Feb. 24, 1882.] 



• KNOWLEDGE * 



353 



tracted by tremendous oscillations, has broken off a mighty 

 fraijmcnt. That fragment formed the moon. 



The date of this occurrence (or, to speak more precisely, 

 the date when we find the moon to have been placed as if 

 this occurrence had happened) we cannot tell. It is certain 

 tliat it must have been more than tifty million years ago — it 

 is probably very much more. The subsequent history of 

 he moon can be traced with comparative certainty. It 

 appears that the critical condition in which the moon was, 

 close to the earth and rapidly rotating around the earth in 

 a period equal to the day, could not last. The case is one 

 (if unstable equilibrium ; either the moon must fall back 

 ai,'ain into the e^rth, or else it must begin to move out- 

 wards from the earth. The fact that the moon exists 

 sliows that the latter alternative was adopted, though it 

 does not seem quite clear why that course rather than the 

 <ither should have been chosen. As the moon receded, 

 the duration of the month increased, its duration at any 

 distance being determined l)y Kepler's laws. The month 

 has increased steaddy from its primitive value of three 

 liiiurs, up to the present time, when the month is over 27 

 days. 



This alteration in the length of the month has entailed 

 a corresponding alteration in the length of the day. As 

 the distance of the moon increased, so the length of the day 

 increased from the primitive three hours up to the present 

 24 hours. The ratio between the day and the month has, 

 however, altered in a manner which must receive careful 

 attention, as it involves consequences of the very deepest 

 interest. In the primitive state of things, the day and the 

 month were equal ; but when they both began to lengthen, 

 the month increased much more rapidly than the day. Of 

 course, it will be understood that we are here speaking of 

 the changes in the ratio of the length of the month to the 

 length of the day at the same epoch. The month gradually 

 became twice the day, it became three times the day, and 

 the ratio gradually increased until the time came when the 

 month was twenty-nine times the day. This time has but 

 lately passed, the ratio of the month to the day was then 

 at its maximum, and the decline has now commenced. 

 After the month was twenty-nine times the day, the ratio 

 gradually sank until the length of the month was twenty- 

 seven times that of the day. This is an epoch of the most 

 special interest — it is the present time. 



The tides have thus guided us in tracing the earth-moon 

 history from the beginning, when the moon was lirst cast 

 off, down to our own days. Nor will the tides now 

 desert us — they will enable us to make a forecast of the 

 distant future. The day will continue to lengthen, the 

 moon will continue to recede, the month will get longer 

 (measured by hours), but the day will lengthen more rapidly 

 than the month. Instead of the month being 27 days, it 

 will in time to come be only 26 days, only 2.5 days, and at 

 some enormously distant epoch the final state of things will 

 have been reached, and the day and the month will be 

 again equal. The first stage of this history and the last 

 stage are in one sense identical. In each case, the day is 

 equal to the month. In the first case, the day and the 

 month are each three hours ; in the last case, the day and the 

 month will each have lengthened to the enormous extent of 

 1,400 hours. The 1,400 hours is no doubt more or less 

 doubtful, but we are assured by the laws of dynamics that 

 there is some magnitude of that kind to which both day 

 and month are tending, and to which they will both 

 ultimately become equal. In another way, also, the first 

 stage of the earth-moon history and the final stage may be 

 compared. The earth turned the same face constantly 

 towards the moon at the beginning. The earth will turn 

 the same face constantly towards the moon at the end. 



FALLACIES ABOUT LUCK. 



By the Editor. 



AS to fallacies alx)ut luck, the supposition that after a 

 great number of heads in tifty tossings, the next fifty 

 would probably show a smaller number, involves precisely 

 thf; same error (diluted by being spread over a larger space, 

 but not diminislied in amount) tliat I dealt with in my 

 former paper. How can the n\imber of heads in one set of 

 fifty tossings affect the number which shall appear in the 

 next I Science says on d, priori grounds, " not at all " ; 

 Experiment repeats as emphatically (it could not say so 

 more emphatically) "not at all." But then, says the 

 querist, how is it that, as science assures us, there is always 

 in the long run an approach to equality in the nuuiber of 

 heads and tails tossed in a great number of trials '! If the 

 balance always tends to the horizontal position, surely a 

 movement of one scale upwards should assnii^ us that 

 presently the other scale will begin to rise. E(iuality is 

 indeed brought about in the long run, but not in the way 

 imagined. Absolutely not the slightest inttuence is pro- 

 duced on the results of one set of, say, a huudi'ed tossings, 

 by the observed results of the next preceding set: (how 

 could there be )). Nor is there any tendency in a very long 

 series of tossings, starting from some particular point, to 

 reduce a discrepancy between heads and tails, which had 

 attained any amount uj) to that point. On the contrary, 

 if wo count from and after that point, as well as if wc 

 count from and after the absolute beginning, we shall find 

 the same tendency to equality in the results of a great 

 number of tossings. The excess of heads over tails, or of 

 tails over heads, may go on increasing, and yet tliere is the 

 tendency to equality which science indicates. This sounds 

 paradoxical, but it is what science teaches and what ex- 

 pierience confirms. It is demonstrable that the greater the 

 niunber of trials of coin tossing, the nearer will the ratio of 

 heads to tails approach to equality, though the actual 

 excess of one over the other may probably be greater, and 

 possibly much greater, than in a smaller number of 

 trials. 



Take a very simple case. Suppose a coin tossed four 

 times, and consider the chance that there will be either two 

 more heads than tails, or two more tails than heads. Tliere 

 are in all 2', or 16 possible events. That there may be two 

 more heads than tails, three heads must be tossed, which 

 can happen manifestly in four difierent ways, for the first, 

 second, third, or fourth toss may give the single tail. So, 

 also, there may be two more heads than tails in four 

 different ways. There are therefore 8 ways (out of IG) in 

 which either heads or tails may show three times as against 

 one of the other kind. The chance is therefore I, or it 

 is an even chance, that there will be this degree of dis- 

 crepancy. On the other hand, there are only 6 ways in 

 which there can be 2 heads and 2 tails, for only 6 pairs 

 can be made out of 4 (the first tossing may be head, as also 

 second, third, or fourth ; the second may be head, as also 

 third or fourth ; the third may be head, as also the fourth ; 

 and these arrangements of 2 heads give also all the 

 arrangements of two tails). Thus the chance of absolute 

 equality is only 6-16ths, or 3-8ths, that is, the odds are 

 .5 to 8 against absolute equality, while the cliance that there 

 will be a difl'erence of 2 exactly between the heads and the 

 tails is },. (The chance that all 4 will be of the same kind 

 is, of course, l-8th. ) 



Now compare with this the results we get when, instead 

 of 4. there are 8, tossings. Here there are 2*, or 256 

 possible events, and it can readily be shown (but I leave 

 this and the general problem to a series of papers which I 

 shall hereafter write on probabilities) that the chances of 



