Chapter XXV: Part 25

Text size

Of magic squares having an even number of places we have hitherto had to deal only with the square of 4. To construct squares of this description having a higher even number of places, different and more complicated methods must be employed than for squares of odd numbers of places. However, in this case also, as in dealing with the square of 4, we start with the natural sequence of the numbers and must then find the complements of the numbers with respect to some other certain number (as 17 in the square of 4) and also effect certain exchanges of the numbers with one another. To form, for example, a magic square of 6 times 6 places, we inscribe in the 12 diagonal cells the numbers that in the natural sequence of inscription fall into these places, then in the remaining cells the complements of the numbers that belong therein with respect to 37, and finally effect the following six exchanges, viz. of the numbers 33 and 3, 25 and 7, 20 and 14, 18 and 13, 10 and 9, and 5 and 2. In this way the following magic square is obtained.

+--+--+--+--+--+--+
| 1|35|34| 3|32| 6|
+--+--+--+--+--+--+
|30| 8|28|27|11| 7|
+--+--+--+--+--+--+
|24|23|15|16|14|19|
+--+--+--+--+--+--+
|13|17|21|22|20|18|
+--+--+--+--+--+--+
|12|26| 9|10|29|25|
+--+--+--+--+--+--+
|31| 2| 4|33| 5|36|
+--+--+--+--+--+--+
]

This square may also be constructed by the method of De la Hire, from two auxiliary squares with the numbers 1, 2, 3, 4, 5, 6 and 0, 6, 12, 18, 24, 30 respectively. In this case, however, the vertical rows of the one square and the horizontal rows of the other must each so contain two same numbers thrice repeated that the summation shall always remain 21 and 90 respectively. In this manner we get the magic square last given above from the two following auxiliary squares:

+--+--+--+--+--+--+
| 1| 5| 4| 3| 2| 6|
+--+--+--+--+--+--+
| 6| 2| 4| 3| 5| 1|
+--+--+--+--+--+--+
| 6| 5| 3| 4| 2| 1|
+--+--+--+--+--+--+
| 1| 5| 3| 4| 2| 6|
+--+--+--+--+--+--+
| 6| 2| 3| 4| 5| 1|
+--+--+--+--+--+--+
| 1| 2| 4| 3| 5| 6|
+--+--+--+--+--+--+
]

and

+--+--+--+--+--+--+
| 0|30|30| 0|30| 0|
+--+--+--+--+--+--+
|24| 6|24|24| 6| 6|
+--+--+--+--+--+--+
|18|18|12|12|12|18|
+--+--+--+--+--+--+
|12|12|18|18|18|12|
+--+--+--+--+--+--+
| 6|24| 6| 6|24|24|
+--+--+--+--+--+--+
|30| 0| 0|30| 0|30|
+--+--+--+--+--+--+
]

It is to be noted in connection with this example that here also as in the case of odd-numbered squares, it is possible so to inscribe six times the numbers from 1 to 6 that each number shall appear once and only once in each horizontal, vertical, and diagonal row; for example, in the following manner:

+--+--+--+--+--+--+
| 1| 2| 3| 4| 5| 6|
+--+--+--+--+--+--+
| 2| 4| 6| 1| 3| 5|
+--+--+--+--+--+--+
| 3| 6| 5| 2| 1| 4|
+--+--+--+--+--+--+
| 5| 3| 1| 6| 4| 2|
+--+--+--+--+--+--+
| 6| 5| 4| 3| 2| 1|
+--+--+--+--+--+--+
| 4| 1| 2| 5| 6| 3|
+--+--+--+--+--+--+
]

But if we attempt so to insert, in a like manner, the other set of numbers 0, 6, 12, 18, 24, 30 in a second auxiliary square, that each number of the first auxiliary square shall stand once and once only in a corresponding cell with each number of the second square, all the attempts we may make to fulfil coincidently the last named condition will result in failure. It is therefore necessary to select auxiliary squares like the two given above. It is noteworthy, that the fulfilment of the second condition is impossible only in the case of the square of 6, but that in the case of the square of 4 or of the square of 8, for example, two auxiliary squares, such as the method of De la Hire requires, are possible. Thus, taking the square of 4 we get

+--+--+--+--+
| 1| 2| 3| 4|
+--+--+--+--+
| 4| 3| 2| 1|
+--+--+--+--+
| 2| 1| 4| 3|
+--+--+--+--+
| 3| 4| 1| 2|
+--+--+--+--+
]

and

+--+--+--+--+
| 0| 4| 8|12|
+--+--+--+--+
| 8|12| 0| 4|
+--+--+--+--+
|12| 8| 4| 0|
+--+--+--+--+
| 4| 0|12| 8|
+--+--+--+--+
]

The reader may form for himself the magic square which these give.

