566 



NATURE 



[October 14, 1897 



effect such a saving of time and trouble that I think they ought 

 to be regularly taught in schools. 



Years ago I had discovered the curious fact that, if you put a 

 "o" over the unit-digit of a given number, which happens to 

 be a multiple of 9, and subtract all along, always putting the 

 remainder over the next digit, the final subtraction gives 

 remainder " o," and the upper line, omitting its final "o," is 

 the "9-quotient" of the given number («"^. the quotient pro- 

 duced by dividing it by 9). 



Having discovered this, I was at once led, by analogy, to the 

 discovery that, if you put a " o" under the unit-digit of a given 

 number, which happens to be a multiple of 11, and proceed in 

 the same way, you get an analogous result. 



In each case I obtained the quotient of a division-sum by the 

 shorter and simpler process of stihiraction : but, as this result 

 was only obtainable in the (comparatively rare) case of the given 

 number being an exact multiple of 9, or of 11, the discovery 

 seemed to be more curious than useful. 



Lately, it occurred to me to examine cases where the given 

 number was not an exact multiple. I found that, in these cases, 

 the final subtraction yielded a number which was sometimes the 

 actual remainder produced by division, and which always gave 

 materials from which that remainder could be found. But, as 

 it did not yield the quotient (or only by a very " bizarre" pro- 

 cess, which was decidedly longer and harder than actual division), 

 the discovery still seemed to be of no practical use. 



But, quite lately, it occurred to me to try what would happen 

 if, after discovering the remainder, I were to put it, instead of a 

 " o," over or under the unit-digit, and then subtract as before. 

 And I was charmed to find that the old result followed : the 

 final subtraction yielded remainder " o," and the new line, 

 omitting its unit-digit, was the required quotient. 



Now there are shorter processes, for obtaining the 9-remainder 

 or the ii-remainder of a given number, than my subtraction- 

 rule (the process for finding the 11 -remainder is another discovery 

 of mine). Adopting these, I brought my rule to completion on 

 September 28, 1897 (I record the exact date, as it is pleasant to 

 be the discoverer of a new and, as I hope, a practically useful, 

 truth). 



(i) Rule for finding the quotient and remainder produced by 

 dividing a given number by 9. 



To find the 9-remainder, sum the digits : then sum the digits 

 of the result : and so on, till you get a single digit. If this be 

 less than 9, it is the required remainder : if it be 9, the required 

 remainder is o. 



To find the 9-quotient, draw a line under the given number, 

 and put its 9-remainder under its unit-digit : then subtract 

 downwards, putting the remainder under the next digit, and 

 so on. If the left-hand end-digit of the given number be less 

 than 9, its subtraction ought to give remainder "o" : if it be 

 9, it ought to give remainder " i," to be put in the lower line, 

 and " I " to be carried, whose subtraction will give remainder 

 "o." Now mark off the 9-remainder at the right-hand end 

 of the lower line, and the rest of it will be the 9-quotient. 



Examples. 9/75309,6 9/ 94613 8 9/5^317 3 

 83677/3' 105 1 26/4' 64797/0' 



_ (2) Rule for finding the quotient and remainder produced by 

 dividing a given number by 11. 



To find the 11 -remainder, begin at the unit-end, and sum the 

 ist, 3rd, &c., digits, and also the 2nd, 4th, &c., digits ; and 

 find the 11 -remainder of the difference of these sum's. If the 

 former sum be the greater, the required remainder is the 

 number so found : if the former sum be the lesser, it is the 

 difference between this numlier and 1 1 ; if the sums be equal, 

 it is "o." 



To find the ii-quotient, draw a line under the given number, 

 and put its ll-remainder under its unit-digit: then subtract, 

 putting the remainder under the next digit, and so on. The 

 final subtraction ought to give remainder "o." Now mark off 

 the 1 1 -remainder at the right-hand end of the lower line, and 

 the rest of it will be the ii-quotient. 



