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Let AC, BC, Fig. 26, be two lines meeting at c, the plane passing through them being the plane of the paper, and let them be viewed by the eyes successively placed at
E"", E", E', and E, at different heights in a plane, GMN, perpendicular to the plane of the paper. Let R be the right eye, and L the left eye, and when at E", let them be strained so as to unite the points A, B. The united image of these points will be seen at the binocular centre D", and the united lines AC, BC, will have the position D" C. In like manner, when the eye descends to E", E', E, the united image D' c will rise and diminish, taking the positions D" C, D'C, DC, till it disappears on the line CM, when the eyes reach M. If the eye deviates from the vertical plane GMN, the united image will also deviate from it, and is always in a plane passing through the common axis of the two eyes and the line GM.
If at any altitude EM, the eye advances towards ACB in the line EG, the binocular centre D will also advance towards ACB in the line EG, and the image DC will rise, and become shorter as its extremity D moves along DG, and, after passing the perpendicular to GE, it will increase in length. If the eye, on the other hand, recedes from ACB in the line GE, the binocular centre D will also recede, and the image DC will descend to the plane CM, and increase in length.
The preceding diagram is, for the purpose of illustration, drawn in a sort of perspective, and therefore does not give the true positions and lengths of the united images. This defect, however, is remedied in Fig. 27, where E, E', E', E""
is the middle point between the two eyes, the plane GMN being, as before, perpendicular to the plane passing through Now, as the distance of the eye from G is supposed to be the same, and as AB is invariable as well as the distance between the eyes, the distance of the binocular
centres o, D, D', D", D"", P from G, will also be invariable, and lie in a circle ODP, whose centre is G, and whose radius is Go, the point o being determined by the forHence, in order to find the
mula GO GD= binocular centres D, D', D", D'", &c., at any altitude, E, E', &c., we have only to join EG, EG, &c., and the points of intersection D, D', &c., will be the binocular centres, and the lines DC, D'C, &c., drawn to c, will be the real lengths and inclinations of the united images of the lines AC, BC.
When GO is greater than GC there is obviously some angle A, or E" G M, at which D'c is perpendicular to GC. This takes place when Cos. A When o coincides with c, the images CD, CD', &c., will have the same positions and magnitudes as the chords of the altitudes A of the eyes above the plane GC. In this case the raised or united images will just reach the perpendicular when the eye is in the plane GCM, for since GC=GO, Cos. A=1 and A = 0.
When the eye at any position, E" for example, sees the points A and B united at D", it sees also the whole lines AC, BC forming the image D" C. The binocular centre must, therefore, run rapidly along the line D'C; that is, the inclination of the optic axes must gradually diminish till the binocular centre reaches c, when all strain is removed. The vision of the image D'c, however, is carried on so rapidly that the binocular centre returns to D" without the eye being sensible of the removal and resumption of the strain which is required in maintaining a view of the united image D' C. If we now suppose A B to diminish, the binocular centre will advance towards G, and the length
and inclination of the united images DC, D'C, &c., will diminish also, and vice versa. If the distance RL (Fig. 26) between the eyes diminishes, the binocular centre will retire towards E, and the length and inclination of the images will increase. Hence persons with eyes more or less distant will see the united images in different places and of different sizes, though the quantities A and AB be invariable.
While the eyes at E' are running along the lines AC, BC, let us suppose them to rest upon the points ab equidistant from c. Join ab, and from the point g, where ab intersects GC, draw the line gE", and find the point d" from the formula gd" – 9 F" × ab. Hence the two points a, b
will be united at d", and when the angle E"GC is such that the line joining D and c is perpendicular to GC, the line joining d'c will also be perpendicular to GC, the loci of the points D"d", &c., will be in that perpendicular, and the image DC, seen by successive movements of the binocular centre from D" to c, will be a straight line.
In the preceding observations we have supposed that the binocular centre D", &c., is between the eye and the lines AC, BC; but the points A, C, and all the other points of these lines, may be united by fixing the binocular centre beyond AB. Let the eyes, for example, be at E"; then if we unite A, B when the eyes converge to a point, A", (not seen in the Figure) beyond G, we shall have G A" and if we join the point A" thus found and c, the line 'c will be the united image of AC and BC, the binocular centre ranging from " to c, in order to see it as one line. In like manner, we may find the position and length of the
image A"c, A'c, and Ac, corresponding to the position of Hence all the united images of A c,
the eyes at E" E and E.
BC, viz., CA", CA", &c., will lie below the plane of ABC, and extend beyond a vertical line NG continued ; and they will grow larger and larger, and approximate in direction to CG as the eyes descend from E" to M. When the eyes
are near to M, and a little above the plane of ABC, the line, when not carefully observed, will have the appearance of coinciding with CG, but stretching a great way beyond G. This extreme case represents the celebrated experiment with the compasses, described by Dr. Smith, and referred to by Professor Wheatstone. He took a pair of compasses, which may be represented by ACB, AB being their points, AC, BC their legs, and c their joint; and having placed his eyes about E, above their plane, he made the following experiment:" Having opened the points of a pair of compasses somewhat wider than the interval of your eyes, with your arm extended, hold the head or joint in the ball of your hand, with the points outwards, and equidistant from your eyes, and somewhat higher than the joint. Then fixing your eyes upon any remote object lying in the plane that bisects the interval of the points, you will first perceive two pair of compasses, (each by being doubled with their inner legs crossing each other, not unlike the old shape of the letter W.) But by compressing the legs with your hand the two inner points will come nearer to each other; and when they unite (having stopped the compression) the two inner legs will also entirely coincide and bisect the angle under the outward ones, and will appear more vivid, thicker, and larger, than they do, so as to reach from your hand to the remotest object in view even