the movements of the human eye

THE MOVEMENTS OF THE HUMAN EYE. BY

PROF., DR. PHIL. K . KROMAN.

If we undertake an investigation as to what the more important physiological works teach us concerning the movements of the eyes, a subject which cannot be considered especially difficult-in any case in so far as its foundamental features are concerned - we shall be surprised to meet with an overwhel- ming number of rather indefinite and somewhat unsatisfactory opinions, which at least superficially seen often appear to contradict each other. Even Helmholtz, who has dealt with the question in both editions of his BPhysiological Optics<, as well as in his three-volume ))Scientific Treatisesa, has been - as far as I can judge - somewhat unfortunate in his choice of expressions, while several of the other writers on this topic have been satisfied by simply repealing his views without any further support. That the eyesight can be directed upwards and downwards, to the right and to the left, as well as it can make all the intermediate movements, on these facts there can, of course, be no disagreement; even Listing’s and Donders’ laws seem to be generally accepted, although on the other hand, these two laws are not always interpreted in the same way. Thus, if it comes to a question, for instance, of the axial rotation of the retina, neither clearness nor final agreement of opinion can be said to exist.

A n attempt, therefore, to gain more accurate information on this subject, does not appear to be at all superfluous.

55

I. In order to facilitate the solution of the problem there can

be no objection to our introducing certain simplifications. For the present let us only consider the single eye, for example the right one. We imagine it as a spherical object tightly enclosed in the eye-socket in such a manner that it is able to turn round about the common centre as were this a fixed point. Through the eye-socket we will draw three rectangular axes, OA to the right, OB upwards, and OC forwards. If the head be kept upright and imnioveable these axes may be considered to be fixed in space. If the sight be directed straight ahead, we shall consider the eye to be in its primary position, and will give it three axes, in this position coincident with those aforementioned, denoted OX, OY and OZ, of which we will take the latter as the polar axis of the eye, coincident with the visual line, which latter may be regarded as passing through the centres of the retina and the cornea.

A s is well-known the movements of the eye take place by means of six extrinsic eye-muscles, the four so-called recti, the external and the internal, the superior and the inferior rectus, and the superior and the inferior oblique. It is only in a somewhat imperfect manner that they correspond in pairs to the three axes. As, however, they constantly work together, certain simplifications are made possible concerning them.

For further consideration let us imagine a larger sphere with the eye-socket’s centre and axes, but with a radius R of 10-20cms. At right angles to the extreme end of its radius OC we will imagine a plane, the ,card(< or ,screen((, upon which, by means of central projections from 0, we can depict the different results which we desire further to illustrate.

XI. We commence our investigations with a little experiment.

We hang upon the wall a quadrilateral sheet of light grey, dull cardboard about 60cms square with the centre at the level of the eyes. Upon this are marked a number of hori-

56

sontal aiid vertical lines, of which the two passing trough the centre are termed the x-axis and the y-axis. A t the centre M we affix a small horizontal and a small vertical strip of red or green coloured paper, and then we take our stand in front of the sheet in such a manner that when in an upriglit position we gaze a t right angles towards the centre, M, and at such a distance that we are able to see up to approximately 45O in an upward, downward or sideward direction, without the visual line passing beyond the sheet.

I t is important throughout the whole expeririient to keep the head immowable, also when the eyes are turned from their central position out towards the edge of the sheet.

After having gazed for some twenty seconds or so at a point in the middle of the small vertical rectangular cross in M , its image will have been formed upon the retina, aiid if we then change the line of sight to the right or the left, upwards or downwards, without, however, changing the position of the head, we will observe -- when the eyes cease moving - a negative after-image, which in the four instances nientioned, provided the axes have been followed, will also he a vertical, rectangular cross. If this should not be the case then we are not in the correct position in relation to the sheet, and we must then either raise or lower the head soniewhat, or move the sheet a little until the experiment is completely successful.

