Let us now see how this phenomenon agrees with our tenets. The eye the nearer it is placed to the point B in the foregoing figures, the more distinct is the appearance of the OBJECT; but as it recedes to O the appearance grows more confused; and at P it sees the OBJECT yet more confused; and so on till the eye being brought back to Z sees the OBJECT in the greatest confusion of all. Wherefore by sect. 21 the OBJECT should seem to approach the eye gradually as it recedes from the point B, that is, at O it should (in consequence of the principle I have laid down in the aforesaid section) seem nearer than it did at B, and at P nearer than at O, and at Q nearer than at P; and so on, till it quite vanishes at Z. Which is the very matter of fact, as anyone that pleases may easily satisfy himself by experiment.
This case is much the same as if we should suppose an Englishman to meet a foreigner who used the same words with the English, but in a direct contrary signification. The Englishman would not fail to make a wrong judgment of the IDEAS annexed to those sounds in the mind of him that used them. Just so, in the present case the OBJECT speaks (if I may so say) with words that the eye is well acquainted with, that is, confusions of appearance; but whereas heretofore the greater confusions were always wont to signify nearer distances, they have in this case a direct, contrary signification, being connected with the greater distances. Whence it follows that the eye must unavoidably be mistaken, since it will take the confusions in the sense it has been used to, which is directly opposed to the true.
This phenomenon as it entirely subverts the opinion of those who will have us judge of distance by lines and angles, on which supposition it is altogether inexplicable, so it seems to me no small confirmation of the truth of that principle whereby it is explained. But in order co a more full explication of this point, and to show how far the hypothesis of the mind's judging by the various divergency of rays may be of use in determining the apparent place of an OBJECT, it will be necessary to premise some few things, which are already well known to those who have any skill in dioptrics.
FIRST, any radiating point is then distinctly seen when the rays proceeding from it are, by the refractive power of the crystalline, accurately reunited in the retina or fund of the eye: but if they are reunited, either before they arrive at the retina, or after they have passed it, then there is confused vision.
SECONDLY, suppose in the adjacent figures NP represent an eye duly framed and retaining its natural figure. In Fig. 1 the rays falling nearly parallel on the eye, are by the crystalline AB refracted, so as their focus or point of union F falls exactly on the retina: but if the rays fall sensibly diverging on the eye, as in Fig. 2, then their focus falls beyond the retina: or if the rays are made to converge by the lens QS before they come at the eye, as in Fig. 3, their focus F will fall before the retina. In which two last cases it is evident from the foregoing section that the appearance of the point Z is confused. And by how much the greater is the convergency, or divergency, of the rays falling on the pupil, by so much the farther will the point of their reunion be from the retina, either before or behind it, and consequently the point Z will appear by so much the more confused. And this, by the bye, may show us the difference between confused and faint vision. Confused vision is when the rays proceedings from each distinct point of the OBJECT are not accurately recollected in one corresponding point on the retina, but take up some space thereon, so that rays from different points become mixed and confused together. This is opposed to a distinct vision, and attends near objects. Faint vision is when by reason of the distance of the object or grossness of the interjacent medium few rays arrive from the object to the eye. This is opposed to vigorous or clear vision, and attends remote objects. But to return.
The eye, or (to speak truly) the mind, perceiving only the confusion itself, without ever considering the cause from which it proceeds, doth constantly annex the same degree of distance to the same degree of confusion. Whether that confusion be occasioned by converging or by diverging rays, it matters not. Whence it follows that the eye viewing the object Z through the glass QS (which by refraction causeth the rays ZQ, ZS, etc., to converge) should judge it to be at such a nearness at which if it were placed it would radiate on the eye with rays diverging to that degree as would produce the same confusion which is now produced by converging rays, i.e. would cover a portion of the retina equal to DC (VID. Fig. 3 supra). But then this must be understood (to use Dr. Barrow's phrase) SECLUSIS PRAENOTIONIBUS ET PRAEJUDICIIS, in case we abstract from all other circumstances of vision, such as the figure, size, faintness, etc. of the visible objects; all which do ordinarily concur to form our idea of distance, the mind having by frequent experience observed their several sorts or degrees to be connected with various distances.
