500 



NATURE 



[September 22, 1898 



which give " heads " will occur about as often as the conditions 

 which give ' ' tails. " 



If you examine any event which occurs by chance, you will 

 find that the fortuitous character of its occurrence always 

 depends upon our ignorance concerning it. 



If we know so little about a group of events that we cannot 

 predict the result of a single observation, although we can pre- 

 dict the result of a long series of observations, we say that 

 these events occur by chance. And this statement seems to 

 me to contain the best definition of chance that can be offered. 



If we used the word chance in this sense, we see at once that 

 our knowledge of animal variations is precisely knowledge of 

 the kind referred to in our definition of chance. We know with 

 some certainty the average characters of many species of animals ; 

 but we do not know exactly the character of the next individual 

 of these species we may happen to look at. So that in the 

 present state of our knowledge it is h />n'ort certain that the great 

 majority of animal variations should occur by chance, in the 

 sense in which we have used the phrase ; and I will show you 

 in a moment illustrations of the fact that they do so occur. 



But before doing so, I would point out the difterence between 

 the sense in which we have used the word chance, and the sense 

 in which it is used by many objectors to the theory of Natural 

 Selection. Such epithets as d/md, lawless, and the like, are 

 constantly applied to chance ; and a kind of antithesis is estab- 

 lished between events which happen by chance, and those 

 which happen in obedience to natural laws. In many Germaii 

 writings, especially, this antithesis between Zzifalligkeit and 

 Gesetzmiissigkeit is strongly insisted upon, whenever organic 

 variation is discussed. 



This view of chance is not supported by experience ; and in- 

 deed, if it could be shown that any thing in human experience 

 were absolutely lawless, if it could be shown that in any depart- 

 ment of nature similar conditions did not produce similar effects, 

 the whole fabric of human knowledge would crumble into chaos, 

 and all intellectual effort would be a profitless waste of time. 

 There is not the slightest reason to believe that any such abso- 

 lutely lawless phenomena do exist in nature ; so that we need 

 pay no further attention to the writers who assume that chance 

 is a lawless thing. 



, But if chance is a perfectly orderly and regular phenomenon, 

 then the question, whether animal variations occur by chance or 

 not, can be settled by direct observation. I ^will now show you 

 one or two examples of events which undoubtedly occur by 

 chance, and then compare these with one or two cases of organic 

 variation. 



As events which occur by chance, I have taken- the results 

 of tossing twelve dice. My wife has spent some time during, the 

 last two months in tossing dice for you, and I will ask you to 

 look at the results. 



Her first record gives the number of dice showing more than 

 three points in each of 4096 throws of twelve dice. There are, 

 of course, six numbers on each of the dice ; so that if all the 

 dice were perfectly symmetrical and similar, the average number 

 of dice with more than three points should be six in each throw 

 of twelve. But dice are not symmetrical and similar. The 

 points on the dice used were marked by little holes, scooped out 

 of their faces ; and the face with six such holes scooped out of 

 it was opposite to the face with only one such hole : so that the 

 face with one point was heavier than the face with six points ; 

 and therefore six was rather more likely to be uppermost than 

 one. In the same way, two was opposite five ; so that the five 

 face was a little more likely to fall uppermost than the face with 

 two points. Therefore, it is a little more likely that you will 

 throw four, five, or six, in throwing dice, than it is that you will 

 throw one, two, or three. 



Accordingly, the average number of dice, in these 4096 

 throws, which had more than three points, was not six, but 

 6-I35. 



To show you that this excess of high points was due to some 

 permanent property of the dice, she threw these twelve dice 

 another 4096 times ; and the average number of dice with more 

 than three points was 6*I39, A third series of trials gave an 

 average of 6"I04, and a fourth gave an average of 6 •116, 



You see that the difference between the highest and the lowest 

 of these determinations is only about one-half per cent. , so that 

 the mean result of such a series of fortuitous events can be 

 determined with great accuracy. 



And just as the mean of the whole series can be determined, 

 so we can know with considerable accuracy how often any 



NO. 1508, VOL. 58] 



possible deviation from the average result will occur. The degree 

 of accuracy with which we can know this may be judged froni 

 Table I. 



Table I. — Frequency with which Dice showing more thanthrcc 

 Points were thrown in each of Four Series of Trials, the 

 number of throws in each Series heijig 2^- = 4096. 



You see that the results of the experiments agree fairly well' 

 with one another, and differ from the results most probable with 

 symmetrical dice, in the way which the structure of the actual' 

 dice would lead one to expect. Throws which give seven, 

 eight, or nine dice with more than three points occur too often, 

 throws in which only two, three, or four dice have more than 

 three points do not occur often enough. You see then that each 

 of these results is orderly and regular, and that the four results 

 agree very fairly among themselves, not only in the mean value 

 of each of them, but in the magnitude and frequency of de- 

 partures from the mean. That they differ from the results which 

 would probably be obtained with symmetrical and similar dice 

 is only to be expected, because the dice used are neither sym- 

 metrical nor- similar. 



You notice that this tabje is very nearly symmetrical ; the 

 most frequent result is that which lies in the middle of the series 

 of possible results ; and the other frequencies would, with perfect 

 dice, be distributed symmetrically on each side of it ; so that 

 with perfect dice one would be as likely to throw five dice out of 

 twelve with more than three points as one would be to throw 

 seven, and so on. 



This symmetry in the distribution of the results is only found 

 when the chance of the event occurring in one trial is even. 

 The next table shows the result of 4096 throws of twelve dice, 



Table II. — Frequency of Sixes in 4096 throws of Twelve Dice. 



in which sixes only were counted. The chance against throwing 

 six with any one of the dice is of course five to one ; so that in 

 throwing twelve dice you are more likely to throw two sixes 

 than to throw any other number. But you see that the chance of 

 throwing only one six is very much greater than the chance of 

 throwing three ; the chance of throwing none is greater than 

 the chance of throwing four, and while there is a chance of 

 throwing five, six, or more, of course it is impossible to throw 

 less than none at all ; so that the diagram is all askew. You 

 see that this time, as before, the freqjjency with which any 



