the initiated, they offer them-and in general successfully-to the inexperienced.

It is often said that a man may so lay his wagers about a race as to make sure of gaining money whichever horse wins the race. This is not strictly the case. It is of course possible to make sure of winning if the bettor can only get persons to lay or take the odds he requires to the amount he requires. But this is precisely the problem which would remain insoluble if all bettors were equally experienced.

It is

Suppose, for instance, that there are three horses engaged in a race with equal chances of success. readily shown that the odds are 2 to 1 against each. But if a bettor can get a person to take even betting against the first horse (A), a second person to do the like about the second horse (B), and a third to do the like about the third horse (C), and if all these bets are made to the same amount-say, 10007.-then, inasmuch as only one horse can win, the bettor loses 10007. on that horse (say A), and gains the same sum on each of the two horses B and C. Thus, on the whole, he gains 10007., the sum laid out against each horse.

If the layer of the odds had laid the true odds to the same amount on each horse, he would neither have gained nor lost. Suppose, for instance, that he laid 10007. to 500l. against each horse, and A won; then he would have to pay 1000l. to the backer of A, and to receive 5007. from each of the backers of B and C. In like manner, a person who had backed each horse

to the same extent would neither lose nor gain by the event. Nor would a backer or layer who had wagered different sums necessarily gain or lose by the race; he would gain or lose according to the event. This will at once be seen, on trial.

Let us next take the case of horses with unequal prospects of success-for instance, take the case of the four horses considered above, against which the odds were respectively 3 to 2, 2 to 1, 4 to 1, and 14 to 1. Here, suppose the same sum laid against each, and for convenience let this sum be 847. (because 84 contains the numbers 3, 2, 4, and 14). The layer of the odds wagers 847. to 567. against the leading favourite, 847. to 421. against the second horse, 847. to 217. against the third, and 847. to 67. against the fourth. Whichever horse wins, the layer has to pay 847.; but if the favourite wins, he receives only 421. on one horse, 217. on another, and 67. on the third-that is 697. in all, so that he loses 157.; if the second horse wins, he has to receive 567., 217., and 67.-or 837. in all, so that he loses 17.; if the third horse wins, he receives 567., 427., and 67.—or 1047. in all, and thus gains 207.; and lastly, if the fourth horse wins, he has to receive 567., 427., and 217. or 1197. in all, so that he gains 35l. He clearly risks much less than he has a chance (however small) of gaining. It is also clear that in all such cases the worst event for the layer of the odds is, that the favourite should win. Accordingly, as professional book-makers are nearly always layers of odds, one often finds the success of a favourite spoken of in the

papers as a great blow for the book-makers,' while the success of a rank outsider will be described as

fortune to backers.'

a mis

But there is another circumstance which tends to make the success of a favourite a blow to layers of the odds, and vice versa. In the case we have supposed, the money actually pending about the four horses (that is, the sum of the amounts laid for and against them) was 1407. as respects the favourite, 1267. as respects the second, 1057. as respects the third, and 901. as respects the fourth. But, as a matter of fact, the amounts pending about the favourites bear always a much greater proportion than the above to the amounts pending about outsiders. It is easy to see the effect of this. Suppose, for instance, that instead of the sums 847. to 567., 847. to 427., 847. to 217., and 847. to 67., a book-maker had laid 84007. to 56007., 8407. to 4207., 847. to 217., and 147. to 17., respectively— then it will easily be seen that he would lose 79587. by the success of the favourite; whereas he would gain 47821. by the success of the second horse, 59371. by that of the third, and 60271. by that of the fourth. We have taken this as an extreme case; as a general rule, there is not so great a disparity as has been here assumed between the sums pending on favourites and outsiders.

Finally, it may be asked whether, in the case of horses having unequal chances, it is possible that wagers can be so proportioned (just odds being given and taken) that, as in the former case, a person backing, or

laying against, all the four shall neither gain nor lose. It is so. All that is necessary is, that the sum actually pending about each horse shall be the same. Thus, in the preceding case, if the wagers 97. to 67., 10l. to 5l., 127. to 37., and 147. to 17., are either laid or taken by the same person, he will neither gain nor lose by the event, whatever it may be. And therefore, if unfair odds are laid or taken about all the horses, in such a manner that the amounts pending on the several horses are equal (or nearly so), the unfair bettor must win by the result. Say, for instance, that instead of the above odds, he lays 81. to 6l., 91. to 57., 117. to 37., and 137. to 17. against the four horses respectively; it will be found that he must win 17. Or if he takes the odds 187. to 11., 207. to 97., 241. to 5l., and 287. to 17. (the just odds being 187. to 127., 207. to 107., 247. to 67., and 287. to 27. respectively), he will win 17. by the race. So that, by giving or taking such odds to a sufficiently great amount, a bettor would be certain of pocketing a large sum, whatever the event of a given race might be.

In every instance, a man who bets on a race must risk his money, unless he can succeed in taking unfair advantages over those with whom he bets. Our readers will conceive how small must be the chance that an unpractised bettor will gain anything but dearly-bought experience by speculating on horse-races. We would recommend those who are tempted to hold another opinion to follow the plan suggested by Thackeray in a similar case-to take a good look at professional and

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practised betting-men, and to decide which of those men they are most likely to get the better of' in turf transactions.

(From Chambers's Journal, July 1869.)

SQUARING THE CIRCLE.

THERE must be a singular charm about insoluble problems, since there are never wanting persons who are willing to attack them. We doubt not that at this moment there are persons who are devoting their energies to Squaring the Circle, in the full belief that important advantages would accrue to science—and possibly a considerable pecuniary profit to themselves -if they could succeed in solving it. Quite recently, applications have been made to the Paris Academy of Sciences, to ascertain what was the Iamount which that body was authorised to pay over to any one who should square the circle. So seriously, indeed, was the secretary annoyed by applications of this sort, that it was found necessary to announce in the daily journals that not only was the Academy not authorised to pay any sum at all, but that it had determined never to give the least attention to those who fancied they had mastered the famous problem.

It is a singular circumstance that people have even attacked the problem without knowing exactly what its nature is. One ingenious workman, to whom the