The existence of these two auxiliary squares furnishes a key to the solution of a pretty problem at cards. If we replace, namely, the numbers 1, 2, 3, 4 by the Ace, the King, the Queen, and the Knave, and the numbers 0, 4, 8, 12 by the four suits, clubs, spades, hearts, and diamonds, we shall at once perceive that it is possible, and must be so necessarily, quadratically to arrange in such a manner the four Aces, the four Kings, the Four Queens, and the four Knaves, that in each horizontal, vertical, and diagonal row, each one of the four suits and each one of the four denominations shall appear once and once only. The auxiliary squares above given furnish the appended solution of this problem:

To fix the solution of the problem in the memory, observe that, starting from the several corners, each suit and each denomination must be placed in the spots of the move of a Knight. If we fix the positions of the four cards of any one row, there will be only two possibilities left of so placing the other cards that the required condition of having each suit and each denomination once and only once in each row shall be fulfilled.

Of magic squares of an even number of places we have up to this point examined only the squares of 4 and of 6. For the sake of completeness we append here one of 8 and one of 10 places. The mode of construction of these squares is similar to the method above discussed for the lower even numbers.

+--+--+--+--+--+--+--+--+
| 1|63|62| 4| 5|59|58| 8|
+--+--+--+--+--+--+--+--+
|56|10|11|53|52|14|15|49|
+--+--+--+--+--+--+--+--+
|48|18|19|45|44|22|23|41|
+--+--+--+--+--+--+--+--+
|25|39|38|28|29|35|34|32|
+--+--+--+--+--+--+--+--+
|33|31|30|36|37|27|26|40|
+--+--+--+--+--+--+--+--+
|24|42|43|21|20|46|47|17|
+--+--+--+--+--+--+--+--+
|16|50|51|13|12|54|55| 9|
+--+--+--+--+--+--+--+--+
|57| 7| 6|60|61| 3| 2|64|
+--+--+--+--+--+--+--+--+
]

+---+---+---+---+---+---+---+---+---+---+
| 1 | 99| 3 | 97| 96| 5 | 94| 8 | 92| 10|
+---+---+---+---+---+---+---+---+---+---+
| 90| 12| 88| 14| 86| 85| 17| 83| 19| 11|
+---+---+---+---+---+---+---+---+---+---+
| 80| 79| 23| 77| 25| 26| 74| 28| 22| 71|
+---+---+---+---+---+---+---+---+---+---+
| 31| 69| 68| 34| 66| 65| 37| 33| 62| 40|
+---+---+---+---+---+---+---+---+---+---+
| 60| 42| 58| 57| 45| 46| 44| 53| 49| 51|
+---+---+---+---+---+---+---+---+---+---+
| 50| 52| 43| 47| 55| 56| 54| 48| 59| 41|
+---+---+---+---+---+---+---+---+---+---+
| 61| 32| 38| 64| 36| 35| 67| 63| 39| 70|
+---+---+---+---+---+---+---+---+---+---+
| 21| 29| 73| 27| 75| 76| 24| 78| 72| 30|
+---+---+---+---+---+---+---+---+---+---+
| 20| 82| 18| 84| 15| 16| 87| 13| 89| 81|
+---+---+---+---+---+---+---+---+---+---+
| 91| 9 | 93| 4 | 6 | 95| 7 | 98| 2 |100|
+---+---+---+---+---+---+---+---+---+---+
]

The magic squares of even numbers thus constructed are not the only possible ones. On the contrary, there are very many others possible, which obey different laws of formation. It has been calculated, for example, that with the square of 4 it is possible to construct 880, and with the square of 6, _several million_, different magic squares. The number of odd-numbered magic squares constructible by the method of De la Hire is also very great. With the square of 7, the possible constructions amount to 363,916,800. With the squares of higher numbers the multitude of the possibilities increases in the same enormous ratio.

MAGIC SQUARES WHOSE SUMMATION GIVES THE NUMBER OF A YEAR.

