THE MAGIC OF NUMBERS by Edward T. Bennett With an Afterword by Romeman This two-part article was published in The Occult Review (also known at times as Rider's Review) in July and August, 1906. The author was Assistant Secretary (1882-1902) to the Society for Psychical Research. Romeman has shown, in the Gannstudygroup booklet on The Law of Vibration, that it is highly probable that W. D. Gann quoted more than once from this article in his 1909 interview with Richard D. Wyckoff in The Ticker and Investment Digest, the so-called Ticker Interview. (Nov. 15, 2013)
This E-Book is not to be sold. It is a free educational service in the public interest published by
Gann Study Group
£5
' A PRIZE OF
is offered to the resder who se\'
the best and best authenticated recqrd of the aP~arance of an animal in visible form after death. · · ' · · As on the· previou~- ~~asiOn the· Editor win consult a ·well-known autp'Ority on P~ychical_Research before making his award . . It is understood that any record wlll be disqualified which has alreadY appeared in prlnt. . . . . . This Competitio-n wi~ remain_ open, until t]:le 1oth of. J_uljr, and the award will-be made·, if possible, in the August.issue. _ All compE1t:itions mu~t Qe. type-written. A nom de· plume must be adopted by the writer,. and his real name enclosed in a -seal_ed envelopebearing the nom de :Plume. outSide. ·" · ThiS:-will not be opened until the award has been -made. · Records sen:t in competitioi;l. :glUSt be- marked on the_ outside " PRIZE €OMPETITION, •I
.
· J:'lle Editor reserves the rigb,t to print any of the communications he may 1·~ceive. Competitors whose records are printed will ·be entitled·, if they Wish, to receive one boUnd volume of the OC0tJLT -REvmw.
OCCULT REVIEW J,JSYCHOMETRIST. THE following Offer is made for· the present month, and may, or may not, be continued. Any re_ader desiring to have questions answered by the OccuLT R~VIEW Psychometrist, must out out, and send up, '_not less than ~our coupons .~uoh as that given below, ~d dis:(>atch t~e copies from which they are Cut to friends who nlay possibly "be interested in the Subjectmatter of the magazine. Each quetent will be entitled· to aSk not more than two questions. ·Any reader- desirous of having his, or her, character and general Condition~ psychically diagriosed, will be required 1;o _send. up not l~ss than eight coupons, and double· space will be allotted t'o these diagno.ses. Readers .who desire to avail themselves of. this. offer should seD.d either a· glove or tie, o! piece of ribbon tliii.t they haVe Worn constantly, or failing this 'an ordinary letter. WhS:tever is sent niust be do"zle up in a sepai:ate parCel, marked with the name or· assUmed name of the inquirer, and sealed. This separate parcel Should be sent :under the same cover. as. the letter containing· the inquiry and .tb~ necessary coupons. Care must be taken tha:t the article ·or letter is. kept away from contact with other influences previous to its dispatch, as these teild to confuse the Psychometrist. -The articles thus sent will not be returned . .
.
PSYCHO COUPON. July, · 1906• .
•
·i!
THE. MAGIC· .OF .NUMBERS BY .11JDWARD T. ·BENNETT··
This appears t~ be a IU.ndamental law of the Universe :·_::...n~mely, that an original impulse of any kind finally resolves itseH into periodic or rhythmical motion."-PROFESs~RW. F. BARRETT, F.R:S. [1870] . 41
.THE. M dgic of Numbm as the Basis of all Diversity and Apparent Fate is the title of a book published at Vienna in tlie year 1882. The prefa~ is dated February, in curiously exact coin.cidence with the formation of the Society for Psychkal Research. The writer is L. B. Hellenbach, usually designated Baron Hellen·bach. · He is. the author...of several .other works. The only one of these which has been translated into English, so far as I know, · is entitled, Birth and Death as a Change of Form of Perception. The Magic of Numbers commences ;with a b_rief Preface of two pages, and a short Introduction of six pages, the object of which is to prevent the reader beingmisled at the outset by the quaint title of the book, artd to indicate its purport. The following paragraphs are not a translation of these eight pages, but are intended, in the form· of an abstract, to convey the ideas of their writer~ as far' as possible in his own phraseology. INTRO:hUCTORY;
·' '
If a man is quietly sitting on a bank in his own garden, he runs less danger of misadventure than if he is engaged in any kind of sport, or if be is taking a voyage orl the ocean, If he ascends in a balloon be will be still more insecure, but be will have a wider outlook. These various e:g:periences cannot be enjoyed at the saine time. My choice is, for my reader and myself, to .ascend in a balloon, but it shall be a ·captive one. #We will not ge't)ost. In order to travel safely we will look out for a fit place to anchor the balloon, and so not to lose our conne~tion with the earth .. We will fasten our. rope securely through the ' early chapters. Possibly after that a steady judgment may serve tis· for a connecting cord. In the later chapters we will rise still higher, till we come to the limit where the .air is too thin to breathe. · There are those who cannot exert their minds, because they
•
8
THE OCCULT. REVIEW
have not the ability. The~e are. those who will n 0t 1 because they consider it needless for them to. do so. There are others who . practically have no time, being engaged in bus,inyss and other occupations. For none of these is this book written.· It ls intended as an incentive to those who are fond of speculation, as a treatise on phenomena which are based on and supported by Numbers. One thing however I must dispose of, which cannot ,conveniently be introduced into the book itself, and yet which nuist be cleared up. ~;: I have been reproached from friendly quarters, that, by the tenor of my writings, I have wilfully estranged both the scientific circle and the Pres$. . It is true that from my. childhood I was taught to be courteous and considerate to '!11, not to hurt any man's feelings, At the same time,. I 'was ~so taught never to submit .t<;> arrogance. It is the case that :the arrogance of the Jhrone and of tl:].e Pulpit has found its way into Scientific Societies, and into the Editorial Chair. I therefore co.nsider it a duty not to allow myself to be led away by any personal considerations, and at the same time to show that although more or less slighted · by scientific. Circles, and hit hard. by the paper blows ofjoumalists, it is still possible ~o find enjoyJp.ent in the universe.. So long ;as men of sCience oppose me, not by aCtual experience, but .from the obscurity of their own a priori arguments, I shall be bound to expand the saying '' errare humanum .est," and to. translate it " to err is human and scientific." · It is incumbent upon me to justify the peculiar title of this book, in order that the reader ,may not proceed to lecture me from a pre-conceived standpoint. The word " Magic " has been used. for the incomprehensible, for that which is outside the kingdom of law. No distinction has been made between the relatively incomprehensible tricks of a conjurer, and that which is supposed to·happen in a supernormal way, through the agency , of good and evil demons. With none of these things have we anything to do. I use the word. Magic" in the sense in which we speak of a Magical glance, or of a Magical will, as of a power, the cause of which we either do not know, or which appears to proceed from something ·.beyond ordinary.