46 



• KNOWLEDGE 



[June 16, 1882. 



fxporiinoiual. We have now Hxcd on a plan wo hope 1 

 Trithout any intorrnptioa for many years. 



ELECTRICAL ANSWERS. 

 W. N. M. 1. I^jt a littlp loss bicliromato, say 2 oii. instead of 

 tj oi. Tho crystallisation is probably duo to super-saturation ; 1 

 presume you made your solution in boiling water ? 2. Tho use of 

 porous pots would inorcasc the internal resistance of tho coll, but 

 this is f.ir more than compensated for by increasing tho constancy. 

 A simple bichromate cell runs down in a few minutes, while one 

 with a porous pot (especially if a little mercury, about 3 oz., is put 

 in the pot containing the zinc) will give a continuous current for 

 sovoral weeks — as long in fact as the constituents of the cell last. 

 3. Tho same current would bo obtained from two zincs and one 

 carbon, as from two carbons and one zinc, but in tho former case 

 there would be a larger, and therefore wasteful, consumption of 

 sine. — George Pesdbill. "Resistance in feet per ohm" is on 

 extremely awkward and clumsy expression. It is intended to 

 mean the number of feet of wire which offer a resistance of 

 one ohm. Thus, in the case cited, a wire of 0,826 feet to the 

 pound, having a resistance in feet per ohm of 3 3, means 

 that 33 feet (about ono mi^tre) of wire offer a resistance of 

 one ohm. It is far better, more logical, and therefore scientific to 

 ■peak of " Resistance in ohms per foot," or whatever length wo 

 may choose as a standard. In telegraphy the standard length is 

 one mile, so that we should say a mile of copper wire, No. 16 b.w.g. 

 •ffere a resistance of 25 ohms. 



<0iir iHatDrmatical Column. 



THE LAWS OP PROBABILITY. 



A FEW cx.imp'es of the application of this rule to questions 

 about hypotheses will serve to illustrate its real simplicity ; 

 for, as is the case with nearly all rules, the verbiage necessary to 

 remove ambiguity has introduced complexity. 



Ex. 1. — A teetntum has four faces, and it '« known that these are 

 either numbered I, 2, 3, i, or 1, 1, 2, 2, or 1, 1, 1, 1 ; but the chances 

 of these several arrangements are, respectively, as 3, 2, and 1; the 

 teetotum is spun tince, and each time 1 is the number turned up. 

 What is now the probability of the several arrangements ? 



If the first arrangement exists, the chance of the observed event 

 is - X 7. o"" ,-^i 'f "'<' second is the actual arrangement, the chance 

 of the observed event is - x -, or - ; and, lastly, if tlio third i.s the 



arrangement, the observed event is a certainty, or 1. Hence our 

 formula informs ns that after the event, the chance that tho teetotum 

 is marked 1, 2, 3, 4, is 



1 



1 1 



3x_ + 2x- + lxl 

 16 -t 



the chance that the teetotum is marked 1, 1, 2, 2, is 2 x 1 over tho 



same denominator ; and the chance that the teetotum is marked 

 1, 1, 1, 1, is 1 y 1 over the same denominator. The denominator is 



That arrangement which wag antecedently the most likely Ih the 

 moat unlikely of all after the observed event ; and the arrangement 

 which was antecedently most unlikely is most likely of all after the 

 observed event. 



It is important to notice how the antecedent probabilities, or the 

 d priori probahilitlcR, as they are called, are modified after tlio 

 observed event in such instances. For example, we may regard 

 the spinning of the teetotum in this case as an observation or 

 experiment, and the illustration shows ns how theories antece- 

 dently more probable may become less likely as observation is ex- 

 tended. The jnst appreciation of this fact is the essence of sound 

 theorising, or, rather, of all science. 



I give next two examples tending to illustrate this point. 



Ex. 2.— .Suf^'»c the antecedent probability of the theory that the 

 earth on vchich we live it at rest and the centre of planetary motions, 



to be a million times greater than the probability that the earth is a 

 planet circling round the sun. Then on the former theo^-y, although 

 a planet travelling round the earth might have a path so looped that 

 tho planet would appear to follow looped paths in the heavens, yet 

 tho chance of this occurriitg might fairly ba regarded as small. But 



set it at -, or an even chance. On the second theory, the jilancts 



would be certain to travel on looped paths, or the probability of their 

 so doing would be represented by unity. .Yoie, there arc 120 plane's, 

 all of which travel on loopid paths. Supposing no other facts known, 

 what is the prohability that tho earth is at rest in the centre of the 

 planetary system ? 



Here we liave two hypotlu 



I ho chances of which are respcc- 

 lie chance of the observed event on 



the former hypothesis 



'•(^r 



our formula gives as tho probability tl 

 centre of the planetary system— 



ioiioooo 



and on the latter i; 

 earth 



ity. Uenco 

 t rest at tho 



^^ 



\2/ looeooi 



which reduces to ^"""^"-" -. 



1000000 + 2 '•» 

 Now, the logarithm of 2 is 0-3010300, and multiplying this expres- 

 sion by 120, wo get 36'1236000; so that 2 "° is a number containing 

 37 digits, the first seven of which are 1328230 (because tho loga- 

 rithm of 1-328230 is 01236001). Putting this value in tho abovo 

 expression, and dividing numerator and denominator by lOUOOOO, 

 we obtain for the probability that tho earth is tho centre of tho 

 planetary system — 



l,32H,23O,000,OUO,00O,OUO,OOO,0OO,OOO,0UO 

 I may bo permitted to select for my next example, a theory of 

 my own. 



" Ex. 3. — Let it be regarded as antecedently a billion times more 

 likely that the stars of our galaxy are spread with a certain general 

 uniformity throughout the space around the sun, than that they are 

 gathered into definite aggregations so marked in their character as to 

 be recognisable in the stutistical distribution of tlie stars. On the 

 latter hypothesis, it may be regarded as an even chance that on 

 comparing the number of stars visible to the naked eye in the northern 

 and southern hemispheres, we should find as great a diaparity as is 

 represented by such a proportion as 7 to 5. But set the odds as 10 

 to 1 against such a result. Noiv it is observid that, as a matter of 

 fact, out of 0,000 stars visible to the naked tye, 3,500 are in the 

 southern hemisphere. Required the probabdity that stars are spread 

 v:ith a certain general uniformity throughout space. 



In a paper which I communicated a few years ago to the Royal 

 Astronomical Society, I showed that if the stars are spread in thia 

 way, tho chance of tlio observed arrangement is 



(;tvJ«6769(T,ooo;odii,boo,ouo,u'uo 



f)n tho hypothesis that there aro definite stellar aggregations, 

 tho chanco of the observed arrangement is 



n ^^^ 



But tho antecedent probability of tho former supposition is set 

 by our question at 



1.000,000, 000,000 /p^ 



l,Oi)O,WU,0U0,(JOr 

 that of the latter at only 



-- ^ (D) 



1,000,000,000,001 



Ilonco tho probability of tho fonner hypothesis in presence of tho 

 observed fact is obtained by multiplying (A) by (C) for a, nume- 

 rator, and adding that product to (C) x (U) for a denominator. It 

 is (very approximately), 



1 



(A) 



nit,'.m>,(m),(m) 



reduces 



00,906,090,011 "' 6,090,5.';3,636 



that is, the odds aro more than six thousand millions to one against 

 tho generally-accepted hypothe 



I do not think I have treated 