Examples. ii/732io 8 11/ 85347 r 11/594263 11/475684 

 66555/3 77588/3 54023/10' 432447o' 



These new rules have yet another advantage over the rule 

 of actual division, viz. that the final subtraction supplies a test 

 of the correctness of the result : if it does not give remainder 

 "o," the sum has been done wrong: if it does, then either it 



NO. 1459, VOL. 56] 



has been done right, or there have been two mistakes — a rare 

 event. 



Mathematicians will not need to be told that rules, analogous 

 to the above, will necessarily hold good for the divisors 99, loi, 

 999, looi, &c. The only modification needed would be to 

 mark off the given number in periods of 2 or more digits, and 

 to treat each period in the .same way as the above rules have 

 treated single digits. Here, for example, is the whole of the 

 working needed for dividing a given number of 17 digits by 999 

 and by looi : — 



999/75410836428139^214 . 

 7548632275o89o/fo4 ' 



1 00 1 /7 54108364281 39 214 

 75335500927212/ 2 



But such divisors are not in common use ; and, for the 

 purposes of school-teaching, it would not be worth while to go 

 beyond the rules for division by 9 and by 11. 



Ch. Ch., Oxford. Charles L. Dodgson. 



Notes on Madagascar Insects. 



Having recently received a small miscellaneous collection of 

 rather typical Madagascar insects, collected by my son on that 

 island, perhaps a modest, non-technical letter concerning them 

 might interest your readers. The climate is .so deadly to Euro- 

 peans that insect-collecting is hazardous in the extreme, and I 

 think not much is known of the insects. Some of the Orthop- 

 tera are highly curious, possessing antennae five and six times 

 the length of their bodies, so as to be able to detect danger 

 afar. The Longicorn beetles appear very similar to ours, but 

 the markings on their elytrje are brighter. The beetles gener- 

 ally are remarkable for the extreme brilliancy in colouring of 

 their under surfaces and legs, while the upper surface is dull. 

 I apprehend, therefore, they are fiot ground feeders. The 

 dragon flies appear similar to some of ours, both in size, colour- 

 ing, and shape. There is a lantern fly, or two, and a mole 

 cricket, much resembling ours. 



The spiders are fiot large, but as ugly and venomous-looking 

 as nature knows how to make them. But, oh, the Centipedes I 

 — gruesome-looking, plated, mailed, jointed, spiny-tailed, and, 

 corneous horrors, half a foot long, with twenty legs (each side) 

 their rapidity in travel must be great, while the large curved fangs, 

 attached to the " business end," suggest the deadliest of grips. 

 Had Milton ever seen one, "Paradise Lost" might have 

 contained another horror. 



I have had mounted with them a praying Mantis (also from 

 the island), to somewhat neutralise their felonious aspect,, 

 though I fear the usual attitude of this most hypocritical of 

 insects (with its extended fore limbs) indicates anything but 

 prayer, or even reverence. 



The brilliancy in colouring of Madagascar butterflies is not 

 remarkable for a sub-tropical region. Many are like our fritil- 

 laries in aspect, while the clouded yellows appear identical. A 

 few of the white butterflies are also like ours. 



Rottingdean, Sussex. E. L. J. Ridsdale. 



Protective Colouring. 



The following instance of apparent consciousness of protec- 

 tive colouring in a young bird seems worth recording. On 

 August 14, while walking in my orchard, which being on a steep 

 slope is terraced with low stone walls, I put up a young Night- 

 jar {Caprhmilgus europcetis) which flew straight to the top of 

 one of the walls and flattened itself down on a broad flat stone. 

 As it was within 6 feet of a hedge on one side, and there were 

 gooseberry bushes, &c. , on the other, there was no lack of cover 

 if it had wished to hide. I left it there, and coming again two 

 hours later found it in the same spot. Its colouring matched 

 the stone on which it was lying so closely that had not one 

 known that it was there, it would probably have been overlooked. 

 On being closely approached it flew to another of the walls 

 higher up, and crouched down in exactly the same way. I then 

 tried to catch it with a butterfly net, when it flew over the 

 hedge to a rough field on the opposite side of the valley from 

 which it had, no doubt, come. Alfred O. Walker. 



Nant-y-Glyn, Colvvyn Bay, October 5. 