· causes. Certain words and certain numbers have been termed magical, because certain properties (lave been ascribed to them whiCh do not belong to their ordinary use. Our object is to search out .the reasons for . such beliefs. The title is thus justified. It is needful to state further that we have nothing to do-with Cabbalistic trifling with numbers. What I mean will be best
P*.·:·
•
THE: M:AGIC OF 'NUMBERS
'.9
!'i>emplified by explaining to the reader the' origin of such trifling. Orientananguages have this peculiarity, that letterS also signify numbers. Hence it arose that all large numbers are also words. Also the reverse; that every word is a number. · Thus it is easy to.see how, in. the Bible· for instance, a deeper meaning may be conceived than appears on the surface, if words are treated as numbers; and o_n the other harid; if numbers are treated as words. '!n ,this way- it was attempted:.....for example-eto show a connection· between Luther and his followers, and Antichrist, from .the letters of the_ words of the original. , . Although the incoherences of the Cabbalists are not inviting, itjs impossible not to be impressed with the philosophic Spirit of the Pythagoreans·, in their view of the importance of numbers·. AriStotle perceived in number 'that which dominates ideas and forms, the measuring staff of tile Divine Architect. The philosophical spirit of this man recognized the essence of the phenomenal diversity which I define as signified by numbers. We know to-day that Musical Tone; Light, Colour, Chemical Combiilation are ind~bted to number for their mal)ifold expression. Aristotle had only a dim pre~conception of it. Butthe dominating power of number is not confined to the perceptions of the senses. Our task will be, in the first place, to show the significance of Number in the different kingdoms of phenomenal diversity. A remarkable analogy will be apparent,. for which we shall have to seek the ~ey. We shall find numberS arid groups of numbers, which for .thousands of years have been almost objects of wor'ship, It will become evident that these numbers do not possess all the significance which has been ascribed to them. But we shall also find: that number-"'orship is not entirely without a rational foundation. The reader _will now see that .the. subject, although it may appear so fantastic, has a serious basis. ''-PERIODICITY.-"
The following considerations will assist in understanding the position taken by the Author in the fir5t four chapters of his book. A wheel.revolving below a certain velocity appears to us stationary. A wheel revolving above_ a certain velocity also appears to us to be at rest. Weare capable of perceiving motion within certain limits only. Beyond, in both directions, there is to us~Rest. ' Sound is limited, on one side, by the 'loWest number of vibrations per second, of the air.whieh produces what is to us audible. It is limited on the other side, by the largest number of vibrations
10 per second which produces. the highest note we can hear.. Beyond these limits, ~. ·both dir~ctions, there is to us-5Uence; Light is limited on ~ne side, by a certain lower limit of the number of vibrations or undulations, per :second, of the ether. It is limited on the· other side by a certain higher number. Beyond these limits, in both directions, there is· to ufl-'-Darkness. Then agail)., all the Poetry of Sound which we call Music""""all Melody-is the result, or perhaps we ought to say is coincident with, certain definite mimerical relationships between the number of the vibrations of successive notes. Here comes in the " Magic of Numbers." · Again, all the Poetry of Light,. which we call Colour, all the gorgeous. colouring of sky and.cloud at sunrise and sunset, all. the 'delicate tints of di.stant mo.l\ntains and of the ocean, are the ·result of, or rather we ought perhaps again to say, arecoincident with, definite numerical relationships between the number of vibrations in the different rays into which the ray of white·Iight is decomposed. Here comes in more of the "Magic of Numbers." Curious correspondences exist between Sound and Lig!!t, between the number. of vibrations producing musical intervals and those which· produce colour. These add to the " Magic." Professor W. F. Barrett, F.R.S., of Dublin, has been kind enough to call ·my attention to a Paper he wrote under the title-'' Light and Sound : An Examination of their reputed Analogy," which appeared in The Quarterly Journal of Sci~nce as long ago as January, 1870. ProfesSor ~arrett w"-s_ the first to. place this analogy on a definite basis. In his paper h11-refers to the vague ideas on this subject which had· been a matter of speculation froin very early times, and then says :~"The following considerations will .show not only that the analogy has sqme foundation, .but that it is far more wonderful than has hitherto been suspected." The fascinating details cannot be quoted here. It must suffice to say that when the wave-lengths 'of the musical intervals are compared ·with the wave-lengths of the colour hitervals, a definite correspondence is'seen t 0 exist. This correspondencedoes not relate to absolute value, the colour waves being infinitely small compared with the. sound waves.· It consists ·in the ratio of one to the other. The proportion between the different lengths of the waves of the notes of the musical scale and of the seven colours of the rainbow is almost identical. Another striking example of the Magic of Numbers. Professor Barrett says further at the end of his Paper:~" A rnusical chord thus becomes both a representative picture and an
