5^8 



NA TUKE 



[October 21, 1897 



LETTERS TO THE EDITOR. 



The Editor does not hold himself responsible for opimons ex- 

 pressed by his correspondents Neither can he undertake 

 to return, or to correspond with t/ie writers of, rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications. '\ 



On the Meaning of Symbols in Applied Algebra. 



On reading the correspondence in Nature (vol. Iv.) on this 

 subject, my sympathies were with the physicists as typified by 

 the Professors Lodge ; but I think that the mathematicians as 

 typified by Messrs. Jackson and Gumming have a legitimate 

 grievance. 



The following statement is " abhorrent " to the mathe- 

 maticians. The horizontal intensity of the earth's magnetic 

 field at a certain point is 



\/ '00220 lb. 



= 3-i: 



(min. /60) v** -0328 ft. 



Vib: 



min. vft. 



The physicist attaches a definite-enough meaning to this state- 

 ment, and to the result of this little piece of generalised 

 arithmetic. This is that if an observer go through the well- 

 known process of finding li with two different sets of instruments : 

 (l) a balance with gramme weights, a clock counting in seconds. 

 and a scale divided to centimetres ; and (2) a balance with lb. 

 weights, a clock counting in minutes, and a scale divided to 

 feet ; then if his results on reduction give II = '200 in the first 

 case, they will give H = 3'ii in the second. 



The mathematician will not for a moment dispute this result, 

 and he will not deny that precisely similar processes will always 

 give correct results. But he is, nevertheless, inclined to take up 

 the position that no meaning can be assigned to the combination 

 (gm.)i sec."^ (cm.)"i And his legitimate grievance is that 

 nobody has placed these convenient processes on a general 

 logical basis. (I believe this last is a fact.) 



There is nothing illogical or mathematically immoral in the 

 following simple assertions. In ordinary algebra there is no 

 meaning attached to a length x another length, or to a length 

 -^ a time. We may, therefore, assert that a length x another 

 length shall mean a certain area, viz. that of a rectangle, two 

 of whose adjacent sides are the lengths ; and a length -^ a time 

 shall mean the velocity of a body which covers the length in 

 the time. We are at perfect liberty to make these definitions, 

 even if it should turn out that the ordinary laws of algebra will 

 not hold for the new kind of multiplication and division. But 

 if, as it turns out is the case, those laws should hold, we have 

 extended the meaning of algebraic results, which is a great 

 gain ; and we have provided ourselves with a new physical 

 instrument of thought, which is a greater gain. 



How to put all such mathematical processes, which the 

 physicist is constantly employing, on one general logical basis? 

 The following definitions hint a sketch of one way of pro- 

 ceeding. 



In the definitions "number" will be taken to mean any 

 real algebraic quantity — positive or negative, rational or, 

 irrational. 



The algebraic definition of variation is applicable equally to 

 numbers and to physical quantities. Let A and B be either 

 two numbers or two physical quantities, possibly of different 

 kinds. The ordinary algebraic definition of variation may be 

 expressed thus : — A cc B if A depends on B in such a way that 

 when B is multiplied by any number, A is multiplied by the 

 same number. For instance, if the base of a triangle be given 

 the area oc the altitude. The first of the following definitions 

 includes the above as a particular case. 



Definition i. A ^ B'^ if A depends on B in such a way that 

 when B is multiplied by any number x, A is multiplied by x^. 

 For instance, , 



Edge of cube oc (volume) ; 



and in a race over a given course 



Runner's average velocity a (his time)-^. 



Definition 2. // -V is determined by. and depends in a specified 

 manner on, the independent physical quantities (or numbers) 

 A, B, C, . . ., in such a way that X oz A" when B, C . . . are 

 constant, and X cc B'' when A, C, . . . are constant, <ifc., then 



X = A« B» C . . . 



The words "in a specified manner" are important. For in- 

 stance, an area can be made to depend on two independent 

 lengths in an infinite variety of ways. The specified manner 

 juight be as follows : — The area X is the area of a triangle 

 whose base is A and altitude B. Then according to the de- 

 finition X = A B. But this is not the conventional specification. 

 For that of course we must read " rectangle" for " triangle." 

 Again an acceleration X may be made to depend in the way 

 described in the definition on an independent length A and 

 time B as follows : — X is the acceleration with which a bofly 

 must move from rest to describe the length A in the time B. 

 According to the definition we should then have X = A B"-. 

 But this again is not the conventional specification. For the 

 latter we must read " X is half the acceleration," for "X is the 

 acceleration." 



With these definitions it is not hard to show (i) that all the 

 laws of ordinary algebra which have any meaning under the 

 new circumstances are true, and (2) that all such laws are true 

 generalisations of the ordinary laws in that the latter are 

 particular cases. Alex. McAulay. 



University of Tasmania, Hobart, June 19. 



NO. 1460, VOL. 56] 



Dog Running on Two Legs. 



The following instance shows how easily and well a fonr- 

 legged animal can adapt itself to run on two legs only. 



Last July a beautiful black and white shepherd's dog, on the 

 Downs farm, near here, was caught amongst the knives of a 

 reaping machine. Both the legs on the dog's right side were 

 dreadfully mangled, and rhe animal almost bled to death. The 

 right hind leg was so torn that one long piece and several small 

 pieces of bone dropped from the wound. The dog lay for some 

 time senseless and practically bloodless and lifeless. The kind- 

 hearted shepherd, however, to whom the dog belonged, would 

 not allow it to be at once destroyed ; he bound up its terrible 

 wounds, put it carefully in a wheelbarrow, wheeled it home^ 

 and nursed it. After two or three weeks the animal had so fir 

 recovered as to be able to crawl and move about on its two left 

 legs with a little assistance from its crushed right fore-leg. 



This dog now lives with the shepherd at Dunstable, and runs, 

 backwards and forwards to Downs farm — a mile off — every day. 

 The greater part of the journey is performed on the two legs of 

 its left side, as the dog can do nothing whatever with its right 

 hind-leg, and the right fore-leg is so damaged as to be only 

 useful as a slight occasional prop. In starting to run, the dog 

 quickly gets up, jerks his ruined right fore-leg over the left leg, 

 balances itself on its two left legs only, and very rapidly hops 

 off in the style of a large agile bird, both right legs hanging 

 useless. With this strange mode of rapid progression it now 

 attends to sheep exactly in the way of an ordinary uninjured 

 dog. It is a most affectionate animal, and is now apparently 

 full of life and health. When I went to see it this morning, it 

 sprang up and happily bounded to me balanced on its two lefs 



legs. ■ WORTHINGTON G. SmITH. 



Dunstable. 



Foraminifera in the Upper Cambrian of the Malverns. 



In the early part of this year, whilst engaged in researches in. 

 the Spluerophthalmus zone of the Upper Lingula Flag Series,. 

 Prof Theodore Groom, of Cirencester, found a shaley limestone 

 which, when examined superficially under a fairly high power, 

 showed indications of Foraminifera. Dr. Groom had a thin, 

 section prepared from this rock, and detected in it undoubted, 

 remains of Foraminifera. This preparation, together with 

 specimens of the rock, he has courteously placed in my bands for 

 further investigation, the results of which will be embodied in 

 an appendix to Dr. Groom's paper on these beds. 



The Foraminifera, for the most part, belong to the genus 

 Spirillina, which has hitherto never been found below Jurassic 

 strata, and these organisms make up at least 20 per cent, of the 

 bulk of some specipiens of the rock. The other genera present 

 appear to be Lagena, Nodosaria {Dentalina), Margintilitia, and, 

 Cristellaria. 



