T3D Re: From Euclid to Wheatstone

[Peter Homer <P.J.Homer@xxxxxxxxxxxxx>]

>(continued from t3d digest 366, 26 Sep 1998)
>
>* Binocular Vision
>In Euclid's "Optics" there are three theorems on binocular vision,
>and two more which have additions on seeing with both eyes.
>I first quote the famous theorem 25 entirely, including its proof:
>
>"When a sphere is seen by both eyes, if the diameter of the sphere
>is equal to the straight line marking the distance of the eyes from
>each other, the whole hemisphere will be seen."
>
>                   G                         E
>                 .-|-.-----------------------|
>              .-'  |  '-.                    |
>            .'     |     '.                  |
>           /       |       \                 |
>          |       A|________|________________| Z
>          |        |        |                |
>          \        |        /                |
>           '       |       .                 |
>            '-.    |    .-'                  |
>               '-._|_._'_____________________|
>                   B                         D
>
>                                                 (ASCII circle copied)
>
>"Let there be a sphere, of which A is the center, and on the sphere
>let the circle BG be inscribed about the center A, and let BG be
>drawn as its diameter, and at right angles from B and G let lines be
>drawn, BD and GE, and let DE be parallel to BG, and upon this (DE),
>let D and E represent the eyes.
>I say that the complete hemisphere will be seen.
>Through A let AZ be drawn parallel to each of the lines, BD and GE;
>then ABZD is a parallelogram. Now, if the inscribed figure is
>revolved and then restored to the same position whence it started,
>AZ remaining in its place, it will start from B and B will come over
>G, and the figure formed under AB will be a circle through the center
>of the sphere. So a hemisphere will be seen by the eyes D and E."
>
>
>* Discussion
>This is ALL Euclid EVER wrote on vision with both eyes. The not quoted
>proofs are again purely geometrical like the proof of theorem 25.
>
>Euclid was a great mathematician, "The Elements" is after the Bible
>the most translated and printed work *of all times*, as model for
>logical reasoning it has probably influenced western science more
>than any other work. Therefore it is a little shocking to see that
>theorem 25 is in error (and 26 inaccurate).
>
>For, let there be an interpupillary distance of 65 mm, and let a
>globe of 65 mm diameter be held so that the plane of the equator is at
>eye level, then exactly half of the equator is  seen with both eyes.
>I say you cannot see the North pole nor the South pole.
>For looking around the object works only horizontally (with
>horizontal eyes). So a hemisphere will not be seen.
>
>This is not to blame Euclid (would be rather ridiculous after 23
>centuries !), but the error is significant with respect to our
>understanding of the evolution of space concepts.
>It is clear that Euclid knew that in looking at rounded objects the
>left eye sees a the part of the surface that the right cannot see,
>and vice versa. So two eyes see MORE of the surface of such an
>object. Note that theorem 23 and 24, which directly precede the
>first 'binocular' theorem, also are on seeing parts of the surface
>of a sphere, but with one eye.
>But seeing more of a surface with two eyes doesn't mean depth
>perception.

The version of Euclids sphere in Lenny Liptons "Foundation of The
Stereoscopic Cinema" p17 is slightly different to the above and includes a
couple of diagonals as well as horizontals and also shows that different
hemispheres are seen by each eye . The text with it also implies
stereoscopic vision  but this version is not direct from Euclid but comes
from Brewster 1856. As Lipton says Sir David Brewster attemted to discredit
the discoverer(Wheatstone) so perhaps Brewster doctored Euclids original
diagram to show a knowledge of stereo vision.
 Besides although Brewsters said "It is therefore ,a fact well known to
every person of common sagacity that the pictures of bodies seen by both
eyes are formed by the union of two dissimilar pictures formed by each". It
is probably more accurate to just say that it is obvious that the views are
different if you view with one eye then the other . A number of people may
have drawn the two views and leonardo-Da-Vinci's double and multiple
versions of the camera obscura would have done that if they were ever used
for drawing . But they would also have to invent the stereoscope or at
least discover free viewing which is what Wheatstone did. According to
Lipton the Greek physician Galen who supplied the first description of left
and right eye perspective in "On The Use Of The parts Of the Human Body" in
the second century AD. Perhaps he is the ancient Greek that the original
post refered to but 200 AD is not normaly considere ancient.
 It is interesting that he says "But if any person does not understand
these demonstrations (he is refering to Euclid) by means of lines." before
going on to his own explanation . So he himself seemed to believe that
Euclid was aware of the difference between the views of each eye.
P.J.Homer



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