February 15, 1894J 



NATURE 



365 



however, that when the origin in glacial time of the grand 

 Norwegian fjords is sufficiently proved, their origin by glacial 

 forces will be more easily granted. The same may certainly be 

 said of the far smaller lake basins in Norway, for which an 

 analogous demonstration can be given. That the fjords now 

 must really be of pleistocene origin is the point I wish to make 

 in this letter. Only if anyone can, in a simple manner, explain 

 how an inland ice could be able to pass the close set row of fjord 

 heads, is it possible to dismiss my argument. 



Andr. M. Hansen, 

 University Library, Kristiania, January 29. 



A FEW words are due from me in reply to the kindly criticisms 

 of my suggestion regarding the erosion of rock basins that have 

 appeared in Nature since its publication on November 9, 

 1893. 



In the first place, I must apologise to Sir H. Howorth for 

 having misunderstood his remarks on the plasticity of ice in 

 his letter of July 13, a misunderstanding due, of course, to my 

 not having had an opportunity of reading the chapter devoted 

 to the subject in his book. Unfortunately the libraries of our 

 small outlying stations in India do not as a rule provide us with 

 works of scientific interest, and the conditions of life of most 

 of us wlio take an interest in such subjects out here force us to 

 content ourselves with the possession of very few books of the 

 kind, and only those that are absolutely necessary for our 

 work. Provided that it is admitted that the plasticity of 

 glacier ice is sufficient to allow motion in the upper layers of a 

 glacier, even v/hen it rests on a nearly level surface, it does not 

 matter, so far as my hypothesis is concerned, whether the 

 bottom layers move or not, for a movement of the upper layers 

 alone is required to enable the " moulins " to transfer their action 

 from place to place, and in time to exert their force on every 

 part of the rock surface beneath that portion of the glacier. 



That the action of the " moulins" is not so restricted as would 

 appear from Prof. Bonney's letter in Nature of November 16, 

 1893, can, I think, hardly be doubtedbyany one who has traversed 

 a Himalayan glacier of the kind I have described, on a hot 

 summer's day. Hundreds of ihem may be seen in action in 

 every direction, and, given sufficient time, their aggregate 

 effect in wearing down the rock surface must be very large. I 

 have noticed the dry shafts mentioned by Prof. Bonney in front 

 of an active " moulin," but do not see why they should not be 

 accounted for by the opening of a new crevasse, without having 

 to suppose that the new crevasse was in the same position as the 

 old one. The crevasses to which I refer are mostly very narrow, 

 easily stepped across in many cases, and do not appear to ex- 

 tend far down into the glacier, so that they are probably due to 

 some other cause than an unevenness of the rocky floor, which 

 would cause them to form in succession at the same point, and 

 their number would give the " moulins " plenty of opportunity 

 to attack the whole surface in course of time. Besides, the 

 wearing away of any inequality that did exist, would surely cause 

 the crevasse to open at some other point, if it were due to that 

 cause, and the " moulin " would thus be enabled to shift its 

 point of attack. The very rarity, too, of such collections of 

 " giant's kettles " as that at Lucerne would seem to show that it is 

 seldom that the " moulins " keep working at one point for any 

 length of time. I did not mean to suggest, of course, that any lake 

 basin had been due to the action of one " moulin" ; the hollow 

 ultimately produced need not bear any relation in form to the 

 individual "giant's kettles" that gave rise to it ; indeed, there 

 is no necessity that a real " giant's kettle " should be formed at 

 any one point. Just as in the case of a drill moved over the 

 surface of a piece of wood, the pattern ultimately produced need 

 bear no relation to the form of the drill. 



If we except the doubtful action of the ice[itself, I do not know 

 of any agent that will produce a rock-enclosed hollow in the 

 course ot a river channel, but falling water, aided by boulders and 

 sediment. Such a hollow may be seen at the foot of any water- 

 fall, even of moderate height. 



In calling attention to the rarity of true rock basins in the 

 Himalayas, an expression that Mr. Oldham takes exception to, 

 I should have said lake basins, that is, lakes lying in true rock 

 basins. As I pointed out, any hollows that may have been 

 formed beneath a pre-existing glacier have been filled with 

 debris, but it is very likely that such hollows do occur beneath 

 the extensive flats found at the foot of the larger glaciers, as in 



NO. 1268, VOL. 49] 



the case of the one shown in the view given in my paper. Of 

 course, where such hollows occur in positions where it is im- 

 possible that glaciers ever existed, as in eastern Baluchistan, 

 they must be accounted for in other ways. My suggestions were 

 not intended to account for all rock basins, but merely to apply to 

 those which occur in now or formerly highly glaciated regions, 

 where it seems possible that there is an intimate connection be- 

 tween the excavation of the basins and the existence of glaciers. 

 Sukkur, January 10. T. D. LaTouche. 



A Plausible Paradox in Chances. 



It seems worth while to record the following pretty statis- 

 tical paradox as a good example of thepitfalls into which persons 

 are apt to fall, who attempt short cuts in the solution of 

 problems of chance instead of adhering to the true and narrow 

 road. It is true that the paradox would excite immediate 

 suspicion in the mind of any one accustomed to such pro- 

 blems, but I doubt if there are many who, without recourse 

 to paper and pen, could distinctly specify off-hand where the 

 fallacy lies. It will be easy for the reader to make the 

 experiment of his own competence to do so after reading to the 

 end of the second of the two following paragraphs. 



The question concerns the chance of three coins turning up 

 alike, that is, all heads or else all tails. The straightforward 

 solution is simple enough ; namely, that there are 2 different 

 and equally probable ways in which a single coin may turn up ; 

 there are 4 in which two coins may turn up, and 8 ways in which 

 three coins may do so. Of these 8 ways, one is all-heads and 

 another all-tails, therefore the chance of being all-alike is 2 to 8 

 or I to 4. 



Against this conclusion I lately heard it urged, in perfect good 

 faith, that as at least two of the coins must turn up alike, and 

 as it is an even chance whether a third coin is heads or tails \ 

 therefore the chance of being all-alike is as i to 2, and not as 

 I to 4. Where does the fallacy lie? 



It lies in omitting one link in the chain of the argument as 

 being unimportant, whereas it is vital. This omitted link i5 

 distinguished by brackets and is numbered (3) below. The 

 reasoning then stands : — 



(i) At least two of the coins must turn up alike, 



(2) It is an even chance whether a third coin is heads or 

 tails. 



[(3) Therefore, it is an even chance whether the third coin is 

 heads or tails. (Here is the error).] 



The true state of the case is seen by writing out the eight 

 several events^ as in the table below. 



No. 2 in the argument is justified by the total number of 

 the /i's in the third column being equal to that of the fs, while 

 No. 3 is obviously not justified. In the particular 8 events 

 with which we are concerned, an h h is associated with a 

 t three times as often as with an //, and a // is associated with an 

 h three times as often as with a /. Hence as the combination 

 hh h is one-third as frequent as that of any 2 h'% and i t, and as 

 tttxi, one-third as frequent as any combination of 2 /'s and I h, 

 and, lastly, as the two classes of combinations are equally 

 frequent, it follows that the frequency of the all-alike cases is to 

 that of the remainder as i to 3, or to that of the total cases as 

 I to 4, which is the result first arrived at. 



I amused myself with testing the theoretical conclusion by 

 making 120 throws of dice, 3 dice in each throw; the odd 



