T3D From Euclid to Wheatstone

[abram klooswyk <abram.klooswyk@xxxxxx>]

* Introduction.
Ray Sokolowski asked (PHOTO-3D Digest 2858, 28 Jul 1998):
>(...) the author mentions that an ancient Greek (can't remember 
>his name) was the first person to notice that each eye saw a 
>slightly different image. 
>They also mention that there were some drawings or paintings done
>before the advent of photography which displayed this phenomena. 
>Does anybody know what these paintings are?

Harold Layer's 1979 article (Ref see P3D Digest 2861, 29 Jul 1998)
gives a time line and an gives an excellent overview of space 
concepts in the history of art.
I should like to add some details on "the ancient Greek". 
(Posted here - there is no Hist-3D list :-))

Sir David Brewster was a great man in science and his virtues for
early stereoscopy are unsurpassed. However, he is the origin of some
serious errors in the history of stereoscopic depth perception that
still can be found in many articles and books today. He apparently
was not fully aware of the changes of the world image through the
centuries. It seems also that rivalry with Wheatstone blurred his
judgement in these matters. 

To be more specific, one of the most widespread errors is the idea
that Euclid knew of binocular depth perception. Two examples:
1) Time magazine once had printed on its front cover the slogan: 
   "3D - Euclid had a word for it".
2) The Web pages of the "Turing institute" (actually a commercial
company) even today have a section on the history of stereo
photography (also to be found on the otherwise so excellent 
3D-CD ROM of Dan Shelley and friends) which says: 
"In 280 A.D., Euclid was the first to recognize that depth
perception is obtained when each eye simultaneously receives one 
of two dissimilar images of the same object."

I have seen dozens of similar statements "all over the place"
(but, for the record, *not* in the Burder & Whitehouse booklet 
"Photographing in 3-D", publ. Stereoscopic Soc. UK).
The erroneous statements are all (directly or indirectly) 
based on Brewster's 1856 book "The Stereoscope" and some of his 
other writings. 

* Euclid.
Euclid, author of "The Elements" (about 300 A.D.) also wrote other
treatises, among which one on "Catoptrics" and one on "Optics". 
Catoptrics is about mirrors, I don't know if it is translated in
English, I have seen a French translation. (by Paul Ver Eecke, 
"Euclide, l'Optique et la Catoptrique", Paris 1959). 
In the Catoptrics a few propositions are on looking in spherical 
concave mirrors with two eyes, but Euclid only discusses whether 
one eye can see the other or not, in different positions of the 
eyes. He doesn't mention binocular vision (looking with both eyes 
at the same object(s)) in that treatise.

An English translation of the "Optics" was published by Harry E. 
Burton in the Journal of the Optical Society of America 1945, 
vol. 35  Nr. 5, pages 357-372, so it is only 16 pages long (A4). 
The "Optics" is in fact a treatise on perspective, it contains many
of the basic concepts which artists would need for using central 
perspective, although it doesn't specifically discuss the projection 
of images on surfaces. Also, the "Optics" is not about lenses or
prisms.
This treatise again is composed of a short list of postulates and 
several propositions (theorems) each followed by geometrical proof. 

In ancient Greece there were several theories of vision, one of them
(advocated by Plato) was that the eyes EMIT visual rays, these rays
mix with the luminous rays from the sun (or other sources), and
give the visual sensation (what exactly was meant remains unclear).
Aristotle had a different theory, but it seems that Plato's ideas
were followed by Euclid, but he doesn't mention any specific theory.

The theory Euclid adhered to seems to follow from a number of phrases.
In one postulate he says "those things upon which the vision falls are
seen". And the first theorem states: "nothing is seen at once in its
entirety", and he explains that the rays of vision diverge and are not
contiguous, but "(the object) seems to be seen all at once because
the rays of vision shift rapidly".

That eyes emit rays may sound strange to adults in this century, but 
most of us do believe in the existence of X-rays, gamma-rays, RADAR, 
ultrasound and radiowaves, all invisible and not directly percepted. 
And when an eye is hit, or with gentle pressure sideways on a closed
eye, don't you actually "see light" which has its origin in the eye 
itself? 
(Disclaimer: this is potentially harmful for the eye - NOT a joke).

Moreover, in recent psychological studies it has been demonstrated 
that many children and even college students (OK, at Ohio State 
University, but still a university :-)) believe that something goes
out of the eyes. A psychologists group at Ohio State has published 
several papers on this subject. In one study for example some of the 
questions were "when people look at something, do you think waves or 
rays or anything else goes out of their eyes?", and also "... into 
there eyes?" and "... both into and out of their eyes?".
Of 67 grade 6 children (about 11.5 years old) only 6 answered 'in', 
18 'out' and 39 'both'. Of 98 college students (mean age 21.8 years) 
58 said 'in', still 4 answered 'out' and no less than 32 'both'!
When they were told in debriefing that there exist no emissions 
from the eyes many children protested vigorously. Embarrassing 
was also that some of the *psychology* students trained to do the 
testing asked to tell them the right answers first.
(Jane E. Cottrell and Gerald A. Winer "Development in the 
Understanding of Perception: The Decline of Extramission Perception 
Beliefs", Developmental Psychology 1994 Vol 30 No 2 pp 218-228)


* On the ONE eye or BOTH eyes issue.
Several theorems of the "Optics" involve seeing different distances, 
but *none* of them mention seeing different distances with *two*
eyes. 
On the contrary, ALL theorems (or their proofs), except the
few I will quote further on, invariably speak of "the eye" (singular).
Euclid's phrasing leaves no doubt: it is always clear whether one eye
or both eyes are meant.

Some examples, also to give an idea of the treatise's contents:
(Theorem 2) "Objects located nearby are seen more clearly than
objects of equal size located at a distance". "Let B represent the 
eye and ...etc". ('Clearly' obviously means 'with higher definition',
the proof says that the closer one of two equal and parallel lines 
is seen by more rays; we would say seen by more retinal cones.)
(5) "Objects of equal size unequally distant appear unequal and the
one lying nearer to the eye always appears larger".
(6) "Parallel lines, when seen from a distance, appear not to be
equally distant from each other". "Let there be two parallel lines,
AB and GD, and let the eye... etc"
(10) "In the case of flat surfaces lying below the level of the eye,
the more remote parts appear higher." And of course:
(11) "In the case of flat surfaces located above the level of the
eye, the more remote parts appear lower." 
(36) "The wheels of the chariots appear sometimes circular, sometimes
distorted." 
"(...) but if the line drawn from the eye to the center is not at
right angles to the plane (...) the diameters appear unequal (...)"
(54) "When objects move at equal speed, those more remote seem to
move slowly". "For let B and K move at equal speed, and from the eye,
A, let rays be drawn ...etc".

A number of theorems state how much is seen of the surface of various 
objects in different cases. 
Theorem 23, on vision with one eye says: 
"Of a sphere seen in whatever way by one eye, less than a hemisphere
is always seen, and the part of the sphere that is seen itself
appears as an arc". (The Burton translation says "arc", Ver Eecke has 
"circonfÈrence de cercle", circle circumference, which seems closer 
to the Greek original which says "kuklou perifereia"). 
The theorem is proven by drawing lines from the eye touching the
sphere, the circle of all contact points cuts off a part from the 
sphere which is less than a hemisphere.
(24)  "When the eye approaches the sphere, the part seen will be
less, but will seem to be more." (A smaller amount of the surface is
seen under a greater visual angle).

Abram Klooswyk

(to be continued)


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