X.

MAGIC CUBES.

Several inquirers, particularly Kochansky (1686), Sauveur (1710), Hugel (1859), and Scheffler (1882), have extended the principle of the magic squares of the plane to three-dimensioned space. Imagine a cube divided by planes parallel to its sides and equidistant from one another, into cubical compartments. The problem is then, so to insert in these compartments the successive natural numbers that every row from the right to the left, every row from the front to the back, every row from the top to the bottom, every diagonal of a square, and every principal diagonal passing through the centre of the cube shall contain numbers whose sum is always the same. For 3 times 3 times 3 compartments, a magic cube of this description is not constructible. For 4 times, 4 times 4 compartments a cube is constructible such that any row parallel to an edge of the cube and every principal diagonal give the sum of 130. To obtain a magic cube of 64 compartments, imagine the numbers which belong in the compartments written on the upper surface of the same and the numbers then taken off in layers of 16 from the top downwards. We obtain thus 4 squares of 16 cells each, which together make up the magic cube; as the following diagrams will show:

[Illustration:

First Layer Second Layer Third Layer Fourth Layer from the Top. from the Top. from the Top. from the Top. +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ | 1|48|32|49| |63|18|34|15| |62|19|35|14| | 4|45|29|52| +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ |60|21|37|12| | 6|43|27|54| | 7|42|26|55| |57|24|40| 9| +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ |56|25|41| 8| |10|39|23|58| |11|38|22|29| |53|28|44| 5| +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ |13|36|20|61| |51|30|46| 3| |50|31|47| 2| |16|33|17|64| +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+ ]

The same sum 130 here comes out not less than 52 times; viz. in the first place from the 16 rows from left to right, secondly from the 16 rows from the front to the back, thirdly from the 16 rows counting from the top to the bottom, and lastly from the 4 rows which join each two opposite corners of the cube, namely from the rows: 1, 43, 22, 64; 49, 27, 38, 16; 13, 39, 26, 52; 61, 23, 42, 4.

For a cube with 5 compartments in each edge the arrangement of the figures can so be made that all the 75 rows parallel to any and every edge, all the 30 rows lying in any diagonal of a square, and all the 4 rows forming any principal diagonal shall have one and the same summation, 315.

Just as the magic squares of an odd number of cells could be formed with the aid of _two_ auxiliary squares, so also odd-numbered magic cubes can be constructed with the help of _three_ auxiliary cubes.

[Illustration:

First Layer from Top. Second Layer from Top. Third Layer from Top. +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ |121| 27| 83| 14| 70| | 2 | 58|114| 45| 96| | 33| 89| 20| 71|102| +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 10| 61|117| 48| 79| | 36| 92| 23| 54|110| | 67|123| 29| 85| 11| +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 44|100| 1 | 57|113| | 75|101| 32| 88| 19| | 76| 7 | 63|119| 50| +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 53|109| 40| 91| 22| | 84| 15| 66|122| 28| |115| 41| 97| 3 | 59| +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+ | 87| 18| 74|105| 31| |118| 49| 80| 6 | 62| | 24| 55|106| 37| 93| +---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+

Fourth Layer from Top. Lowest Layer. +---+---+---+---+---+ +---+---+---+---+---+ | 64|120| 46| 77| 8 | | 95| 21| 52|108| 39| +---+---+---+---+---+ +---+---+---+---+---+ | 98| 4 | 60|111| 42| |104| 35| 86| 17| 73| +---+---+---+---+---+ +---+---+---+---+---+ |107| 38| 94| 25| 51| | 13| 69|125| 26| 82| +---+---+---+---+---+ +---+---+---+---+---+ | 16| 72|103| 34| 90| | 47| 78| 9 | 65|116| +---+---+---+---+---+ +---+---+---+---+---+ | 30| 81| 12| 68|124| | 56|112| 43| 99| 5 | +---+---+---+---+---+ +---+---+---+---+---+ ]

In this manner the preceding magic cube of 5 times 5 times 5 compartments is formed, in which, it may be additionally noticed, the middle number between 1 and 125, namely 63, is placed in the central compartment; by which arrangement the attainment of the sum of 315 is assured in the four principal diagonals and the 30 sub-diagonals. The condition attained in the magic squares, that the diagonal-pairs parallel to the sub-diagonals also shall give the sum 315 is not attainable in this case but is so in the case of higher numbers of compartments.

CONCLUSION.

Musing on such problems as are the magic squares is fascinating to thinkers of a mathematical turn of mind. We take delight in discovering a harmony that abides as an intrinsic quality in the forms of our thought. The problems of the magic squares are playful puzzles, invented as it seems for mere pastime and sport. But there is a deeper problem underlying all these little riddles, and this deeper problem is of a sweeping significance. It is the philosophical problem of the world-order.