On the other hand, if after having fixed the eyes for some time on the cross we direct the line of sight into one of the four quadrants of the sheet, that is to say simultaneously sidewards and upwards or downwards, an altogether different result will he obtained. The two arms of the cross will now be found to have more or less approached each other. In the upper right-hand quadrant, for example, the horizontal arm will be observed to hqve lifted its extreme end, while the vertical arm will have become inclined outwards somewhat towards the edge of the sheet. In other words the horizontal arm has been turned a few degrees to the left, while the vertical arm has been turned a few degrees to the right and it follows, therefore, that the image of the cross is neither upright

57

nor rectangular; effects corresponding symmetrically with this will be obtained in each of the other quadrants to which the sight is directed. The two arms of the cross will more and more approach each other the further the sight is directed from the centre of the sheet. Fig. 1 illustrates these results of our experi- in e n t s.

A third result will be obtained if we rotate the cross in M so that one of its arms lies in the direction in which we intend to turn the sight, the other arm being

rt: Fig. 1 .

at right angles to that direction. In this case neither angular nor directional alterations will take place ; the cross will remain rectangular. Fig. 2.

Finally, a fourth result may he added to the three we have already described. If we carry out each of these experiments several times, we will discover that it does not matter by which way we go outwards from 31 to a given ex-

Fig. 2. treme position. For each extreme position, as long a s the cross in JM remains

unchanged, there is a corresponding constant position of the arms of the cross quite independent r f the manner in which this extreme position is reached.

)( I 111.

In endeavouring to find the explanation of the phenomena decribed above it will be advisable to bear in mind two important points.

It is, for example, possible that the projection of the after- image on the sheet or screen is quite capa~ble of explaining everything we saw, and i t is also possible that the eye-movements concerned have played a more or less essential rdle. The last supposition, however, is the least probable as it is rather difficult to conceive the movement of the eye at the same time turning the one arm of the cross to the left, and the other in the opposite direction. It will be safe, however, to investigate both suppositions.

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Let u s first look at the conditions governing tbe projection of the image. Curiously enough, most inrestigators appear to have given these conditions no attention at all or a t most very little attention. It has obiously been taken for granted that it was the retinal-picture from tbe centre of the screen with which we had to deal, and that it was, therefore, hardly possible for projcction to take place. This conclusion, however, has been arrived at too hastily. If I construct a cross of wood o r cardboard, and an assistent stands before a door with the cross in his hand in such a way that ils frontal-side faces me, while I myself stand in a position in which I gaze obliquely a t the door, then I shall not see the cross projected o n the door in any altered form: I notice a t once, and I notice all the time, that thc assistent is holding it in his hand, that its front side is turned towards me, and that I - if one may so describe it- see the cross a s it actually is. With regard to the small after-image aforementioned, however, the facts are quite otherwise. In that case I see it clearly and distinctly as a spot or an iniage on lhe screen. As I a m not accustomed to observe such pictures hovering about in the air I automatically place it 011 the screen, and just as automatically I perceive what ensues upon such a placing. It will be of no use to raise any objection to the premiss that we hare a t one and the same time an image of the upright and reclangular cross 011 the retina, as it is well known that the image on the retina and what one actually sees are quite different things. If, e. g., a person stands before me, the piclure I have upon my retinae is that of two persons standing upon their heads, though what I see in reality, is one person who stands quite naturally on his feet.

It seems, therefore, quite safe to assert that a n investigation of the conditions under which prqjeclion takes place is most advisable. The only writer who has given this matter some attention is W. Wundt, though even he - so far a s I can gather- has not paid sufficient attention to it when formulating his theories.

Let us now, therefore, proceed further into the question of projection.

We will imagine the aforementioned globe with a radius H

59

equal to 10ceiitirnetres, and a t right angles to its OC axis in M , we will imagine our sheet or screen placed. If the eye is in its primary position, that is to say, if the visual line is in O M , the z-axis of the screen will be in a line with the horizontal meridian of the eye, while its y-axis will cover its vertical meridian.

Let us conceive the suggested vertical and horizontal lines on the screen as being fixed for every tenth degree of the sphere in such a way that on the screen their respective distances from the axes will be R tan (10, 20, 30, 40 45)O. Finally, we will imagine round the visual line, the OZ axis of the eye, a narrow, circular cone with its apex at 0 and possessing a small angular radius i. If the eye is in its primary position the cone will cut the screen in a small circle round M , and the two arms of the cross will correspond to a horizontal and a vertical diameter in the circle. But if the line of sigbt be directed towards a point P, for example, in the first quadrant of the screen, then the cone will cut the screen obliquely, and the circle will be converted into an ellipse. This is illustrated in Fig. 3 and 4 .

