Natural science is the attempt to understand nature by means of exact concepts.
 According to the concepts through which we comprehend nature, our perceptions are supplemented and filled in, not simply at each moment, but also future perceptions are seen as necessary. Or, to the degree that the conceptual system is not fully sufficient, future perceptions are determined beforehand as probable; according to the concepts, what is "possible" is determined (thus also what is "necessary," and conversely, impossible). And the degree of possibility (of "probability") of each individual event which is seen as possible, in light of these concepts, can be mathematically determined, if the concepts are precise enough.
 To the extent that what is necessary or probable, according to these concepts, takes place, then this confirms the concepts, and the trust that we place in these concepts rests on this confirmation through experience. But if something takes place that is unexpected according our existing assumptions, i.e., that is impossible or improbable according to them, then the task arises of completing them or, if necessary, reworking the axioms, so that what is perceived ceases to be impossible or improbable. The completion or improvement of the conceptual system forms the 'explanation' of the unexpected perception. Our comprehension of nature gradually becomes more and more complete and correct through this process, simultaneously penetrating more and more behind the surface of appearances.
 The history of causal natural science, in so far as we can trace it back, shows that this is, in fact, the way our knowledge of nature advances. The conceptual systems that are now the basis for the natural sciences, arose through a gradual transformation of older conceptual systems, and the reasons that drove us to new modes of explanation can always be traced back to contradictions and improbabilities that emerged from the older modes of explanation.
 The formation of new concepts, in so far as this process is accessible to observation, therefore takes place in this way.
 Herbart furnished the proof that concepts that allow us to comprehend the world-those whose origin we can trace neither in history nor in our own development, because they are delivered to us unnoticed through our language-can be derived from this source, in so far as they are more than mere forms combining simple sense images; and therefore these concepts need not be derived from some special constitution of the human mind which precedes all experience (such as Kant's categories).
 This proof of their origin in our ability to comprehend that which is given to us by sense perception, is important for us, because it is only in this way that their meaning can be determined in a manner satisfactory for science... .

 After the concept of things existing in themselves has been formed, then in reflecting on the process of change, which contradicts the concept of things existing in themselves, the task arises of maintaining this already proven concept as far as possible. From this problem arise simultaneously the concepts of continuous change and causality.
 All that is observed is the transition of a thing from one state into another, or, to speak more generally, from one mode of determination to another, without a sudden jump being perceived in the transition. In order to complete the observations, we can either assume that the transition occurs through a very great, but finite, number of leaps imperceptible our senses, or that the thing goes continuously through all of the intermediate steps, taking it from one state to the other. The strongest reason for the latter conception is the demand to maintain as far as possible, the already proven concept of the existence of the thing in itself. Of course, it is not possible to actually represent such a transition through all intermediate steps, which, however, as noted, is valid, strictly speaking, for all concepts.
 At the same time, however, according to the concept of the thing in itself, formed earlier and proven by experience, the thing would remain what it is, unless something else intervened. This creates the impulse to seek a cause for every change.

  I. When is our comprehension of the world true?
 "When the relations among our conceptions correspond to the relations of things."
 The elements of our picture of the world are completely distinct from the corresponding elements of the reality which they picture. They are something within us; the elements of reality are something outside of ourselves. But the connections among the elements in the picture, and among the elements of reality which they depict, must agree, if the picture is to be true.
 The truth of the picture is independent of its degree of fineness; it does not depend upon whether the elements of the picture represent larger or smaller aggregates of reality. But, the connections must correspond to one another; a direct action of two elements upon each other may not be assumed in the picture, where only an indirect one occurs in reality. Otherwise the picture would be false and would need correction. If, however, an element of the picture is replaced by a group of finer elements, so that its properties emerge, partly from the simpler properties of the finer elements, but partly from their connections, and thus become in part comprehensible, then this increases our insight into the connection of things, but without the earlier understanding having to be declared false.

  II. How do we find the relations among things?
 "From the connections of phenomena"
 The representation in determinate space-and-time relations of things of the senses is something met with in deliberate reflection on nature or is given in that reflection. However, as we well know, the quality of the characteristics of things of the senses, color, sound, tone, smell, taste, heat or cold, is something merely derived from our own sensations and does not exist outside of ourselves.
 The relations among things must therefore become known to us from quantitative relations, the spatial and temporal relations of things of the senses and the relative intensities of their characteristics and their qualitative differences.
 Knowledge of the connections among things must arise from reflection on the observed relations of these relations of magnitudes.

 I. What an action strives to accomplish must be determined through the concept of the action; its acting cannot be dependent upon anything else than the action's own being.
 II. This demand is satisfied when the action strives to maintain or restore itself.
 III. Such an action is not conceivable, if the action is a thing, a being; but only if it is a state or a relationship. If a striving exists, to maintain or restore something, then deviations from this something must also be possible - and indeed in different degrees. And in so far as this striving conflicts with other strivings, it will in fact be maintained or restored only to the extent possible. But there is no gradation of being; a difference of degrees is conceivable only for states or relationships. If therefore, an action strives to maintain or restore itself, it must be a state or a relationship.
 IV. Obviously, such action can only occur in those things that can assume such a state. But in which of these things it occurs, and whether it occurs in them at all cannot be determined from the concept of the action.