The magic squares which we have so far considered contain only the natural numbers from 1 upwards. It is possible, however, easily to deduce from a correct magic square other squares in which a different law controls the sequence of the numbers to be inscribed. Of the squares obtained in this manner, we shall devote our attention here only to such in which, although formed by the inscription of successive numbers, the sum obtained from the addition of the rows is a determinate number which we have fixed upon beforehand, as _the number of a year_. In such a case we have simply to add to the numbers of the original square a determinate number so to be calculated, that the required sum shall each time appear. If this sum is divisible by 3, magic squares will always be obtainable with 3 times 3 spaces which shall give this sum. In such a case we divide the sum required by 3 and subtract 5 from the result in order to obtain the number which we have to add to each number of the original square. If the sum desired is even but not divisible by 4, we must then subtract from it 34 and take one fourth of the result, to obtain the number which in this case is to be added in each place. If, for example, we wish to obtain the number of the year 1890 as the resulting sum of each row, we shall have to add to each of the numbers of an ordinary magic square of 4 times 4 spaces the number 464; in other words, instead of the numbers from 1 to 16 we have to insert in the squares the numbers from 465 to 480. As the number of the present year 1892 is divisible by 11, it must be possible to deduce from the magic square constructed by us at the conclusion of Section III a second magic square in which each row of 11 cells will give the number of the year 1892. To do this, we subtract from 1892 the sum of the original square, namely 671, and divide the remainder by 11, whereby we get 111 and thus perceive that the numbers from 112 to 232 are to be inscribed in the cells of the square required. We get in this way the preceding square, from which _one and the same sum, namely 1892, can be obtained 44 times_, first from each of the 11 horizontal rows, secondly from each of the 11 vertical rows, thirdly from each of the two diagonal rows, and fourthly twenty additional times from each and every pair of any two rows that lie parallel to a diagonal, have together 11 cells, and lie on different sides of the diagonal, as for example, 196, 122, 158, 205, 131, 167, 214, 140, 187, 223, 149.

+----+----+----+----+----+----+----+----+----+----+----+
| 112| 124| 136| 148| 160| 172| 184| 196| 208| 220| 232| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 147| 159| 171| 183| 195| 207| 219| 231| 122| 123| 135| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 182| 194| 206| 218| 230| 121| 133| 134| 146| 158| 170| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 217| 229| 120| 132| 144| 145| 157| 169| 181| 193| 205| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 131| 143| 155| 156| 168| 180| 192| 204| 216| 228| 119| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 166| 167| 179| 191| 203| 215| 227| 118| 130| 142| 154| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 190| 202| 214| 226| 117| 129| 141| 153| 165| 177| 178| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 225| 116| 128| 140| 152| 164| 176| 188| 189| 201| 213| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 139| 151| 163| 175| 187| 199| 200| 212| 224| 115| 127| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 174| 186| 198| 210| 211| 223| 114| 126| 138| 150| 162| = 1892
+----+----+----+----+----+----+----+----+----+----+----+-------
| 209| 221| 222| 113| 125| 137| 149| 161| 173| 185| 197| = 1892
+----+----+----+----+----+----+----+----+----+----+----+
1892 1892 1892 1892 1892 1892 1892 1892 1892 1892 1892
]

VI.

CONCENTRIC MAGIC SQUARES.

The acuteness of the mathematicians has also discovered magic squares which possess the peculiar property that if one row after another be taken away from each side, the smaller inner squares remaining will still be magical squares, that is to say, all their rows when added will give the same sum. It will be sufficient to give two examples here of such squares, (the laws for their construction being somewhat more complicated,) of which the first has 7 times 7 and the second 8 times 8 places. The numbers within each of the dark-bordered frames form with respect to the centre smaller squares which in their own turn are magical.

+--+---+---+--+--+---+---+
| 4| 5| 6 |43|39| 38| 40|
+--++================++--+
|49||15| 16|33|30| 31|| 1|
+--||--++========++--||--+
|48||37||22|27|26||13|| 2|
+--||--||--+--+--||--||--+
|47||36||29|25|21||14|| 3|
+--||--||--+--+--||--||--+
| 8||18||24|23|28||32||42|
+--||--++========++--||--+
| 9||19| 34|17|20| 35||41|
+--++================++--+
|10| 45| 44| 7|11| 12| 46|
+--+---+---+--+--+---+---+
]

+---+---+---+---+---+---+---+---+
| 1 | 56| 55| 11| 53| 13| 14| 57|
+---++=====================++---+
| 63||15| 47| 22| 42| 24|45|| 2 |
+---||--+==============++--||---+
| 62||49||25| 40| 34|31||16|| 3 |
+---||--||--+---+---+--||--||---+
| 4 ||48||28| 37| 35|30||17|| 61|
+---||--||--+---+---+--||--||---+
| 5 ||44||39| 26| 32|33||21|| 60|
+---||--||--+---+---+--||--||---+
| 59||19||38| 27| 29|36||46|| 6 |
+---||--++=============++--||---+
| 58||20| 18| 43| 23| 41|50|| 7 |
+---++=====================++---+
| 8 | 9 | 10| 54| 12| 52| 51| 64|
+---+---+---+---+---+---+---+---+
]

In the first of these two squares the internal square of 3 times 3 places contains the numbers from 21 to 29 in such a manner that each row gives when added the sum of 75. This square lies within a larger one of 5 times 5 spaces, which contains the numbers from 13 to 37 in such a manner that each row gives the sum of 125. Finally, this last square forms part of a square of 7 times 7 places which contains the numbers from 1 to 49 so that each row gives the sum of 175.