In Fig. 3 OM represents the line of sight in its primary position, OP the line of sight after it has performed a turn through the angle u. The distance M P is equal to R tan u, and this we will call e. The line M P is the screen as seen from above. We observe how the image of the small sight-cone has had its angular radius ,8 in the direction of e prolonged to a, i. e. nearly multiplied by sec u, while the distance acros e remains unchanged.

In Fig. 4 the effects of the aforementioned prolongation are indicated. If 4 forms the angle s with the horizontal x-axis of the screen, the two arms of the cross facing the quadrant will in M form the angles s and t (= 90°-s) with e. If now all the distances in the direction e are multiplied by sec u,

M Q

0 Fig, 3.

60

both these two angles will also have their cosines multipled by sec u, and will in consequence become two lesser angles,

m and n, while tan m will equal cos 11 tan s, and tan IZ will equal cos u

tan t , in that cos u = --. The two

origina! angles have thus been dimi- nished by the angle I and the angle f, i. e . by precisely such two angles which indicate an elevation of the horizontal and an outward inclination of the vertical a rm of the cross.

If we imagine the angular radius of the small sight-cone i as a final quantity, then our results will only be approximate; but if on the other hand we - as what follows will justify - regard i as an unlimited small quantity, our results will be found to be quite precise *).

For the sake of brevity let us put:

18 a

Fig. 4.

1 cos 11 = - = z . cf

Then we obtain the ensuing equations for the two last-named angles, I and f :

I n precisely the same manner and wilhout any difficulty we find a symmetrical correspondence so far as the three remaining quadrants are concerned.

Moreover, it will be noted that these two equations yield results which qualitatively correspond accurately with all the observations which we have made in regard to the after-image.

*) The exact equations are as follows; a = R sec ZI sin i sec (u + i), p - R

sec 11 tan i, or 2- = ____ , and as i may be considered vanishing

it follows that cos i becomes 1 while cos ( 1 1 + i) = cos 11, yielding as a

result

cos i p cos ; ( I + i)

= sec 11 and f? =cos 11. B

If the line of sight was moved from A2 to the left or righl along the x-axis no change in the angles toDk place. But in this case the angle s equals 0, and the two equations then give as well tan I as tan f = 0. If the visual line followed the y-axis neither did any diminution of the angle between the arms of the cross take place; but then is L s = 90° and the equations give also then as well L I as L f = 0. If the cross was placed in such a manner that one of its arms ran along e and the other across, there was also no diminution. But also then the two equations give 1 and f equal 0. When the last named arm OP the cross falls across e it forms a right angle with Q . There will then be neither cosines nor prolon- gations of cosines.

And as the projection for each individual point is obviously decided by the point itself, without regard to the route by means of which the said point is reached, it follows that the last-named results of our observations can be explained as a simple effect of this projection.

In order to be able to decide whether this projection also corresponds quantitatively with the results SO far achieved we must calculate the extent of the angular changes involved, as well as draw the oblique or slopiiig lines on the screen.

The calculation of the angle f which follows, may be regarded a s sufficient.

Table of L f.

20"

30°

40°

450

100 ~ 200 i 300 I 4 0 0 - I

l o o I 0°,86 1 1°,74 I 2C',G5 I 3O,61 ~ 4",11

1 O,74 1 3",36 5",14 1 7O,00 ~ 7",99

2"65 5O,14 7@,25 9O,95 1 11°,41

3O,61 7O,00 9O,95 12O,20 1 14O,07

4O,ll I 7O,99 , 11°,41 14O,07 15O,00

I

I

A s a matter of fact the ingress-figures should properly read R tan loo, R tan 20° etc, as they correspond to the various points of intersection between the horizontal and the vertical lines on the card. It will be seen, moreover, that half of the table is not required, as the ingress loo, 20° yield precisely

62

the same results as the ingress 20°, loo, etc. In the same way it will not be found necessary to deal with the other quadrants as the conditions in them correspond symmetrically to those in the one quadrant we are considering. Neither need we estimate I , as if we turn the screen one-quarter to the left, then I will become the vertical arm in the upper left-hand quadrant and will, therefore, have the same values as f in the original upper right-hand quadrant. Only in a left quadrant the outward inclination is naturally to the left. while in a tight hand quadrant it is to the right.

