Newtonian mechanics mv2. In this the dependence of energy on the system of reference appears obvious. But since the t-axis can now be taken in the direction of any time-vector, the law of energy, on the other hand, formulated for every possible system of reference, contains the entire system of equations of motion. This fact retains its significance in the above-mentioned limiting case for c=0, also for the deductive development of the Newtonian mechanics, and in this sense it has already been noted by J. R. Schütz."

We can from the start so determine the relation of unit length to unit time, that the natural limit of velocity becomes c=1. If we then introduce V-1.t s in place of t the quadratic differential expression becomes

dr2-dx2-dy2 - dz2 - ds2

thus completely symmetrical in x, y, z, s, and this symmetry now enters into every law which does not contradict the world-postulate. Accordingly we can express the essence of this postulate very significantly in the mystical formula:

300,000 kilometers V-1 second.

V.

Perhaps the advantages secured by the world-postulate are nowhere show more impressively than in stating the effect according to the Maxwell-Lorentz theory of a point-charge moving at will. Let us consider the world-line of such a paint-electron with the charge e and introduce the characteristic time from any initial point. To obtain the field determined by the electron at any worldpoint P, we construct the past cone P, (Fig. 4). This meets the infinite world-line of the electron at a single point P because its directions are everywhere those of a time vector. We construct the tangent at P to the world-line and through P, the normal P, Q to this tangent. Let the scalar of P, Q be r. Then, according to the definition of a past cone we must take the scalar value of PQ as r/c.

1

Now the vector in the direction PQ of length e/r represents in its components along the x-, y-, z-axes the vector potential multiplied by c, and in the component along the t-axis the scalar potential

J. R. Schütz, "Das Prinzip der absoluten Erhaltung der Energie" in Nachrichten der k. Gesellschaft der Wissenschaften zu Göttingen (mathematisch-physikalische Klasse), 1897, p. 110.

of the field produced by e for the world-point P1. This is the basis of the fundamental laws established by A. Liénard and E. Wiechert.

In the description of the field itself produced by the electron it is clearly seen that the separation of the field into electric and magnetic forces is a relative one depending on the time axis of reference. Both forces can be described together most luminously after the analogy, however imperfect, of a force screw in mechanics. e

I shall now describe the ponderomotive effect of one pointcharge moving at will on another point-charge moving at will. Let us take the world-line of the second point-electron of charge e1, passing through the world-point P1. Let us determine P, Q, r as before, then (Fig. 4) construct the center M of the hyperbola of

8 A. Liénard, "Champ électrique et magnétique produit par une charge concentrée en un point et animée d'un mouvement quelconque," in L'éclairage électrique, Vol. XVI, 1898, pp. 5, 53, 106; E. Wiechert, "Elektrodynamische Elementargesetze" in Archives néerlandaises des sciences exactes et naturelles (2), Vol. V, 1900, p. 549.

curvature at P, and finally the normal MN from M to a straight line through P parallel to QP1. Let us next determine with P as origin a system of reference with the t-axis in the direction of PQ, the x-axis in the direction of QP,, the y-axis in the direction of MN, so that finally the direction of the z-axis is determined as normal to the t-, x-, y-axes. Let the acceleration vector at P be x, y, z, t, and the velocity-vector at P, be x, y, z, t. Now the action of the moving force-vector of the first electron e moving at will on the second electron e, moving at will at P, is formulated thus:

1

in which the three relations between the components Kx, Ky, Kz, Kt, of the vector K are: cKt — Kx=1/r2, Ky=ÿ/er, Kz=0 and lastly, this vector K is normal to the velocity-vector at P1 and through this circumstance alone is dependent on the latter velocity-vector.

If we compare this statement with the previous formulation of the same fundamental law of the ponderomotive effect of moving point-charges on each other, we cannot but grant that the relations here coming under observation do not manifest their intrinsic character of utter simplicity except in four dimensions, but throw a very complicated projection upon a tri-dimensional space preimposed upon them.

In mechanics reformed according to the world-postulate the disagreements which have caused friction between the Newtonian mechanics and modern electrodynamics disappear of their own accord. I shall touch upon the relation of the Newtonian law of attraction to this postulate. I shall assume that when two point masses m and m, describe their world-lines a moving force-vector acts from m on m, just as in the above expression in the case of electrons, except that now mm, is to be substituted for - ee,.