In the second square the inner central square of 4 times 4 places contains the numbers from 25 to 40 in such a manner that each row gives the sum of 130. This square is the middle of a square of 6 times 6 places which so contains the numbers from 15 to 50 that each row gives the sum 165. Finally, this last square is again the middle of an ordinary magic square composed of the numbers from 1 to 64.

VII.

MAGICAL SQUARES WITH MAGICAL PARTS.

If we divide a square of 8 times 8 places by means of the two middle lines parallel to its sides into 4 parts containing each 4 times 4 spaces, we may propound the problem of so inserting the numbers from 1 to 64 in these spaces that not only the whole shall form a magic square, but also that each of the 4 parts individually shall be magical, that is to say, give the same sum for each row. This problem also has been successfully solved, as the following diagram will show.

+--+--+--+--++--+--+--+--+
| 1| 4|63|62|| 5| 8|59|58|
+--+--+--+--++--+--+--+--+
|64|61| 2| 3||60|57| 6| 7|
+--+--+--+--++--+--+--+--+
|42|43|24|21||34|35|32|29|
+--+--+--+--++--+--+--+--+
|23|22|41|44||31|30|33|36|
+===========++===========+
|13|16|51|50|| 9|12|55|54|
+--+--+--+--++--+--+--+--+
|52|49|14|15||56|53|10|11|
+--+--+--+--++--+--+--+--+
|38|39|28|25||46|47|20|17|
+--+--+--+--++--+--+--+--+
|27|26|37|40||19|18|45|48|
+--+--+--+--++--+--+--+--+
]

The 4 numbers in each row of any one of the sub-squares here, gives 130; so that the sum of each one of the rows of the large square will be 260.

Finally, in further illustration of this idea, we will submit to the consideration of our readers a very remarkable square of the numbers from 1 to 81. This square, which will be found on the following page (Fig. 28), is divided by parallel lines into 9 parts, of which each contains 9 consecutive numbers that severally make up a magic square by themselves.

+---+---+---++---+---+---++---+---+---+
| 31| 36| 29|| 76| 81| 74|| 13| 18| 11|
+---+---+---++---+---+---++---+---+---+
| 30| 32| 34|| 75| 77| 79|| 12| 14| 16|
+---+---+---++---+---+---++---+---+---+
| 35| 28| 33|| 80| 73| 78|| 17| 10| 15|
+===========++===========++===========+
| 22| 27| 20|| 40| 45| 38|| 58| 63| 56|
+---+---+---++---+---+---++---+---+---+
| 21| 23| 25|| 39| 41| 43|| 57| 59| 61|
+---+---+---++---+---+---++---+---+---+
| 26| 19| 24|| 44| 37| 42|| 62| 55| 60|
+===========++===========++===========+
| 67| 72| 65|| 4 | 9 | 2 || 49| 54| 47|
+---+---+---++---+---+---++---+---+---+
| 66| 68| 70|| 3 | 5 | 7 || 48| 50| 52|
+---+---+---++---+---+---++---+---+---+
| 71| 64| 69|| 8 | 1 | 6 || 53| 46| 51|
+---+---+---++---+---+---++---+---+---+
]

Wonderful as the properties of this square may appear, the law by which the author constructed it is equally simple. We have simply to regard the 9 parts as the 9 cells of a magic square of the numbers from I to IX and then to inscribe by the magic prescript in the square designated as I the numbers from 1 to 9, in the square designated as II the numbers from 10 to 18, and so on. In this way the square above given is obtained from the following base-square:

+----+----+----+
| IV | IX | II |
+----+----+----+
| III| V | VII|
+----+----+----+
|VIII| I | VI |
+----+----+----+
]

VIII.

MAGIC SQUARES THAT INVOLVE THE MOVE OF THE CHESS-KNIGHT.