THE'MAGIC OF NUMBERS
11
acoustic painting, while the rimsical sca:le is literally a rainbow· of sou:nd. It is· hardly too much to say that we might possibly translate into a musical melody, a sunset, a flower, or a painting by a Rubens or a Raphael." · . . In four chapters dealing with what Baron Hellenbach calls Periodicity (Die Periodicitiit) he ,discusses at length the facts on which the above paragraphs' are based. · He believes that he also finds a similar Periodic system· h:i. the science of chemistry. To what extent the r~searches ()f the last twenty-five years into the molecular constitution of matter, and into the nature· of the ether, have strengthened or otherwise the views which Baron 'Hellenbach puts forth with regard to Chemistry, I cannot say. Iri the concluding paragraphs of this section of his book he makes ·some interesting remarks, the· substance· of which I will endeavour to give. In Music, or rather-in .Melody, we have found ~ anchor for our speculations which' strengthens· our argument for the existence of Periodicity. We find a Periodicity which is the necessary antecedent of definite diversity. We are brought to the convic· tion that the diversity in phenomenal Nature, in its different kingdoms, is most intimately associated with numerical relatioriship. The numbers are not intermixed, chaotically and acci· dentally, but are subject to a regular periodicity. The changes and developments are also seen to be; in many cases, undulatory. We will now make the attempt, as far as.our powers will permit, to soar into a transcendental world, iri order to obtain a wider view of this Periodicity. · To look into a transcendental world does not imply looking into another world, but only to make a change in our way of regarding the world. There is only one world, but there are .many ways of looking at it. Only thus can lhe diversity of the world be realized. Music is the most beautiful type of Monism in Nature, and as we shall see later, of the course of human life I What is it in its essence ? Nothing more than an undulatory movement of the air. If we give the vibrations, which are per- .. ceived from our phenomenal standpoint, a certain definite intensity; they are perceived as tones. The number of·vibrations In a given space of time, of two or more tones, determines. whether they are in harmmiy or discord. All· the innumerable varieties of ·Melody are the result of relations between numbers which evoke analogous vibrations deep in the interiorof our own selves. We can express orlr perceptions in terms of vibrations' of tone, ·nnd, vice versii, vibrations of tone will call forth certain percep·
~
}
.
~~-.,~
/
12
.·1'1fE ·OCCULT REVIEW
tions in us. What immense diven;ity, intimately related to our own life, is ~ailed ·forth from not]ling by the. Magic of Numbers ! With colol)r .it is the same. Whether the number of vibrations represents a reality, or only what we may call a phenomenal eoin of reckoning, whether the~e are l)ndulations of ether, or whether the attractive power of substance is the cause, the whole splendour of colour is a diversity which proceeds from a very simple origin. All that we have been speaking of stands .in a peculiar manner . under .the sway of certain numbers and combinations of numbers.
<
THE Mi\.GIC OF NUMBERS IN ANTIQUITY.
Following the chapters in Jielleribach's bookwhtch we have just dealt with comes one with the above title. It is at this stage that the. writer speaks, in the Introduction,· of dispensing with the rope which anchon; his captive balloon to the solid earth, But it is .quite needless to do that at present. I do not propose to enter upon the speculative or mystical part of this section of his bo.ok, but to confine our attention to the portion dealing with definite facts in relation to certain number$, facts, some of which are very curious, but which belong to arithmetic, or to Easy Mathematics, to borrow the title of a recent book by Sir Oliver Lodge, facts about which there is not necessarily anything occult . or mystical. . What has been called a ~· Tetragram '' or " Magic Square " is as old or older than Pythagoras. Hellenbach describes it thus :.c._It is a ·square composed of smaller squares,. similar to a chessboard. In each small square. a number .is placed. 'These numben; .are so grouped that all the vertical and horizontal rows of figures, and the twO' diagonals, add up to the same sum. This is the principal and most obvious feature of a " Tetragram." There are others, as we shall see later. A Tetragram of this nature may be constructed with any one of .the arithmetical ~erjes of numbers as .a "root." .It is convenient to make two series, one of odd and one of even numbers, thus :-r, 3; 5, 7,. g, etc., and 2; 4, 6, 8, ro, etc. The mode of construction, as well as the 'properties of the Tetragrams of the two series, differ somewhat. TheTetragram consists of a square composedof as many smaller squares, as the square· of the· number chosen for the "root." Suppose,. to take ·one of the simplest examples, we choose Jhe .number 3 as the root or base; Three times three is nine. That is to say, the· square of three is nine. Let us draw
··------~-1
NUM]l~RS
THE MAGIC OF
18
·a square composed of nine smaller squares. Fig·. L_ Then hisert the numbers from one to nine in- the small 'sq\lares as in Fig. II. There are eight rows of three figures each' Three hOrizontal, three vertical, and two diagonal. Each and all add up fifteen. Another,of the obvious peculiarities is that all the pairs of numbers opposite the centre ofthe square add up alike. 9 +I, 7 + 3, II.
I,
~
t±±j
--~-4: -.--9i. 2 8 6 7
'
--
. , a· 1: 6
8+z, 6+4, all-ro •. Aiwther peculiarity is' that if two circles be drawn, _with the centre of the central square as centre, and the circumferences passing throl'lgh the centres bf the four ' . . squares at the centre of each side, and the 'centres of the four corner squares, the sum of the numbers hi the squares will be the same. 9+7+3+I, and 8+6+4+2, both == :zo. These three features are, with slight modifications, common to all Tetragrams. It will be interesting to spend a little time on one of the methods of constructing a Tetragram or "Magic Square." With a small number such as three, it would be easy to construct Fig. II, by a_ series of trials. But it is better to .find a system or law. Let tis again take the n)imber three as an example. HaVing drawn a square composed of nine smaller squares-Fig. !-,enlarge it by extending a small square in the centre of each side, producing Fig. III as the result. Insert the numbers r--g as. in
two.