Bu t if we draw the sloping lines involved as accurately a s possible on the said screen and if we thereafter repeat- also as accurately as possible - the after-image experiments which we have described, endeavouring to place the head and eye in just the correct position in relation to the screen, then most probably we will discover that the conditions under which projection takes place correspond accurately, both qualitatively and quantitatively, with the results of our observations, and thus exhaustively explain these to us.

Of the many different persons whom 1 have caused to make these experiments none have expressed any doubt as to a complete agreement on these points between theory and practice.

IV. I f we examine a card upon which are a number of the small

oblique lines which we have described earlier, we at once arrive at the conclusion that if they were present in sufficiently great num- bers they would form a multitude of continous curves situated symmetrically round the x- and y-axis. These curves, have often been regarded as hyperbolae. This, however, as we shall see, is not quite the case. Each curve nevertheless constitutes the one branch of a hyperbola, whose corresponding branch pas- ses through 41 and does not concern our problem. This can be prooved as follows.

According to page 59 and fig. 5 for the angle n between an arbitrary small oblique line and its radius vector g we have

Fig. 5.

- M [dr

63

e cis R de i/R' + . e s

tan n.= -- = cots cos u = cot s ____

or d.cos s Rde = tall sds = - --. F m q cos s

____.

If now we put + 4' = p , then p2= IP + e 2 , pdp = ede, p 2 - Rs = es , and we get

RpdP =-.- RdP =- Rdp =---- 4 (IP 3dP eaw+T p2-RR4 ( p - l I ( ) ( p + H ) p - R p + R

and, 1 denoting the natural logarilm,

or

lP--R=-lcoss+K and F+T- R cos s = k, e e

If .the coordiriates z and y of point P be fully introduced we obtain by

-__ e = i/xz + yz,

k*y' = (1 - k') X* - 2 R k ~ ,

That is the equation for a round the x-axis symmetrical 2Rk curve of second order, which for y=O has x=O and x=-

l-kz' Rk Rk itscentralpoint, C,inx=---, and its first semi-axis a=-- -

1 - kr 1-k'

64

If C be taken as the origin of the coordinates, and if we term the new abscissce z, then z = z1 + - Rk and from the

fourth of our equations we get 1 -k'

But this is the equation for an hyperbola round the z-axis with its centre in C and the semi-axes:

Rk a , 1 - k'

1 U k becomes - V U f f h e while ils eccentricity e =

Its left branch has its top i n 111, and has no connection with our problem. Its right branch, which corresponds to the sinall oblique line which we have used, has this line as a tangent in the point I' and has its top in z = 2a.

We need, therefore, only to choose a series of values for 2a, corresponding, for instance, to the 2-distances froni M: R tan (10, 20,. - .45)O in order to obtain a share of curves over the + z-axis. The remaining axes (- z, + y, - !I) may be treated in the same manner.

Froni the sixth equation we get:

r - R 2a

k = - -

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From the fourth equation we get:

But according to equations 6 and 8 is

that is r - R y 2 = 0 xz - 2 ax - ~ 2R

and

By introducing into this formula y = R tan (20, 30, 45)O or by using more 9-values, and thereby computing x we obtain a number of points through which the curves in question can be drawn and thus the accuracy of direction of the iso- lated oblique lines be controlled.

If we desire to find the curve which touches the outward- sloping oblique line for a certain point (x , y) we first ascertain by means of equation (3) its k, then with (6) its a, and then we are able to draw it, with the help of (9), which here may advantageously be transformed to :

As all the oblique lines are sloped outwards in relation to two halfaxes of the card, the formula (10) may be used generally.