What one of our readers does not know the problems contained in the recreation columns of our magazines, the requirements of which are to compose into a verse 8 times 8 quadratically arranged syllables, of which every two successive syllables stand on spots so situated with respect to each other that a chess-knight can move from the one to the other? If we replace in such an arrangement the 64 successive syllables by the 64 numbers from 1 to 64, we shall obtain a knight-problem made up of numbers. Methods also exist indeed for the construction of such dispositions of numbers, which then form the foundation of the construction of the problems in the newspapers. But the majority of knight-problems of this class are the outcome of experiment rather than the product of methodical creation. If however it is a severe test of patience to form a knight-problem by experiment, it stands to reason that it is a still severer trial to effect at the same time the additional result that the 64 numbers which form the knight-problem shall also form a magic square.

This trial of endurance was undertaken several decades ago, by a pensioned Moravian officer named Wenzelides, who was spending the last days of his life in the country. After a series of trials which lasted years he finally succeeded in so inscribing in the 64 squares of the chess-board the numbers from 1 to 64 that successive numbers, as well also as the numbers 64 and 1, were always removed from one another in distance and direction by the move of a knight, and that in addition thereto the summation of the horizontal and the vertical rows always gave the same sum 260. Ultimately he discovered several squares of this description, which were published in the _Berlin Chess Journal_. One of these is here appended:

+--+--+--+--+--+--+--+--+
|47|10|23|64|49| 2|59| 6|
+--+--+--+--+--+--+--+--+
|22|63|48| 9|60| 5|50| 3|
+--+--+--+--+--+--+--+--+
|11|46|61|24| 1|52| 7|58|
+--+--+--+--+--+--+--+--+
|62|21|12|45| 8|57| 4|51|
+--+--+--+--+--+--+--+--+
|19|36|25|40|13|44|53|30|
+--+--+--+--+--+--+--+--+
|26|39|20|33|56|29|14|43|
+--+--+--+--+--+--+--+--+
|35|18|37|28|41|16|31|54|
+--+--+--+--+--+--+--+--+
|38|27|34|17|32|55|42|15|
+--+--+--+--+--+--+--+--+
]

The move of the knight and the equality of the summation of the horizontal and vertical rows, therefore, are the facts to be noted here. The diagonal rows do _not_ give the sum 260. Perhaps some one among our readers who possesses the time and patience will be tempted to outdo Wenzelides, and to devise a numeral knight-problem of this kind which will give 260 not only in the horizontal and vertical but also in the two diagonal rows.

IX.

MAGICAL POLYGONS.

So far we have only considered such extensions of the idea underlying the construction of the magic square in which the figure of the square was retained. We may however contrive extensions of the idea in which instead of a square, a rectangle, a triangle, or a pentagon, and the like, appear. Without entering into the consideration of the methods for the construction of such figures, we will give here of magical polygons simply a few examples, all supplied by Professor Scheffler:

1) The numbers from 1 to 32 admit of being written in a rectangle of 4 × 8 in such a manner that the long horizontal rows give the sum of 132 and the short vertical rows the sum of 66; thus:

+--+--+--+--+--+--+--+--+
| 1|10|11|29|28|19|18|16|
+--+--+--+--+--+--+--+--+
| 9| 2|30|12|20|27| 7|25|
+--+--+--+--+--+--+--+--+
|24|31| 3|21|13| 6|26| 8|
+--+--+--+--+--+--+--+--+
|32|23|22| 4| 5|14|15|17|
+--+--+--+--+--+--+--+--+
]

2) The numbers from 1 to 27 admit of being so arranged in three regular triangles about a point which forms a common centre, that each side of the outermost triangle will present 6 numbers of the total summation 96 and each side of the middle triangle 4 numbers whose sum is 61; as the following figure shows:

26 3 6 10 24 27
20 9 11 21
18 2
16 17
15 8
22 5
12
7 13
4 23
19
1 14

25
]

3) The numbers from 1 to 80 admit of being formed about a point as common centre into 4 pentagons, such that each side of the first pentagon from within contains two numbers, each side of the second pentagon four numbers, each of the third six numbers, and each side of the fourth, outermost pentagon eight numbers. The sum of the numbers of each side of the second pentagon is 122, the sum of those of each side of the third pentagon is 248, and that of those of each side of the fourth pentagon 254. Furthermore, the sum of any four corner numbers lying in the same straight line with the centre, is also the same; namely, 92.

