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An Essay Towards a New Theory of Vision


by

George Berkeley (1685-1753)



CONTENTS


Sect.

    1   Design
    2   Distance of itself invisible
    3   Remote distance perceived rather by experience than by sense
    4   Near distance thought to be perceived by the ANGLE of the
        OPTIC AXES
    5   Difference between this and the former manner of perceiving
        distance
    6   Also by diverging rays
    7   This depends not on experience
    8   These the common accounts, but not satisfactory
    9   Some IDEAS perceived by the mediation of others
   10   No IDEA which is not itself perceived, can be the means of
        perceiving another
   11   Distance perceived by means of some other IDEA
   12   Those lines and angles mentioned in optics, are not themselves
        perceived
   13   Hence the mind does not perceive distance by lines and angles
   14   Also because they have no real existence
   15   And because they are insufficient to explain the phenomena
   16   The IDEAS that suggest distance are, 1st, the sensation arising
        from the turn of the eyes
   17   Betwixt which and distance there is no necessary connection
   18   Scarce room for mistake in this matter
   19   No regard had to the angle of the OPTIC AXES
   20   Judgment of distance made with both eyes, the result of EXPERIENCE
   21   2ndly, Confusedness of appearance
   22   This the occasion of those judgments attributed to diverging rays
   23   Objection answered
   24   What deceives the writers of optics in this matter
   25   The cause why one IDEA may suggest another
   26   This applied to confusion and distance
   27   Thirrdly, the straining of the eye
   28   The occasions which suggest distance have in their own nature
        no relation to it
   29   A difficult case proposed by Dr. Barrow as repugnant to
        all the known theories
   30   This case contradicts a received principle in catoptrics
   31   It is shown to agree with the principles we have laid down
   32   This phenomenon illustrated
   33   It confirms the truth of the principle whereby it is explained
   34   Vision when distinct, and when confused
   35   The different effects of parallel diverging and converging rays
   36   How converging and diverging rays come to suggest the same
        distance
   37   A person extreme purblind would judge aright in the
        forementioned case
   38   Lines and angles, why useful in optics
   39   The not understanding this, a cause of mistake
   40   A query proposed, by Mr. Molyneux in his DIOPTRICS, considered
   41   One born blind would not at first have any IDEA of distance by
        sight
   42   This not agreeable to the common principles
   43   The proper objects of sight, not without the mind, nor the images
        of any thing without the mind
   44   This more fully explained
   45   In what sense we must be understood to see distance
        and external things
   46   Distance, and things placed at a distance, not otherwise perceived
        by the eye than by the ear
   47   The IDEAS of sight more apt to be confounded with the IDEAS
        of touch than those of hearing are
   48   How this comes to pass
   49   Strictly speaking, we never see and feel the same thing
   50   Objects of SIGHT twofold, mediate and immediate
   51   These hard to separate in our thoughts
   52   The received accounts of our perceiving magnitude by sight, false
   53   Magnitude perceived as immediately as distance
   54   Two kinds of sensible extension, neither of which is
        infinitely divisible
   55   The tangible magnitude of an OBJECT steady, the visible not
   56   By what means tangible magnitude is perceived by sight
   57   This further enlarged on
   58   No necessary connection between confusion or faintness
        of appearance, and small or great magnitude
   59   The tangible magnitude of an OBJECT more heeded than the visible,
        and why
   60   An instance of this
   61   Men do not measure by visible feet or inches
   62   No necessary connection between visible and tangible extension
   63   Greater visible magnitude might signify lesser tangible magnitude
   64   The judgments we make of magnitude depend altogether on experience
   65   Distance and magnitude seen as shame or anger
   66   But we are prone to think otherwise, and why
   67   The moon seems greater in the horizon than in the meridian
   68   The cause of this phenomenon assigned
   69   The horizontal moon, why greater at one time than another.
   70   The account we have given proved to be true
   71   And confirmed by the moon's appearing greater in a mist
   72   Objection answered
   73   The way wherein faintness suggests greater magnitude illustrated
   74   Appearance of the horizontal moon, why thought difficult to
        explain
   75   Attempts towards the solution of it made by several, but in vain
   76   The opinion of Dr. Wallis
   77   It is shown to be unsatisfactory
   78   How lines and angles may be of use in computing apparent
        magnitudes
   79   One born blind, being made to see, what judgment he would make
        of magnitude
   80   The MINIMUM VISIBLE the same to all creatures
   81   Objection answered
   82   The eye at all times perceives the same number of visible points
   83   Two imperfections in the VISIVE FACULTY