If we do not desire to draw the curve through the given point (x, y ) , but only wish to determine the angle of outward- sloping it will possess at this point, that is the angle f of page 61, we have from analyt. geom. the well-known formula:

ba x tan u = cot f = - Y

aL 11 here also in accordance with (5)

5

66

or

We have, therefore, only to use equations 3 and 6 to deter-

If we do not wish to determine a but only f it will be mine k and a.

simpler to employ the following:

(12) k2y

-~ - - kZ Y tan f.= _I 1 - ka x - Rk ( l - k k e ) x - R k ' 1 - k 2

V. On a basis of the results so far gained we now propose to

consider in more detail the eye itself and its movements. It may he taken for granted that all the six extrinsic ocular

muscles play a part in every one of the eye-movements, and as it is not within our province to investigate the contribu- tion which each individual muscle makes, we can, without any drawback, for the moment simplify the question and, for instance, accept it as a fact that the three pairs of muscles correspond to the rotations about the three axes OA, OB and OC.

At first sight, however, it would appear that such an assump- tion leads us to a somewhat surprising result. The four recti should serve to give the eye almost a kind of Cardan's suspen- sion, or, when taken together, act as a universal joint. Most of us have noticed that if two bars be united by such a joint, it is possible to hold one of the bars firm while at the same time moving the other in all possible directions within a hemisphere or more, just as in fact a compass-cup is able to rock in all directions. According to this the four recti should be able unaided to perform all the requisite and beneficial movenients of the eye, and the oblique muscles should act as a kind of superfluous and unnecessary adjunct only able

67

to carry out rotations round the visual axis itself, serving altogether to disturb our sense of sight. It would almost appear as if nature, which, according to the general conception, in all its operations is continually striving towards suitability, had in this particular case adopted an absolutely opposite method. This hardly seems credible, and it is, therefore, highly important to take care, that in some way or other we have not perhaps formed a wrong conception of these phenomena.

It seems appropriate, therefore, to begin with a more detailed investigation of the term rotation or iurning, as

this confusing term manner is very in often the literature used in a of vague the move- and ;? __.___ c ments of the eyes.

Let AB, fig. 6, be a thin bar, which rotates upon its vertical axis at a constant angular speed w, for example towards the right, (positive), as seen from B. DC is a similar bar,

with AB, that we can give the angle u between them, at will, any number of degrees between 0 and 180. If u equals O o then it is obvious that DC will possess the same rotation round its vertical axis as AB has round its axis, while if u equals 90° it follows that DC will rotate round one of its transverse axes, but not at all of course round its vertical axis. If u, however, becomes an acute angle, what will then happen? As DC con- stantely turns the same side toAI3, it might be assumed that it will now possess the same rotation as AH, and this is correct in so far as in reality DC, with an angular speed w describes a conic surface round AB. But if we wish for information more particularly concerning the rotation of DC about itself or its own longitudinal axis, the answer must be a different one. If, for instance, we imagine a small circular disc EF fastened at right angles to DC at C, and if its radius r equals CE, then it follows that the point E - the extreme end of r - will remain 'stationary during the rotation of AB, while the centre of the disc C, will be rotated inwards to us at a speed of o r cos u, i. e. at an angular speed of w cos u. This is, however, merely a different expression denoting that

which by means of a joint in D is so connected A Fig. '.

5*

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the disc will rotate round C with the positive angular speed w cos u, and this will, therefore, be the actual rotation of DC about its longitudinal axis. If the angle u equals Oo, then the cosine of u will be 1, and the rotation of DC round its longitudinal axis will also be as before, that is w, while if u equals 90° the cosine of u will be 0, and again we have the same result as before, namely that no rotation round DC’s longitudinal axis will take place.

If the angle u is more than 90°, then the cosine of u will be negative, and quite the same will occur in regard to the longitudinal rotation ofIlC, as this now forms an acute angle with AB as seen from A, and both turnings become left- rotations seen from this point. Helmholtz has called such a rotation round the longitudlnal axis, where the thin bar entirely and constantly remains, or could remain, in its own place, a wheel-rolation. Using this term we now can put forward the following law:

If a straight line rotates round a fixed axis at right angles to it, then the rotation takes place in a plane, and there will he no wheel rotation of the line; if, however, the line forms an oblique angle with the axis, it will desribe a conic surface round the latter, and obtains at the same time a wheel-rotation, determined by the turning and the angle, w cos u. For the sake of brevity lhis may be re-stated as follows: If a straight line rotates with the angular speed w round a fixed axis, with which it forms an angle u, then its wheel roiation will be w cos u.