1
26 54
31 49
10 15 80
76 36 44 9
50 70 72 32
55 71 16 66 27
5 45 25 65 37 2
11 61 60 24 14
30 20 17 53
40 56 59 43
35 21 64 48
69 57 58 73
6 79
77 75 62 23 67 8
46 41 19 22 63 18 38 33
51 12 39 68 74 42 13 28
4 29 34 7 78 47 52 3
]

4) The numbers from 1 to 73 admit of being arranged about a centre, in which the number 37 is written, into three hexagons which contain respectively 3, 5, and 7 numbers in each side and possess the following pretty properties. Each hexagon always gives the same sum, not only when the summation is made along its six sides, but also when it is made along the six diameters that join its corners and along the six that are constructed at right angles to its sides; this sum, for the first hexagon from within, is 111, for the second 185, and for the third 259.

1 5 6 70 60 59 58
63 8
62 19 53 46 22 45 9
61 20 24 64
2 48 31 42 38 49 57
3 47 39 40 44 56
67 51 41 37 33 23 7
66 50 34 35 54 11
65 25 36 32 43 26 12
10 30 27 13
17 29 21 28 52 55 72
18 71
16 69 68 4 14 15 73
]

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:

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.

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:

I am sorry to see that Mr. Spencer, instead of frankly acknowledging his errors, has taken refuge in discrediting the translations, which might very easily have been examined either by himself or by friends of his; especially as the German original of the most important passages, wherever any doubt might arise, and also of those expressions on the misconception of which Mr. Spencer bases his unfavorable opinion of Kant, were added in foot-notes.

II.

But Mr. Spencer adduces, as if it were a fact, an instance of my grave mistakes. He says that I failed to distinguish between “consciousness” and “conscientiousness.” Mr. Spencer makes much of a small matter, which, if it were as he assumes, would have to be considered as a misprint.

Mr. Spencer’s statement is so positive that it must make on any reader the impression of being indubitably true. However, in the whole first article of mine, and indeed in both articles, “conscientiousness” is nowhere mentioned and it would be wrong to replace the word “consciousness” in any of the passages in which it occurs by “conscientiousness.”

I should be glad if Mr. Spencer would kindly point out to me the passage which he had in mind when making his statement, for since there is not even so much as an occasion for confounding consciousness and conscientiousness, I stand here before a psychological problem. Mr. Spencer’s statement is a perfect riddle to me. Either I have a negative hallucination, as psychologists call it, so that I do not see what is really there, or Mr. Spencer must have had a positive hallucination. That which Mr. Spencer has read into my article, was never written and it is not there. The alleged fact to which he refers, does not exist.

This kind of erroneous reference into which Mr. Spencer has inadvertently fallen is a very grievous mistake. It appears more serious than a simple slip of the pen, when we consider that Mr. Spencer uses the statement for the purpose of incrimination. He justifies upon this exceedingly slender basis his doubt concerning the correctness of the translations of the quoted passages, and Mr. Spencer’s doubt concerning the correctness of these translations is his main argument for rejecting my criticisms _in toto_.

It is not impossible, indeed it is probable, that Mr. Spencer meant “conscience” instead of “conscientiousness.” There is one passage in which a superficial reader might have expected “conscience” in place of “consciousness.” However that does not occur in any of the translations, but in a paragraph where I speak on my own account. This passage appears in the appended reprint on page 23, line 14. Whatever anybody might have expected in that passage, I certainly intended to say “consciousness,” and only a hasty reader, only he who might merely read the first line of the paragraph, would consider the word “consciousness” a mistake.

To avoid any equivocation, however, even to hasty readers, and to guard against a misconstruction such as Mr. Spencer possibly has given to the sentence, I propose to alter the passage by adding a few words as follows:

“It is quite true that _not only conscience, but_ every state
of consciousness is a feeling,” etc.

The italicised words are inserted, simply to show that here I mean “consciousness,” and _not_ “conscience.” For the rest, they do not alter in the least the sense of the sentence. In this passage as throughout the whole article the terms “consciousness,” and “conscience” have been used properly.

* * * * *

Observing that Mr. Spencer appears to have committed the same mistake for which he erroneously blames me, I do not mean to say that he “failed to distinguish between” conscientiousness and conscience. I should rather regard it as trifling on my part if I drew this inference from what is either a slip of the pen or an oversight in proof-reading. But it strikes me that that knavish rogue among the fairies whom Shakespeare calls Puck and scientists define as chance or coincidence played in a fit of anger and perhaps from a sentiment of pardonable irony a humorous trick upon Mr. Spencer. The moral of it is that when an author censures his fellow authors with undue severity for things that might be mere misprints, he should keep a close eye on his own printer’s devil.

III.

Mr. Spencer discredits my knowledge of Kant. He says of me:

“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.”