   84   Answering to which, we may conceive two perfections
   85   In neither of these two ways do microscopes improve the sight
   86   The case of microscopical eyes, considered
   87   The sight, admirably adapted to the ends of seeing
   88   Difficulty concerning erect vision
   89   The common way of explaining it
   90   The same shown to be false
   91   Not distinguishing between IDEAS of sight and touch, cause
        of mistake in this matter
   92   The case of one born blind, proper to be considered
   93   Such a one might by touch attain to have IDEAS of UPPER and LOWER
   94   Which modes of situation he would attribute only to things
        tangible
   95   He would not at first sight think anything he saw, high or low,
        erect or inverted
   96   This illustrated by an example
   97   By what means he would come to denominate visible OBJECTS,
        high or low, etc.
   98   Why he should think those OBJECTS highest, which are painted on
        the lowest part of his eye, and VICE VERSA
   99   How he would perceive by sight, the situation of external objects
  100   Our propension to think the contrary, no argument against
        what has been said
  101   Objection
  102   Answer
  103   An object could not be known at first sight by the colour
  104   Nor by the magnitude thereof
  105   Nor by the figure
  106   In the first act of vision, no tangible thing would be suggested
        by sight
  107   Difficulty proposed concerning number
  108   Number of things visible, would not at first sight suggest
        the like number of things tangible
  109   Number the creature of the mind
  110   One born blind would not at first sight number visible things
        as others do
  111   The situation of any object determined with respect only to
        objects of the same sense
  112   No distance, great or small, between a visible and tangible thing
  113   The not observing this, cause of difficulty in erect vision
  114   Which otherwise includes nothing unaccountable
  115   What is meant by the picture being inverted
  116   Cause of mistake in this matter
  117   Images in the eye, not pictures of external objects
  118   In what sense they are pictures
  119   In this affair we must carefully distinguish between ideas
        of sight and touch
  120   Difficult to explain by words the true Theory of Vision
  121   The question, whether there is any IDEA common to sight
        and touch, stated
  122   Abstract extension inquired into
  123   It is incomprehensible
  124   Abstract extension not the OBJECT of geometry
  125   The general IDEA of a triangle, considered
  126   Vacuum, or pure space, not common to sight and touch
  127   There is no idea, or kind of idea, common to both senses
  128   First argument in proof hereof
  129   Second argument
  130   Visible figure and extension, not distinct IDEAS from colour
  131   Third argument
  132   Confirmation drawn from Mr. Molyneux's problem of a sphere
        and a cube, published by Mr. Locke
  133   Which is falsely solved, if the common supposition be true
  134   More might be said in proof of our tenet, but this suffices
  135   Further reflection on the foregoing problem
  136   The same thing doth not affect both sight and touch
  137   The same idea of motion not common to sight and touch
  138   The way wherein we apprehend motion by sight, easily collected
        from what hath been said
  139   QU. How visible and tangible IDEAS came to have the same name
        if not of the same kind
  140   This accounted for without supposing them of the same kind
  141   OBJ. That a tangible square is liker to a visible square than
        to a visible circle
  142   ANS. That a visible square is fitter than a visible circle,
        to represent a tangible square
  143   But it doth not hence follow, that a visible square
        is like a tangible square
  144   Why we are more apt to confound visible with tangible IDEAS,
        than other signs with the things signified
  145   Several other reasons hereof, assigned
  146   Reluctancy in rejecting any opinion, no argument of its truth
  147   Proper objects of vision the language of nature
  148   In it there is much admirable, and deserving our attention
  149   Question proposed, concerning the object of geometry
  150   At first view we are apt to think visible extension the object
        of geometry
  151   Visible extension shown not to be the object of geometry
  152   Words may as well be thought the object of geometry,
        as visible extension
  153   It is proposed to inquire, what progress an intelligence that
        could see, but not feel, might make in geometry
  154   He cannot understand those parts which relate to solids, and
        their surfaces, and lines generated by their section
  155   Nor even the elements of plane geometry
  156   The proper objects of sight incapable of being managed as
        geometrical figures
  157   The opinion of those who hold plane figures to be the immediate
        objects of sight, considered
  158   Planes no more the immediate objects of sight, than solids
  159   Difficult to enter precisely into the thoughts of the
        above-mentioned intelligence
  160   The object of geometry, its not being sufficiently understood,
        cause of difficulty, and useless labour in that science




