Magic Squares and Pythagorean Numbers

Browne, C. A., Jr, Magic Squares and Pythagorean Numbers , Monist, 16 (1906) p.422-433

CRITICISMS AND DISCUSSIONS. MAGIC SQUARES AND PYTHAGOREAN NUMBERS. “T have compiled this discourse, which asks for your consideration and pardon not only because the matter itself is by no means easy to be handled, but also because the doctrines herein contained are somewhat contrary to those held by most of the Platonic philosophers.” Plutarch. The fascinating series of articles upon “Magic Squares’ bv Mr. W. S. Andrews and the interesting “Reflections” upon the same by the Editor in recent numbers of The Monist have induced me to make a few comments upon a subject in which I have long been interested,—that of the relationship between magic squares and certain Pythagorean numbers. The mysterious relationships of numbers have attracted the minds of men in all ages. The many-sided Franklin, whose 200th anniversary the philosophical, scientific, and literary worlds have recently celebrated, used to amuse himself with the construction of magic squares and in his memoirs has given an example of his skill in this direction, by showing a very complicated compound square with the comment that he believes the same to be the most magical magic square yet constructed by any magician. I wouid therefore attribute the discovery of compound magic squares to Franklin rather than to Professor Schubert as suggested by Mr. Andrews. That magic squares have had in centuries past a deeper meaning for the minds of men than that of simple mathematical curios we may infer from the celebrated picture by Albert Dürer entitled “Melancolia,” engraved in 1514. The symbolism of this engraving has interested to a marked degree almost every observer. The figure of the brooding genius sitting listless and dejected amid her uncompleted labors, the scattered tools, the swaying balance, the flow-

CRITICISMS AND DISCUSSIONS. ing sands of the glass, and the magic square of 16 beneath the bell, these and other details reveal an attitude of mind and a connection — of thought, which the great artist never expressed in words, but left for every beholder to interpret for himself. Y, 4 y o \ à r } DSDOGFOO |4scaìi 2Lo Yilfa‘#DBt yÙ Efil I so + i MELANCHOLY. The discovery of the arrangement of numbers in the form of magic diagrams was undoubtedly known to the ancient Egyptians and this may have formed part of the knowledge which Pythagoras brought back from his foreign travels. We have no direct evidence

THE MONIST. that the Pythagorean philosophers in their studies of the relationship of numbers ever combined them into harmonic figures, yet the supposition that they did so is not at all improbable. Such diagrams and their symbolic meanings may well have formed part of the arcana Of the esoteric school of Pythagoras, for similar facts were accounted by ancient writers as constituting a part of the aporrheta of the order and the story is told of an unworthy disciple who revealed the secret of the construction of the dodecahedron inscribed within a sphere, this being a symbol of the universe. Among the best expositions of the Pythagorean philosophy are sections of the “Timaus” and “Republic” of Plato. These dialogues were written after Plato’s return from Magna Grecia, where from contact with Archytas of Tarentum and other philosophers, he imbibed so much of the Italian school that his whole system of philosophy became permeated with Pythagorean ideas. It is even suggested that he incorporated into these dialogues parts of the lost writings of Philolaus, whose works he is known to have purchased. No portions of the dialogues named have been more puzzling to commentators than the vague references to different numbers, such as the number 729, which is chosen to express the difference between the kingly man and the tyrant, or the so-called number of the State in the “Republic,” or the harmonic number of the soul in the “Timaus” of which Plutarch said that "it would be an endless toil to recite the contentions and disputes that have from hence arisen among his interpreters.” Either our text of these passages 1s corrupt or Plato is very obscure, throwing out indirect hints which would be intelligible only to those previously informed. Plato states himself in the “Phadrus” that “all writings are to be regarded purely as a means of recollection for him who already knows,” and he, therefore, probably wrote more for the benefit of his hearers than for distant posterity. It is upon the principle of a magic square that I wish to interpret the celebrated passage in the “Republic” referring to the number 729, proceeding from this to a discussion of certain other numbers of peculiar significance in the Pythagorean system. My efforts in this direction are to be regarded as purely fanciful ; the same may be said, however, of the majority of other methods of interpretation. The passage from the “Republic” referred to (Book IX, § 587-8. Towett’s translation) reads as follows: Socrates. “And if a person tells the measure of the interval which separates the king from the tyrant in truth of pleasure, he

