sentation, which consists in supposing a vertical transparent plane between the eye and the object, called the perspective plane. As rays are supposed to proceed from every p int of the object to the eye, if these rays leave traces on the vertical plane, it is evident that they will there produce the same effect on the eye as the object itself, since they will paint the same image on the retina. The traces made by these rays are called the perspective image.

Let A C (fig. 16 pl. 4) then be the diameter of the circle on the horizontal plane A CP, perpendicular to the perspective plane; QR a section of the perspective plane, and PO a plane perpendicular to the horizon and to the line A P, in which it is required to find the point O, where, if the eye be placed, the representation ac, of the circle A C, shall be also a circle.

If PO be made a mean proportional between AP and CP, the point O will be the one required.

For, if P A be to PO, as PO to PC, the triangles PAO and PCO will be similar, and the angles PA O and C O P will be equal: the angles PAO and CcQ, or PAO and RO, will also be equal; hence it follows that in the small triangle ac O, the angle at c will be equal to the angle OA C, and the angle at O being common to the triangles AOC and a Oc, the other two, A CO and ca O, will be also equal: AO then will be to CO, as c O to a O; hence the oblique cone A CO will be cut in a sub-contrary manner, or sub-contrary position, by the vertical plane QR, and consequently the new section will be a circle, as is demonstrated in conic sections.

PROBLEM IX.

Why is the image of the sun, which passes into a darkened apartment through a square or triangular hole, always circular?

THIS problem was formerly proposed by Aristotle, who gave a very bad solution of it; for he said it arose from the rays of the sun affecting a certain roundness, which they resumed when they had surmounted the restraint imposed on them by the hole being of a different figure. This reason is entirely void of foundation.

To account for this phenomenon, it must be observed that the rays proceeding from any object, whether luminous or not, which pass through a very small hole into a darkened chamber, form there an image exactly similar to the object itself; for these rays, passing through the same point, form beyond it a kind of pyramid similar to the first, and having its summit joined to that of the first, and which, being cut by a plane parallel to that of the object, must give the same figure, but inverted.

This being understood, it may be readily conceived that each point of the triangular hole, for example, paints on paper, or on the floor, its solar image round; for every one of these points is the summit of a cone of which the solar disk is the base.

Describe then on paper a figure similar and equal to that of the hole, whether square or triangular, and from every point of its periphery, as a centre, describe equal circles: while these circles are small, you will have at first a triangular figure with rounded angles; but if the magnitude of the circles be

increased more and more, till the radius be much greater than any of the dimensions of the figure, it will be observed to become rounder and rounder, and at length to be sensibly converted into a circle.

But this is exactly what takes place in the darkenened apartment; for when the paper is held very near to the triangular, hole, you have a mixed image of the triangle and the circle; but if it be removed to a considerable distance, as each circular image of the sun becomes then very large, in regard to the diameter of the hole, the image is sensibly round. If the disk of the sun were square, and the hole round, the image at a certain distance would, for the same reason, be a square, or in general of the same figure as the disk. The image of the moon therefore, when increasing, is always, at a sufficient distance, a similar crescent, as is proved by experience.

PROBLEM X.

To make an object which is too near the eye to be distinctly perceiv.d, to be scen in a distinct manner, without the interpofition of any glass.

MAKE a hole in a card with a needle, and without changing the place of the eye or of the object, look at the latter through the hole; the object will then be seen distinctly, and even considerably magnified.

The reason of this phenomenon may be deduced from the following observations: When an object is not distinctly seen, on account of its nearness to the eye, it is because the rays proceeding from each of its points, and falling on the aperture of the pupil, do not converge to a point, as when the object is

at a proper distance: the image of each point is a small circle, and as all the small circles, produced by the different points of the object, encroach on each other, all distinction is destroyed. But, when the object is viewed through a very small hole, each pencil of rays, proceeding from each point of the object, has no other diameter than that of the hole; and consequently the image of that point is considerably confined, in an extent which scarcely surpasses the size it would have, if the object were at the necessary distance; it must therefore be seen distinctly.

PROBLEM XI.

When the eyes are directed in such a manner as to see a very distant object; why do near objects appear dou ble, and vice versa?

THE reason of this appearance is as follows. When we look at an object, we are accustomed, from habit, to direct the optical axis of our eyes towards that point which we principally consider. As the images of objects are, in other respects, entirely similar, it thence results that, being painted around that principal point of the retina at which the optical axis terminates, the lateral parts of an object, those on the right for example, are painted in each eye to the left of that axis; and the parts on the left are painted on the right of it. Hence there has been established between these parts of the eye such a correspondence, that when an object is painted at the same time in the left part of each eye, and at the same distance from the optical axis, we think there is only one, and on the right; but if by a

forced movement of the eyes we cause the image of an object to be painted in one eye, on the right of the optical axis, and in the other on the left, we see double. But this is what takes place when, in directing our sight to a diftant object, we pay attention to a neighbouring object situated between the optical axes it may be easily seen that the two images which are formed in the two eyes are placed, one to the right and the other to the left of the optical axis; that is to say, on the right of it, in the right eye, and on the left of it in the left. If the optical axis be directed to a near object, and if attention be at the same time paid to a distant object, in a direct line, the contrary will be the case. By the effect then of the habit, above mentioned, we must by one eye judge the object to be on the right, and the other to be on the left; the two eyes are thus in contradiction to each other, and the object appears double.

This explanation, founded on the manner in which we acquire ideas by sight, is confirmed by the following fact. Cheselden relates that a man having sustained a hurt in one of his eyes by a blow, so that he could not direct the optical axes of both eyes to the same point, saw all objects double; but this inconvenience was not lasting the most familiar objects gradually began to appear single, and his sight was at length restored to its natural state.

What takes place here in regard to the sight, takes place also in regard to the touch; for when two parts of the body which do not habitually correspond, in feeling one and the same object, are employed to touch the same body, we imagine it to be double. This is a common experiment. If one of the fingers be placed over the other, and if any