VI. We will now return to the four straight muscles in order

further to investigate what they wsuld be capable of carrying out, if .they were alone. We will regard the plane AOC of the eye-socket - or rather the larger globe’s AOC-plane - as the equator-plane with OB as its polar axis; then, with the aid of the well-known geographical terms longitude and Itrfitzzde, we will be able in a brief and concise manner to determine each of the positions of the visual line. Its primary position thus corresponds, for example, to the point of intersection between the equator and the zero-meridian. Let us term the northern latitude + b and the southern

latitude - b, the right-longitude + 1 and the left-logitude - l * ) . We will consider the rotations about the axes OA and OB from the outer ends of the axes, buth the eventual wheel-rotation of the visual line 02 from its inner end 0, and we will call rotations to the right positive and those to the left negatiue. For the present it will not be necessary to go beyond the sphere to the screen, although such a step would be very easy, as 1 on the sphere will simply become R tan1 on the screen, and b on the sphere = Rsec 1 tan b on the screen. A circle of latitude on the sphere will therefore be represented by a curved line on the screen with its distance from the equator increasing proportionately with sec I; however, we need not deal furlher with this line, as the above-mentioned formula R sec 1 tan b obviously places us in possession of everyone of its points.

If the line of sight through one or other way has turned from the primary position to a point ( I , b) on the sphere, we will attempt to ascertain how far two rotations round OA and OB may lead to this position. If for instance, a turn of an angle - b be made round the A-axis it will lead up to a point in latitude b a t longitude 0, while a following turn of an angle 1 about the B-axis will bring the line in the desired place. The first rotation took place at right angles between the A-axis and the line of sight and did not, therefore, give any wheel-rotation to the line. But during the second turning the line possessed an angle of (90 - b)O with the B-axis, and it has thus got a wheel-rotation 1 cos (90 - b)O = I sin b, as seen from its outer extremity, Z, or - 1 sin b, as seen from 0.

If the desired position had been ( I , - b), in the second quadrant, it would have been necessary to begin with a rotation, b, to the right round the A-axis, which would have pro- duced no wheel-rotation. But the subsequent rotation 1 round the B-axis under (90 + b)O between the axis and the line of sight gives a wheel-rotation 1 cos (90 + b)O = 1 sin (- b), as seen from Z, or - 1 sin (- b) = 1 sin b, as seen from 0.

*) We avoid using the terms east and west longitude as, contrary to the geographer we regard the sphere from its interior.

70

1st Quadrant . . . . 2nd D . . . . 3rd n . . . . 4th D . . . .

Using precisely the same methods it now becomes possible to investigate, what takes place in the remaining quadrants. We will find that the line of sight in the third quadrant gets a wheel-rotation, as seen from 0, of - lsin b, and in the fourth quadrant one of 1 sin b.

These results are collected in the table which follows:

- b + 1 - 1 sin b t b + I + I sin b t b - 1 - 1 sin b - b - 1 + 1 sin b

Wheel Rotation Round OA Round OB I of oz

In order to facilitate the study of the movements of the eyes a number of ingenious instruments have been contrived, although many of them do not appear to have been of much practical value. A simple little apparatus, however, which anyone can construct for himself, will serve to elucidate and confirm some of the results which we have here put forward. See fig. 7.

Upon a small block of wood a little gallows is erected, in which a ring can be rotated round a vertical diameter, and in the ring a disc can be turned round a horizontal diameter. This disc at its central point, which coincides with that of the ring, is penetrated at right angles by a thin bar repre- senting the line of sight, and upon this bar at its furthermost extremity as seen by the observer, and at

Fig. 7. right angles to it, is a cross-bar fixe'd, which can be adjusted either

vertically or horizontally,when ,the visual linecc is in its primary position.