CRITICISMS AND DISCUSSIONS. will find him, when the multiplication is completed living 729 times more pleasantly, and the tyrant more painfully by this same interval.” Glancon. Socrates. “What a wonderful calculation.” “Yet a true calculation and a number which closely concerns human life, if human life is concerned with days and nights and months and years.” The number 729 is found to be of great importance all through the Pythagorean system. Plutarch states that this was the number belonging to the sun, just as 243 was ascribed to Venus, 81 to Mercury, 27 to the moon, 9 to the earth, and 3 to Antichthon (the earth opposite to ours). These and many similar numbers were derived from one of the progressions of the Tetractys,—1:2::4:8 and 1:3 ::9:27. The figures of the above proportions were combined by Plato into one series I, 2, 3, 4, 9, 8, 27. (“Timeeus, § 35). Plutarch in his “Procreation of the Soul,” which is simply a commentary upon Plato’s “Timaus,” has represented the numbers inthe form of a triangle; the interior numbers, 5, 13, and 35, representing the sums of the opposite pairs, were also of great importance. The deep significance of the Tetractys in the system of Pythagoras may be inferred from a fragment of an oath contained in the “Golden Verses.” 8 Nai pù Tov duerepov Yuxd mupadorra TETPAKTOV Ilayav, devaov dicews Pilar” Exovoav. “Yea, by our Tetractys which giveth the soul the fount and source of ever flowing nature!” Odd numbers were especially favored by the Pythagoreans and of these certain ones such as 3 and its higher powers were considered to have a higher significance than others and in this way, perhaps, arose the distinction between expressible and inexpressible or ineffable numbers (dpiuoì fyroi Kai äßpyro). Numbers which expressed some astronomical fact also held high places of honor, as may be seen from a statement by Plutarch (loc. cit.) in reference to the Tetractys. “Now the final member of the series, which is 27, has this peculiarity, that it is equal to the sum of the preceding numbers (1+2+3-+4+9-+8) : it also represents the periodical num-

