December 1, 1891.] 



KNOW I^ EDGE 



229 



cube is made much easier by knowing the following rule : — 

 "If the sum of the digits in its simplest form equals any 

 of the following numbers it cannot be a perfect square, 

 i.e., 2, 8, 5, 6 and 8 ; and the number can only be a 

 perfect cube when the sum of its digits, reduced to its 

 simplest form, equals one of the numbers 1, 8 or 9." It 

 is also an interesting fact that the sum of the digits of the 

 squares of consecutive numbers recur in the following 

 order — 1 49779419, and similarly the sum of the 

 digits of perfect cubes recur in the order 1 — 8—9. 

 Explanatory Table. 



Yours faithfully, 

 London, 3rd November, 1891. A. W. Gordon. 



[Mr. Gordon's note may interest some of our readers. 

 His test for cube numbers is so obvious from theory that 

 he is probably mistaken in regarding it as a fresh discovery. 

 The test is useful, as it enables the cube hunter to 

 discard two-thirds of the numbers which may come before 

 him, and Mr. Gordon's test for square numbers enables him 

 to discard five-ninths of the numbers he may have to deal 

 with, but it is not as easy to remember as the well-known 

 test already referred to in Knowledge by Mr. Christie 

 and Mr. Barrett, viz., that numbers which are perfect 

 squares cannot end wdth the digits 2, 3, 7 or 8, but both of 

 the tests for square numbers may be applied successively. 



Mr. Gordon's test for cube numbers follows from the 

 fact that all perfect cubes are either multiples of 9, 

 or when divided by 9 a remainder which is either 1 

 or 8 is left. When the number is divisible by 9 the sum of 

 its digits is equal to 9, and when the remainders 1 or 8 

 are left the sum of its digits must be either 1 or 8. 

 The rule with regard to the division of cube numbers 

 by 9 follows directly from the fact that a number which 

 is divisil)le by 3 must when cubed be divisible by 9, while 

 a number which is divisible by 3, leaving a remainder 1, 

 must when cubed and divided by 9 leave a remainder 1, for 



(3;i 4- 1)3=3' «■' + 3. 33. n- + 3. 3. n + 1 

 which must be a multiple of 9 plus 1, and a number which 

 is divisible by 3, leaving a remainder 2, must when cubed 

 and divided by 9 leave a remainder 8, for 



(3 n + 2)3 = 33 rt" + 3 . 2 . 3- i> ■' + S . 2^ 3 . n + 2 • 

 which is a multiple of 9 plus 8. 



Mr. Gordon's test for square numbers follows from the 

 fact that to pass from the square of any number n to the 

 square of the number immediately above it we must add 

 2 n -f 1, or what amounts to the same thing, the square 

 of any number n may be found by adding together the first 



« terms of the series l + 3-)-5-|-7-f9-t- 11 -t-&c. The sums 

 of the digits of consecutive numbers recur in the order 

 1, 2, 3, 4, 5, 6, 7, 8, 9 ; therefore, if we pass along this 

 recurring series with steps of increasing length, beginning 

 at 1 and stepping 3 inteiwals at the first step, 5 at the 

 second, and so on, we shall .step from the sum of the digits 

 of one square to the sum of the digits of the next, and shall 

 always avoid the numbers 2, 3, 5, 6 and 8, for they are 

 avoided in the first eight steps, and afterwards the same 

 numbers are avoided because the subsequent steps are 

 similar to the first eight steps, except that they are 

 each eighteen intervals longer, and eighteen intervals 

 corresponds to two complete circuits of the recurring series. 

 Thus the sums of the digits of the squares of successive 

 numbers are marked with a point above them : — 



1234 567891234567891234567891234567891234 



5678 9123 4 5678 9123 4567 89i 2345 6789 1234 56789 123 



A. C. Ranvakd.] 



SNAKE POISON. 

 To the Editor of Knowledge. 



Sir, — Two years and a half ago you were good enough 

 to open your paper to a discussion between Mr. Field and 

 myself on the subject of experimenting on animals with 

 snake poison. Mr. Field, who had performed in France 

 some mild experiments on mice in this line of research, 

 took occasion in writing for you on the " Common Adder," 

 to make the following remarks : — 



" Wheu we thiuk of tlie thousands of our fellow men who die 

 annually by reason of our want of knowledge with respect to snake 

 jjoisons, the importanee of experimenting on living animals (Jor by 

 these means alone can an antidote he foumi ) in this case cannot be 

 over-estimated. . . We may truly say then, that he who 



hinders the (irogress of such investigations eommits a sin against 

 mankind." — Kxowlkdge. February, 1SS9. 



Naturally I protested against this view, and asserted 

 that " he who, for the sake of remote and doubtful physical 

 benefits to our race, encourages a practice which im- 

 questionably must stitle the impulses of compassion in the 

 human soul, is the real sinner against mankind." I also 

 quoted the l^iici/iloiurditi IUit((n>ucc(. vol. xxii., p. 191, to 

 contradict Mr. Field's statement, "an antidote has been 

 discovered,'' the Enciji-hqKrdiii elaborately explaining that 

 "?«) antidote is known capable of counteracting or 

 neutralizing the action of snake poison." 



After two years my contention has been so remarkably 

 confirmed that 1 must ask your permission to refer your 

 readers, who may take interest in the controversy, to no 

 less an authority than the columns of the I.nnctt (October 

 24th, p. 960), for the fullest satisfaction on the subject. 

 They will there learn that " a country practitioner at the 

 Antipodes discovered the antidote and for years practised 

 it with unfailing success, when Feoktistow, miaUd hij hix 

 expii-itiuntx, rejected it." ^Mueller's theory, derived from 

 careful observation with no viviseetious experiments. Las 

 proved a true and vast benefit to humanity, while those 

 experiments on animals which Mr. Field asserted were the 

 " only means " of discovering an antidote, not only failed 

 to result in such a discovery, but actually put the investi- 

 gator on a wrong scent, and pnirntrd him from finding 

 what was before his eyes ! 



I am, Hir, truly yours, 



Fkancks Power Cobbe. 



[I do not agree with Jliss Cobbe's interpretation ot the 

 paragraph in the l.nnrvt, but I will ask ^Ir. Field to allow 

 the lady to have the last word, as the controversy is not 

 suitable for the pages of Kxowlkdgk., and must now come 

 to an end.— A.C.R.J 