If we now hold this little apparatus up before the eyes, and rotate the bar with its vertical cross-bar from the

7 1

primary position outwards towards one or other angle of latitude and longitude, the cross bar will clearly reveal that the bar has got a wheel-rotation similar to those we have already mentioned. This will be readily understood. Because if the bar be raised an angle b from the primary position, then the cross-bar will point an angle b obliquely inwards towards the vertical axis of the ring, the B-axis, and if further-more the bar be rotated an angle 1 sideways it will describe a conic surface round the B-axis, while the cross-bar will describe anolher conic surface about the same axis and thereby clearly demonstrate, that the bar, i. e. the visual line, has got a wheel-rotation.

From this we see that while the four presupposed straight eye-muscles are able, without help, to turn the visual line in all directions (within a certain area), they can, however, only do this at a cost of giving the line of sight, including the retina, and therefore the complete eye, a very considerable wheel- rotation. And if we exchange the four artificial muscles by four more correct ones, the result will remain rather unchanged.

It would seem, therefore, that the oblique eye-muscles are scarcely the superfluous and hurtful accessories which - in accordance with our preliminary suppositions - we had almost considered them to be. In fact, we may now almost take it for granted that they exist for, in a greater or lesser degree, to act against, or even entirely to prevent, the unfortunate wheel-rotation. In order to obtain further information on this point it is necessary to consider this said wheel-rotation somewhat more accurately.

It is evident from the formulae that this wheel-rotation in- creases as well with 1 as with b. The eye, it is true, under normal conditions only performs small turnings, as most people prefer to turn the head to moving the eye. It may, however, at the cost of some effort, raise I as well as b to an angle of about 45O, corresponding to a wheel-rotation of approximately 320. When 1 = 400 and b = 30° there would be a corresponding wheel-rotation of 20 O. This implies, that quite considerable wheel-rotations might at times occur.

72

It will be swings from quadrant into

observed, moreover, that this wheel-rotation positive to negative or vice-rersa for each new which the visual line is cast. Just as this wheel-

rotation is zero for 1 or b = 0, and consequently along all the four axes & 1, k b, or on the screen x, y, thus for a constant distance from the primary position, it will reach its maximum in the middle of each quadrant. Especially it should be noted that it is negative in the first quadrant, positive in the second quadrant, etc. as very significant con- sequences result herefroni.

This-wheel rotation is namely not only in the first quadrant, but also in all the other quadrants, opposed to the turning which the projection through the angle f gave to the vertical arm of the cross, and at the same time so considerable that if it actually took place it would completely alter the observations which the after-image experiment yielded us. We may, therefore, immediately take it for granted that in reality such a rotation does not occur.

The oblique muscles must have counteracted, nay, even totally prevented this rotation. Hence we perceive both the use and the necessity of these muscles.

And it should especially be noted here that just as certainly as we are able to prove that the after-image experiments gave us precisely the results we expected in accordance with the laws of simple geometrical projection under the presupposilion that the eye carries the after-image unturned outwards into the field of vision in order to project it on the screen, just as certainly we must admit that the said rotation has been totally annihilated.

Only the one experiment, that an arbitrarily-directed strip, placed in M, of which the after-image is carried out in the same direction, will give a projection-image of precisely the same direction. proves decidedly, that the retina has not in its extreme position been turned the least round its axis.

Many physiologists assert that the visual line is able to turn from the primary position out into the field of vision along the x- or y-axis, without producing any wheel-rotation of the retina, but along no other radius unless wheel-rotation

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occurs. This assertion, however, is contrary to the above mentioned result, for, if the retina had reached its extreme position wheel-rotated, the projection-image of the strip had also been turned.

According to Listing’s law, iT the visual line be radially directed from the primary position outwards, the movement takes place as if the line had turned on a fixed axis at right angles to itself and the plane in which it moves; and this law expresses precisely what we have maintained above. For, if there had been the slightest wheel-rotation of the line of sight, it is obvious that an axis at right angles to it would also have rotated and been unable to remain fixed. Of course we refer here only to the natural and customary movements of the visual line, and of the eye generally. If we try how far we can force the visual line out from ill, then it is quite possible that disturbances may occur in the cooperation of the muscles.

The just menlioned little experiment, therefore, proves the correctness of Lisling’s law.