THE MONIST. ber of days in which the moon completes her monthly course: the Pythagoreans have made it the tone of all their harmonic intervals.” This passage indicates sufficiently the supreme importance of the number 27. If we construct a magic square 2727 upon the plan of a 35213911 326/439 468 1413] 27) 3531379/414 4401466 274 /303/248]613 6421587) 7001 729,674) 5351 66416091 1 18147) 92 [205 234 (249 27513011588 (6141 6401675) 701 172715101536 562) 1890 119 148| [179] 40 180/206 2321116) [69| 14 41 1167 3801325 3541467) 412/441 302/24 71276) 641] 586 61 5172816731 702 (6631 5081537) 146 1911112012331 17812071 68 [131 42 77) 30612611355 (384) 3294331462 407/638 (567, 612/66] 645 1690|694 119316680143) 72 [NTI 121 [1501 95 11091228173 262/278 30413301356 ¡387408 434, 460|51:31 539 1660159110171 64316691695 1721] 18 [441] 70 [951] 1221148! 17412001226] 3051 250/279] 383/398] 357/461 409/435) 566) 5111 540/6441589 (6 81722 66716964 71 16 11451 14911941 12312271 1721201 436 (465141012711 3001245) 358119871 3321697) 72616711592) 561 606/61 9648 593} 202/231) 176] 4111497 (4631246 12491 1371 66 1111111241153) 98 2981333 359 9856721698 724 (507) 533 5691594 (620/646, 117 203/2291 12 ['38 64 [991 125 161] 45414091438 1299 94419173) 386/33 113601725 670/699/560/505 5346471592621] 2301751 204,65) 10 113911152 9711126 127) 156110412 14/2431 188149. 78 123 361) 390; 335 1448) 477 420) 283 912 267156051624 569|682|(11 656 |517]546 491 10211281154 [189 215 241) 24 (50. 76 [336 3623881423 440,475 19581284 31015701596 6221657 683 1709) 492 518 544 5110011291242 1871216077) 22 1.511389 33413631476 421 1450| 31 1126612851623 568/597/P71015551 6841545490519 52 081) 26 [1301591041208 2371182 |286:315 2601364 2931338442 471 14161520 /5491494 5981627 15721676 17 05/650 2711 69 [191 105113//157[183:208/235/261|287|313339/385| 3914417 |443,469 495 521/54715731599 625 165116771703 180 11251 54 [158 103,1321236,181 210[314 9592881390 2971366 4701415 14441 548/403 522626, 571 160017041649 678 »211 240/185) 46 [175% 20 [133/162 1074445 4741419 1280 909 2541367 396 3411679 70816531514 (5431488 [601 6301675 fl fiat 186 313 238 1811, 47 |, 73-|108;134) 1601420) 446/472 2551281307 342! 368/394) 654/680, 7061489 515/541/5 76/602, 628. Aa HEY] 2391840219) 74 119: 48 [161106 1361473 418 447 [308,253 282] 395/340) 3691707) 652/681|542)48| 51616241674 603 ARE ciro id pere ii ST HALL Du eat {pie El ¡604633 578 6911 720/665) 526.555 500[ 100 138 83196 225,170) 31)| 60 [511370 399 i : t 3441457 486 431129213211 266 566,892 718501, 527,553] 84 110136 [1211 197|223] e 132 58 13451371 [397] 432 458/48 267 293/319 ii '632/577:606 '719:664:693/554 499.528|[137 82 |1111224,160 59 19859] 4 [1331398343 372[485 430146911320 1265 294 529.558:5031507/536 581|685 7141659/ 34 [63 8 [112,141 86 190,219 1641996 3241269373 402 347 5114801495) pepe TTT Ta aya alla ; Hi 504 530 556/582 608 6341660 686471201914 35 È etl rT hm mR > " [611 | i Wh YY dl ‘i in DI i 87 1113| 1391465) 191 191712702961 322/348, 374 4001428 aR 4521478| 1 ! DE : mp | ri 635:580 6091113 65816871 62 [7:1: 36 [140, 85 |1141218|1631192]323.268120714011346 3754479: 494 453 ate ate 529.662 4071610 1639 5841931222 [1674 28 (57 2 |11611441189114541483142819881 318 2631376 405) 350 498 541650 585 6111630168 1941220) 13) 29 115511 90 |1116/142|499|455 4811 26410901316 136113771403 Thee 716 664,690 6511 496) 6251638 } ips dt 7 | 583161202241 1681195] 66 jit, 30 [143] 88 1117114821427 45613171262 2911404 340 378 Alta pi checker-board—arranging the numbers I to 729 first in numerical order, then shifting the 9 largest squares (9X9) into the positions indicated in the familiar 3X3 square, repeating the process with the subdivisions of the 9Xg squares and so on down—we will arrive at the following combination. It will be noted that we have 365 white squares or days and * This method of constructing compound magic squares is, so far as I know, original with the writer. It bears some resemblance to the method of Schubert (Monist, XV, p. 566); the numbers of each square, however, increase in periods of threes instead of by sequence.

CRITICISMS AND DISCUSSIONS. 364 dark squares or nights—a veritable “checkerboard of nights and days.” The number 365, the days of the solar year, very appropriately occupies the centre of the system. The columns, horizontals, and diagonals of the central square 3X3 foot up 1095, or the days of a 3 year period, those of the larger center square 9X9 foot up 3285 the days of a 9 year period, while those of the entire combination 27X27 foot up 9855,” the days of a 27 year period,— in other words, periods of years corresponding to the Tetractys 1, 3,9, 27. We may with safety borrow the language of Plato and say that the above arrangement of numbers “is concerned with days and nights and months and years.” The interpretation of the other passage referred to in the “Republic’—the finding of the number of the State—(Book VIII, § 546) has been a subject of the greatest speculation and by consulting the various editions of Plato it will be found that scarcely any two critics agree upon a solution.” As Jowett remarks, it is a puzzle almost as great as that of the Beast in the Book of Revelation. Unfortunately we have no starting-point from which to begin our calculations; this and the very uncertain meanings of many of the Greek terms have caused many commentators to give up the solution of the problem in sheer despair. Aristotle, who was a hearer of Plato’s, writes as if having a full knowledge of the mystery; Cicero, however, was unable to solve the riddle and his sentiment became voiced in the proverb numeris Platonicis nıhil obscurius. By taking a hint from our magic square and starting with the number 27, I believe we may arrive at as good a solution of the problem as any that I have seen suggested. The following interpretation of the Greek terms is offered. avfyosıc dvvduevai te Kai the square of the numdvraorevduevat ber times its root, pele ATOOTAOELC increased by thrice the first terms (of 27° X iN 27= 2187 the Tetractys) “errapac de porc Zagotand ca four (1+2+3+4+9)X3= times whole series 57 the (1+2+3+4+9+8+27)x4= 216 “Not only the perpendiculars, horizontals, and diagonals of this large square foot up 9855, but there are an almost indefinite number of zig-zag lines, which give the same footing. ® Schleiermacher, Donaldson, and Schneider suggest 216, and much may be said in favor of this number. Jowett gives 8000 as the possible solution. Others suggest 951, 5040, 17,500, 1728, 10,000, etc.