주 : These are valid only if the effect is to be ascribed to a simple real cause.
 If two things a and b are connected through an external cause, then a consequence c can be ascribed either to the connection, the process of being connected itself, or else to a change in the degree of the connection. The simplest assumption is that the consequence c can be ascribed to the process of being connected.
 It is unnecessary to to take these considerations further. Their principle consists in holding to the thesis: "What an action strives to effect must be determined from the concept of the action"; but this thesis must be applied, not as Leibniz or Spinoza did, to beings with a manifold of determinations, but ralher to real causes of the greatest possible simplicity.

 In German, one tends to translate 'actio' as well as 'effectus' by "Wirkung [effect]." Since the word occurs in the latter sense more commonly, unclarity easily arises if it is used for 'actio', as, for example, with the standard translation of "actio aequalis est reactioni [action and reaction are equal]," or "principium actionis minimae [principle of least action]." Kant seeks to remedy this by adding the Latin expressions 'actio' and "actio mutua" In parenthesis to "Wirkung" and "Wechselwlrkung [interaction]." One could perhaps write, "die Kraft is gleich der Gegenkraft [the force is equal to the opposing force]," "Satz vom kleinsten Kraftaufwande [the principle of least expenditure of force]." Since, in fact, we lack a simple expression for 'agere', a striving directed toward something else, I may be permitted the use of the foreign word[agens, action].

  Kant quite rightly note that we can neither discover the existence of a thing, nor that it is the cause of something else, merely from analysis of the concept of the thing; so that the concept of being and causality cannot be derived from analysis but only from experience. When, however, he later believes himself compelled to assume that the concept of causality precedes all experience, this is tantamount to throwing the baby out with the bath; because this implies that the mind would be preconditioned to accept any perception, given by experience, as a cause, if it could be connected to any other arbitrary one as effect, according to a rule of mere sequence. (Of course, we must derive the relationships of causality from experience, but we must not dispense with correcting and completing our comprehension of the data of experience through reflection.)

  The word hypothesis now has a somewhat different meaning than with Newton. We are now accustomed to understand by hypothesis all that is added by thought to phenomena. Newton was far from the absurd thought that the explanation of phenomena could be gained by abstraction from experience.
 Newton: [In Latin from the General Scholium of Principia Mathematica] "And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to natural philosophy. [ ... ] But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses. "
 Arago, Oeuvres Completes, Vol. 3, 505:
 [In French] "Once and once only did Laplace rise into the realm of conjecture. His conception at that time was nothing less than a cosmogony. "
 Laplace in response to Napoleon' question, why the name God did not occur in his Celestial Mechanics: [in French] "Sire, I have no need for that hypothesis. "

  The distinction that Newton makes between laws of motion, or axioms, and hypotheses, does not seem tenable to me. The law of inertia is the hypothesis: If a material point were present alone in the world and moved in space with a definite velocity, then it would constantly maintain this velocity.

