Chapter III: Appendix

The elements of the geometry of spherical space are most easily obtained by putting for space of four dimensions the equation for the sphere

_x_² + _y_² +_z_² + _t_² = _R_² (1.)

and for the distance _ds_ between the points (_x_, _y_, _z_, _t_) and [(_x_ + _dx_) (_y_ + _dy_) (_z_ + _dz_) (_t_ + _dt_)] the value

_ds_² = _dx_² + _dy_² + _dz_² + _dt_² (2.)

It is easily found by means of the methods used for three dimensions that the shortest lines are given by equations of the form

_ax_ + _by_ + _cz_ + _ft_ = 0 }
} (3.)
_αx_ + _βy_ + _γz_ + _φt_ = 0 }

in which _a_, _b_, _c_, _f_, as well as _α_, _β_, _γ_, _φ_, are constants.

The length of the shortest arc, _s_, between the points

(_x_, _y_, _z_, _t_), and (_ξ_, _η_, _ζ_, _τ_) follows, as in the sphere, from the equation

cos_s_ (_xξ_ + _yη_ + _zζ_ + _tτ_)
------ = --------------------------- (4.)
_R_ _R_²

One of the co-ordinates may be eliminated from the values given in 2 to 4, by means of equation 1, and the expressions then apply to space of three dimensions.

If we take the distances from the points

_ξ_ = _η_ = _ζ_ = 0

from which equation 1 gives _τ_ = _R_, then,

( _s_₀ ) _σ_
sin ( ---- ) = -----
( _R_ ) _R_

in which

____________________
_σ_ = √(_x_² + _y_² + _z_²)

or,

( _σ_ ) ( _σ_ )
_s_₀ = _R_ . arc sin( --- ) = _R_ . arc tang( --- ) (5.)
( _R_ ) ( _t_ )

In this, _s_₀ is the distance of the point _x_, _y_, _z_, measured from the centre of the co-ordinates.

If now we suppose the point _x_, _y_, _z_, of spherical space, to be projected in a point of plane space whose co-ordinates are respectively

( _Rx_ ) ( _Ry_ ) ( _Rz_ )
χ = ( ---- ) ϒ = ( ---- ) ζ = ( ---- )
( _t_ ) ( _t_ ) ( _t_ )

_R_²_σ_²
χ² + ϒ² + ζ² = _r_² = ---------
_t_²

then in the plane space the equations 3, which belong to the straightest lines of spherical space, are equations of the straight line. Hence the shortest lines of spherical space are represented in the system of χ, ϒ, ζ, by straight lines. For very small values of

_x_, _y_, _z_, _t_ = _R_, and χ = _x_, ϒ = _y_, ζ = _z_

Immediately about the centre of the co-ordinates, the measurements of both spaces coincide. On the other hand, we have for the distances from the centre

( _r_ )
_s_₀ = _R_ . arc tang( ± ---- ) (6.)
( _R_ )

In this, _r_ may be infinite; but every point of plane space must be the projection of two points of the sphere, one for which _s_₀ < ½_R_π, one for which _s_₀ > ½_R_π. The extension in the direction of _r_ is then

_ds_₀ _R_²
----- = -------------
_dr_ _R_² + _r_²

In order to obtain corresponding expressions for pseudospherical space, let _R_ and _t_ be imaginary; that is, _R_ = ℛ_i_, and _t_ = τ_i_. Equation 6 gives then

_s_₀ _r_
tang ------ = ± ------
_i_ℛ _i_ℛ

from which, eliminating the imaginary form, we get

ℛ + _r_
_s_₀ = ½ℛ log. nat. ---------
ℛ - _r_

Here _s_₀ has real values only as long as _r_ = R; for _r_ = ℛ the distance _s_₀ in pseudospherical space is infinite. The image in plane space is, on the contrary, contained in the sphere of radius _R_, and every point of this sphere forms only one point of the infinite pseudospherical space. The extension in the direction of _r_ is

_ds_₀ ℛ²
----- = -----------
_dr_ ℛ² - _r_²

For linear elements, on the contrary, whose direction is at right angles to _r_, and for which _t_ is unchanged, we have in both cases

_____________________
√_dx_² + _dy_² + _dz_² _t_ τ _σ_
---------------------- = ----- = ---- = -----
_____________________ _R_ ℛ _r_
√_d_χ² + _d_ϒ² + _d_ζ²

__________________
√_x_² + _y_² + _z_²
= -------------------
_____________
√χ² + ϒ² + ζ²

ON THE RELATION OF OPTICS TO PAINTING.