THE MONIST. OLOLOUVT WY avoof numbers unlike yet potovvrwv Kai avsévTwy Te Kai bearing the same ra- Kai bLvörvur [carried over from last page] 2460 tio whether increasing or decreasing (i. e. 1:2::4:8 or 8:4: :2:1 It may also refer to the ascending and descending figures of the triangle. 8, 4, 2, 1, 3,9, 27) mävra mpoonyopa kai pyrà makes roùç dàAna arépyvav the sum mensurable pressible comand exin all its sum== 2460 parts. (i. e 2460 is easily divisible by 1, 2, 3, 4, 5, 6, 10, 12, etc.) OV Emirpiros meu, this sum increased by 2460X1%— 3280 2 meumadı ovarvyeic and adding 5 3280+5= 3285 rpie ausmdeig is multiplied by 3 3285 X 3== 9855 This solution of the problem, 9855, it will be noted, brings us again but by a different route to the magic number of our large square. The second part of the passage contains a description of the number by which the above calculation may be verified. dio dpuoviac rapéxera: (the Tv uÈv LOTV LOGK:S, one number) yields two harmonic parts, of which is a square 3X3= ÉKATÔV TOCAUTÁKIC, multiplied by roo: ry dè loounKn pèv, the other has one side 9 gX100== equal to the square 900 3 sum== 9855 The remainder of the passage describes the length of the oblong which we have shown above to be 2985: Exatov uev apiGuav aro (the diausrpar TEUT ADOC, oblong) times is 100 the side of a rectangle having diagonals of 5. ‘i. e onto deouévuv Evor ékaohaving sides of 3 and 4.) less of one each of the zw. expressible parts, i. e 4 and 5 Copyright (c) 2007 ProQuest LLC

CRITICISMS AND apphrwv dé Öveiv, DISCUSSIONS. and z ofthe inexpressi- 429 300—(5+4+3+3)= 285 ble erarov de kùBwv tpiddoc plus 100 times the cube of 3 Xx 100= 2700 (3) sum= 2985 Plato states that the number of the State “represents a geometrical figure which has control over the good and evil of births. For when your guardians are ignorant of the right seasons and unite bride and bridegroom out of due time, the children will not be goodly and happy.” The number 9855, expressing a period of 27 years, might thus represent the dividing line between the ages when men and women should begin to bear children to the State, — 20-27 years for women, 27-34 years for men. (See also “Republic,” Book V, § 460). Aristotle in his “Politics” (V, 12. 8) says in reference to the number of the State that when the progression of number is increased by */, and 5 is added, 2 harmonies are produced giving a solid diagram. This, as may be seen from our analysis of the first part of the passage, may have reference to the number 3285, which, being represented by 37365, may be said to have the dimensions of a solid. In the January number of The Monist the Editor gave some very striking examples of the relationship between magic squares and the musical figures of Chladni. I would like to touch before concluding upon a closely related subject and show certain connections which exist between the magic square, which we have constructed, and the numbers of the Pythagorean harmonic scale. This scale had, however, more than a musical significance among the Greek philosophers; it was extended to comprehend the harmony of planetary movements and above all else to represent the manner in which the “soul of the universe” was composed. It is especially in the latter sense that Plato employs the scale in his “Timæus.” In a treatise by Timæus the Locrian upon the “Soul of the World and Nature,” we find the following passage: “Now all these proportions are combined harmonically according to numbers, which proportions the demiurge has divided according to a scale scientifically, so that a person is not ignorant of what things and by what means the soul is combined; which the deity has not ranked after the substance of the body...., but he made it older by taking the first of unities which is 384. Now of these the first being assumed it is easy to reckon the double and triple; and all the terms, with

