January 1, 1898.] 



KNOWLEDGE. 



17 



freely acknowledged." I gave him the case of fhe work on 

 the growth of fishes, referred to in a cursory manner which 

 deprives it of any importance. Of course, Mr. Masterman 

 may be permitted to have an opinion of his own as to 

 what work is important, but marine biologists are also at 

 liberty to challenge it. 



With regard to the hermaphroditism of ^fl|.l•ille, the 

 quotation is from a paper by Mr. Masterman himaelf, and 

 the words used are ; " We may cite Nansen's observation 

 of the protandric hermaphrodite cDndition of Mi/xine." 

 This certainly gives the idea that the hermaphroditism 

 was discovered by Dr. Nansen. 



As to Nature, Mr. Masterman distinctly ascribed "par- 

 tiality" to the editor in tlie matter of the review of 

 his book. He knows that Mr. Cunnmgham's work was 

 treated in exactly the same way that the work of Dr. 

 Mcintosh atfd himself was treated, and yet he has not the 

 good grace to withdraw his charge of partiality. I cannot 

 say that Mr. Canningham sent a reply to the criticism of 

 his own work, but I know that no reply was published in 

 Natui-f, any more than was 'Sir. Masterman's reply to 

 Prof. Lankester's review. It is a pity that there are 

 authors like Mr. Masterman ever ready to resent fair 

 criticism and impugn editorial actions. — The Reviewer.] 



TESXINO- MULTIPLICATION AND DIVISION. 

 To the Editors of Knowledge. 



SiKs, — The properties of " the mystic numbsr three," 

 and its square 0, referred to by your correspondents on 

 page 292 of your migazine, bring 0113 back to the 

 earliest recollec lions of one's school days, when the multi- 

 plication sums were tested by "casting out the nines," as 

 it was then called. The property that the sum of the 

 digits of any integer, divided by .9, gives the same 

 remainder as if the number itself were divided by 9, is a 

 natural consequence of our decimil notation. Had the 

 notation been duodecimal, 11 would have possessed the 

 same property; and, even in the decimal notation, 11 

 may be used with almost equal ease, and is a safer test. 

 Beginning with the units, add the alternate figures, and, 

 carryiug to the tens, add the other alternate figures ; then 

 add what is over to the units of the sum. If the number 

 thus obtained be divided by 11, the remainder is the 

 same as if the whole number were divided by 11. 

 This can be easily shown from the obvious fact that every 

 even number of nines is divisible by 11. These pro- 

 perties are well known, and hardly need illustration. 

 They are mentioned as introductory to what follows. 



Some years since, I had to do with the multiplication 

 and division of very large numbers, consisting of sixty 

 figures and upwards. Finding that neither '-) nor 11 was 

 a sufficiently reliable test, I was led to seek for something 

 safer. The numbers, for distinctness, were arranged in 

 periods of five figures each, beginning of course at the 

 right. My test was to be adapted to tliis arrangement, 

 and I soon found that 11111, consequently 99999, is 

 divisible by 41 ; therefore 99999 99999, etc. It follows 

 that if the sum of the periods, taken as separate numbers 

 and carrying what is over to the units' place, be divided by 

 41, the remainder is the same as if the whole number were 

 divided by 41. 



The division by 41 may be abridged thus : — 

 Let the sum of the periods, found 



as above, be 37529 



Subtract the largest multiple of 



mil contained therein ... 33333 



41)4196 remainder 14, 



showing that, if the whole number were divided by 41, the 

 remainder would be 14. 



Years afterwards I was requested by the late Prof. 

 Cayley to verify some results, involving also very large 

 numbers, but arranged in periods of three figures each. 

 To this also it was judged expedient to use a test specially 

 adapted to the arrangement. I saw that 111, therefore 

 999, is divisible by 37 ; and consequently that if the sum of 

 the periods of three figures, taken as above, be divided by 

 37, the remainder is the same as if the whole number were 

 divided by 37. 



Take, for example, any numbar at random, say — 

 45 286 507 613 941 

 The sum of the periods is 2 392 



and, adding 2, the unit of the second period in the sum, to 

 the first period, we obtain finally 394 



Subtract the largest multiple of 111 therein 333 



The number thus obtained 61 divided 



by 37, leaves the same remainder, 24, as if the whole 

 number were divided by 37. 



A. Graham. 

 Cambridge Observatory, 



14th December, 1897. 



ARTIFICIAL SUNSPOTS. 

 To the Editors of Knowledge. 



Sirs,— With regard to Mr. East's experiments and your 

 remarks in the Dacember Number of Knowledge, is it 

 possible that in the sun's surface we have anything similar"? 

 —that is, are the rice grains really the only partial con- 

 solidated matter that we see ?— all the interior of the sun 

 being in a gaseous state, under such conditions of extreme 

 heat and pressure as to make chemical combination and 

 luminosity impossible. I should like to see some remarks 

 in your journal on this head. It seems to me one of 

 impossibility ; the photosphere would then be the very 

 first stages of a crust formation on the sun. 



December lOih, 1897. Thos. .1. Haddy. 



BARE BIRDS. 

 To the Editors of Knowledge. 



Sirs, — Your issue for December contains an account of 

 the shooting of four rare birds. Most probably each of 

 these rare birds had mates and would have continued the 

 race if let alons, but the collector comes with his gun and 

 endeavours to make the rare bird an extinct bird. Of 

 course, his specimen would become more valuable if this 

 species of bird became extinct in this country, while if he 

 allowed the bird to escape he would have no specimen at 

 all. But is this a suflicient reason for shooting a bird that 

 is doing no harm and is not intended to be eaten, and 

 whose only crime is that very few Uke it are to be found in 

 this country ? Our object should be to preserve — not to 

 destroy — such rare specimens. This would be admitted if 

 they were domestic animals. It is only when an animal 

 is wild that he is shot because there are few Uke him. If 

 the shooting goes on there will soon be none. 



It is time that we had a society for the preservation of 

 rare animals and birds — unless, of course, they are mis- 

 chievous like the wolf, which has now died out in the 

 British Islands. 



Every zoologist will admit that utility is not the only 

 thing to be looked to as regards the preservation or 

 destruction of a race of animals; and, if there is no reason 