We shall now consider especially the particular case where the acceleration-vector of m is constantly zero, in which case we can so introduce t that m is conceived of as at rest, and the motion of m1 depends only on the moving force-vector proceeding from m. If we modify this vector first by the factor

t1 = √1 − v2/c2,

• K.Schwarzschild, Nachrichten der k. Gesellschaft der Wissenschaften zu Göttingen (mathematisch-physikalische Klasse), 1903. p. 132. H. A. Lorentz, Enzyklopädie der mathematischen Wissenschaften, Vol. V, Art. 14, p. 199.

which, up to quantities of the order 1/c2 is equal to 1, then it follows1o that for positions 1, y1, 21 of m, and their corresponding time-positions, Kepler's laws would again obtain, except that in place of the times t, the characteristic time T, of m would be substituted. On the basis of this simple observation we can see that the proposed law of attraction in conjunction with the new mechanics would be no less suitable for explaining astronomical observations than Newton's law of attraction in conjunction with the Newtonian mechanics.

The fundamental equations for electromagnetic processes in ponderable bodies are likewise in complete harmony with the worldpostulate. Even the derivation of these equations, as taught by Lorentz, on the basis of conceptions of the electron theory need not for this end by any means be abandoned, as I shall show elsewhere.11

The universal validity of the world-postulate is, I should believe, the true core of an electromagnetic world-picture; first discovered by Lorentz, then further developed by Einstein, it is now clearly discernible. In the future development of its mathematical consequences enough indications will be found for experimental verification of the postulate to reconcile by the idea of a pre-established harmony between pure mathematics and physics even those to whom a surrender of old accustomed view-points is uncongenial or painful. HERMANN MINKOWSKI.

SUGGESTIONS FOR A NEW LOGIC.

The world of logic is in a state of disturbance. A new logic is wanted and anxiously sought after. The logisticians are active and non-Aristotelian thinkers are presenting solutions. Among those dissatisfied with both the traditional and modern logic there is one man of particular originality and distinction. It is Dr. Charles Mercier of Charing Cross Hospital, London, and we take pleasure in presenting a review of his work.

DR. MERCIER'S LOGICAL WORK.

Dr. Charles A. Mercier is a physician whose specialty is mental

10 H. Minkowski, Ges. Abhandlungen, II, p. 403.

11 This idea is developed in the paper: "Eine Ableitung der Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern vom Standpunkte der Elektronentheorie. Aus dem Nachlass von Hermann Minkowski bearbeitet von Max Born in Göttingen. Mathematische Annalen, Vol. LXVIII, 1910, p. 526; Ges. Abhandlungen, Vol. II, p. 405.

pathology. Believing that in mental as in bodily disorder the study of order is indispensable to the study of disorder, and that in mental disease the power of logical reasoning is often impaired, he thinks the knowledge of an adequate and correct logic very important and necessary. For this purpose he has found the traditional logic defective in every particular. Nor does he agree with Mill and the modern logicians who follow him, although they too found fault with the traditional school. He has frankly confessed that he does not understand their logic, and the symbolic logic of the logicians is even less satisfactory to him for this he considers no logic at all but "mathematics gone mad." Therefore he has written A New Logic of his own in a volume of 422 pages1 which, though not complete and of course not wholly new, yet is different enough from all previous expositions to warrant the name. He regards his system as an organized and coherent body of doctrine, covering the whole field of reasoning, growing naturally from a single root and forming a harmonious and interdependent whole. And he hopes it will prove of great practical value in clarifying the ideas of the thoughtful and intelligent public with regard to the laws of reasoning. It is impossible to give any comprehensive summary of his method here, for he takes up the defects of the time-honored system one by one, and then in each case works out his own corresponding theory. But we can include here his sweeping indictment of traditional logic, which will give some idea of the scope of his contemplated reform. He says: "In my opinion, its concepts of the composition of the proposition, and of the constituent parts of the proposition, are erroneous; its doctrines of quantity and quality are wrong; its immediate inferences are but a poor few out of multitudes that may be obtained by an adequate logic; the few immediate inferences it does obtain are faulty; its doctrine of the syllogism is artificial and mistaken; the rules of the syllogism are all wrong; there are multitudes of mediate inferences that cannot be reached by the syllogism;....in short, its whole system is insufficient, defective, and erroneous from beginning to end."

Dr. Mercier takes issue with Mill and subsequent writers when they devote a chapter of their logic to the subject of causation and insist that causation lies at the root of induction, for he believes that causation no more belongs to the subject matter of logic than rotation or imitation; that it is neither a principle nor method, but 1 London: Longmans Green and Co.; Chicago: Open Court Publishing Company, 1912.