I did not give Mr. Spencer any occasion for making this personal reflection. I do not boast of any extraordinary familiarity with Kant’s writings. There are innumerable German and also English and American scholars and philosophers who know Kant almost by heart. But the question at issue is not what I conceive Kant’s ideas to be, but what Kant has really said, and I was very careful in letting Kant speak for himself.

My criticism of Mr. Spencer’s conception of Kant consisted almost exclusively in collating and contrasting Mr. Spencer’s views of Kant with quotations from Kant’s works. How can I read anything into some of Kant’s expressions, if I present translations of the expressions themselves, adding thereto in foot-notes the original whenever doubts could arise? And the general drift of the quotations alone suffices to overthrow Mr. Spencer’s conception of Kant.

The truth is that Mr. Spencer committed the mistake himself, for which he censures me unjustly. “Mr. Spencer has read into some of Kant’s expressions meanings which they do not rightly bear.”

IV.

But Mr. Spencer adduces a fact, which, if it were as Mr. Spencer represents it, would show an inability on my part of making important distinctions. He says of me:

“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.”

This is a double mistake: (1) Kant and his translators did not make the distinction of which Mr. Spencer speaks, and (2) the quotation Mr. Spencer makes from my article is represented to mean something different from what it actually means in the context.

Before I speak for myself as to what I actually said, let us state the facts concerning Kant’s usage of the terms “intuitions” and “forms of intuition.”

Kant defines in § 1 of his “Critique of Pure Reason” what he understands by “Transcendental Æsthetic.” He distinguishes between “empirical intuition” (_empirische Anschauung_) and “pure intuition” (_reine Anschauung_). He says:

“That sort of intuition which relates to an object by means of
sensation, is called an empirical intuition.”

Representations contain besides that which belongs to sensation some other elements. Kant says:

“That which effects that the content of the phenomenon can be
arranged under certain relations, I call its _form_.”

And later on he continues:

“This pure form of sensibility I shall call pure intuition.”

These are Kant’s phrases in J. M. D. Meiklejohn’s well known translation. The term “pure intuition” is repeated again and again, and we find frequently added by way of explanation the phrases “as a mere form of sensibility,” “the mere form of phenomena,” “forms of sensuous intuition,” and also (as Mr. Spencer emphasises as the only correct way) “forms of intuition.”

Kant says:

1) “_Diese reine Form der Sinnlichkeit wird auch selber reine
Anschauung heissen._ § 1.

2) “_Zweitens worden wir von dieser (der empirischen
Anschauung) noch alles abtrennen, damit nichts als reine
Anschauung und die blosse Form der Erscheinungen übrig bleibe._
§ 1.

3) “_Raum ... muss ursprünglich Anschauung sein._ § 3.

4) “_Der Raum ist nichts anderes als nur die Form aller
Erscheinungen äusserer Sinne._ § 3.

5) “_Der Raum aber betrifft nur die reine Form der Anschauung._
(This passage appears in the first edition only, the paragraph
containing it is omitted in the second edition.) § 3.

6) “_Die Zeit ist ... eine reine Form der sinnlichen
Anschauung...._ § 4.

7) “_Es muss ihr[71] unmittelbare Anschauung zum Grunde
liegen._ § 4.

8) “_Die Zeit ist nichts anderes als die Form des inneren
Sinnes._ § 6.

9) “_... dass die Vorstellung der Zeit selbst Anschauung sei._
§ 6.

10) “_Wir haben nun ... reine Anschauung a priori, Raum und
Zeit._ § 10. _Beschluss der transcendentalen Æsthetik._”

These quotations do not pretend to be exhaustive, nor is that necessary for the present purpose.

Kant, as we learn from these quotations, makes no distinction between _reine Anschauung_ and _Form der Anschauung_. He uses most frequently the term _reine Anschauung_ and designates in several places Space and Time simply as _Anschauung_. (See the quotations 3, 7, and 9.) So far as I can gather from a renewed perusal, the expression proposed by Mr. Spencer, “form of intuition,” _Form der Anschauung_, occurs only once and that too in a passage omitted in the second edition.

It is almost redundant to add that the English translators and interpreters of Kant follow the original pretty closely. Accordingly it is actually incorrect “that they have everywhere(!) described Kant as maintaining that Space and Time are _forms_ of intuition.” In addition to the quotations from Meiklejohn, I call Mr. Spencer’s attention to William Flemming’s “Vocabulary of Philosophy” (4th ed., edited by Henry Calderwood) which reads _sub voce_ “Intuition,” p. 228 with reference to Kant’s view:

“Space and time are _intuitions_ of sense.”

