"The surface of the ball of the eye is polished, and covered with a thin moisture, which renders the gloss more perfect. When we look upon an open eye, we see a bright point upon its surface. of great lustre, hut of very limited extent, whose position depends upon the situation of the observer and the direction of the object. Were the sin face of the eye perfectly spherical, it might turn on its vertical axis without in the least affecting the position of the brilliant point : but the surface being lengthened in the direction of the axis of vision, the position of the point is changed every time that the eve moves upon its vertical axis. Long experience having made us familiar with this change, our judgment, as to the direction of the eye, is con siderably biassed by it. By the difference of position in the bright points upon the balls of the two eyes of a person, we chiefly judge whether he squint, or not ; whether he look towards us ; and when he does not, to which side his attention is directed.
" We do not pretend to infer from this example, that it is indispensable, in a picture. that the position of the brilliant point upon the ball of the eye be geometrically defined ; our intention is merely to demonstrate how trifling errors as to this position may produce considerable distortion in the apparent form of the object, though in other respects the tracing of its apparent outline may remain the same.
" We now proceed to the determination of planes tangent to curved surftees, drawn through points taken on the out side of them.
"The surface of the sphere is one of the most simple that can fall under our consideration; it has common generations with a great number of different surfaces ; we may, for example, class it among revolving surfaces, and say nothing particular relative to it. But its regularity is productive of remarkable results, some of which are curious from their novelty, and with them, in the first instance, we are now about to be occupied, not so much on their own account, as to acquire, by the observation of the three dimensions, a habit, of which we shall stand in need, for more general and useful subjects.
" First Question.—Throug,h a given right line to draw a tangent plane to the surface of a given sphere.
"Solution.—First method. 1G. Let A, a, be the two projections of the centre of the sphere; a c n, the pro jection of the great horizontal circle; E F, e f, the two indefinite projections of the given right line. Through the centre of the sphere, imagine a plane perpendicular to the right line, and eonstruet. by the methods given under l'igare
6, the projections o, y, of the point of coineidenee of the right line with the plane.
"From this position, it is evident, that from the given right line two tangent planes may be drawn to the sphere, the first on one side, the second on the other, and, conse quently, that the sphere will be placed between them : this indicates two different points of contact, whose projections we must now construct.
" If from the centre of the sphere, a perpendicular be con ceived to fall upon both the tangent planes, they will each be bounded, at the point of contact with the surface of the sphere, by the corresponding planes ; and will both he in the plane perpendicular to the given right line : therefore the two points of contact will be in the section of the sphere by the perpendicular plane ; a section which must be the cir cumference of one of the great circles of the sphere. and to which the two sections made in the tangent planes by the same plane will be tangent.
" If in the perpendicular plane, and through the centre of the sphere, an horizontal line be imagined. whose vertical projection may be obtained by drawing the horizontal line ale, and its other projection by letting the perpendicular A II fall upon E F ; and if the perpendicular plane he conceived to turn upon this horizontal line, like a hinge, till it become horizontal itself; it is evident that its section with the sur face of the sphere would be lost in the circumference B C that the two points of contact would then be upon this cir cumference, and that were the point J constructed. in which, by this movement, the perpendicular plane would meet the given right line ; the tangents .1 c, J D, drawn to the circle • C D, would determine these two points of contact to the position in which they then appear. It is easy to construct the point a, or, which is tantamount, to find its distance from the point a ; for the horizontal projection of this distance is G u, and the difference of the vertical heights of its extremi ties is q therefore by transferring the distance o Il upon the horizontal tine a h, from g to h, the hypothenuse h g will be the amount of this distance: and by transferringy It upon E F, from it to j, and drawing the two tangents .T C. J D, the two points of contact. c, D, will he determined to the position they assumed, whilst the perpendicular plane was laid upon the horizontal one.