shall obtain a square of 6, which will be "associated," but in which two lines or columns will be erratic, one showing a correct summation -I and the other a correct summation +1. The following equations (VII) may be used for the columns: the last being an inequality. Fig. II shows the complementary pairs of natural numbers 1 to 36 with their whole differences, which in this case are used in the equations (VII) and (VIII) instead of the half differences, because these differences can not be halved without involving fractions. Fig. 12 is the square derived from equations (VII) and will be found correct in the columns. Fig. 13 is the square formed from equations (VIII) and is correct in the 1st, 2d, 5th, and 6th rows, but erratic in the 3d and 4th rows. The finished six-square made by combining Figs. 12 and 13 is shown in Fig. 14 which is associated or regular, and which gives correct summations in all the columns and rows excepting the 3d and 4th rows which show -I and +1 inequalities respectively. Fig. 14, like Fig. 10, could not probably be produced by any other method than the one herein employed, and both of these squares therefore demonstrate the value of the methods for constructing new variants. Fig. 14 can be readily converted into a continuous or pan-diagonal square by first interchanging the 4th and 6th columns and then, in the square so formed, interchanging the 4th and and 6th rows. The result of these changes is given in Fig. 15 which shows correct summations in all columns and rows, excepting in the 3d and 6th row which carry the inequalities shown in Fig. 14. This square has lost its property of association by the above change but has now correct summation in all its diagonals. It is a demonstrable fact that squares of orders 4n + 2, (i. e., 6, 10, 14 etc.) cannot be made perfectly magic in columns and rows and at the same time either associated or pandiagonal when constructed with consecutive numbers. Dr. Planck also points out that the change which converts all even associated squares into pan-diagonal squares may be tersely expressed as follows: Divide the square into four quarters as shown in Fig. 16. The inverse change from pan-diagonal to association is not necessarily effective, but it may be demonstrated with the "Jaina" square given by Dr. Carus in Magic Squares and Cubes, p. 125, which is here reproduced in Fig. 17. This is a continuous or pan-diagonal square, but after making the above mentioned changes it becomes an associated or regular square as shown in Fig. 18. 4432 53 24331 54 58 19 39 14 57 20 4013 38 15 59 18 37 16 60 17 29 56 4 41 30 55 342 23 62 10 35 24 61936 11 34 22 63|12|33|21|64 Fig. 19. Magic squares of the 8th order can however be made to combine the pan-diagonal and associated features as shown in Fig. 19 which is contributed by Frierson, and this is true also of all larger squares of order 8n. W. S. ANDREWS. SCHENECTADY, N. Y. CRITICISMS AND DISCUSSIONS. HEINRICH HERTZ'S THEORY OF TRUTH. A CONTRIBUTION TO CRITIQUE OF COGNITION. In a paper read before the American Philosophical Association in 1902,1 I explained critique of cognition to be the examination of systems of cognition according to principles and briefly sketched out these principles. They were formulated as conditions which a system must satisfy, and formed four groups the last of which was the group of the conditions of truth. This group has the remarkable property that it requires the fulfilment of all the other conditions. We can therefore say that critique of cognition has for its problem the determination of the content of truth of the systems of cognition. Its principles can be considered as principles of truth, and so it is clear that the problem of truth is the most important and deepest which the establishment of the principles of critique of cognition. offers. The first condition of this last group determines the truth of a system with relation to its generating problem; a second condition determines the truth of the generating problem itself and therewith the truth of the system not relatively to its own problem but with respect to the system of cognition. The conditions of truth have been formulated differently in the different schools. We can distinguish two large groups and call them (1) the group of the external conditions of truth and (2) the group of the internal conditions of truth. The first, in determining the truth of a system B, takes another system A as given; the truth of the system B is then determined with respect to A as "agreement" or "correspondence" of B with A. It goes beyond the system B to another system A, the "object," or "nature," or the "things." It is the theory of truth of realism or dualism. The other does not take |