straight line be drawn from the focus, to
any point in the directrix, the perpendicular which bisects it will touch the parabola; also, every perpendicular to it, which cuts the curve, will be nearer to
the focus than to the point in the directrix. CoR. 3.—The parabola is concave towards the axis. .
For (Cor. 2) the tangent lies between the curve and the directrix; the curve is therefore convex towards the directrix, or concave towards the axis.
PROPOSITION IV. Every straight line, drawn through the focus of a parabola, except the axis, meets the curve in two points.
Let FQ be a line passing through the focus, FD perpendicular to it, DA, DE, each equal to DF; AG, EH, parallel to CF, and intersected by A__ D_C_E _ B FQ, in G, H; and let the points AF be joined. Then, because DA is equal to DF, the angles DAF, DFA, are equal ; and these being taken from the right angles DAG, DFG, the remainders, the PLV angles GAF, GFA, are equal. Whence the sides GA, GF, are also equal, and therefore G a point in the curve (I. 1, Cor. 1). In the same manner, it may be shown that H is a point in the curve. COR.-A straight line, making an indefinitely small angle
with the axis, being produced, meets the curve; hence
the rate of divergency must be very small. For let MN be the given line. Through the focus draw GH parallel to it, then it will cut the curve in two points. Through one of these points, as G, draw the tangent GP, which must be inclined to GH, and therefore to MN, and will meet MN, if produced, in some point P; and therefore MN must cut the curve.
PROPOSITION V. If, from any point in a parabola, a straight line be drawn, not parallel to a diameter, nor bisecting the angle formed by two lines drawn from that point, the one to the focus, and the other perpendicular to the directrix, it will meet the curve in one other point, and not in more than one.
Let H be a point in the curve, and HG a line not parallel to CF, nor bisecting the angle EHF; then HG shall meet the curve in another point.
For, let FL be perpendicular to GH, and it will meet AB (Pl. Ge. I. 29, Cor), since GH is not parallel to CF; also, the point D, in which a MD E C' BT it meets ĀB, will be different from E; since, if D, E, were the same, it would follow that HG bisects the angle EHF, contrary to the hypothesis (1.3).
Let DA = DE, and AG parallel to EH, meet HG in Va G, G will be a point in the I curve. For about the centre H, with the radius HE or HF, let a circle be described intersecting FD in K, and let another circle be described through the three points A, K, F: Then, because AB touches the circle EKF in E (Pl. Ge. III. 16), the rectangle FD.DK =DE(Pl. Ge. III. 36)= DA?; hence DA is a tangent to the circle AKF (Pl. Ge. III. 37), and therefore AĞ passes through its centre ; but HL, which bisects the cord FK, at right angles, also passes through its centre (Pl. Ge. III. 3); consequently G is the centre of the circle AKF. Whence GA = GF, and G a point in the parabola (I. 1, Cor. 1). If GH were to meet the curve in another point, that point would be the centre of a circle passing through F, K, and touching the line AB in a point different from A or E, which is impossible ; for if it touch AB in another point, as M, then FD • DKDM;