M

evidently a plane SA'B', the intersection AB of which with the primitive is a straight line. When the line, as AD, is directed to S, its projection is the point A. COR. 2. The perspective of a straight line parallel to the primitive is parallel to the original.

COR. 3.-The perspective of a

A

B

R A

plane figure, whose plane passes through the point of sight, is a straight line.

For the projecting surfaces of its boundaries evidently lie in one plane.

PROPOSITION V.

The projection of a circle inclined to the primitive is an ellipse, unless it be a subcontrary section of the projecting conical surface.

The perspective ACBD of the inclined circle A'C'B'D' is an ellipse, unless it be a subcontrary section. For ACBD is a section of the projecting conical surface SA'B', and it is therefore an ellipse, unless it be a subcontrary section, in which case it is a circle (Conic Sections IV. 3).

PROPOSITION VI.

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M

B

B

с

A

O

R

The perspectives of parallel lines converge towards their vanishing point.

Let A'B' and C'D' be two parallel lines, and V their vanishing point, their projections AB and CD converge towards V.

For since SV and A'B' are parallel, the

D

B

A

projecting plane of A'B' will pass through SV, and therefore through V; and hence the perspective of A'B' passes

through V. Similarly it is shown that the perspective of C'D' passes through V.

COR. 1.-The perspectives of lines perpendicular to the primitive converge towards its centre.

For the centre is their vanishing point.

COR. 2.-The perspectives of lines parallel to the primitive are parallel.

For the vanishing point in this case is infinitely distant. It is evident also from Cor. 2 to Prop. IV.

COR. 3.-The perspectives of parallel horizontal lines converge towards a point in the horizontal line.

COR. 4.-The perspectives of parallel lines that are also parallel to the vertical plane, converge towards a point 'in the vertical line.

COR 5.-The perspectives of horizontal lines, whose inclination to the primitive is half a right angle, converge towards the point of distance.

PROPOSITION VI.

Given the perspective of a straight line and its vanishing point, to find a point in it, which is the perspective of a point in the original line, that divides it in a given ratio.

Let AB be the given perspective, V the vanishing point, and P, Q, two lines in the given ratio.

From V draw any line VS in the plane of the primitive, and take any point S in it; M join S, A, and S, B, and produce these lines. Through A draw AH parallel to SV, and divide AH in G, so that AG: GHP: Q; join S, G, and the point of intersection C is the point required.

A

A Α'

D

E

G

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Β'

For through C draw DE parallel to SV; then, from similar triangles AC: AV=DC: SV, and AC SV AV DC. Also CB: BV = CE: SV, and CB SV = BV CE. Also DC: CE=P: Q, and AV: BV AV: BV; therefore (Pl. Ge. VI. 23, Cor. 1) AV DC: BV CEP AV: QBV. But AC SV:

=

CB SV AV DC: BV CE; and therefore AC SV: CB SVP AV: Q BV; or AC: CBP AV: Q·BV. Let X and Y be the sides of two squares respectively equal to the given rectangles PAV and Q BV, and let Z be a third proportional to X and Y. Then AC : CB = X2 : Y2 = X: Z. Hence divide AB in C in the ratio of X to Z.

The position of C, thus determined, is independent of the length and direction of SV; and therefore SV may be drawn of any length and in any plane. Hence let S be the point of sight on the farther side of the primitive MR, and SV will then be parallel to the original line, the extremities of which will lie in SA and SB produced. Let A'B' be the original line, then DE and AH are parallel to A'B' or SV, and these lines, with the given line AB, lie in one plane. Divide A'B' in C', so that A'C': C'B' = P:Q; then since DC: CEP: Q, if SC and CC' be drawn, they will lie in one straight line, and therefore C is the perspective of C'.

PROPOSITION VII.

Given the perspective of a straight line which is divided into segments having a given ratio and its vanishing point; to find those segments of its perspective, that are respectively the perspectives of the segments of the original line.

Let AB be the given perspective, and V the vanishing point, and X, Y, Z, three lines in the ratio of three segments into which the original line is divided.

Draw VS parallel to the ground line PR, and take in it any convenient point S; draw SA, SB, and produce them to E and F in PR. Divide M

EF in G and H similarly to the original line; join S, G, and S, H; then AC, CD, DB, are the required seg

ments.

PE

D B

Ꮐ H

R F

For (Pr. IV. 6), since SV and EF are parallel, and EF is similarly divided to the original line; therefore C, D, are the perspectives of the points of section, and consequently AC, CD, DH, are those of the segments of the original line.

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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 required perspective.

For, join the point of sight S and V, and draw AP' parallel to SV, and AP' will therefore lie in a horizontal plane containing the base of the given line, and it will also lie in the

H

B

N

R

A

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: VPAB: PN; and therefore AB P'N', and P'N' is the given line, and therefore PN is its perspective.

1

CONIC SECTIONS.

FIRST BOOK.

PARABOLA.

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.

5. An ordinate to a diameter is a straight line, terminated on b 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.

8. A subtangent is that part of a diameter which is intercepted by a tangent, and an ordinate to that diameter, from the point of contact.

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.

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t

1

Let GVN be a parabola, whose directrix is AB, vertes V, and focus F; and let PA be a point without the curve, that is, on the same side of the curve with the directrix : then, if PF be joined, and PQ be drawn perpendicular to AB, PF will be greater than PQ.

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E

N

For, as PF necessarily cuts the curve, let G be the point of section, GD perpendicular to AB, and P, D, joined. Then, because GF 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).

T

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

H