To say “Time and Space are forms of intuition” is quite correct according to Kantian terminology. No objection can be made to Mr. Spencer on that ground. But to say “Time and Space are intuitions” is also quite correct, and Mr. Spencer is wrong in censuring the expression.

Why does Mr. Spencer rebuke me so severely on a point which is of no consequence? He appears confident that I have betrayed an unpardonable misconception of Kant’s philosophy. But having pointed out by quotations from Kant that this is not so, I shall now proceed to explain why the quotation which Mr. Spencer makes from my article, although the eight words in quotation marks are literally quoted, is a misquotation. It is torn out of its context. I did not blame the English translators of Kant at all, but I blamed his interpreters, among whom the English interpreters (not all English interpreters, but certainly some of them) are the worst, for “mutilating Kant’s best thoughts, so that this hero of progress appears as a stronghold of antiquated views”; and as an instance I called attention to the misconception of Kant’s term _Anschauung_, saying:

“How different is Kant’s philosophy, for instance, if his
position with reference to time and space is mistaken! ‘Time
and Space are our _Anschauung_,’ Kant says. But his English
translators declare ‘Kant maintained that space and time
are intuitions.’ What a difference it makes if intuition
is interpreted in the sense applied to it by the English
intuitionalist school instead of its being taken in the
original meaning of the word _Anschauung_.”

The word “intuition” implies something mysterious; the word _Anschauung_ denotes that which is immediately perceived, simply, as it were, by looking at it. So especially the sense-perceptions of the things before us are _Anschauungen_.

Mr. Spencer, believing that he had caught me in making unawares a blunder, tears the passage out of its context, ignores its purport, makes a point of an antithesis which had nothing in the world to do with the topic under discussion, only to throw on me the opprobrium of incompetence. Even if Mr. Spencer’s antithesis of “intuition” and “forms of intuition” were of any consequence (as, unfortunately for Mr. Spencer, it is not), it would count for nothing against me because I did not speak of “forms” in the passage referred to, I simply alluded to one misinterpretation of the term _Anschauung_, which is quite common among English Kantians. It was not required by the purpose I had in view, to enter into any details as to what kind of _Anschauung_ I meant, and an allusion to “form” or to any other subject would have served only to confound the idea which I intended to set forth in the paragraph from which Mr. Spencer quotes.

Misquotation of this kind, into which Mr. Spencer was inveigled by a hasty reading, should be avoided with utmost care, for it involves an insinuation. It leads away from the main point under discussion to side issues, and it misrepresents the author from whom the quotation is made. It insinuates a meaning which the passage does not bear and which was not even thought of in the context out of which it is torn.

Mr. Spencer quotes the passage as if I had preferred the term “intuition” to the term “form of intuition,” or at least, as if I had no idea that Kant conceives Time and Space as “forms.” Yet Mr. Spencer in trying to make out a point against me betrays his own lack of information. Kant insisted most emphatically on calling the forms of our sensibility (i. e. space and time) “_Anschauungen_.”

But Mr. Spencer’s case is worse still. While he insists upon the statement that according to the translators of Kant space and time are “forms of intuition,” which is at least correct, he uses twice in the very same paragraph the expression that according to Kant “space and time are forms of thought,” which is incorrect. The forms of thought according to Kantian terminology are not space and time but the domain of the transcendental logic. Anyone who confounds the two terms “forms of intuition” and “forms of thought” proves himself unable to form a correct opinion on Kant’s philosophy. That is just characteristic of Kant that he regards time and space not as thought, nor as forms of thought, but as _Anschauungen_ and in contradistinction to sense-intuitions (i. e. sensations) he calls them _reine Anschauungen_ or _Formen der Anschauung_.

Mr. Spencer commenting upon his criticism of Kant’s idea of a Good Will, says:

“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.”

Kant’s idea of a good will has nothing to do with evolution, and we can abstain here from discussing whether or not Kant was an evolutionist. Whether evolution is true or not, what difference does it make to the proposition, that a good will is the only thing which can be called good without further qualification (_ohne Einschränkung_)? Pleasure is good, but it is not absolutely good, there are cases in which pleasure is a very bad thing. We must qualify our statement and limit it to special cases. A good will, however, says Kant, is in itself good under all circumstances.

Did Mr. Spencer prove the baselessness of Kant’s proposition by proving evolution? Is it inconsistent to believe in evolution and at the same time to regard a good will as absolutely good, as good without reserve or limitation? I think not!

VI.

Previous Next

Comments

The Monist, Vol. 2, 1891-1892Chapter XXV: Part 25

0%35 min left in chapter