IV.
Fig. III., Then fill in the numbers in_ the centre and four corner squares, as they stand, and transfer the numbers from the four outside squares to the vacant square on the opposite side. The result will be Fig. lV, which will be seen. to be identical With Fig. II. The " Magic Square " with the root five may be similarly constructed,_ ·· · ·
14
. .THE OCCULT REVIEW
As a. peculiar interest attaches to· the number seven, it may be worth while to give the details of .the.'' Magic Squar~ " of that number, especially as the three chief features. above allud~d to
'·I I
v. I
8
I
i51 I 9 22
29
.
1
I8 .
10
16
23.
I2
17
4 5
11
: ..'
~
80 87
r:--
18
24
81
12
0
1b
19
25
6
~
. .82
88
26 ~
45
89
2'1
88
84
40
46
\4~
-
41
14
20
-
21 ~
28
85
0
come out much more strikingly than with a very small number. Draw a large square composed of forty~nine small squares, and enlarge it by extensions, to form Fig. V. · Insert the nun'ibers VI.
22 47
16. 41
10 85
•4
-.
6 28 48 17 42 11 29· 80
6
24 49118 86 12
18
81
7
25
48
88
14 82
1
26 44 20
21
89
8
88
2
27
46
15 40 .9
84
3
~
19 87
46 28
r-49 as in. Fig. V. Then, in Fig. VI, fili in. the. numbers as they stand, .in the twenty-five occupied squares in. the middle of Fig. V, Then transfer the four outside g·roups of numbers to the corre-
0
•
THE' 'MAGIC 01<' NUMBERS
15
spending vacant squares on the opposite side. An easy way -of doing this accurately is to cut off the four. outside blocks of nin~ small squares each and place them on the opposite side of the centre square. All the vacant squares will be found to be occu" pied. The -result will be Fig. V:I. IX:
VII.
i' 1\_2
s
4
·7'
8 r-..9
18
14' 15
16
19 .2~ 21
2~
2& ';;.- ~6
81 Vs2
o/
t' ;-.!!~
~2
8
8~
/6
8' r-ea
2'!f "it /7
18
t9
zf
t6 Vt4 24
r-?B 24
t8
t~ at
-
17
~7 ~
2}
N!O
1:Jt 26
84'
s'r!-
86
8i Va
~s
84
8~
11).- 11
'
6
t6
t8
22 t--.20
Vg
til'
4
88
r-e9' ~5 5' ~6
·.
Let us test the_ three features we observed in the " Magic Square " of the number three, Fig; lV. _ (r) There are sixteen row~· of seven numbers each~ seven vertically, seven horizontally, and two diagonally .. The sixteen rows . all-add up 175. · (2) The pail'S of opposite numbers, taking -the four corner squares, and the centre squares on each side, _all add up alike. 2:;1 +28, 4+46, VIII. 1'r-..a
8
4
I~
X
a
·7,1
9'-~o ~1 1211s 1~ 15 17 1~ 1~20 I21,Y'22 ~s
s C{6
r-..t~S 69 1~il' ~4
5
4
62 68
41 28'!-!9' ~6144 f'2,.J(tsl48
-
26
aiq 28 I29 Vso
s1
s2
82 34 88
ss
B4
s51 ssls7 ~
B9
40
40
26 813t 86 37
17
~ v48/ 21 20
4~4414'~6 ~7 48 4~ 5~Ci1 lo2l&s o.i'~o ~6 .57 l/wl5sl6o 161 62 6~~4 42/
6~*'
24
25
41
62 6§-(s /
28129 l7so 39 85 81
88
f\.48~
24
27
6~ !15~ 114
12l1s
1~6 ~~
8
61 lao
6
11'!.-Y7
1
2
64
4I+9. IJ+37, all = so. More than this, theadjacent pairs of numbers, taken 0bliquely across, all add up to the same figure. For instance, 47+3, 5+45 •. 35+IS, zg+zr, all =so. And not only so, but_ this also applies to the inner squares of numbers. (3) In Fig. VI concentric circles may be drawn, the common centre
'l'HE OCCULT REVIEW. being the centre of the whole square. The circumferences of the · successive Circles go through the· centres of successive rings of squates~six rings of four squares each, and three rings of eight squares each.. The sum ofthe numbers irt•each.ring of'four squares is roo, and the sum in each ring of eight squares is 200. We will now turn to the series of even numbers, 2, 4, 6, 8, ro, etc. Let us take 6 and 8.as examples and construct their "Magic Squares." Draw two,sqtjares consisting respectively Of ,36 and 64 small squares. fill in tl;le numbers consecutively; and draw lines diagonally, as indicated, producing Figs, VII and VIII. The ,numbers in. squares, the centres of "Yhieh ar~ within' the diagonally drawn lines, remain in their existing positions,· In respect of the. six squares outside the: diagonals on'. each side, 24 fn all, in Fig. VII, one remains in the same section, me!J'ly' shifting its pia~e, atid tl(e other five are transferred.'to the 'opposite section. In respect of the squares outsid~. the diagpl)als in Fig.VIII, 48 in all, two remain. in the same section merely ·shifting their places, and the bthers are transferred to the opposite section. The law governing the redistribution in the case of even n.umbe
(To be contznued;)
TilE
MA{j~C
OF NpMBERS
BY El>WARD .T. BENNETT
PART II I . MATHEMATICAL.
