mathematicians, as to require demonstration before it can be believed. Nor is the demonstration at all difficult, if the reader will have patience to enter but a little into the mathematical consideration of visible figure, which we shall call the geometry of visibles.

In this geometry, the definitions of a point; of a line, whether straight or curve; of an angle, whether acute, or right, or obtuse; and of a circle—are the same as in common geometry. The mathematical reader will easily enter into the whole mystery of this geometry, if he attends duly to these few evident principles.

1. Supposing the eye placed in the centre of a sphere, every great circle of the sphere will have the same appearance to the eye as if it was a straight line; for the curvature of the circle being turned directly toward the eye, is not perceived by it. And, for the same reason, any line which is drawn in the plane of a great circle of the sphere, whether it be in reality straight or curve, will appear straight to the eye.

2. Every visible right line will appear to coincide with some great circle of the sphere; and the circumference of that great circle, even when it is produced until it returns into itself, will appear to be a continuation of the same visible right line, all the parts of it being visibly in directum. For the eye, perceiving only the position of objects with regard to itself, and not their distance, will see those points in the same visible place which have the same position with regard to the eye,

* How does this differ from a doctrine of Perspective ?-At any rate, the notion is Berkeley's. Compare "New Theory of Vision," §§ 153-159.-H,

how different soever their distances from it may be. Now, since a plane passing through the eye and a given visible right line, will be the plane of some great circle of the sphere, every point of the visible right line will have the same position as some point of the great circle; therefore, they will both have the same visible place, and coincide to the eye; and the whole circumference of the great circle, continued even until it returns into itself, will appear to be a continuation of the same visible right line.

Hence it follows--

3. That every visible right line, when it is continued in directum, as far as it may be continued, will be represented by a great circle of a sphere, in whose centre the eye is placed. It follows

4. That the visible angle comprehended under two visible right lines, is equal to the spherical angle comprehended under the two great circles, which are the representatives of these visible lines. For, since the visible lines appear to coincide with the great circles, the visible angle comprehended under the former must be equal to the visible angle comprehended under the latter. But the visible angle comprehended under the two great circles, when seen from the centre, is of the same magnitude with the spherical angle which they really comprehend, as mathematicians know; therefore, the visible angle made by any two visible lines is equal to the spherical angle made by the two great circles of the sphere which are their representatives.

5. Hence it is evident, that every visible right-lined triangle will coincide in all its parts with some spherical triangle. The sides of the one will appear equal to the sides of the other, and the angles of the one to the angles of the other, each to each: and, therefore, the whole of the one triangle will appear equal to the whole of the other. In a word, to the eye they will be one and the same, and have the same mathematical properties. The

properties, therefore, of visible right-lined triangles are not the same with the properties of plain triangles, but are the same with those of spherical triangles.

6. Every lesser circle of the sphere will appear a circle to the eye, placed, as we have supposed all along, in the centre of the sphere; and, on the other hand, every visible circle will appear to coincide with some lesser circle of the sphere.

7. Moreover, the whole surface of the sphere will represent the whole of visible space; for, since every visible point coincides with some point of the surface of the sphere, and has the same visible place, it follows, that all the parts of the spherical surface taken together, will represent all possible visible places-that is, the whole of visible space. And from this it follows, in the last place

8. That every visible figure will be represented by that part of the surface of the sphere on which it might be projected, the eye being in the centre. And every such visible figure will bear the same ratio to the whole of visible space, as the part of the spherical surface which represents it, bears to the whole spherical surface.

The mathematical reader, I hope, will enter into these principles with perfect facility, and will as easily perceive that the following propositions with regard to visible figure and space, which we offer only as a specimen, may be mathematically demonstrated from them, and are not less true nor less evident than the propositions of Euclid, with regard to tangible figures.

Prop. 1. Every right line being produced, will at last return into itself.

2. A right line returning into itself, is the longest possible right line; and all other right lines bear a finite ratio to it.

3. A right line returning into itself, divides the whole of visible space into two equal parts, which will both be comprehended under this right line.

4. The whole of visible space bears a finite ratio to any part of it.

5. Any two right lines being produced, will meet in two points, and mutually bisect each other.

6. If two lines be parallel—that is, everywhere equally distant from each other-they cannot both be straight.

7. Any right line being given, a point may be found, which is at the same distance from all the points of the given right line.

8. A circle may be parallel to a right line—that is, may be equally distant from it in all its parts.

9. Right-lined triangles that are similar, are also equal.

10. Of every right-lined triangle, the three angles taken together, are greater than two right angles.

11. The angles of a right-lined triangle, may all be right angles, or all obtuse angles.

12. Unequal circles are not as the squares of their diameters, nor are their circumferences in the ratio of their diameters.

This small specimen of the geometry of visibles, is intended to lead the reader to a clear and distinct conception of the figure and extension which is presented to the mind by vision; and to demonstrate the truth of what we have affirmed above-namely, that those figures and that extension which are the immediate objects of sight, are not the figures and the extension about which common geometry is employed; that the geometrician, while he looks at his diagram, and demonstrates a proposition, hath a figure presented to his eye, which is only a sign and representative of a tangible figure that he gives not the least attention to the first, but attends only to the last; and that these two figures have different properties, so that what he demonstrates of the one, is not true of the other.

It deserves, however, to be remarked, that, as a small part of a spherical surface differs not sensibly from a

plain surface, so a small part of visible extension differs very little from that extension in length and breadth, which is the object of touch. And it is likewise to be observed, that the human eye is so formed, that an object which is seen distinctly and at one view, can occupy but a small part of visible space; for we never see distinctly what is at a considerable distance from the axis of the eye; and, therefore, when we would see a large object at one view, the eye must be at so great a distance, that the object occupies but a small part of visible space. From these two observations, it follows, that plain figures which are seen at one view, when their planes are not oblique, but direct to the eye, differ little from the visible figures which they present to the eye. The several lines in the tangible figure, have very nearly the same proportion to each other as in the visible; and the angles of the one are very nearly, although not strictly and mathematically, equal to those of the other. Although, therefore, we have found many instances of natural signs which have no similitude to the things signified, this is not the case with regard to visible figure. It hath, in all cases, such a similitude to the thing signified by it, as a plan or profile hath to that which it represents; and, in some cases, the sign and thing signified have to all sense the same figure and the same proportions. we could find a being endued with sight only, without any other external sense, and capable of reflecting and reasoning upon what he sees, the notions and philosophical speculations of such a being, might assist us in the difficult task of distinguishing the perceptions which we have purely by sight, from those which derive their origin from other senses. Let us suppose such a being, and conceive, as well as we can, what notion he would have of visible objects, and what conclusions he would deduce from them. We must not conceive him disposed by his constitution, as we are, to consider the visible appearance as a sign of something else it is no

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