THE MONIST. their complements and eights must amount to 114,695.” lation by Burge.) (Trans- Plato’s account of the combination of the soul is very similar to the above, though he seems to have selected 192, (384/2) for the first number. Plutarch in his commentary makes no mention of Timæus, but states that Crantor* was the first to select 384, for the reason that it represented the product of 8*6, and is the lowest number which can be taken for the increase by eighths without leaving fractions. Another very possible reason, which I have not seen mentioned, is that 384 is the harmonic ratio of 27?/2 or 364.5, a number which expresses very closely the days of the year. 243 :256 : :364.5 :384. The proportion 243:256(3°:4*) was employed by the Pythagoreans to mark the ratio® which two unequal semitones of the harmonic scale bear to one another. Batteux has calculated the 36 terms of the Pythagorean scale starting with 384 and his series must be considered correct, for it fulfils the conditions specified by Timæus,—the numbers all footing up 114,695: A few of the numbers of this harmonic scale marking the “first unity” and several of the semitones will be given. Ist octave E 384 € 486 | F 729 i | 2nd octave fc (For Batteux' full series and method 972 l E 1458 fe 1944 . of reader is calculation translation of Plato Vol. 3rd octave VI P. 171). | B-flat 2187 4th octave the referred to Burge's B-flat 4374 By referring to our magic square it will be noted that the first of unities,” 384, constitutes the magic nmber of the small 3x3 square beginning with the number 100. If we arrange the magic numbers of the 81 squares (33) in the order of their magnitudes we find that they fall into 9 series of 9 numbers, each series beginning as follows: I II III IV V VI VII VIII IX 87 330 573 816 1059 1302 1545 1788 2031 *Crantor lived nearly 100 years after Timæus the Locrian. The treatise upon the “Soul of the World and Nature,’ which bears the latter’s name probably belongs to a much later period. ‘for further references to this ratio see Plato’s “Timeus,” § 36, and Plutarch’s “Procreation of the Soul,” § 18.

CRITICISMS AND DISCUSSIONS. The intervals between these series are worthy of note. INTERVALS. Between I and II 243 the first member of the ratio 243:256. 0 I“ III 486 Cof the rst octave di 1“ IV 729 Fo" «st “ di I ‘1 V 972 C te et and di ti I ‘6 VII 1458 F vi di and di ts I te IX 1944 C ti da 3rd di If we arrange the magic numbers of the large squares (9X9) in the same way, it will be found that they fall into 3 series of 3 numbers, each series beginning I II III 1017 3204 5391 Interval between I and 11 = 2187 54 ot I Y III = 4374 B-flat of the 3rd octave. B-flat .. .. 4th ti Numerous other instances might be given of the very intimate connection between magic squares and various Pythagorean numbers, but these must be left for the curious-minded to develop for themselves. Such connections as we have noted are no doubt in some respects purely accidental, being due to the intrinsic harmony of numbers and therefore not implying a knowledge by the ancients of magic squares as we now know them. The harmonic arrangement by the Greeks of numbers in geometrical forms both plane and solid may, however, be accepted, and Plato’s descriptions of various numbers obscure and meaningless as they were to succeeding generations, may have been easily comprehended by his hearers when illustrated by a mathematical diagram or model.f Differences between the methods of notation in ancient and modern times have necessarily produced differences in the conception of numerical relations. The expression of numbers among the Greeks by letters of the alphabet was what led to the idea that every name must have a numerical attribute, but the connection of the jetters of the name was in many cases lost, the number being regarded as a pure attribute of the object itself. A similar confusion of symbols arose in the representation of various concepts by geometrical forms, such as the five letters of YTEIA and the symboliza° The description of the number of the State in the “Republic” and that of the Soul in the “Timæus” render such a mode of representation almost necessary. Plutarch (“Procreation of Soul,” §12) gives an illustration of an harmonic diagram 5X7 containing 35 small squares “which comprehends in its subdivisions all the proportions of the first concords of music.”

