292 



KNOWLEDGE. 



[December 1, 1897. 



reply of Mr. Cunningham to the criticism of his work on 

 " British Marketable Fishes " was published, so that he, 

 on the one side, may feel just as aggrieved as Mr. Master- 

 man on the other. In any case, it is hardly in good 

 taste for Mr. Masterman to make charges of partiality 

 when both Mr. Cunningham's work and the one just 

 published have been similarly treated in our contem- 

 porary.— The Ee-\tewer.] 



THE INTERNATIONAL CONQ-RESS ON THE PROTECTION 



OP BIRDS. 



To the Editors of Knowledge. 



Sirs, — The International Congress on the Protection of 

 Birds, inaugurated at Aix-en-Provence on November 9th, 

 under the auspices of the Ligue Ornithophile Fran^aise, of 

 which M. Louis Adrien Levat is the able president, was 

 concluded on Saturday last. The protection of insecti- 

 vorous birds useful to agriculture was the chief matter 

 treated of, and it was decided to forward to the Govern- 

 ments of Europe through the French Minister for Foreign 

 Affairs the resolutions which were formulated. Public 

 educational bodies are also to be memorialized in order 

 to obtain, if possible, the serious consideration of this 

 important subject by schoolmasters and Government school 

 inspectors. Numerous French and Italian agricultural, 

 horticultural, and sporting societies were represented at 

 the Congress, and delegates from the Selborne Society and 

 the Society for the Protection of Birds were also present. 



Among the French societies which sent delegates were 

 the Societe d'Horticulture de France, Association Pomo- 

 logique de I'Ouest, Societe Centrale des Chasseurs, 

 Societe Protectrice des Animaus, and Societe Protectrice 

 des Oiseaux. Mr. Lemon and I had the honour of 

 representing the Selborne Society and the Society for the 

 Protection of Birds. 



Hillcrest, Redhill, Surrey, M. L. Lejion. 



17th November, 1897. 



" THE MYSTIC NUMBER THREE." 

 To the Editors of Knowledge. 

 Sirs, — In connection with this subject, may I point out 

 some remarkable peculiarities in the multiples of the 

 number nine? If these are written down in columns up to 

 nine times one hundred, as in the following table, a regular 

 sequence of numbers appears, the figvures on the right hand 

 of each column being in reverse order to those on the left, 

 and the central figures in each multiple of three figures 

 also run in regular sequence, reading either up, down, or 

 across the table. In fact, the table may be rapidly con- 

 structed by writing down the consecutive numbers from 

 one to ninety in ten columns, repeating the number at the 

 foot of each column at the top of the follovring one, and 

 then placing the numerals from the cypher to nine opposite 

 to them in reverse order, thus : — 



9: 99 18 9 I 27 9 ?6 9 ! 45 9 54 9 6.3 9 ! 72 9 ' 81 9 



1 8 ! 10 8 19 8 1 28 8 : 37 8 i 46 8 55 8 6i 8 73 8 82 8 



2 7 11 7 20 7 29 7 38 7 47 7 56 7 65 7 74 7 83 7 



3 6 12 6 21 6 30 6 39 6 48 6 57 6 66 6 75 6 84 6 



4 5 13 5 22 5 31 5 40 5 49 5 58 5 67 5 76 5 85 5 



5 4 14 4 23 4 32 4 41 4 50 4 59 4 68 4 77 4 86 4 



6 3 f 15 3 24 3 33 3 42 3 51 3 60 3 69 3 78 3 87 3 



7 2 16 2 25 2 34 2 43 2 52 2 61 2 70 2 79 2 88 2 



8 1 17 1 26 1 35 1 I 44 1 53 1 j 62 1 71 1 80 1 89 1 



9 18 27 , 36 45 51 | 63 ! 72 ; 81 90 



As a consequence of this sequence the figures, when read 

 diagonally on the above square (omitting from the multiples 

 of three figm-es the units, which are, of course, consecutive), 

 an arrangement may be produced in which the figures 

 reappear in numerical order, as in the following table : — 



Diagonals from left to right commencing at the top— 



" 10 20 30 40 50 60 70 80 90 



1 U 21 31 41 51 61 71 81 



2 12 22 32 42 .52 62 72 



3 13 23 33 43 53 63 



4 14 24 34 44 54 



5 15 25 35 45 



6 16 26 36 



7 17 27 



8 18 

 9 



The figures in each product in the first table, when 

 added together, produce nine or some multiple of nine, 

 however far the table may be carried. 



A. G. Moncreiff Grahame. 



2, Walpole Terrace, Brighton. 



To the Editors of Knowledge. 



Sirs,— In the article on " The Mystic Number Three" 

 in your current issue you refer to the "mysterious and 

 unaccountable properties of the number," described in a 

 paper published by the Asiatic Society of Bengal, and 

 quote as an instance the fact that " when any number is 

 multiplied by three, or any multiple of three, as six, nine, 

 twelve, etc., the separate figures in the result, if added 

 together, give a total of three or a multiple of three " ; or, 

 in other words, that when a number is divisible by {i.e., a 

 multiple of) three, the sum of the digits of the number is 

 also divisible by three. 



Your mathematical readers will know that this is not 

 quite so "unaccountable " as it appears, and to others the 

 following may be of interest as throwing some light on 

 this particular instance of the " mysterious." 



Suppose a h c d to represent any number in the ordinary 

 system of notation, i.e., a, b, c, and d each represents a 

 single digit, the whole being equivalent to 

 a X 1000 + i X 100 + (■ X 10 -H '/ = a 10' + hl02 + clO + d. 



Now, it is obvious that any power of ten may be expressed 

 by writuig as many nines in succession as there are imits 

 in the number representing the power in question and 

 adding unity ; thus, 10= = 99999 -fl. 



Hence, a 10^ -f- l> 10= + c 10 -I- (/ may be written— 

 a (999 + l) + b{9d+l) + c{9 + 1) + d 

 = rt 999 + i 99 -f c 9 -f « + b + c + d. 



Obviously, the first three terms in this expression are 

 divisible by three, each being a multiple of nine. 



Hence the whole expression will be divisible by three if, 

 and only if, the remaining portion — that is, the sum of the 

 four digits of the number — is divisible by three. 



As this method of reasoning is applicable to any number 

 whatever, it is, I think, fairly obvious why, when a number 

 is divisible by three, the sum of its digits is also so divisible. 



November 16th, 1897. G. Haecourt Hill. 



Mr. F. Enock, whose name is so familiar to readers of 

 this magazine, will deliver a Christmas course of lectures 

 at the London Institution. The subject of his discourses 

 is — " Insects at Home, at Work, and at Meals." 



Dr. Fewkes, explorer for the American Bureau of Eth- 

 nology, has described the site of a city that once belonged 

 to the Funii tribe — a city probably as populous as that of 

 Rome under Severus, and built as one huge house con- 

 structed in tiers like the cells of a honeycomb, forming 

 the walls and fortifications of the city on the outside, the 

 windows and doors opening towards an inner court. 