READERS of a mathematical tu:in may be interested in a more abstract method of exhibiting the main features of the " Magic. Squares " of the arithmetical series of numbers-I, 2, 3, 4, 5, 6,_ 7, ·8, etc. . · let nrepresel).t any number. We will call it the root number of tl).e square. The following formulre will then be found to h~ld• good:~
=The Root Number,, . . = The number of _small squares forming the " Magic Square." .. · =The sum of pairs· of opposite numbers. n• +I (n° + n) + 2 = The sum of the 11Uil).bers in l!llY'one row. (n' + n•) + 2 = The sum of \)J1 the numbers in the square. · (n 2 +I) X 2 = The sum of the numbers in eac)l concentric . · circle. (n -I) x (n + 2) =The "LeadingNumber" ("DieLeitzahl"). ' These seven formulre hold _good for hoth odd and even numbers. When n is an odd number, the following also hold good :-. (n• + I) + 2 =The centre number of the square; ' (n• + I) + 2 also =The number of inside nu.mbers (in forming the square). :.... The number of outside numbers (in form(n•- I) + 2 ing the square). When n is an even number the first of these last three formulre is substituted by :(n• + I) x 2 =The sum of the four centre numbers of the square. In this case also, the second and third of these last three formulre have no application. . Purely mathematical considerations have probably been carried as far as most readers will.care to follow, but there are other developments which may possess some practical interest. From what Hellenbach himself says, and from his references to antiquity, it might be supposed that the number of "Magic
"
.
-'.~-
';'
·.•.
j'
I
'l
THE MAGIC OF NUMBERS
79
Squares " which can conveniel).tly be studied is very small. lt would seem as if there were only nine " Magic Squares " consistingof IOOsmallsquareS Of less, each with one single set of numbers, the "roots" being the arithmetical series z-ro. Beyond this the squares of successive numbers increase so rapidly as to. become impracticable. · Hellenbach tells us that one enthusiast worked out the Tetragrams up to that for 26 ( This involves 6J6 small squares and as .many different numbers. If the number of XI.
1
1 1 r;-1
8
... ,..., •
+1
. r :1_ ~
~-
• ,a
·a :-a
••
~!}- 1
~
11
lls,
•• •
•
•
•
7."
8
H
8
~ ,.
17
16
8
•
10 12 18
• -10 -8 _,.
••. • 2
'1- .
.
• - -•' 9
8 8
1
8
'8 .
• • .18
-
11
~
•• • ,. • • • • '" •
16
• -••
' 0
6
- -•-
16
.~
18
-.
·~
10 6
11
8
10 14 6
10 11
18
8
.' +-+
4
• • • •
6.
•
17
. -!--'-+
16
+---+
.
NOTE.-Each arrow ~rl t~is an~ in the two following figures senes w1th an mfimte number of terms.
indicates a
" Magic Squares " of reasonable dimensions were very few, the facilities for study would be small, aild symbolic interpretation would be restricted to very small compass .. .·Therejs, however, no such limitation. The number of " Magic ~quares/' even of the smaller dimensions, is infinite. Let us tak:e a base line starting· from o and representing simple arithmetical succession. ·Along it 'we. can construct our series of "Magic Squares." At o ei:ect'a vertical line with an ascending successio'n of numbers, as in. Fig. XV· ·orr each horizontal line above the
. TI'IEOCCULT· REVIEW
80
base line <;onstruqt a series of squares and fill th¢ni with nuffibers as shown in the diagram, that is, adding one, everywhere. Silties of '" Magic Squares " are thus obtained which 111.ay be continued infinitely. ' Again, instead of placing the arithmetical series of numbers along our ba.Se line, suppose we start with the " Magic Square '' of3, and multiply the number in each s.mall square· by 2,.3, 4, etc.; we get a series of .new " Magic Sq';lares " along our base. line
i;
XII,
1 1
1 I~ ~~...!..
+al-----j--"'"iL9!..\.~18~!1!_11119
& 1&
t&
ao,
18
~~
at
91
8
81
T
19
r-
••
•
•
.18
18
•
•
9
8
8
&
'II-
8
10 14
8
1
8
18
8
·8
x'B
11 8'1
18
....
•• •
19
88
12 8'1
8.
18
9
16 at
84
8
"I~
18
1-
9
.I ..
18 81 89
. I
I
91
I 1•.
+ti------~--~~~~~L-1~11~~ 1 11 16 f- 1~ 18 88 111
1
I
I 18 88
~
1• aa 81 8&
T
I 9
19 .. tt
9·
~
6
8
11 :_oo 88 ·"'4
"
~
84
"I·
(Fig; XII.). If now we combine what we have done in Fig. XI. with this, we get .new ascending Series of squares, each series continuing of the same size, and possessing the satne properties. Again, if instead of taking longer steps, we take shorter ones along the. base line, if we _advance by, tenths instead of whole nUl)lbers, and combine what we have done in Fig. XI., we :get Fig. XIII. Elimmate the decimal point by multiplying all the numbers by ten, and w~ obtain a whole set of new series. · ·
''
TIJE MAGIC OF NUMBERS
81
Bqth these \a.St va,riations may be multiplied by taking other multiples and other fractions., The same processes inay also 'be o::arried out with the' original " Magic Squares " of other numbers~ Thus without exceeding squares, of say .25 small squares, .and Without ·getting to large .indiVidual n~mbers, the . number of "Magic Squares" which may be obtail)e~. is illimitable, Some curious repetitions of the same numbers. will occur, but these are few and far betwe~il~ With a little ingenuity, a square With any XIII.
1
1 '
'1'1 11'1 6'1
1
'1'.9 12'2 &·~
7'8 12'8 6'8
~
... +'8
6'1 .8'1 10'1 11'1 4'1 9'1
I 8"1.- 11'1 4'1
6'1
· ..
_7"':1_
9'1
1m a.·t,F
r- -
--
-••• r •••
tt·.a
to·a
I +I
o·1 8'1 7'1
r-
-
1 6 8 -r--
-
4'1 8'1 8'1
--. r--
8'1 6'1 1'1 8'1 1'1 6'1
+, ..