In the same way the repetition of the image-experiments proves the correctness of Donders’ law, which states that the results of the image-projection at each special point will be precisely the same, no tilatter at which route the extreme position has been reached.

A curious contradictory theory in regard to the movements of the eyes has gained wide currency. A number of physiologists who all believe in the laws of Listing and Doiiders’ proclaim at the same time, that the visual line may certainly move radially from M to an arbitrary extreme position, and, in accordance to Donders’ law, also through other ways without getting any wheel-rotation. But nevertheless they are of the opinion that such a wheel-rotation will always occur when the sight is directed from one outer position to another or, as it is sometimes termed, from a secondary to a tertiary position. This is a very curious assertion, inasmuch as it is clear that the visual line cannot arrive at any position outside M without passing through innumerable intermediate ouier posilions, whether it moves radially forward, or it follows

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other lines. This belief concerning the wheel-rotation cannot, therefore, be maintained without abandoning both Listing’s and Donders’ laws, and doing this we clash with the above-named experimental results. I t would, moreover, be opposed to what we generally assume regarding the great adjustability of our muscles, if we assumed that the oblique eye-muscles were only in such an inperfect condition to carry out the func- tions which we may cluite naturally assigne to them.

Only one single phenomenon which has most decidedly played a prominent part in originating the just mentioned error may still briefly be commented, although it has already been treated before.

Sometimes a conception has been put forward which has been termed the irregular, spurious, or secondary wheel-rotation (e. g. 0. Zofh in Nugel’s Handbuch, vol. 111, indirectly originated by Helmholtz) :

If the sight moves radially (corresponding to a great circle on the sphere, or a straight line on the screen) from M outwards to the right, with a pitch, u, then, in accordance with Listing’s law, the line of sight will rotate round an imaginary fixed axis in the AB-plane of the eye socket at right angles to the line of sight and its plane of movement, in other

, words, parallel with the diameter C C , of the retina (fig. 8) which forms an angle (90 -u)O

c@c with the original horizontal diameter of the retina H H , . But while the angle C O H at the beginning of the movement is situated in a vertical plane, it will, after a turning of the line of sight through no, be situated in a plane which has been rotated no, round the .axis C C , , which latter has retained its

direction. The horizontal diameter H H , of the retina is therefore now situated at a smaller angle vertically above C C , , and has, so far, rotated round 0, although the line of sight has had no ordinary wheel-rotation.

Nearly all these statements are correct, but nevertheless we have here got introduced a foundamental error.

HI

V, Fig. 8.

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As the retina is situated at right angles to the line of sight it is obvious that it will incline whenever the visual line inclines. It is most important, however, to note here that the diameter CC, of the retina, during the whole of the movement, has retained its direction, just as the angle COH on the retina has preserved its magnitude, and has not been changed in the least by the movement, consequently like every other radius of the retina has had no rotation round OZ. It is only when we arbitrarily project the angle C O H on a vertical plane, that i t is exchanged for another angle. But this is in no way different from the results which we put forward when we ccnsidered the question of the projection of the after-image. H H , and VV, (fig. 8) together correspond to the vertical rectangular cross in M , the image of which, it is true, when carried obliquely outwards in a sideways direction and projected upon the upright screen, lost both its vertical position and its rectangular shape. This fact, however, has no connection whatever with wheel-rotation. It is wholly and entirely a phenomenon of projection, and as such quite variable, being in fact dependent on the surface of projection we choose, or have given to us.

There is, therefore, hardly any real ground for assuming any wheel-rotation cooperating in our sensation of sight, as long as we keep within the bounds of the natural, customary and normal movements of the eye. And this supposition holds good not only for the movements of each single eye, upon which we have concentrated our attention, but also for the movements of both eyes in unison, so long as their axes, by reason of the sufficient distance of the object concerned, are able to maintain an approximate parallelism. I f the object approaches too near the organ of sight and causes the axes to converge to any noticeable extent, different irregularities in the coordination of the eye-muscles are inevitable, as is in fact the case in all strained or forced actions of this kind. It is not, however, our purpose to enlarge upon this matter here.

the movements of the human eye

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