AN ESSAY TOWARDS A NEW THEORY OF VISION


1. My design is to show the manner wherein we perceive by sight the
distance, magnitude, and situation of OBJECTS. Also to consider the
difference there is betwixt the IDEAS of sight and touch, and whether
there be any IDEA common to both senses.

2. It is, I think, agreed by all that DISTANCE, of itself and
immediately, cannot be seen. For DISTANCE being a Line directed end-wise
to the eye, it projects only one point in the fund of the eye, which
point remains invariably the same, whether the distance be longer or
shorter.

3. I find it also acknowledged that the estimate we make of the distance
of OBJECTS considerably remote is rather an act of judgment grounded on
EXPERIENCE than of SENSE. For example, when I perceive a great number of
intermediate OBJECTS, such as houses, fields, rivers, and the like, which
I have experienced to take up a considerable space, I thence form a
judgment or conclusion that the OBJECT I see beyond them is at a great
distance. Again, when an OBJECT appears faint and small, which at a near
distance I have experienced to make a vigorous and large appearance, I
instantly conclude it to be far off: And this, it is evident, is the
result of EXPERIENCE; without which, from the faintness and littleness I
should not have inferred anything concerning the distance of OBJECTS.

4. But when an OBJECT is placed at so near a distance as that the
interval between the eyes bears any sensible proportion to it, the
opinion of speculative men is that the two OPTIC AXES (the fancy that we
see only with one eye at once being exploded) concurring at the OBJECT do
there make an ANGLE, by means of which, according as it is greater or
lesser, the OBJECT is perceived to be nearer or farther off.

5. Betwixt which and the foregoing manner of estimating distance there is
this remarkable difference: that whereas there was no apparent, necessary
connection between small distance and a large and strong appearance, or
between great distance and a little and faint appearance, there appears a
very necessary connection between an obtuse angle and near distance, and
an acute angle and farther distance. It does not in the least depend upon
experience, but may be evidently known by anyone before he had
experienced it, that the nearer the concurrence of the OPTIC AXES, the
greater the ANGLE, and the remoter their concurrence is, the lesser will
be the ANGLE comprehended by them.

6. There is another way mentioned by optic writers, whereby they will
have us judge of those distances, in respect of which the breadth of the
PUPIL hath any sensible bigness: And that is the greater or lesser
divergency of the rays, which issuing from the visible point do fall on
the PUPIL, that point being judged nearest which is seen by most
diverging rays, and that remoter which is seen by less diverging rays:
and so on, the apparent distance still increasing, as the divergency of
the rays decreases, till at length it becomes infinite, when the rays
that fall on the PUPIL are to sense parallel. And after this manner it is
said we perceive distance when we look only with one eye.

7. In this case also it is plain we are not beholding to experience: it
being a certain, necessary truth that the nearer the direct rays falling
on the eye approach to a PARALLELISM, the farther off is the point of
their intersection, or the visible point from whence they flow.

8. I have here set down the common, current accounts that are given of
our perceiving near distances by sight, which, though they are
unquestionably received for true by MATHEMATICIANS, and accordingly made
use of by them in determining the apparent places of OBJECTS, do,
nevertheless seem to me very unsatisfactory: and that for these following
reasons:--

9. FIRST, It is evident that when the mind perceives any IDEA, not
immediately and of itself, it must be by the means of some other IDEA.
Thus, for instance, the passions which are in the mind of another are of
themselves to me invisible. I may nevertheless perceive them by sight,
though not immediately, yet by means of the colours they produce in the
countenance. We often see shame or fear in the looks of a man, by
perceiving the changes of his countenance to red or pale.

10. Moreover it is evident that no IDEA which is not itself perceived can
be the means of perceiving any other IDEA. If I do not perceive the
redness or paleness of a man's face themselves, it is impossible I should
perceive by them the passions which are in his mind.