""'--
•••
6'8 11'8
1'2, 9'2
r- ,e,·s. 1'8
8'8 8'8
10'8
g·~
s·s·
-
o:a a•a: 1'8
1-
•••
•••
10'8
4'8 6'8 6'8
. 0:0· 8'8 -:;;a
I
I
4'8 8'8 2'8
4'8 9'8 a·8
"- 8'8 o·2
6'8 1'2 1'8
+ .•
·,
8'8
_l
8'8 10'8 8'8 ,;;,;; 4'8 o•a, 8'8
I :
8'8 10'8
I
...
I a:-1
4'1 0'1 8\1
•• o a 8 -.-r.;-
I
•••
11,"8 4'8 9'8
11"2 ~8 9'2
.. ~~~
8'1 10'1
r-
6'8 8·a tO'
•••
r-
••• 6'8 8'8 118
7~8
•••
+·a·
desired number oi numbers in. the centre, .or with any ·desired nuinber as the leading one (Leitzahl) may he obtained. The search for symbolism, and the discovery of what Hellenbach c!llls Periodicity, if it ~xists,would thus· be, greatly facilitated. . ' Beyond this· a,gain, if we imagine our horizdn.tal and veuical lines extended in both direCtions, otlr 'zero poiilt in Fig. XL forming a <;entre, and if, instead of a,dding, we subtract. along the tw() lines leftward and downward, we have a fourfold diagram, one
THE OCCULT REVIEW
82
q~arter only o£ .which has been included in our previous conside~ ations. An idea of. it is. given in Fig. XIV. In the. quarter }l)arked I we have to deal with the numbers of simple arithrrietic only. In. II and III we have a soniewh,at confusing. mixture of + and - quantities, and in IV wehave entered into a wholly negative region. · But, .in all four, an infinite number of serieS stretch out before us, all with an infinite nurriber of terms, con• sisting of squares of numbers possessing the same principal features. XIV.
+3-1 +2II~
.I..
,
+1
~s
J
-2
~
I
-1IY
II
-2--'
.-~r · NOTE.-It would be. easy, on a !?heet of draWing-paper, with tl_l.e help of the earHer Figures to construct a diagram with a portion of each quarter filled .in, in detail, ·
We cannot, in this article, go further in these directions. The claim, however, seems almost to"be justi!i,ed, Jhat the idea of number may thus be exp;u~.ded from that i>f a one-dimensional infinitl, line, into tl)at of a two-dimensional infinite suif8.<;e; Baron Hellenbach in his book gives no indicatioll-'that eithedi.e, · or tlui ancientsto whom he' refers, were acquainted with these extensions of the construction of "Magic Squares;" or that an infinite number of " Magic Squares " of moderate size can thus . . be constructed ·and studied With facility.
THE MAGIC l . .dF NUMBERS . -. ,.
~
83
The literature of " Magic Squares " is scattered i\nd. not very. accessible. In vol. xv. of theEncyolopaedia Britannica (nint!l edit. 1:883) is au article in which references are given to various writers on the subject, mostly ancient. References to several modem writers Will be. foqnd in the Subject Catalogue in the British Museum. Major P. A. MacMahon,. F.R.S., read a Paper before the. Royal Society in rgo4, in which he defines " Magiy Squares," and describes their properties. An abstract of the PaR~r is given in the Proceedings of.the Society. A sm,Ul .volume o:f, thirty-two · pages, entitled M agio Squares, by Tll.omas S. Barrett, a Fellow of the Royal Stati~tical Society, and a Member of the London Mathematical Society, was publis)led in r8g3. It _contains much ,as to the different modes of their formation, and ·the varied properties .they possess. .But so far as I have been able to ascertain, nothing has been published as to their construction ~so simple as. the preceding quotations from Hellenbach's book. Arid I hav;e not met with any indications of .the ways in which the number of such squares of small dimensions may be multiplied indefinitely; illustrated more clearly than by Figs. XI. to XIII. Additional importance is 11:owseen to be given to the "remarkable alternative" with whiCh.Hellenbach maintains we-are confronted, and which is stated in his own words in the next paragraph. THE LATER CHAPTER!! •.