 The free movement of a system of material points $m_1, m_2$ ... with rectangular coordinates $x_1, y_1; x_2, y_2, z_2$; ..., on which forces $X_1, Y_1, Z_1; X_2, Y_2, Z_2$; ... act in parallel to the three axes, takes place according to the equations
(1) $m_i \frac{d^2x_i}{d t^2}=X_i, \,m_i \frac{d^2y_i}{d t^2}=Y_i,\, m_i \frac{d^2z_i}{d t^2}=Z_i,$
This law can also be expressed as follows: The accelerations are so determined that
   $\sum m_i((\frac{d^2x_i}{d t^2} - \frac{X_i}{m_i})^2+(\frac{d^2y_i}{d t^2} - \frac{Y_i}{m_i})^2+(\frac{d^2z_i}{d t^2} - \frac{Z_i}{m_i})^2)$
becomes a minimum; for this function of the accelerations takes its smallest value 0 if the accelerations collectively are determined in accordance with equation (1), that is, the magnitudes $\frac{d^2 x_i}{dt^2}- \frac{X_i}{m_i}$ ... collectively = 0, and they also take the minimum value only then; for, were one of these magnitudes, for example, $\frac{d^2 x_i}{dt^2}- \frac{X_i}{m_i}$ not equal to $0$, then $\frac{d^2 x_i}{dt^2}$ could continuously change so that the absolute value of this magnitude and consequently its square would decrease. The function would thus become smaller if all the other accelerations were simultaneously left unchanged.
 This function of the accelerations is distinguished from
  $\sum m_i((\frac{d^2x_i}{d t^2})^2+(\frac{d^2y_i}{d t^2})^2+(\frac{d^2z_i}{d t^2})^2)-2 \sum (X_i\frac{d^2x_i}{d t^2} +Y_i\frac{d^2y_i}{d t^2}+Z_i\frac{d^2z_i}{d t^2})$
only by a constant, that is, by a magnitude independent of the accelerations.
 If the forces between points result only from attraction and repulsion, which are functions of distance, and the $i$th point and the $i'$th point at a distance $r$ repulse one another with a force $f_{i, i'}(r)$ or attract one another with the force -$f_{i, i'}(r)$ then, as is known, the components of the forces can be expressed through the partial derivatives of a function of the coordinates of all the points
  $P=\sum_{i, i'} F_{i, i'}(r_{i, i'})$,
where $F_{i, i'}(r)$ is a function with derivative $f_{i, i'}(r)$, and for $i$ and $i'$ two different indices are set for each.
 If these values of the components
  $X_i=\frac{\partial P}{\partial x}, \,Y_i=\frac{\partial P}{\partial y},\, Z_i=\frac{\partial P}{\partial z}$
are substituted into the above function of the accelerations and are multiplied by $\frac{dt^2}{4}$, through which the positions of their maxima and minima are not changed, then we obtain an expression which is distinguished from
  $\frac14 \sum((d \frac{dx_i}{dt})^2+(d \frac{dy_i}{dt})^2+ (d \frac{dz_i}{dt})^2) -P_{(t+dt)}$
only by a magnitude which is independent of the accelerations. If the position and the velocities of the points at time $t$ are given, then this position is determined at time $t + dt$ such that this magnitude becomes as small as possible. Accordingly, there is a striving for this magnitude to become a minimum.
 This law can be explained on the basis of actions which strive to make the individual terms of this expression as small as possible if we assume that the strivings working against one another are so equalized that the sum of the magnitudes which the individual actions strive to maintain at a minimum, becomes itself a minimum.
 If we assume that the masses of the points $m_1, m_2,....m_n$ behave like the whole numbers $k_1, k_2...., k_n$, so that $m_t=k_t\mu$, then the expression, which becomes as small as possible, consists of the sum of the magnitudes
  $\frac{\mu}{4} ((d \frac{dx_i}{dt})^2+(d \frac{dy_i}{dt})^2+ (d \frac{dz_i}{dt})^2)$
for the totality of material particles $\mu$ and of magnitude -$P_{t+dt}$. If we therefore, with Gauss, consider the magnitude
  $(d \frac{dx_i}{dt})^2+(d \frac{dy_i}{dt})^2+ (d \frac{dz_i}{dt})^2$
as the measure of the deviation of the state of motion of mass $\mu$ at time $t + dt$ from its state of motion at time $t$, then the analysis of the total action in relation to each mass yields an action which strives to make the deviation of its state of motion at time $t + dt$ as small as possible relative to its state of motion at time $t$, or an effort to preserve its state of motion, and, additionally, an action which strives to keep the magnitude -$P$ as small as possible.
 The latter action can be analyzed into efforts to keep the individual terms of the sum $\sum_{i, i'} F_{i, i'}(r_{i, i'})$ as small as possible, that is, into attractions and repulsions between any two points, and this would lead us back to the customary explanation of the laws of motion from the law of inertia and of attraction and repulsion; but it can also lead us back, for all natural forces known to us, to the forces that act between contiguous spatial elements, as will be explained in the following article on gravitation.

 Although the title of this essay will hardly create a favorable impression on most readers, nonetheless it seems to me to best express the overall direction of the essay. Its purpose is to penetrate beyond the foundations of astronomy and physics laid by Galilei and Newton, into the interior of nature. For astronomy, certainly these speculations cannot immediately have any practical use, but I hope that this circumstance will not cause any diminution of interest in the eyes of the readers of this publication...
 The foundation for those general laws of the motion of measurable bodies that are presented at the beginning of Newton's Principia lies in the internal state of these bodies. Let us attempt to form an analogy between these and our own inner mode of perception. New image masses constantly arise in us and very rapidly disappear again from our consciousness. We observe a constant activity of our mind. Every mental act is based upon something enduring, which is manifest (through memory) on certain occasions, without exerting a lasting influence on the phenomena. Thus (with every act of thinking) something enduring continually enters our mind, which does not however, exert a lasting influence upon the world of phenomena. Every mental act, therefore, is based upon something enduring, which enters our mind with the act, but at the same moment completely disappears from the world of phenomena.
 Guided by this fact, I form the hypothesis that there is a kind of space-filling substance which continually flows into measurable atoms and there disappears from the world of phenomena (the corporeal world).

2024.7.22: 세상은 hologram
태양 빛은 8분 전에 출발한 거고 전자기장의 retarded time을 그렇게 많이 봤는데... 이제야 깨닫는군. 실체가 발산하는 기가 감각기관에 도달해야 존재 인식. 따라서 인간이 육체 유지하기 위해 음식물 섭취하듯, 물질 또한 외부로부터 구조 유지 및 존재를 알리는 외부 유출용 기를 공급받을 수 밖에 없다
disappear하는 게 아니라 3차원 물질 세계에서 고차원 세계로 가는 거