THE MONIST. tion of health by the Pythagoreans under the form of the pentalpha or five-pointed star. It was the great defect of the Greek schools that in their search for truth, methods of experimental research were not cultivated. Plato in his “Republic” (Book VII, § 530-531) ridicules the empiricists, who sought knowledge by studying the stars or by comparing the sounds of musical strings, and insists that no value is to be placed upon the testimony of the senses. ‘Let the heavens alone and train the intellect” is his constant advice. If the examples set by Pythagoras in acoustics and by Archimedes in statics had been generally followed by the Greek philosophers, our knowledge of natural phenomena might have been advanced a thousand years. But as it happened there came to prevail but one idea intensified by both Plato and Aristotle, and handed down through the scholastics even to the present time, that knowledge was to be sought for only from within. Hence came the flood of idle speculations which characterized the later Pythagorean and Platonic schools and which eventually undermined the structure of ancient philosophy. But beneath the abstractions of these schools one can discover a strong undercurrent of truth. Many Pythagoreans understood by number that which is now termed natural law. Such undoubtedly was the meaning of Philolaus when he wrote “Number is the bond of the eternal continuance of things,” a sentiment which the modern physicist could not express more fittingly. As the first study of importance for the youth of his “Republic” Plato selected the science of numbers; he chose as the second geometry and as the third astronomy, but the point which he emphasized above all was that these and all other sciences should be studied in their “mutual relationships that we may learn the nature of the bond which unites them.” “For only then,” he states, “will a pursuit of them have a value for our object, and the labor, which might otherwise prove fruitless, be well bestowed.” Noble utterance! and how much greater need of this at the present day with our complexity of sciences and tendency towards narrow specialization. In the spirit of the great master whom we have just quoted we may compare the physical universe to an immense magic square. Isolated investigators in different areas have discovered here and there a few seemingly restricted laws, and paying no regard to the territory beyond their confines, are as yet oblivious of the great pervading and unifying Bond which connects the scattered parts

CRITICISMS DISCUSSIONS. and binds them into one harmonious system. 433 Omar, the astronomer-poet, may have had such a thought in mind, when he wrote: “Yes; and a single Alif were the clue— Could you but find it—to the treasure-house And peradventure to the Master too; Whose secret presence, through creation’s veins Running quicksilverlike eludes your pains;” etc. When Plato’s advice is followed and the “mutual relationships between our sciences’ are understood we may perchance find this clue, and having found it be surprised to discover as great a simplicity underlying the whole fabric of natural phenomena as exists in the construction of a magic square. C. A. BROWNE, JR. New ORLEANS, LA. THOUGHTS ON TIME, SPACE, AND EXISTENCE.* I. All existence is one. Every thing that is, is part of the All. That all of the parts are adjusted to each other, and work in harmony, proves the relationship of all the parts to the whole, and the unity of the All. Existence extends, and existence endures. While it endures there is an unending and unceasing succession of events occurring. When we mentally think away all of the features of existence except the feature of extension, that feature which remains when all the rest are thought away, forms, in consciousness, the conception we term space. Space, then, is abstracted mentally from reality, and in so far is a mental existence. But in reality itself, space, or the property of extension can not be separated from existence. We think of it as a thing in itself, or by itself, but this is not the * A great many of the ideas expressed in this article have been obtained by a study of the philosophical works of Dr. Paul Carus, and I wish to give proper credit for numerous expressions and quotations which I can not well avoid using. Prior to my study of his writings, my philosophical studies had been confined mostly to the works of Herbert Spencer. While I had in a certain degree noticed that he overlooked the importance of the forma! and the subjective features of existence, my ideas along these lines were very hazy and undeveloped. I supposed, that in his philosophy, finality had indeed been reached, However, when I began to study the works of Dr. Carus, a new world was opened to my view.