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lelogram. Hence NE IH, and consequently FIG is projected into an equal circle KNL, whose centre is E. COR. The radius of the projection is the cosine of the parallel's distance from the primitive, or the sine of its distance from the pole of the primitive.

PROPOSITION III.

An inclined circle is projected into an ellipse, whose transverse axis is the diameter of the circle.

Case 1. Let ELF be a great circle, inclined to the primitive EBF, and EF their line of common section.

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From the

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centre C, and any other point K, in EF, let the perpendiculars CB, KI, be raised in the plane of the primitive, and CL, KN, a in the plane of the great circle, meeting the circumference in L2 N. Let LG, ND, be perpendicular to CB, KI; then G, D, are the projections of L, N. And because the triangles LCG, NKD, are equiangular, CL2: CG2 = NK2: DK2, or EC2 : CG2 EKF: DK2; therefore the points G, D, are in the curve of an ellipse, of which EF is the transverse, and CG the semi-conjugate axis. COR. 1.-In a projected great circle, the semi-conjugate axis is the cosine of the great circle's inclination to the primitive.

COR. 2.-Perpendiculars to the transverse axis intercept corresponding arcs of the projection, and the primitive. COR. 3. The eccentricity of the projection is the sine of the great circle's inclination to the primitive. For LG2LC2 - CG2 EC2

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-CG2; LG is therefore equal to the eccentricity, and it is also Sin LCG.

Case 2. Let AQB be a small circle, inclined to the primitive, and let the great circle LBM, perpendicular to both, intersect them in the lines AB, LM. From the centre O, and any other point N, in the diameter re

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AB, let the perpendiculars TOP, NQ, be drawn in the plane of the small circle, to meet its circumference in T, P, Q. Also, from the points A, N, O, B, let AG, NI, OC, BH, be drawn perpendicular to LM, and from P, Q, T, PE, QD, TF, perpendicular to the primitive; then G, I, C, H, E, D, F, are the projections of these points. Because OP is perpendicular to LBM, and OC, PE, being perpendicular to the primitive, are in the same plane, the plane COPE is perpendicular to LBM. But the primitive is perpendicular to LBM. Therefore the line of common section EC is perpendicular to LBM, and to LM. Hence CP is a parallelogram, and EC = OP. In like manner, FC, DI, are proved perpendicular to LM, and equal to OT, NQ. Thus, ECF is a straight line, and equal to the diameter PT or AB. Let QR, DK, be parallel to AB, LM; then RO = NQ = DI= KC, and PR RT EK · KF. But AO: CG = NO: CI; therefore AO2: CG2QR2: DK2, or EC2 : CG2 EK KF: DK2.

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COR. 1.-The transverse axis is to the conjugate, as radius to the cosine of the circle's inclination to the primitive.

COR. 2.-Half the transverse axis is the cosine of half the sum of the greatest and least distances of the small circle from the primitive.

COR. 3.-The extremities of the conjugate axis are in the line of measures distant from the centre of the primitive, by the cosines of the greatest and least distances of the small circle from the primitive.

COR. 4.-If, from the extremities of the conjugate axis of any elliptical projection, perpendiculars be raised (in the same direction, if the circle do not intersect the primitive, but, if otherwise, in opposite directions), they will intercept an arc of the primitive, whose cord is equal to the circle's diameter.

PROPOSITION IV.

The projected poles of an inclined circle are, in its line of measures, distant from the centre of the primitive, the sine of the circle's inclination to the primitive.

Let ABCD be a great circle, perpendicular, both to the

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primitive and the inclined circle, and intersecting them in the diameters AC, MN. Then ABCD passes through the poles of the inclined circle; let these be P, Q, and let Pp, Qq, be per- M pendicular to AC; p, q, are the projected poles, and it is evident that po Sin BP, or sin MA, the inclination.

COR. 1.The centre of the primitive, the centre of projection, the projected poles,

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and the extremities of the conjugate axis, are all in one and the same straight line.

COR. 2.-As radius is to the sine of a small circle's inclination to the primitive, so is the cosine of its distance from its own pole to the distance of the centre of its projection from the centre of the primitive.

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EXERCISES.

1. To describe the projection of a small circle parallel to the primitive, its distance from the pole of the primitive being given.

2. To project an inclined circle, whose distance from its pole and inclination to the primitive, are given.

3. To find the poles of a given projection.

4. To measure any part of a given projection.

15. In any inclined circle, two diameters, which cut each other at right angles, are projected into conjugate diameters of the ellipse.

6. In every elliptical projection, half the transverse axis is to the eccentricity, as radius to the sine of the circle's inclination to the primitive; and half the conjugate axis is to the eccentricity, as radius to the tangent of the same.

7. Through two given points, in the plane of the primitive, to describe the projection of a great circle.

8. Given the distance of a circle from its pole, and the projection of that pole, to describe the projection of the circle. 1

19. Through a given point, in the plane of the primitive, to describe the projection of a great circle, having a given inclination to the primitive.

THIRD BOOK.

PERPENDICULAR PROJECTION.

DEFINITIONS.

1. In Perpendicular Projection, the position of the point of sight is at an indefinitely great distance in the axis of the primitive.

2. The perpendicular projection of any point is called the seat of that point.

3. Two planes at right angles are sometimes assumed for planes of projection, and they are then called co-ordinate planes.

4. When the two co-ordinate planes are a horizontal and a vertical plane, the projections on them are respectively called the horizontal and vertical projection.

5. In the case of an object of a regular figure, such as a house, the horizontal projection is called the plan; and the vertical, the elevation. When the projection is that of a section, it is called a section.

COR. 1.-A projecting line in perpendicular projection is a line drawn from the original point perpendicular to the primitive (Def. 1).

COR. 2.-If the seats of a point on any two planes, not parallel, be given, the position of the point in space may be found; for it will be the point of intersection of the two projecting lines drawn from the two seats. COR. 3.-If the projections, on two planes, of all the remarkable points and lines of an object, be given, the figure and dimensions of the object may be determined. This method of projection is employed in the construction of plans and sections of houses, the plans of philosophical apparatus, and of various objects in engineering and carpentry.

PROPOSITION I.

The projection of a line inclined to the primitive is less than the line in the ratio of radius to the cosine of its inclination.

Let A'B' be a line inclined to the primitive MR, and AB its projection, and A'C parallel to AB, then A'B': AB = Radius : cosine B'A'C.

For, the projecting lines A'A, and B'B, being perpendicular to the primitive, are parallel; and the angles at A and B are right angles, and A'C is pa

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rallel to AB; therefore AC is a rectangle, and A’C = AB; and the inclination of A'B' to A'C is the same as its inclination to AB or to the primitive. Also, A'B': A'C= Radius: cosine B'A'C; or A'B': AB = Radius: cosine of inclination.

COR. 1.-If a line parallel to the primitive join any two projecting lines, and if a line on the primitive parallel to it join the same lines, these lines, with the intercepted portions of the projecting lines, form a rectangle. For AC is a rectangle.

COR. 2. The projection of a line parallel to the primitive, is equal to the original.

For the projection of A'C is AB, and they are equal. COR. 3. The projection of a line perpendicular to the primitive, is a point.

For the projection of the line B'C is the point B.

PROPOSITION II.

The projection of a plane figure parallel to the primitive, is a figure similar and equal to the original.

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Let ABCDE be the projection of A'B'C'D'E', which is parallel to the D primitive, these figures are similar and equal.

For the sides AB, BC, &c. are equal to A'B', B'C', &c. (Pr. III. 1, Cor. 1). Also, if any plane perpen

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dicular to the primitive cut the two figures in lines B’Eʼ,

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