_Being the substance of a series of Lectures delivered in Cologne, Berlin, and Bonn._

I fear that the announcement of my intention to address you on the subject of plastic art may have created no little surprise among some of my hearers. For I cannot doubt that many of you have had more frequent opportunities of viewing works of art, and have more thoroughly studied its historical aspects, than I can lay claim to have done; or indeed have had personal experience in the actual practice of art, in which I am entirely wanting. I have arrived at my artistic studies by a path which is but little trod, that is, by the physiology of the senses; and in reference to those who have a long acquaintance with, and who are quite at home in the beautiful fields of art, I may compare myself to a traveller who has entered upon them by a steep and stony mountain path, but who, in doing so, has passed many a stage from which a good point of view is obtained. If therefore I relate to you what I consider I have observed, it is with the understanding that I wish to regard myself as open to instruction by those more experienced than myself.

The physiological study of the manner in which the perceptions of our senses originate, how impressions from without pass into our nerves, and how the condition of the latter is thereby altered, presents many points of contact with the theory of the fine arts. On a former occasion I endeavoured to establish such a relation between the physiology of the sense of hearing, and the theory of music. Those relations in that case are particularly clear and distinct, because the elementary forms of music depend more closely on the nature and on the peculiarities of our perceptions than is the case in other arts, in which the nature of the material to be used and of the objects to be represented has a far greater influence. Yet even in those other branches of art, the especial mode of perception of that organ of sense by which the impression is taken up is not without importance; and a theoretical insight into its action, and into the principle of its methods, cannot be complete if this physiological element is not taken into account. Next to music this seems to predominate more particularly in painting, and this is the reason why I have chosen painting as the subject of my present lecture.

The more immediate object of the painter is to produce in us by his palette a lively visual impression of the objects which he has endeavoured to represent. The aim, in a certain sense, is to produce a kind of optical illusion; not indeed that, like the birds who pecked at the painted grapes of Apelles, we are to suppose we have present the real objects themselves, and not a picture; but in so far that the artistic representation produces in us a conception of their objects as vivid and as powerful as if we had them actually before us. The study of what are called illusions of the senses is however a very prominent and important part of the physiology of the senses; for just those cases in which external impressions evoke conceptions which are not in accordance with reality are particularly instructive for discovering the laws of those means and processes by which normal perceptions originate. We must look upon artists as persons whose observation of sensuous impressions is particularly vivid and accurate, and whose memory for these images is particularly true. That which long tradition has handed down to the men most gifted in this respect, and that which they have found by innumerable experiments in the most varied directions, as regards means and methods of representation, forms a series of important and significant facts, which the physiologist, who has here to learn from the artist, cannot afford to neglect. The study of works of art will throw great light on the question as to which elements and relations of our visual impressions are most predominant in determining our conception of what is seen, and what others are of less importance. As far as lies within his power, the artist will seek to foster the former at the cost of the latter.

In this sense then a careful observation of the works of the great masters will be serviceable, not only to physiological optics, but also because the investigation of the laws of the perceptions and of the observations of the senses will promote the theory of art, that is, the comprehension of its mode of action.

We have not here to do with a discussion of the ultimate objects and aims of art, but only with an examination of the action of the elementary means with which it works. The knowledge of the latter must, however, form an indispensable basis for the solution of the deeper questions, if we are to understand the problems which the artist has to solve, and the mode in which he attempts to attain his object.

I need scarcely lay stress on the fact, following as it does from what I have already said, that it is not my intention to furnish instructions according to which the artist is to work. I consider it a mistake to suppose that any kind of æsthetic lectures such as these can ever do so; but it is a mistake which those very frequently make who have only practical objects in view.