 Both hypotheses can be replaced by the one, that in all measurable atoms, substance from the corporeal world continuously enters into the world of mind. The reason the substance disappears there is to be sought in the thought matter which was formed in the immediately preceding period; and the pon derable bodies are accordingly the place where the world of mind engages the corporeal world. (주: At every instant, a definite quantity of substance, proportional to the gravitational force, enters into every measurable atom, and disappears there.
 It is a consequence of the psychology based on Herbart's work, that substantiality accrues not to the mind but to every individual image formed within it.)
 The effect of universal gravitation, the first thing to be explained by this hypothesis, is well known to be fully determined for every part of space, if the potential function $P$ of all measurable mass for this part of space be given, or, which is the same thing, there is a function of position $P$, such that the measurable masses contained within the closed surface $S$, are $\frac{1}{4\pi}\int{\frac{\partial P}{\partial p}}dS$
 If we now assume that the substance that fills space is an incompressible homogeneous fluid, without inertia, and that an amount proportional to the mass of any given atom flows into it during equal times, then obviously, the pressure exerted on the measurable atom (will be proportional to the velocity of the substance at the site of the atom(?))
 Thus the effect of universal gravitation on a measurable atom can be expressed through (and thought of as dependent upon) the pressure of this space-filling substance in the immediate neighborhood of the atom.
 It necessarily follows from our hypothesis that the space filling substance must propagate the vibrations that we perceive as light and heat.
 If we consider a simple polarized beam, and designate as $x$ the distance of an indeterminate point of this beam from a fixed origin, and $y$ it displacement at a time $t$, then the following equation must be at least very nearly satisfied, since the velocity of propagation of the vibrations in space free of measurable atoms is under all conditions very nearly constant (= $\alpha$):
 $y = f(x + \alpha t) + \varphi (x - \alpha t)$
 For it to be strictly satisfied,
   $\frac{\partial y}{\partial t}= \alpha ^2 \int^t \frac{\partial ^2 y}{\partial^2x}d\tau $
would have to apply; obviously, however, for the sake of experiment, we can be satisfied with the equation
   $\frac{\partial y}{\partial t}= \alpha ^2 \int^t \frac{\partial ^2 y}{\partial^2x} \varphi(t-\tau)d\tau $
even if $\varphi(t - \tau)$ is not equal to 1 for all positive values of $t-\tau$(which decreases ad infinitum with increasing $t-\tau$), as long as for a sufficiently long period of time it remains very close to 1. . . .
 Let the positions of the points of the substance at a given time $t$ be expressed by a rectangular coordinate system and let the coordinates of an indeterminate point $0$ be $x, y, z$. Similarly, let the coordinate of a point $0'$ be $x', y', z'$, also with regard to a rectilinear coordinate system. Then $x', y', z'$ are functions of $x, y, z$, and $ds'^2={dx'}^2+{dy'}^2+{dz'}^2$ will be equal to a homogeneous quadratic expression of $dx, dy, dz$. According to a well-known theorem, the linear expressions of $dx, dy, dz$
  $\alpha_1 dx+\beta_1 dy+ \gamma_1 dz=ds_1$
  $\alpha_2 dx+\beta_2 dy+ \gamma_2 dz=ds_2$
  $\alpha_3 dx+\beta_3 dy+ \gamma_3 dz=ds_3$
can now always in one and only one way be determined, such that
   ${dx'}^2+{dy'}^2+{dz'}^2=G_1^2{ds_1}^2+G_2^2{ds}^2+G_3^2{ds_3}^2$
while
   $ds^2={dx}^2+{dy}^2+{dz}^2={ds_1}^2+{ds_2}^2+{ds_3}^2$
The magnitudes $G_1-1, G_2-1, G_3-1$ then signify the major deformations for the particle of substance at $0$, in the transition from the former form to the latter. I indicate them by $\lambda_1, \lambda_2, \lambda_3$).
 Now I assume that a force results from the difference between the earlier forms of the particle of substance and its form at time $t$, which strives to change it; and, other things being equal, that the influence of an earlier form will become the less the longer the time prior to $t$ when it occurred. Thus there is a limit before which all earlier forms can be ignored. I further assume that those states that still manifest a detectable influence differ so slightly from the state at time $t$, that the deformations may be regarded as infinitely small. The forces that strive to make $\lambda_1, \lambda_2, \lambda_3$ small can then be regarded as linear functions of $\lambda_1, \lambda_2, \lambda_3$; and indeed, because of the homogeneity of the aether for the total moment of these forces (the force which strives to make $\lambda_1$ small must be a function of $\lambda_1, \lambda_2, \lambda_3$), which remains unchanged when we exchange $\lambda_2$ with $\lambda_3$, and the remaining forces must follow from it, when $\lambda_2$ is exchanged with $\lambda_1$ , and $\lambda_3$ with $\lambda_1$) we obtain the following expression:
   $\delta \lambda_1(a \lambda_1+b \lambda_2+b \lambda_3) +\delta \lambda_2(b \lambda_1+a \lambda_2+b \lambda_3)+\delta \lambda_3(b \lambda_1+b \lambda_2+a \lambda_3)$
or with a somewhat changed meaning of the constants:
   $\delta \lambda_1(a (\lambda_1+\lambda_2+\lambda_3)+b\lambda_1) +\delta \lambda_2(a (\lambda_1+\lambda_2+\lambda_3)+b\lambda_2)+\delta \lambda_3(a (\lambda_1+\lambda_2+\lambda_3)+b\lambda_3)$