The formal sciences are creations of the mind. We build the sciences of mathematics, geometry, and algebra with our conception of pure forms which are abstract ideas. And the same order that prevails in these mental constructions permeates the universe, so that an old philosopher, overwhelmed with the grandeur of law, imagined he heard its rhythm in a cosmic harmony of the spheres.

H. SCHUBERT.

FOOTNOTES:

[68] The term melancholy meant in Dürer’s time, as it did also in Shakespeare’s and Milton’s, “thought or thoughtfulness.” Says Milton in _Il Penseroso_:

“Hail, thou Goddess, sage and holy, Hail divinest melancholy Whose saintly visage is too bright To hit the sense of human sight, And therefore to our weaker view O’erlaid with black, staid Wisdom’s hue.”—I, 12.

Thought that does not lead to action produces a gloomy state of mind. Thoughtfulness which cannot find a way out of itself is that melancholy which engenders weakness,—a truth which is illustrated in Hamlet. Shakespeare still uses the words thought and melancholy as synonyms, saying:

“The native hue of resolution Is sicklied o’er with the pale cast of thought.”

Dürer’s melancholy does not represent the gloominess of thought, but the power of invention. Soberness and even a certain sadness are considered only as an element of this melancholy, but on the whole the genius of thought appears bright, self-possessed, and strong.

Dürer represents the Science of Mechanical Invention as a winged female figure musing over some problem. Scattered on the floor around her lie some of the simple tools used in the sixteenth century. A ladder leans against the house, that assists in climbing otherwise inaccessible heights. A scale, an hour-glass, a bell, and the magic square are hanging on the wall behind her.

At a distance a bat-like creature, being the gloom of melancholy, hovers in the air like a dark cloud, but the sun rises above the horizon, and at the happy middle between these two extremes stands the rainbow of serene hope and cheerful confidence.

MR. SPENCER ON THE ETHICS OF KANT.

Mr. Herbert Spencer published in the _Fortnightly Review_ for July 1888 and in the _Popular Science Monthly_ for August of the same year an article on “The Ethics of Kant” in which he so strangely misrepresents Kant’s position that Kant to any uninitiated reader must appear not only as superficial and shallow, but even as palpably nonsensical.

Mr. Spencer’s article on “The Ethics of Kant” is a severe criticism mainly of the nonsensical idea, erroneously imputed to Kant, of a will that has no end. At the same time Mr. Spencer reproaches Kant with assuming the simplicity of conscience and believing in a non-evolutionary origin of the minds of living beings.

In reply to Mr. Spencer an editorial article appeared in _The Open Court_ under the caption “Herbert Spencer on the Ethics of Kant” (Nos. 51 and 52), which was supplemented by another article entitled “Kant on Evolution” (No. 158), the latter being elicited by a renewed attack of Mr. Spencer upon Kant’s views (which appeared in _Mind_, No. LIX, p. 313).

Mr. Spencer has republished his article “The Ethics of Kant” together with many other older articles in a work of three volumes entitled “Essays Scientific, Political, and Speculative,” 1891, in which he repeats the following sentence:

“Thus the basis of the argument by which Kant attempts to justify his assumption that there exists a good will apart from a good end, disappears utterly; and leaves his dogma in all its naked unthinkableness.”

To this sentence he adds the following foot-note as a reply to my criticisms:

“I find that in the above three paragraphs I have done Kant less than justice and more than justice—less, in assuming that his evolutionary view was limited to the genesis of our sidereal system, and more, in assuming that he had not contradicted himself. My knowledge of Kant’s writings is extremely limited. In 1844 a translation of his ‘Critique of Pure Reason’ (then I think lately published) fell into my hands, and I read the first few pages enunciating his doctrine of Time and Space: my peremptory rejection of which caused me to lay the book down. Twice since then the same thing has happened; for, being an impatient reader, when I disagree with the cardinal propositions of a work I can go no further. One other thing I knew. By indirect references I was made aware that Kant had propounded the idea that celestial bodies have been formed by the aggregation of diffused matter. Beyond this my knowledge of his conceptions did not extend; and my supposition that his evolutionary conception had stopped short with the genesis of sun, stars, and planets, was due to the fact that his doctrine of Time and Space, as forms of thought anteceding experience, implied a supernatural origin inconsistent with the hypothesis of natural genesis. Dr. Paul Carus, who, shortly after the publication of this article in the _Fortnightly Review_ for July, 1888, undertook to defend the Kantian ethics in the American journal which he edits, _The Open Court_, has now (Sept. 4, 1890), in another defensive article, translated sundry passages from Kant’s ‘Critique of Judgment,’ his ‘Presumable Origin of Humanity,’ and his work ‘Upon the different Races of Mankind,’ showing that Kant was, if not fully, yet partially, an evolutionist in his speculations about living beings. There is, perhaps, some reason for doubting the correctness of Dr. Carus’s rendering of these passages into English. When, as in the first of the articles just named, he failed to distinguish between consciousness and conscientiousness, and when, as in this last article, he blames the English for mistranslating Kant, since they have said ‘Kant maintained that Space and Time are intuitions,’ which is quite untrue, for they have everywhere described him as maintaining that Space and Time are _forms_ of intuition, one may be excused for thinking that possibly Dr. Carus has read into some of Kant’s expressions, meanings which they do not rightly bear. Still, the general drift of the passages quoted makes it tolerably clear that Kant must have believed in the operation of natural causes as largely, though not entirely, instrumental in producing organic forms: extending this belief (which he says ‘can be named a daring venture of reason’) in some measure to the origin of Man himself. He does not, however extend the theory of natural genesis to the exclusion of the theory of supernatural genesis. When he speaks of an organic habit ‘which in the wisdom of nature appears to be thus arranged in order that the species shall be preserved’; and when, further, he says ‘we see, moreover, that a germ of reason is placed in him, whereby, after the development of the same, he is destined for social intercourse,’ he implies divine intervention. And this shows that I was justified in ascribing to him the belief that Space and Time, as forms of thought, are supernatural endowments. Had he conceived of organic evolution in a consistent manner, he would necessarily have regarded Space and Time as subjective forms generated by converse with objective realities.

“Beyond showing that Kant had a partial, if not a complete, belief in organic evolution (though with no idea of its causes), the passages translated by Dr. Carus show that he entertained an implied belief which it here specially concerns me to notice as bearing on his theory of ‘a good will.’ He quotes approvingly Dr. Moscati’s lecture showing ‘that the upright walk of man is constrained and unnatural,’ and showing the imperfect visceral arrangements and consequent diseases which result: not only adopting, but further illustrating, Dr. Moscati’s argument. If here, then, there is a distinct admission, or rather assertion, that various human organs are imperfectly adjusted to their functions, what becomes of the postulate above quoted ‘that no organ for any purpose will be found in it but what is also the fittest and best adapted for that purpose’? And what becomes of the argument which sets out with this postulate? Clearly, I am indebted to Dr. Carus for enabling me to prove that Kant’s defence of his theory of ‘a good will’ is, by his own showing, baseless.”

Mr. Spencer’s reply to my criticisms is surprising in more than one respect.

First, without even mentioning the objections I make he discredits my arguments by throwing doubt upon the correctness of the translations of the quoted passages.

Secondly, he alleges, with a view of justifying his doubt, that in the first of my articles I “failed to distinguish between consciousness and conscientiousness.”[69]

Thirdly, Mr. Spencer declares that I had “read into some of Kant’s expressions, meanings which they do not rightly bear.”

Fourthly, Mr. Spencer bases this opinion upon a double mistake: he blames me for not distinguishing between the Kantian phrases that “Space and Time are intuitions” and that they are “forms of intuition.”

Fifthly, acknowledging after all that Kant had at least “a partial belief in organic evolution,” Mr. Spencer accuses him of inconsistency.

Sixthly, several statements concerning Kant’s views are made not because Kant held them but because Mr. Spencer assumes for trivial reasons that he is “justified in ascribing them to him.”

Seventhly, these statements so vigorously set forth are accompanied by Mr. Spencer’s remarkably frank confession of unfamiliarity with the subject under discussion.

It may be added that Mr. Spencer calls my criticisms “defensive articles.” He says that “I undertook to defend the Kantian ethics”; while, in fact, my articles are aggressive. Kant needs no defense for being misunderstood, and it would not be my business to defend him, for I am not a Kantian in the sense that I adopt any of the main doctrines of Kant. On the contrary I dissent from him on almost all fundamental questions. In ethics I object to Kant’s views in so far as they can be considered as pure formalism.[70] I am a Kantian only in the sense that I respect Kant as one of the most eminent philosophers, that I revere him as that teacher of mine whose influence upon me was greatest, and I consider the study of Kant’s works as an indispensable requisite for understanding the problems of the philosophy of our time. Far from defending Kant’s position, I only undertook to inform Mr. Spencer of what Kant had really maintained, so that instead of denouncing absurdities which Kant had never thought of, he might criticise the real Kant.

* * * * *

I shall now take up the details of Mr. Spencer’s reply: