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conocive, that an apparatus may be so contrived, with a pannel of glass, and a proper sight-hole fixed at a convenient distance for looking through, as will enable us to draw or trace on the glass every object that may be presented to view from that point of sight.
And, for the more easy ascertaining the apparent place of any part of an original object, the glass frame may be reticulated, or divided into a great number of very small squares, and numbered from 0 at one of the angular points of the frame on each side of it, so that by carrying the eye from that small square, through which that part of the object is seen along the other squares in lines parallel to the sides of the frame, and noting the numbers on the sides, the situation ob. served may be transferred to a corresponding square on paper
reticulated in the same manner.
This mode of drawing is, however, merely mechanical, and will not enable us to represent objects which we may either design, or that may be geometrically drawn from their real dimensions. It is the business of perspective, to investigate such principles as will convincę us of the possibility of accomplishing the end of our desires, and to point out the rules for representing justly all manner of rectilinear objects. As to Landscape Drawing, we must not look for any assistance from the rules of perspective to accomplish their delineation; the reticulated frame, which we have already described, will furnish the best means for obtaining such representations ; for the art of perspective drawing is only applicable to rectilinear objects, or to solids, whose geometrical construction is perfected, known by a section taken parallel or perpendicular to some fixed or given plave.
Perspective is, therefore, applicable to all regularly-formed solids, and, consequently, to architecture, machinery, and furniture.
No just idea can be formed of any complex building, machine, or other object formed by geometrical solids, without a perspective representation of it. Language is altogether incapable of describing a complex body, and even in describing very simple ideas ;-it is difficult to put words so clearly together, as shall convey the construction or figure of the object, so as entirely to prevent niisrepresentation. Oral description is perhaps best, and most haply applied to geometrical definition where the forms are the most simple figures that nature can afford. So that perspective, as supplying the imperfections of language, is in this respect invaluable.
We must not, however, confine the uses of perspective to such objects as are capable of being measured and geometrically delineated ; we shall also find its extensive use in the delineation of natural objects, where the parts are too far extended to admit of measurement.
For, if we know of any part or parts of the field we are about to represent are geometrically formed, and can judge of their proportions, we shall find considerable assistance from the rules of perspective.
SPHERE IN PLANO.
Definitions. 1. Projection of the sphere is the representation of its surface upon a plane, called the plane of projection.
2. Orthographic projection, is drawing the circles of the sphere upon the plane of a great circle, by the means of lines perpendicular to that plane transmitted from all the points of the circle to be projected.
3. Stereographic projection, is drawing the circles of the sphere upon the plane of one of its great circles, by lines drawn from the pole of that great circle to all the points of the circle to be projected.
4. The gnomical projection, is drawing the circles of an hemis. phere upon a plane touching it in the vertex, by lines diverging from the centre of the hemisphere to all the points of the circle which is to be projected.
5. The primitive circle, is that on the plane of which the sphere is projected; and the pole of this circle is called the pole of projection; and the point from whence the projecting right line proceeds, is the projecting point.
6. The line of measures of any circle, is the common intersection of the plane of projection, and of any other plane passing through the eye, and is perpendicular both to the plane of projection, and also to the plane of the circle.
N.B. There are other kinds of projections of the sphere, such as the Cylindrical; Scenographic, which belongs properly to Peixpective; the Globical, which belongs to Geography; and Mercator's, which belongs to Navigation.
Ar. The place of any visible point of the sphere upon the plare of projection, is where the projecting line cuts that plane.
Cor. If the projecting point be supposed to be the place of the eye, it will view all the circles of the spbere, and all parts of them in the projection, just as they appear from thence in the sphere itself.
N.B. The projection of the sphere, is only the shadow of the circles of the sphere upon the plane of projection, the light being supposed to be in the place of the eye or projecting point. Explanation of some additional characters for the sake of brevity.
6 is meant to denote an angle.
The Orthographic Projection of the Sphere. Prof. 1. If a right line AB is projected upon a plane, it is projected into a right line ; and its length will be to the length of the projection, as the radius is to the co-sine of its inclination above the plarie. For, let fall the perpendiculars Aa, Bb, upon the
B plane of projection, then ab will be the line into a which it is projected, let Ao be drawn parallel to ab; then, by trigonometry, as AB : A0 or ab :: the radius to the sine of B, or co-sine oAB.
Cor. 1. If a riglit. line be projected upon a plane to which itself is parallel, it is projected into a right-line cqual and parallel to itself.
Cor. 2. If an angle be projected upon a plane which is parallel to the two lines forming the angle, it is projected into an angle equal to itself.
Cor. 3. Any plain figure projected upon a plane parallel to itself, is pro jected into a figure similar and equal to itself.
Cor. 4. Hence the area of any plane figure is to the area of its projection as the radius to the co-sine of its elevation or inclination.
Prop. 2. A circle which is perpendicular to the plane of its projection, is projected into a right-line equal to its diameter.
For projecting lines being drawn from all the points of the circle, fall in the common intersection of the plane of that circle and the plane projection, which is a right line and equal to the diameter of the circle. Q. E. D.
Cor. Hence, any plain figure, which is perpendicular to the plane of projection, is projected in:o a riglit line.
Prop. 3. A circle parallel to the plane of projection is projected into a circle equal to itself, and concentric with the primitive. Let BOD be the original circle, I its centre, C
A the centre of the sphere, the points I, B, O, D, are projected into the points C, L, F, G; and, consequently, OICF and BICL, are rectangular parallelograms; therefore, LC = BI = OI = FC.