  $=\frac12 \delta (a (\lambda_1+\lambda_2+\lambda_3)^2 +b (\lambda_1^2+\lambda_2^2+\lambda_3^2))$.
 Now the moment of the force that strives to change the form of the infinitely small particle of substance at $0$, can be regarded as resulting from forces that strive to change the length of the line elements ending at $0$. We therefore arrive at the following law of action: If $dV$ is the volume of an infinitely small particle of substance at point $0$ and time $t$, and $dV'$ the volume of the same particle at time $t'$, then the force resulting from the difference in the two states of the substance, which strives to elongate $ds$, is expressed by
  $a \frac{dV-dV'}{dV}+b \frac{ds-ds'}{ds}$
 The first part of this expression derives from the force with which a particle of substance resists a change in volume without a change of form, the second from the force with which a physical line element resists a change in length.
 Now there is no reason to assume that the effects of both causes change with time in accordance with the same law; thus if we sum the effects of all earlier form of a particle of substance upon the change of the line element $ds$ at time $t$, then the value of $\frac{\delta ds}{dt}$, which they strive to determine, becomes
  $=\int_{-\infty}^t \frac{dV'-dV}{dV}\psi(t-t')\delta t'+ \int_{-\infty}^t \frac{ds'-ds}{ds}\varphi(t-t')\delta t'$.
How then must the functions $\psi$ and $\varphi$ be constituted such that gravitation, light, and radiant heat may be propagation through the substance of space?

2024.7.30: 자연은 전자기장처럼 규칙적으로 움직이는데, 소위 '자유의지' '기'들이 몰이 당하면서 생긴 사태가 작금의 현상?

  The effects of measurable matter upon measurable matter are:
 (1) Attractive and repulsive forces inversely proportional to the square of the distance.
 (2) Light and radiant heat.
 Both classes of phenomena can be explained if we assume that the entirety of infinite space be filled with a homogeneous substance and that every particle of that substance acts directly only upon its immediate neighborhood.
 The mathematical law in accordance with which this occurs can be thought of as divided into
 (1) the resistance of a particle of substance to a change in volume, and
 (2) the resistance of a physical line element to a change in length.
 Upon the first part are founded gravitation and electrostatic attraction and repulsion; upon the second, the propagation of light and heat, and electrodynamic or magnetic attraction and repulsion.

 The Newtonian explanation of gravitational motion and the motions of celestial bodies consists in the assumption of the following causes:
 1. There exists an infinite space with the properties which are assigned to it by geometry, and there exist measurable bodies which change their positions within this space only continuously.
 2. At every mass-point, there is at every moment a cause determined by magnitude and direction, by virtue of which cause the mass-point has a determinate motion (matter in a determinate state of motion). The measure of this cause is velocity. (주: Every material body, if alone in space, would either not change its position in space or would move in a straight line with constant velocity.
 This law of motion cannot be explained by means of the Principle of sufficient Reason: That the body continues its motion, must have a cause, which can only be sought in the internal state of the matter.)
 The phenomena to be explained here do not yet lead to the assumption of different masses for measurable bodies.
 3. At every point of space, there exists at every moment a cause (accelerating force), determined by magnitude and direction, which communicates a determinate motion to every mass point present, and indeed, the same motion to each, which combines geometrically with the motion that it already has.
 4. At every mass-point in space, there exists a cause (absolute gravity) determined by magnitude, which combines geometrically with all other accelerating forces present there. By virtue of this cause, at every point of space an accelerating force exists, inversely proportional to the square of its distance from this mass-point and directly proportional to its gravitational force.(주: The same mass point would undergo changes in motion between two points, whose directions coincide with the directions of the forces and whose magnitudes are proportional to the forces.
 The force divided by the change in motion, therefore, always gives the same quotient for the same mass-point. This quotient is different for different mass-points and is called their mass.)
 The cause, determined according to magnitude and direction (accelerating gravitational force), which, according to 3., is found at every point in space, I seek in the form of motion of a substance that is continuously spread through all infinite space, and, indeed, I assume that the direction of the motion is equal to the direction of the force from which it is to be explained, and the velocity is proportional to the magnitude of the force. This substance can therefore be represented as a physical space whose points move in geometrical space.
 According to this assumption, all effects caused by measurable bodies on measurable bodies through empty space must be propagated by this substance. Therefore also the forms of motion of which light and heat consist, which celestial bodies transmit to one another, must be forms of motion of this substance. These two phenomena, however, gravitation and the motion of light through empty space, are the only ones that must be explained purely by means of the motions of this substance.
 Now I assume that the actual motion of the substance in empty space is combined from the motion which must be assumed for explanation of gravitation and that which must be assumed for the explanation of light.
 The further development of this hypothesis can be divided into two parts in that the following are to be sought:
 1. The laws of motion of the substance which must be assumed for the explanation of the phenomena.
 2 . The causes by means of which these motions can be explained.
 The first subject is mathematical, the second, metaphysical. In reference to the latter, I note in advance that the goal will not be considered to be any explanation on the basis of causes that strive to change the distance between two points of the substance. This method of explanation by means of attractive and repulsive forces owes its general application in physics not to any direct evidence (or specific conformity to reason), nor, apart from electricity and gravity, to its particular facility, but on the contrary, to the circumstance that the Newtonian law of attraction, in contradiction to the opinion of its discoverer, has so far been considered to need no further explanation.(주: [In English] Newton says: "That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it." See the third letter to Bentley.)