11. Now from sect. 2 it is plain that distance is in its own nature
imperceptible, and yet it is perceived by sight. It remains, therefore,
that it be brought into view by means of some other IDEA that is itself
immediately perceived in the act of VISION.

12. But those LINES and ANGLES, by means whereof some MATHEMATICIANS
pretend to explain the perception of distance, are themselves not at all
perceived, nor are they in truth ever thought of by those unskilful in
optics. I appeal to anyone's experience whether upon sight of an OBJECT he
computes its distance by the bigness of the ANGLE made by the meeting of
the two OPTIC AXES? Or whether he ever thinks of the greater or lesser
divergency of the rays, which arrive from any point to his PUPIL? Everyone
is himself the best judge of what he perceives, and what not. In vain
shall all the MATHEMATICIANS in the world tell me, that I perceive certain
LINES and ANGLES which introduce into my mind the various IDEAS of
DISTANCE, so long as I myself am conscious of no such thing.

13. Since, therefore, those ANGLES and LINES are not themselves perceived
by sight, it follows from sect. 10 that the mind doth not by them judge
of the distance of OBJECTS.

14. Secondly, the truth of this assertion will be yet farther evident to
anyone that considers those LINES and ANGLES have no real existence in
nature, being only an HYPOTHESIS framed by the MATHEMATICIANS, and by them
introduced into OPTICS, that they might treat of that science in a
GEOMETRICAL way.

15. The third and last reason I shall give for rejecting that doctrine is,
that though we should grant the real existence of those OPTIC ANGLES,
etc., and that it was possible for the mind to perceive them, yet these
principles would not be found sufficient to explain the PHENOMENA of
DISTANCE, as shall be shown hereafter.

16. Now, it being already shown that distance is suggested to the mind by
the mediation of some other IDEA which is itself perceived in the act of
seeing, it remains that we inquire what IDEAS or SENSATIONS there be that
attend VISION, unto which we may suppose the IDEAS of distance are
connected, and by which they are introduced into the mind. And FIRST, it
is certain by experience that when we look at a near OBJECT with both
eyes, according as it approaches or recedes from us, we alter the
disposition of our eyes, by lessening or widening the interval between
the PUPILS. This disposition or turn of the eyes is attended with a
sensation, which seems to me to be that which in this case brings the
IDEA of greater or lesser distance into the mind.

17. Not that there is any natural or necessary connection between the
sensation we perceive by the turn of the eyes and greater or lesser
distance, but because the mind has by constant EXPERIENCE found the
different sensations corresponding to the different dispositions of the
eyes to be attended each with a different degree of distance in the
OBJECT: there has grown an habitual or customary connection between those
two sorts of IDEAS, so that the mind no sooner perceives the sensation
arising from the different turn it gives the eyes, In order to bring the
PUPILS nearer or farther asunder, but it withal perceives the different
IDEA of distance which was wont to be connected with that sensation; just
as upon hearing a certain sound, the IDEA is immediately suggested to the
understanding which custom had united with it.

18 Nor do I see how I can easily be mistaken in this matter. I know
evidently that distance is not perceived of itself. That by consequence
it must be perceived by means of some other IDEA which is immediately
perceived, and varies with the different degrees of distance. I know also
that the sensation arising from the turn of the eyes is of itself
immediately perceived, and various degrees thereof are connected with
different distances, which never fail to accompany them into my mind,
when I view an OBJECT distinctly with both eyes, whose distance is so
small that in respect of it the interval between the eyes has any
considerable magnitude.

19. I know it is a received opinion that by altering the disposition of
the eyes the mind perceives whether the angle of the OPTIC AXES is made
greater or lesser. And that accordingly by a kind of NATURAL
GEOMETRY it judges the point of their intersection to be nearer
or farther off. But that this is not true I am convinced by my own
experience, since I am not conscious that I make any such use of the
perception I have by the turn of my eyes. And for me to make those
judgments, and draw those conclusions from it, without knowing that I do
so, seems altogether incomprehensible.

20. From all which it follows that the judgment we make of the distance
of an OBJECT, viewed with both eyes, is entirely the RESULT OF
EXPERIENCE. If we had not constantly found certain sensations arising
from the various disposition of the eyes, attended with certain degrees
of distance, we should never make those sudden judgments from them
concerning the distance of OBJECTS; no more than we would pretend to
judge a man's thoughts by his pronouncing words we had never heard
before.