GH Q. E. D.
Cur. Tie radius CL or CF is the co-sine of the projected circle's distance from the primitive circle, for it is the sine of AB.
Prop. 4 An inclined circle is projected into an eilipsis, the major axis of which is the diameter of the circle. Let AGDBH be the in
N clined circle, P its centre;
В. and suppose it projected into agdbh; draw the plane ABFCa through the centre
D C of the sphere, perpendicular to the plane of the given circle and plane of projection, to intersect them in the
EL linas AB, ab; draw GPH and DE perpendicular, and DQ
parallel to AB; then, because the line GP, and the plane of projection are both perpendicular to the plane ABF, therefore, GH is both parallel to the plane of projection, and also to the line gh projected on it.
In the circle ADB, DQ?=GQH=gqh; and BP=GP=gpo; and BP : EP, or DQ :: bp : ep or dq; and BPS : DQP :: bpe : dq? ; that is, gpo : gqh :: bp? : dq”; and, therefore, adgbh is an ellipsis, the major axis gh of which is the diameter of the circle. Q. E. D.
Cor. 1. Since ab is perpendicular to gh, ab is the minor axis, and is double the length of the sine of the angle ABb to the radins gp ; that is, the major asis is equal to twice the co-sine of the inclination to the radins of the . circle.
Cor. 2. The major axis is equal to twice the co-sine of its distance from its parallel great circle, for gh=H=2AP= twice the sine of AK.
Cor. 3. The extremities of the conjugate axis are distant from the centre of the primitive circle, by the sines of the circles of the least and greatest distance from the axis or pole of the primitive circle. Thus, aC is the sine of AN, and IC the sine of BN.
Cor. 4. Hence it is plain, also, that the conjugate axis passes invariably through the centre C of the primitive circle, and is always in the line of measures of that circle.
N.B. Every circle in the projection represents two equal circles, which are parallel and equi-distant from the primitive circle. Every right line represents two semi-circles, one towards the eye, and the other on the opposite side. Every ellipsis represents two equal circles, only having a contrary nclination AB, CD, the one being above the primitive circle, the other below it.
And vow having shewed the principles of the theory, it will be proper to (educe some short practical rules, by way of example
Prop. 5. To project a circle parallel to the primitive.
Take the complement of the distance of the circle to be projected from the primitive circle, and set it off from A to E; and, with the centre C, and a radius, cqual the perpendicular line A
FD EF, describe the circle DgG.
To do the sume by the Plain Scale. Take the sine of the circle's distance from the pole of the primitive circle, describe the desired circle with this radius and she centre C.
Prop. 6. To project a right circle, 'iz one which is perpendicular the plane of projection.
Through the centre C of the primitive circle draw the diameter AB, and iake the dis
D tance from its parallel great circle, and set it from A to E, and from B to D, and draw ED for the
B circle required.
To do the same by the Scale. Take the sine of the distance of the circle froin its parallel great circle AB, and at that distance draw a parallel ED for the circle required.
Prop. 7. To project a small oblique circle.
Draw the line of measures AB, and take
E the nearest distance of the circle from the primitive, and set it from B to D upwards, if above the primitive, and downwards if below
A А it, also take its greatest distance, and set it
CH from A to E, and draw ED, and let fall the perpendiculars EF, DG ; and bisect FG in H, and erect the perpendicular KHI, making KH=HI= half ED ; then describe an ellipsis (by the conic sections), the major axis of which is JK, aud the minor axis FG, and the curve will represent the given circle.
To do the same by the Scale. Draw the line of measures AB ; and take the sines of the least and greatest distances of the circle from the pole of the primitive, and set them off from the centre C to F and G both ways, if the periphery encompass the pole, but the same way if it lie on one side of it; bisect FG in H, and erect HK and HI perpendicular to FG, and equal the radius of the circle given, or the sine of its distance from its own pole ; about the Axes FG, KI, describe the ellipsis, which completes it.
Prop. 8. To find the pole of a given ellipse,
Through the centre of the primitive C draw the conjugate axis FG of the ellipsis; on the points F, G, erect the perpendiculars FE, GD, or set off the transverse axis IK from E to D, and bisect ED in R, and let fall RP perpendicular to AB, then P is the pole. Which may be found also hy the Scale.
R Thus, take CF and CG and apply them to the
D sines, and find the corresponding degrees of the supplenients ; then take the sine of kalf the sum of these degrees, if F and G be both one side of C,
A or the sine of half the difference if they are situated
CIPG on contrary sides, and set it off from the point C to the pole P
K Prob. 9. To measure an arch of a parallel cir: le, or to set off any number of degrees upon it.
With the radius of the parallel circle, and one foot of the compasses in C, describe the
9 circle Gg ; draw CGB and Cgb; then Bb will measure the given arch Gg; or Gg will contain the given number of degrees set or from B to b; hence, one being ascertained determines the other.
Prop. 10. To measure any part of a right circle.
Ia the right circle ED, let EA=AD; and let AB be required to be measured ;
B make CF=AE ; and, from the centre F, KAG with the radius AB, describe the arc GI ;
D draw CGK touching the circle in G; then FI HK will be the measure of AB. For FG cqual the sine of the angle, HGK to the radius CF or AE, and BA is the same, by Corollary Proposition 3.