 Expressing the position of a point in space by means of rectangular coordinates $x_1, x_2, x_3$, I designate the velocity components-parallel to the coordinates at time $t$ -of the motion that causes the gravitational phenomena as $u_1, u_2, u_3$, and those of the motion that causes the phenomena of light as $w_1, w_2, w_3$, and those of the actual motion as $v_1, v_2, v_3$, so that $v=u+w$. As will emerge from the laws of motion themselves, the substance, if it is everywhere equally dense at one point in time, maintains this same density everywhere at all times. I will therefore assume this to be everywhere equal to $1$ at time $t$.

 The gravitational force is determined at every point by the potential function $V$, whose partial derivatives $\frac{\partial V}{\partial x_1}, \frac{\partial V}{\partial x_2}, \frac{\partial V}{\partial x_3}$ are the components of the gravltatlonal force, and this $V$ is in turn determined through the following conditions (disregarding an additional constant):
 1. $dx_1 dx_2 dx_3 (\frac{\partial^2 V}{\partial x_1^2} +\frac{\partial^2 V}{\partial x_2^2}+\frac{\partial^2 V}{\partial x_3^2})$ outside the attracting body = 0, and has for every measurable material element a constant value. This is the product of -4$\pi$ in the absolute magnitude of the attractive force, which according to the theory of attraction must be assigned to it, and will be designated as $dm$.
 2. If all attracting bodies are within a finite space, $r\frac{\partial V}{\partial x_1}, r\frac{\partial V}{\partial x_2}, r\frac{\partial V}{\partial x_3}$ at an infinite distance $r$ from a point in this space infinitely small.
 Now according to our hypothesis, $\frac{\partial V}{\partial x}=u$ and consequently
    $dV=u_1 dx_1 + u_2 dx_2+u_3 dx_3$
This include the conditions
(1) $\frac{\partial u_2}{\partial x_3} - \frac{\partial u_3}{\partial x_2}=0$, $\frac{\partial u_3}{\partial x_1} - \frac{\partial u_1}{\partial x_3}=0$, $\frac{\partial u_1}{\partial x_2} - \frac{\partial u_2}{\partial x_1}=0$ (* $\nabla \times u=0$)
(2) $(\frac{\partial u_1}{\partial x_1} + \frac{\partial u_2}{\partial x_2}+ \frac{\partial u_3}{\partial x_3})dx_1dx_2dx_3=-4\pi dm$(* $\nabla \cdot u \, dx_1dx_2dx_3 = -4\pi dm$)
(3) $ru_1=0, ru_2=0, ru_3=0$ for $r=\infty$
Conversely, the magnitudes $u$, if they satisfy these conditions, are equal to the components of the gravitational force. Since the conditions (1) contain the possibility of a function $U$ from which arises the differential $dU=u_1dx_1+u_2dx_2+u_3dx_3$ and thus the derivatives $\frac{\partial U}{\partial x}=u$, and the others then yield $U=V$+constant.*

주: This function $U$ is therefore given through observation (from relative motions) by means of the general laws of motion, but only without taking account of a linear function of the coordinates, because we can only observe relative motions.
 The determination of this function is based on the following mathematical theorem: A function $V$ of position is determined within a finite space (ignoring a constant) if it is not said to be discontinuous along a surface, and for all of Its elements $(\frac{\partial^2 V}{\partial x_1^2}+\frac{\partial^2 V}{\partial x_2^2}+\frac{\partial^2 V}{\partial x_3^2})dx_1dx_2dx_3$ at the limit, either $V$ or its derivative is given for an inward change of position, perpendicular to the limit(boundary). Of which it should be noted:
 1. If this derivative at the bounding element $ds$ is designated by $\frac{\partial V}{\partial p}$, then in the latter case $\int \sum\frac{\partial^2 V}{\partial x^2}dx_1dx_2dx_3$ must be equal to -$\int \frac{\partial V}{\partial p}ds$ through the entire space because of its bound; otherwise, in both cases, all of the determining elements can be taken arbitrarily and are therefore necessary to the determination.
 2. For a spatial element where $\sum\frac{\partial^2 V}{\partial x^2}$ becomes infinitely large, the product of the two is to be substituted by -$\int \frac{\partial V}{\partial p} ds$ in relation to the limit of this element.
 3. If $\sum\frac{\partial^2 V}{\partial x^2}$ has a value other than zero only within a finite space, then the boundary condition can be substituted by the statement that at an infinite distance $R$ of a point in this space $R\frac{\partial V}{\partial x}$ becomes infinitely small.