It plainly follows from what hath been said that a person perfectly purblind (i.e. that could not see an object distinctly but when placed close to his eye) would not make the same wrong judgment that others do in the forementioned case. For to him greater confusions constantly suggesting greater distances, he must, as he recedes from the glass and the object grows more confused, judge it to be at a farther distance, contrary to what they do who have had the perception of the objects growing more confused connected with the idea of approach.
Hence also it doth appear there may be good use of computation by lines and angles in optics; not that the mind judgeth of distance immediately by them, but because it judgeth by somewhat which is connected with them, and to the determination whereof they may be subservient. Thus the mind judging of the distance of an object by the confusedness of its appearance, and this confusedness being greater or lesser to the naked eye, according as the object is seen by rays more or less diverging, it follows that a man may make use of the divergency of the rays in computing the apparent distance, though not for its own sake, yet on account of the confusion with which it is connected. But, so it is, the confusion itself is entirely neglected by mathematicians as having no necessary relation with distance, such as the greater or lesser angles of divergency are conceived to have. And these (especially for that they fall under mathematical computation) are alone regarded in determining the apparent places of objects, as though they were the sole and immediate cause of the judgments the mind makes of distance. Whereas, in truth, they should not at all be regarded in themselves, or any otherwise, than as they are supposed to be the cause of confused vision.
The not considering of this has been a fundamental and perplexing oversight. For proof whereof we need go no farther than the case before us. It having been observed that the most diverging rays brought into the mind the idea of nearest distance, and that still, as the divergency decreased, the distance increased: and it being thought the connexion between the various degrees of divergency and distance was immediate; this naturally leads one to conclude, from an ill-grounded analogy, that converging rays shall make an object appear at an immense distance: and that, as the convergency increases, the distance (if it were possible) should do so likewise. That this was the cause of Dr. Barrow's mistake is evident from his own words which we have quoted. Whereas had the learned doctor observed that diverging and converging rays, how opposite soever they may seem, do nevertheless agree in producing the same effect, to wit, confusedness of vision, greater degrees whereof are produced indifferently, either as the divergency or convergency and the rays increaseth. And that it is by this effect, which is the same in both, that either the divergency or convergency is perceived by the eye; I say, had he but considered this, it is certain he would have made a quite contrary judgment, and rightly concluded that those rays which fall on the eye with greater degrees of convergency should make the object from whence they proceed appear by so much the nearer. But it is plain it was impossible for any man to attain to a right notion of this matter so long as he had regard only to lines and angles, and did not apprehend the true nature of vision, and how far it was of mathematical consideration.
Before we dismiss this subject, it is fit we take notice of a query relating thereto, proposed by the ingenious Mr. Molyneux, is his TREATISE OF DIOPTRICS, [Par. I. Prop. 31, Sect. 9.] where speaking of this difficulty, he has these words: 'And so he (i.e. Dr. Barrow) leaves this difficulty to the solution of others, which I (after so great an example) shall do likewise; but with the resolution of the same admirable author of not quitting the evident doarine which we have before laid down, for determining the LOCUS OBJECTI, on account of being pressed by one difficulty which seems inexplicable till a more intimate knowledge of the visive faculty be obtained by mortals. In the meantime, I propose it to the consideration of the ingenious, whether the LOCUS APPARENS of an object placed as in this 9th section be not as much before the eye as the distinct base is behind the eye!' To which query we may venture to answer in the negative. For in the present case the rule for determining the distance of the distinct base, or respective focus from the glass, is this: as the difference between the distance of the object and focus is to the focus or focal length, so the distance of the object from the glass is to the distance of the respective focus or distinct base from the glass. [Molyneux Dioptr., Par. I. Prop. 5.] Let us now suppose the object to be placed at the distance of the focal length, and one half of the focal length from the glass, and the eye close to the glass, hence it will follow by the rule that the distance of the distinct base behind the eye is double the true distance of the object before the eye. If therefore Mr. Molyneux's conjecture held good, it would follow that the eye should see the object twice as far off as it really is; and in other cases at three or four times its due distance, or more. But this manifestly contradicts experience, the object never appearing, at farthest, beyond its due distance. Whatever, therefore, is built on this supposition (VID. COROL. I. PROP. 57, IBID.) comes to the ground along with it.