It Is quite impossiQle, within. the limits ·of this article, even if It would be interesting and prolitable-concerriing which thei:e would be very diverse opinions,-to follow Baron· Hellenbach through the remaining chapt~rs of his book. Some o;>f the titles are.: "The Growth of Man and the Number Seven," "Apparent Fate," "The Periodic .Oscillations of the Soul," "Phenomenal and· Transcendental Measure of Time." All I can do is to quote, or give an abstract of, a few sentences, .illustrating the tone of the . writer's r)J.ind, and the drift.of his arguments. I11 the Tetragqun, or ''M
·THE OCCULT REVIEW pre)lensible ):)ow they can have attained· such knqwledge in the then state of sdenee)_;_ar; they had no knowledge ofit ; and in the :latter case, the Tetragrammust possess othe_r properties, of whicl:i we ·are igrlorant." · · · _ .· ·Pursuing this idea, Hellenbach says that the belief has been forced upon him that the n)llllber .nine an
(
' !\ ' . i
\
i
I
·.~
THE MAGIC OF· NUMBERS
85
which are founded on the symbolism of the Tetragram vain and unfruitful, at least in most cases, the question-' Why am I in this world ? ' is by no means idle one. I have already referred to Schqpenhauer's dictum, that Fate may possess an objective correspondence,· and a subjective conformabllity to that which exists. In my view, every man '1\as the· power of determining to work iri both directions. The preponderance in different individuals may be in one direction, or it may bein the other. . · . . " The labourer who works in a coal-mine, or throws up a dike, or cultivates the land, renders .a service to the community. And if he cares for the welfare o'f his descendants, he will still find sufficient scope for the development of his character, and his life on this planet will n:othave been purposeless. The position of the more intelligent classes is surrounded with greater difficulties. In the. office;· in trade, on the Exchange, or as an Editor, thework is not always such as brings blessings with it. But, on the other hand, the opportunity exists, by the ver}r coinp!ication and variety of the activity, to gain rich experiences. · Worst off of all, is the rich man of whom the Scripture speaks,.who finds it hard to enter the Kingdom of Heaven. Evensucha man may lead a v~ryuseful .life as a landed proprietor, or .a merchant. But the chief 9ccupation of these so-called happy mortals, or at .least of the majority .of them, is to devise means for kiJling ~ime. Late going to bed, a late breal~fast; some kind of exercise. in order to get an appetite for a late dinner, then the. theatre, then some ho~rs of the night devoted to cards-how can anything that is worthy crystallize from such a round ? .How many fine talents of mind and spirit have I not seen perish in this way ! It is only the changes of fate, .and the lessons of experience, that enable . man to find his way in the kingdom of symbolism, to discover the hidden me.aning, .and to act in accordance therewith.:' . As has been already remarked, it is no part of the purpose of this article to enter into the region of mere speculation. The object is to bring forward facts, and to allude. to, or to indicate the possibility, that•others may be discovered affording. a basis for legitimate speculation, and for the scientific use of the imagination. It is conceivable that we ma:y be thus led into fields of knowledge that we are as yet unacquainted with. Somi.d, light and colour, and to some extent the fac~s of chemistry also, bear certain analogous and numerical relationships to each other, which it is im•. possible to look upon as accidental, and as to which our knowledge is, to say the least, very· imperfect.
an
I
THE OCCULT REVIEW
86
IN CoNCLUSION.
The considerations which have engaged our attention lead. to a realization of the truth of the four imniortallines in Alexander Pope's Essay on Man, in a more definite manner than possibly the poet himself ever .consciously intended. "AU_ Nature is but .Art unkhown to -thee."
The word Nature must be interpreted in·the widest possible sense, as 'including all phenomena. The correspondence which we find existing between different classes of phenomena leads us to a belief in bne great Artist. " All Chance, Direction- which th~u canst nOt see/'.
Hellenbach calls special attention to his use of the phrase "Apparent Fate" ("Die Sclleinbare Fatum "). He evidently believes that what we call Fate and Chance are the results of our limited point of view. He adduces various illustrations to show that this must almost certainly be so. · · ''All Discord, Harmony not understood.''
. This is more difficult to realize a~d to believe. But it is still . more difficult to believe that Harmony and Discord are two realities of equal nature, and 'not that their actual relationship is akin to that ·between Light and Darkness. Light and Darkness, and Harmony ·and Discord in sound, ar:e the results of differences in the rapidity of vibration. · In the fourth line we approach a higher kingdom which we shall not venture to. enter : · · . I ·" All partial Evil, universal Good." .
(
.
c .
.
I· i
..
.. . These four lines carry us on to a magnificent conclu~ion, very far removed froin a cold Pantheism> Modern Scie.nce and Philosophy are making more and more apparent every day, that·: ·'
All a:re but parts of one StupendoUs ~le, ·Whose· Bodr Nature is, and God the Soul." .
1
·~·
Afterword by Romeman The New American Cyclopædia, ed. by G. Ripley and C.A. Dana, p. 518, published in 1859, states: "Arithmomancy depended upon the secret operation of numbers and magical squares, and the numerical value of letters; it was practised by the Chaldeans, and formed a part of the doctrine of the Pythagoreans, Neo-Platonists, and cabalists." http://books.google.com/books?id=1WIIAAAAQAAJ&pg=PA518 The book entitled The Occult Sciences by the Rev. Edward Smedley, W. Cooke Taylor, the Rev. Henry Thompson, and Elihu Rich, published in 1855, elaborates on these points as follows: ARITHMOMANCY, To which head belongs the magical operation of numbers and magical squares, is derived from the doctrines of the Pythagoreans and Platonists. In estimating these doctrines, it must be remembered that all movement, proportion, time, and, in a word, all idea of quantity and harmony, may be represented by numbers: hence, whatever may be attributed to the latter, may also be expressed by numbers, as the signs of occult virtues and laws. It is known to philosophers that the movements of nature are rhythmical; physicians have observed this in the periodicity of diseases; and the appointment of the seventh day as a Sabbath, has added a religious obligation to this law of nature. The three, the ten, and the twelve are also members of well known import, and one is the most divine of all, as expressing the unity of God, and the comprehension of all things in perfect harmony. The use of numbers in divination has assumed many curious forms. It may suffice to mention here the Gematria, or first division of the Cabbala, which teaches how to cast up the letters of particular words as numerals, and to form conclusions from the proportion between the sum of one