21. Secondly, an OBJECT placed at a certain distance from the eye, to
which the breadth of the PUPIL bears a considerable proportion, being
made to approach, is seen more confusedly: and the nearer it is brought
the more confused appearance it makes. And this being found constantly to
be so, there ariseth in the mind an habitual CONNECTION between the
several degrees of confusion and distance; the greater confusion still
implying the lesser distance, and the lesser confusion the greater
distance of the OBJECT.

22. This confused appearance of the OBJECT doth therefore seem to be the
MEDIUM whereby the mind judgeth of distance in those cases wherein the
most approved writers of optics will have it judge by the different
divergency with which the rays flowing from the radiating point fall on
the PUPIL. No man, I believe, will pretend to see or feel those imaginary
angles that the rays are supposed to form according to their various
inclinations on his eye. But he cannot choose seeing whether the OBJECT
appear more or less confused. It is therefore a manifest consequence from
what bath been demonstrated, that instead of the greater or lesser
divergency of the rays, the mind makes use of the greater or lesser
confusedness of the appearance, thereby to determine the apparent place
of an OBJECT.

23 Nor doth it avail to say there is not any necessary connection between
confused VISION and distance, great or small. For I ask any man what
necessary connection he sees between the redness of a blush and shame? And
yet no sooner shall he behold that colour to arise in the face of
another, but it brings into his and the IDEA of that passion which hath
been observed to accompany it.

24. What seems to have misled the writers of optics in this matter is
that they imagine men judge of distance as they do of a conclusion in
mathematics, betwixt which and the premises it is indeed absolutely
requisite there be an apparent, necessary connection: but it is far
otherwise in the sudden judgments men make of distance. We are not to
think that brutes and children, or even grown reasonable men, whenever
they perceive an OBJECT to approach, or depart from them, do it by virtue
of GEOMETRY and DEMONSTRATION.

25. That one IDEA may suggest another to the mind it will suffice that
they have been observed to go together, without any demonstration of the
necessity of their coexistence, or without so much as knowing what it is
that makes them so to coexist. Of this there are innumerable instances of
which no one can be ignorant.

26. Thus, greater confusion having been constantly attended with nearer
distance, no sooner is the former IDEA perceived, but it suggests the
latter to our thoughts. And if it had been the ordinary course of Nature
that the farther off an OBJECT were placed, the more confused it should
appear, it is certain the very same perception that now makes us think an
OBJECT approaches would then have made us to imagine it went farther off.
That perception, abstracting from CUSTOM and EXPERIENCE, being equally
fitted to produce the IDEA of great distance, or small distance, or no
distance at all.

27. Thirdly, an OBJECT being placed at the distance above specified, and
brought nearer to the eye, we may nevertheless prevent, at least for some
time, the appearances growing more confused, by straining the eye. In
which case that sensation supplies the place of confused VISION in aiding
the mind to judge of the distance of the OBJECT; it being esteemed so
much the nearer by how much the effort or straining of the eye in order
to distinct VISION is greater.

28. I have here set down those sensations or IDEAS that seem to be the
constant and general occasions of introducing into the mind the different
IDEAS of near distance. It is true in most cases that divers other
circumstances contribute to frame our IDEA of distance, to wit, the
particular number, size, kind, etc., of the things seen. Concerning
which, as well as all other the forementioned occasions which suggest
distance, I shall only observe they have none of them, in their own
nature, any relation or connection with it: nor is it possible they should
ever signify the various degrees thereof, otherwise than as by EXPERIENCE
they have been found to be connected with them.

29. I shall proceed upon these principles to account for a phenomenon
which has hitherto strangely puzzled the writers of optics, and is so far
from being accounted for by any of their THEORIES OF VISION that it is,
by their own confession, plainly repugnant to them; and of consequence,
if nothing else could be objected, were alone sufficient to bring their
credit in question. The whole difficulty I shall lay before you in the
words of the learned Dr. Barrow, with which he concludes his optic
lectures:--