 The motion that must be assumed in empty space for the explanation of the phenomena of light can be considered (following a theorem) as composed of plane waves, that is, of such motions where the form of motion is constant along each plane of a family of parallel planes(wave planes). Each of these wave systems consists then (in accord with observation) of motions parallel to the wave plane that are propagated perpendicular to the wave plane with a constant velocity $c$ that is the same for all forms of motion (types of light).
 If $\xi_1, \xi_2, \xi_3$ are the rectangular coordinates of a point in space for such a system of waves, the first being perpendicular, the others parallel to the wave plane, and $\omega_1, \omega_2, \omega_3$ are the components of velocity at this point parallel to the coordinates at time $t$, then we have
  $\frac{\partial \omega}{\partial \xi_2} =0, \frac{\partial \omega}{\partial \xi_3} =0$.
According to observation, first
   $\omega_1=0$
second, the movement is composed of motions with velocity $c$, one propagating from the positive side of the wave plane, and one propagating from the negative side. If the velocity components of the first are $\omega'$ and that of the latter are $\omega''$, then the $\omega'$ remain unchanged if $t$ increases by $dt$ and $\xi_1$ increases by $cdt$, and the $\omega''$ are unchanged, if $t$ increases by $dt$ and $\xi_1$ by -$cdt$, and we have $\omega=\omega'+\omega''$. From this it follows that
  $(\frac{\partial \omega'}{dt}+c\frac{\partial \omega'}{d\xi_1})dt=0, (\frac{\partial \omega''}{dt}-c\frac{\partial \omega''}{d\xi_1})dt=0$
 $\frac{\partial^2 \omega '}{\partial t^2}= -c \frac{\partial^2 \omega '}{\partial \xi_1\partial t}=cc\frac{\partial ^2 \omega '}{\partial \xi_1^2}$, $\frac{\partial^2 \omega ''}{\partial t^2}= c \frac{\partial^2 \omega ''}{\partial \xi_1\partial t}=cc\frac{\partial ^2 \omega ''}{\partial \xi_1^2}$
and thus
  $\frac{\partial^2 \omega}{\partial t^2} =cc\frac{\partial ^2 \omega}{\partial \xi_1^2}$
These equations give the following symmetrical results:
  $\frac{\partial \omega_1}{\partial \xi_1}+\frac{\partial \omega_2}{\partial \xi_2}+\frac{\partial \omega_3}{\partial \xi_3}=0$,
  $\frac{\partial ^2 \omega}{\partial t^2} = cc(\frac{\partial ^2 \omega}{\partial \xi_1^2} + \frac{\partial ^2 \omega}{\partial \xi_2^2} + \frac{\partial ^2 \omega}{\partial \xi_3^2})$
which, expressed in the original coordinate system, become equations of the same form, that is,
(1) $\frac{\partial w_1}{\partial x_1}+\frac{\partial w_2}{\partial x_2}+\frac{\partial w_3}{\partial x_3}=0$,
(2) $\frac{\partial ^2 w}{\partial t^2} =cc(\frac{\partial ^2 w}{\partial x_1^2} + \frac{\partial ^2 w}{\partial x_2^2} + \frac{\partial ^2 w}{\partial x_3^2})$
These equations are valid for every plane wave passing through the point $(x_1, x_2, x_3)$ at time $t$ and consequently also for the combined motion of all such plane waves.

 From the conditions established for $u$ and $w$, the following conditions follow for $v$ or laws of motion of the substance in empty space:
(I) $\frac{\partial v_1}{dx_1}+\frac{\partial v_2}{dx_2}+\frac{\partial v_3}{dx_3}=0$,
(II) $(\partial^2 t - cc(\partial ^2 x_1+ \partial ^2 x_2++\partial ^2 x_3)) (\frac{\partial v_2}{dx_3} -\frac{\partial v_3}{\partial x_2})=0$
   $(\partial^2 t - cc(\partial ^2 x_1+ \partial ^2 x_2++\partial ^2 x_3)) (\frac{\partial v_3}{dx_1} -\frac{\partial v_1}{\partial x_3})=0$
   $(\partial^2 t - cc(\partial ^2 x_1+ \partial ^2 x_2++\partial ^2 x_3)) (\frac{\partial v_1}{dx_2} -\frac{\partial v_2}{\partial x_1})=0$,
as is easily derived if the operations are carried out.
 These equations show that the motion of a point of the substance only depends on motions in contiguous regions of space and time, and their (complete) causes can be sought in the effects in their neighborhood.
 Equation (I) proves our earlier assertion that the density of the substance remains unchanged during its motion; since
   $\frac{\partial v_1}{dx_1}+\frac{\partial v_2}{dx_2}+\frac{\partial v_3}{dx_3}dx_1dx_2dx_3dt$,
which as a result of this equation is equal to $0$, expresses the mass of the substance which flows into the spatial element $dx_1 dx_2 dx_3$ in time element $dt$, and therefore the mass of the substance contained in it remains constant.
 Conditions (II) are identical with the condition that
   $(\partial^2 t - cc(\partial ^2 x_1+ \partial ^2 x_2++\partial ^2 x_3)) (v_1dx_1+v_2dx_2+v_3dx_3)$
be equal to a complete differential $dW$. Now
  $(\partial^2 t - cc(\partial ^2 x_1+ \partial ^2 x_2++\partial ^2 x_3)) (w_1dx_1+w_2dx_2+w_3dx_3)=0$
and consequently
  $dW = (\partial^2 t - cc(\partial ^2 x_1+ \partial ^2 x_2++\partial ^2 x_3)) (u_1dx_1+u_2dx_2+u_3dx_3)$
  = $(\partial^2 t - cc(\partial ^2 x_1+ \partial ^2 x_2++\partial ^2 x_3))dV$
or, since $(\partial ^2 x_1+\partial ^2 x_2+\partial ^2 x_3)dV=0$,
  = $d\frac{\partial ^2 V}{\partial t^2}$