text and the sum of another. This method converts the Bible into a book written solely by numbers, and some curious results are obtained, probably as near the truth as the rabbinical astrology (see note to Geomancy). Some curious properties of perfect, amicable, and other numbers have been elucidated by the late Platonist, Thomas Taylor. The most valuable remains of antiquity connected with this subject are contained in the 'Chaldean Oracles' of Zoroaster. For the various arrangements of magic squares we may refer to a curious work entitled ' Qanvon-E-Islam [sic]; or, the Customs of the Moosulmans of India,' by Jaffur Sharreef. The Pythagorean doctrine is noticed by Ennemoser, who quotes some interesting passages from Plato on this subject. http://books.google.com/books?id=8DXXAAAAMAAJ&pg=PA328 GEOMANCY. Geomancy, from two Greek words, ge, the earth, and manteia, divination, is an art connected with astrology, and is called by an old writer on the subject, "the daughter, and abbreviation thereof." An ancient method of practising it was by casting pebbles on the ground, from which conjectures were formed much the same as from the chance lines or dots made on paper; in later times, scratches made in the earth were found to answer the same purpose. The
Arabian Geomancy, said to have been first practised by Almadul, was more recondite, being founded on the effects of motion under the crust of the earth, the chinks thus produced, and the noises or thunderings heard; its foundation was the dogma of Aristotle, that "the moving of the heaven is everlasting, and is the beginning and cause of all inferior movings." The essential principle of geomancy, in whatever form practised, is the lot or chance; it is fully described by Cornelius Agrippa, and as it determines the scheme of the heavens without the necessity of astronomical observation, it may be considered a royal road to astrology. A famous professor of Geomancy, in the sixteenth century, was one 'Maister Christopher Cattan,' a translation of whose work was published by Sparry in 1591. In the following century the art is graced by the name of William Oughtred, a distinguished mathematician, and minister of the Church of England, who died in 1660. For a general idea of the method, we may refer to the well-known 'Book of Fate,' said to have been in the possession of Napoleon, and translated (the title-page avers), "from an ancient Egyptian MS. found in the year 1801 by M. Sonnini, in one of the royal tombs near Mount Lybicus, in Upper Egypt." The geomantic figures obtained by inspecting the chance lines or dots were supposed to represent a certain situation of the stars, and the diviner then proceeded as in astrology, as if the configuration of the stars really was such. Another mode of divination by stars, differing both with astrology and geomancy, was practised by the Cabalists. The stars vertical over a city or nation, were so united by lines, as to form resemblances of the Hebrew letters, and thus words which were deemed prophetic. This was the rabbinical astrology, and it was a very plausible adaptation of the occult meaning attributed to Hebrew characters and roots, united to the traditional belief that the stars were themselves gods, or the abodes of deceased heroes. The "star of your God" occurs in the prophet Amos, (v. 26), and Burder remarks that the rise of a new star, or the appearance of a comet, was thought to portend the birth of a great person; also, that the gods sent stars to point out the way to their favourites, as Virgil shows, and as Suetonius and Pliny actually relate in the case of Julius Caesar. As for the Hebrew letters, they constantly figure in schemes of magic, e. g. the potent which confers its spell on the magical knife, and on the shield of David. http://books.google.com/books?id=8DXXAAAAMAAJ&pg=PA314
For Thomas Taylor's work on numbers, see his Theoretic Arithmetic, 1816 http://books.google.com/books? id=VuY3AAAAMAAJ&printsec=frontcover Thomas Taylor was also the translator of The Chaldean Oracles. For these, see The Classical Journal, in which they were printed in three parts: Part I (December 1817) http://books.google.com/books?id=Voo_AAAAYAAJ&pg=PA333 Part II (March 1818) http://books.google.com/books?id=TFY0dOZSWYoC&pg=PA128
Part III (June 1818) http://books.google.com/books?id=43g_AAAAYAAJ&pg=PA243 Or see W. Wynn Wescott's version of The Chaldean Oracles, based on Taylor's translation http://www.sacred-texts.com/eso/coz/index.htm . Qanoon-E-Islam: Or, The Customs of the Moosulmans of India by Jaffur Shurreef is available here http://books.google.com/books?id=l0Imdq2Cdk0C&printsec=frontcover or in a later translation entitled Qanoon-E-Islam, or The Customs of the Mussulmans of India by G. A. Herklots http://books.google.com/books? id=61AbAAAAYAAJ&printsec=frontcover . Joseph Ennemoser's History of Magic is in two volumes, translated by William Howitt: Vol. I: http://books.google.com/books?id=1TtVAAAAMAAJ&printsec=frontcover Vol. II: http://books.google.com/books?id=zgXuzN2ZRtoC&printsec=frontcover Geomancy is described in the Fourth Book of Occult Philosophy, which is attributed to Henry Cornelius Agrippa (whose Three Books of Occult Philosophy, issued in recent years in a modern edition by Donald Tyson with commentary, are on W. D. Gann's List of Books for Sale); the fourth book is online here: https://archive.org/details/cu31924006718757 Scans of The Geomancie of Maister Christopher Cattan Gentleman are available at: http://www.scribd.com/doc/24134718/The-Geomancie-of-Maister-Christopher-Cattan-Gentleman http://books.google.com/books?id=Et7mSaCuSJAC&printsec=frontcover The Oracle or Book of Fate by Hermann Kirchenhoffer is online at Google books: http://books.google.com/books?id=ir81AAAAMAAJ&printsec=frontcover .
As regards sources mentioned in the two-part article by Edward T. Bennett above, the following are available online: L. B. Hellenbach's Die Magie der Zahlen, also mentioned by Bennett, is here in the original German at: http://books.google.com/books?id=3SkCAAAAQAAJ&printsec=frontcover http://books.google.com/books?id=yn3_JDQ3n6kC&printsec=frontcover . W. F. Barrett's article on "Light and Sound" in The Quarterly Journal of Science, January, 1870, mentioned in the two-part article by Edward T. Bennett, is online here: http://books.google.com/books?id=u1EEAAAAYAAJ&pg=PA1 . Maj. P. A. MacMahon's "Magic Squares and other Problems on a Chess-Board" was published in Vol. 17 of Notices of the Proceedings at the Meetings of the Members of the Royal Institution of Great Britain http://books.google.com/books? id=e8cNAQAAIAAJ&pg=PA50 . [Note to Reader: This file has been formatted in "landscape" view (with pages short up and down and long going across), rather than the traditional or "portrait" view in order to accommodate the format of the scanned images. The books linked above may be seen by readers in the United States, where they are in the public domain. Readers outside the United States may find that Google books blocks display of some scans, which is a policy they adopt to avoid possible copyright infringement in places where such books may still be in copyright.]
