7^ 



NATURE 



[November 26, 1891 



LETTERS TO THE EDITOR. 



[ The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intended for this or any other part of Naturk. 

 No notice is taken of anonymous communications.] 



Warning Colours. 



In the experiments on "Comparative Palatability," recorded 

 in Nature of November 19 (p. 53), Mr. E. B. Titchener refers 

 to the unpalatability of the brimstone butterfly. The insect 

 was "fairly seized several times," but " was always rejected," 

 by a frog. Some of your readers may not be aware that Mr. 

 F. Gowland Hopkins, of Guy's Hospital, has recently shown 

 that the yellow pigment of this butterfly, and of several others of 

 its allies, is due to a substance formed as a urinary pigment ; it 

 is also known that the colours of other butterflies, and other 

 animals, bear a relation to the urinary pigments. These 

 substances may be in many cases of a disagreeable flavour. 

 Dr. Eisig, of the Naples Zoological Station, has suggested 

 that if intense and varied coloration is primarily due to a 

 great quantity and variety of such bitter-tasting pigments, we 

 do not need to assume that the brilliant coloration has been 

 brought about in order to advertise the nauseous taste. The 

 bright and varied colour will be, in fact, a consequence of the 

 deposition in the integument of bitter pigments. This view — 

 ■which has for the most part escaped the attention of those who 

 have written upon animal colours, owing doubtless to its having 

 been put forward in a special monograph upon a group of worms 

 (Capitel/ida) — better explains how it is that brightly-coloured 

 unpalatable creatures are in so many (? the majority of) cases 

 tasted before being refused. I have laid some stress upon this 

 view of warning coloration in a forthcoming book upon 

 " Animal Coloration," which is to be published by Messrs. 

 Swan Sonnenschein and Co. Frank E. Beddard. 



Zoological Society's Gardens, Regent's Park. 



The Salts in Natural Waters. 



The communication of Mr. Lyons, in Nature of Novem- 

 ber 12 (vol. xlv. p. 30), giving an analysis of the water of the 

 salt lake of Aalia Paakai, aftords a suitable opportunity for 

 asking a question, to which, I trust, some chemist among the 

 readers of Nature will be able to give a satisfactory answer. 



The usual analysis of the "solid constituents" of a given 

 specimen of natural water only directly determines, I believe, 

 the respective quantities of the metallic bases — sodium, calcium, 

 &c. — and of the non-metallic constituents — chlorine, carbonic 

 acid, &c. — contained in the "total solids." How does the 

 chemist, then, proceed to mate these two classes of constituents 

 together, so as to be able to state with confidence what salts, 

 and in what quantities, are held in solution in the water ? The 

 problem itself would appear to be an indeierniinaie one, at any 

 I ate if there are more than two of either class of constituents. 

 What additional considerations are introduced to render the 

 problem deteiminate? Are they definite chemical conditions; 

 or is there more or less arbitrariness in the assumptions made, 

 so that two chemists would not necessarily arrive at the same 

 result ? 



In the case of the Honolulu lake, there are, according to 

 Mr. Lyons's analysis, three non-metallic constituents [chlorine, 

 bromine, sulphuric acid) — (is not the abfence of carbonic acid 

 remarkable?) — and four metals [sodiutn, potassium, calcitim, mag- 

 nesium) ; the quantities of which have, 1 suppose, been obtained 

 by direct analysis. From the twelve possible combinations of 

 these constituents to foim simple salts, five have been excluded, 

 the sulphates of sodium and potassium, and the bromides of 

 these metals and calcium also, thus reducing the number to 

 seven, the quantities of which can, of course, be definitely de- 

 termined from the seven direct data of the analysis. Is it certain, 

 however, on assured chemical grounds, that none of these ex- 

 cluded salts are contained in the water, and if not, on what 

 principle has thtir possible existence been ignored ? 



I write, as is evident, with but very slight knowledge of 

 chemical analysis ; and possi'oly answers to my questions are 

 to be found in some text-book. If so, I should be obliged by a 

 reference to any easily accessible work in which the question is 

 discussed. R. B. H. 



NO. I 152, VOL. 45] 



Mental Arithmetic. 

 The following method of multiplying large numbers together 

 mentally, if new, may interest some of your readers. If it has 

 been published before, I should be glad to learn where it may 

 be found. The process is so simple tha^, though I have no 

 special gift for mental arithmetic, I was able, almost v\iihout 

 practice, to multiply together correctly seven figures by seven, 

 and to write down the result from left to right. 



Suppose it is desired to multiply 123 by 456. The sum is 

 usually written thus : — 



123 

 456 



738 

 615 

 492 



56088 



If, instead of completing each step in the multiplication as we 

 arrive at it, and carrying the tens to the left, the digits are 

 merely connected by the multiplication sign and written down 

 in their proper places, the result is : — 



2x4 



1x6 



2x5 

 3x4 



2x6 

 3x5 



3x6 



1x4 (1x5 + 2x4) (1x6+2x5-1-3x4) (2x6-f-3x5) 3x6 



If the figures in the lowest line are multiplied out and the tens 

 carried to the left in the usual way, the result is, of course, the 

 same as that given by the ordinary procedure. Thus, to obtain 

 the first figure, beginning at the right, we say : " 3 x 6 = 18 ; — 

 8 and carry i." To obtain the second figure : "2 x 6 = 12 ; 

 3 X 5 = 15 ; 12 -f 15 -f I (which has been carried) = 28 ;— 8 

 and carry 2." And so on. Thus each figure of the answer can 

 be obtained by multiplying together certain digits of the multi- 

 plier and multiplicand, and adding the amount to be carried 

 from the calculation of the previous figure, without the strain of 

 remembering all the horizontal rows of results and their relative 

 positions vertically. It remains only to show which digits of 

 the multiplier and multiplicand must be combined. A con- 

 sideration of the example worked out above will show that, to 

 obtain the first figure of the answer, we multiply the 1st digit 

 (from the right) of the multiplier (6) by the ist of the multi- 

 plicand (3). To obtain the second, we multiply the 1st of the 

 multiplier by the 2nd of the multiplicand and the 2nd of the 

 multiplier by the ist of the multiplicand {i.e. ihe first two of 

 each line) crosswise, and add the products. Similarly, the third 

 figure is obtained by multiplying the first three digits of each 

 line crosswise, i.e. ist by 3rd, 2nd by 2nd, and 3rd by ist, and 

 adding the products. The number of digits employed in the 

 process is now at a maximum, and begins to diminish. To 

 obtain the fourth figure, we multiply together crosswise all the 



digits except the first of each line, i.e. the group . And to 



obtain the last figure, we umltiply all except the first two of 



each line, i.e. the group . 



If the number of digits in the multiplier is less than that in 

 the multiplicand, the procedure is the same till all the digits of 

 the multiplier are used in the combination. For each successive 

 figure, the group of digits in the multiplicand to be used shifts 

 along one place to the left (ill it comes to the end. The group 

 then diminishes as before, by dropping the right-hand digit in 

 each line. For example : the groups, the digits of which are 

 multiplied together crosswise, in multiplyingi23456 by 789, are 

 as follows : — 



Digits— 81I1. 7th. 6th. 5th. 4th. 3rd. 2nd. ist. 



I 12 123 234 345 456 56 6 

 7 78 789 789 789 789 89 9 



It will be found, on trial, that this method is quite easy, and 

 can be accomplished by anyone who can add together in his 

 head the products of two digits, and can remember the string 

 of figures which form the answer. This is most easily done by 



