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in such cases, the lines of construction have enibarrassed the work, and rendered the operation altogether intricate and confused.
A facility of drawing lines to a point, of any remote given distance whatever, has therefore been very justly considered as a desideratuin in Perspective.
In order to accomplish this object, Mr. Peter Nicholson, the Editor of ihis work, has invented two instruments, called Centrolineads, for which he has been honoured with the approbation of the Society of Arts. For one of these instruments, which is particularly adapted to the purposes of engraving, a premium was awarded to him in the year 1814, and, for the other, and his facile method of adjusting it, the Society voted him their Silver Medal in the following year.
This latter Centrolinead may be readily applied to perspective delineation, and performs iis office with as much facility as the parallel and perpendicular lines may be drawn by a common square upon the edge of a drawing-board, and even with still greater accuracy; since the latter depends on the coincidence of the edge of the board with the stock of the square. This instrument is not entirely confined to the drawing of lines to a point, but it will also draw parallels with equal facility.
In order that the instrument should be accurately made, Mr. Nicholson has employed Mr. Doland, of St. Paul's Church Yard, whose fame in optical and mathematical instruments has been long and justly established.
A familiar idea of perspective may be obtained by laying a thin wash of gum-water òr isinglass on a window, and fixing ihe eye steadily within reach of it, trace over the surface of the glass with a black-lead or red pencil, so that the tracing-point may always mark its progress, and appear to cover the corresponding part of the object, and proceeding until it has gone over all the extremities of the surfaces, then the impression will exhibit the true outline of the perspective representation when viewed from that point only.
Another familiar i'ea of perspective may be obtained of an object, by placing it above a level plane, and holding a torch-light, or lighted candle, at a proper distance above the object, then the shadow on the plane will be a perspective representation of the contour or outline of the object, and if the impression could be supposed to remain, and the eye placed in the point from which the light was supposed to emanate, and to observe the parts on the plane that the object would appear to cover, the shadow would not be perceived, and the contcur would apparently cover the edge of the shadow.
The shadow of an object occasioned by the sun, may also be looked upon as the perspective representation of the contour of an object seen at a vast distance from the plane of projection.
In this case, parallel lines will throw shadow's nearly parallel in all positions in wlich they may be situated, in respect of the projecting plane.
The glass frame is, however, more immediately to our purpose in concriving the representations of objects in perspective. It is easy to conceive, 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 O 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 observed 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 convince us of the possibility of accomplishing the end of our desires, and to point out the rules for epresenting 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 plaue.
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 hemisphere 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 Perspective; 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 sphere, 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 lighit 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 plane. For, let fall the perpendiculars Aa, Bb, upon the
B plane of projection, then ab will be the line into
А. 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 right-line be projected upon a plane to which itself is parallel, it is projected into a right-line cqnal and parallel to itself.
*Cor. 2. If an angle be projected upon a plane which is parallel to the two lines forining the angle, it is projected into an angle equal to itself.
Cor. 3. Any plain figure projected upon a plane parallel to itse If, is pro jected into a ligure similar and equal to itself.
Cor. 4. Hence the area of any plane figure is to the area of its projection as the radons 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 right 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, B are projected into the points C, L, F, G ; and, consequently, OICF and BICL, are rectangular parallelograms; therefore, LC = BI = OI = FC. EL
GH Q. E. D.
T Cor. 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 ellipsis, the major axis of which is the diameter of the circle. Let AGDBH be the in
N clined circle, P its centre;
B and suppose it projected in
E to agdbh; draw the plane ABFCa through the centre C of the sphere, perpendicular to the plane of the given circle and plane of projec
Ho tion, to intersect them in the Fl
C C lines AB, ab; draw GPHand DE perpendicular, and DQ