  The laws of these phenomena can be summed up by the condition that the variation of the integral
 $\frac12 \int [\sum(\frac{\partial \eta_i}{\partial t})^2 -cc [(\frac{\partial \eta_2}{\partial x_3} - \frac{\partial \eta_3}{\partial x_2})^2+(\frac{\partial \eta_3}{\partial x_1} - \frac{\partial \eta_1}{\partial x_3})^2+(\frac{\partial \eta_1}{\partial x_2} - \frac{\partial \eta_2}{\partial x_1})^2]dx_1dx_2dx_3]dt \\+ \int V(\sum \frac{\partial ^2 \eta_i}{\partial x_i \partial t} dx_1 dx_2 dx_3 + 4\pi dm)dt \\ + 2 \pi \int dm \sum (\frac{\partial x_i}{\partial t})^2 dt$
becomes zero under appropriate boundary conditions. In this expression, the first two integrals extend over the entire geometrical space, the latter over all elements of measurable bodies, but the coordinates of every element of measurable bodies are to be so determined as functions of time, and $\eta_1, \eta_2, \eta_3, V$ as functions of $x_1, x_2, x_3$ and $t$, that a variation satisfying their boundary conditions produces only a variation of the second order of the integral.
 Then the quantities $\frac{\partial \eta}{\partial t}(=v)$ are equal to the velocity components of the motion of the substance and $V$ is equal to the potential at time $t$ at point $(x_1, x_2, x_3)$.

Translator's Notes

1 . The German expression is Gelstesmasse. It had earlier appeared in the correspondence between Schiller and Goethe (personal communication of George Gregory).
2. The expression form of motion(Bewegungsform), which begins to appear here early in the fragments, appears as "forms of motion (types of light)" in one late occurrence in which the subject is electromagnetic radiation.
This suggests that form of motion refers to wavelength or frequency.
3. In the fragments on psychology and metaphysics, Riemann refers to the Erdseele. The literal translation is earth mind or earth soul. We have instead used the expression biosphere. It will be helpful to the reader to keep in mind all the possibilities suggested by biosphere, earth mind, and earth soul, in the four instances where biosphere appears in the translation.
 The German Seele(soul or mind) is the equivalent of the Greek psyche. The Greek word also carries the meaning, that which enables life. In his Hannonices Mundi, Kepler used anima-the nearest Latin equivalent of psyche -as a metaphor for universal gravitation. The translator thanks George Gregory for these observations on the Greek and Latin terms and their use.
4. See the first of the three paragraphs marked "1" immediately preceding, which begins "1. The higher ..."
5. The German word is Denkprocess.
6. Not the paragraphs 2. and 3. immediately preceding, but the earlier pair following the paragraph that reads, "Emplrically, the extemal conditions of living processes in the range of phenomena accessible to us are:"
7. "Characteristics of mind" is used for Beseeltheit.
8. Here Riemann addresses the question of the space-filling substance, which he also calls "the aether" In one instance. In this translation, it Is also referred to in the expression "particle of substance", and sometimes as simply 'substance', after the concept of space-filling substance has been introduced. These expressions for space-filling substance are thus distinct from "measurable atoms", "measurable mass" or "measurable bodies".
9. The question mark and both pairs of parentheses appear in the German without explanation. Are they Riemann's marks, or do they indicate an uncertain reading of the manuscript?

  As is well known, geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms. The relationship between these presuppositions [the concept of space, and the basic properties of space] is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori, whether any connection between them is even possible.

  From Euclid to the most famous of the modern reformers of geometry, Legendre, this darkness has been dispelled neither by the mathematicians nor by the philosophers who have concerned themselves with it. This is undoubtedly because the general concept of multiply extended quantities, which includes spatial quantities, remains completely unexplored. I have therefore first set myself the task of constructing the concept of a multiply extended quantity from general notions of quantity. It will be shown that a multiply extended quantity is susceptible of various metric relations, so that space constitutes only a special case of a triply extended quantity. From this however it is a necessary consequence that the theorems of geometry cannot be deduced from general notions of quantity, but that those properties which distinguish Space from other conceivable triply extended quantities can only be deduced from experience. Thus arises the problem of seeking out the simplest data from which the metric relations of Space can be determined, a problem which by its very nature is not completely determined, for there may be several systems of simple data which suffice to determine the metric relations of Space; for the present purposes, the most important system is that laid down as a foundation of geometry by Euclid. These data are - like all data - not logically necessary, but only of empirical certainty, they are hypotheses; one can therefore investigate their likelihood, which is certainly very great within the bounds of observation, and afterwards decide upon the legitimacy of extending them beyond the bounds of observation, both in the direction of the immeasurably large, and in the direction of the immeasurably small.