'I have here delivered what my thoughts have suggested to me concerning
that part of optics which is more properly mathematical. As for the other
parts of that science (which being rather physical, do consequently
abound with plausible conjectures instead of certain principles), there
has in them scarce anything occurred to my observation different from what
has been already said by Kepler, Scheinerus, Descartes, and others. And
methinks, I had better say nothing at all, than repeat that which has
been so often said by others. I think it therefore high time to take my
leave of this subject: but before I quit it for good and all, the fair
and ingenuous dealing that I owe both to you and to truth obligeth me to
acquaint you with a certain untoward difficulty, which seems directly
opposite to the doctrine I have been hitherto inculcating, at least,
admits of no solution from it. In short it is this. Before the double
convex glass or concave speculum EBF, let the point A be placed at such a
distance that the rays proceeding from A, after refraction or reflection,
be brought to unite somewhere in the AxAB. And suppose the point of union
(i.e. the image of the point A, as hath been already set forth) to be Z;
between which and B, the vertex of the glass or speculum, conceive the
eye to be anywhere placed. The question now is, where the point A ought
to appear? Experience shows that it does not appear behind at the point
Z, and it were contrary to nature that it should, since all the
impression which affects the sense comes from towards A. But from our
tenets it should seem to follow that it would appear before the eye at a
vast distance off, so great as should in some sort surpass all sensible
distance. For since if we exclude all anticipations and prejudices, every
OBJECT appears by so much the farther off, by how much the rays it sends
to the eye are less diverging. And that OBJECT is thought to be most
remote from which parallel rays proceed unto the eye. Reason would make
one think that OBJECT should appear at yet a greater distance which is
seen by converging rays. Moreover it may in general be asked concerning
this case what it is that determines the apparent place of the point A,
and maketh it to appear after a constant manner sometimes nearer, at
other times farther off? To which doubt I see nothing that can be
answered agreeable to the principles we have laid down except only that
the point A ought always to appear extremely remote. But on the contrary
we are assured by experience that the point A appears variously distant,
according to the different situations of the eye between the points B and
Z. And that it doth never (if at all) seem farther off, than it would if
it were beheld by the naked eye, but on the contrary it doth sometimes
appear much nearer. Nay, it is even certain that by how much the rays
falling on the eye do more converge by so much the nearer doth the OBJECT
seem to approach. For the eye being placed close to the point B, the
OBJECT A appears nearly in its own natural place, if the point B is taken
in the glass, or at the same distance, if in the speculum. The eye being
brought back to O, the OBJECT seems to draw near: and being come to P it
beholds it still nearer. And so on little and little, till at length the
eye being placed somewhere, suppose at Q, the OBJECT appearing extremely
near, begins to vanish into mere confusion. All which doth seem repugnant
to our principles, at least not rightly to agree with them. Nor is our
tenet alone struck at by this experiment, but likewise all others that
ever came to my knowledge are, every whit as much, endangered by it. The
ancient one especially (which is most commonly received, and comes
nearest to mine) seems to be so effectually overthrown thereby that the
most learned Tacquet has been forced to reject that principle, as false
and uncertain, on which alone he had built almost his whole CATOPTRICS;
and consequently by taking away the foundation, hath himself pulled down
the superstructure he had raised on it. Which, nevertheless, I do not
believe he would have done had he but considered the whole matter more
thoroughly, and examined the difficulty to the bottom. But as for me,
neither this nor any other difficulty shall have so great an influence on
me as to make me renounce that which I know to be manifestly agreeable to
reason: especially when, as it here falls out, the difficulty is founded
in the peculiar nature of a certain odd and particular case. For in the
present case something peculiar lies hid, which being involved in the
subtilty of nature will, perhaps, hardly be discovered till such time as
the manner of vision is more perfectly made known. Concerning which, I
must own, I have hitherto been able to find out nothing that has the
least show of PROBABILITY, not to mention CERTAINTY. I shall, therefore,
leave this knot to be untied by you, wishing you may have better success
in it than I have had.'

30. The ancient and received principle, which Dr. Barrow here mentions as
the main foundation of Tacquet's CATOPTRICS, is that: 'every visible point
seen by reflection from a speculum shall appear placed at the intersection
of the reflected ray, and the perpendicular of incidence:' which
intersection in the present case, happening to be behind the eye, it
greatly shakes the authority of that principle, whereon the
aforementioned author proceeds throughout his whole CATOPTRICS in
determining the apparent place of OBJECTS seen by reflection from any kind
of speculum.

31. 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.

32. This case is much the same as if we should