vice. The fact is, that we are so used to taking the average in simple cases that we look upon it as self-evident. If two men, after having measured the length of a wall, give us respectively fifty-one and fifty-four feet, we should settle the discordance by taking the mean, namely, fifty-two and one half. And many would clinch this by an à priori argument that it must be so, for, having no possible reason to prefer one result to the other, it would be irrational to do any thing else than put them on precisely equal terms by just halving their difference. But, as regards this argument, compare the following case: Two sailors. from a ship are sent to steal into a fort and take the calibre of a cannon. We may suppose that the fort is going to be taken and used against the enemy, and that it is therefore of importance to provide shot of the right calibre. One sailor says he made the calibre eight inches, the other made it nine. What should we say of the wisdom of the captain who should go there provided with shot of eight and one half inches, there being, let us say, no such calibre in the service? It would be as if a man who was uncertain whether or not he had been asked to stop to dinner somewhere were to meet the case half-way by going in dress trousers and waistcoat, but in a flannel shirt and check coat. In the case of the cannon, it would be better far to toss up than to try to "combine" the observations.'

1 A rather curious case of this kind has recently fallen under our notice. It is an attempt to determine the length of certain ancient measures, such as the cubit, simply from the examination and measurement of ancient buildings (we ought to premise that we only know the work in question at second-hand, not having had an opportunity of consulting it). At first sight, the attempt may seem absurd; for why, it may be asked, should any part of a building be of one particular length rather than of another? The only assumption with which we start is that fractions of cubits are less likely to present themselves than whole ones, and irregular fractions less likely than such simple ones as halves and quarters. In modern buildings, of course, the exigencies of space in a crowded city are paramount; but in more roomy times it seems plausible enough that builders would have preferred unbroken units to work with, when possible. Starting with this assumption, the problem becomes feasible. We collect a large quantity of measures of lengths and heights, etc., of these ancient buildings; we examine their differences and ratios, and ascertain whether these show a tendency to yield frequent multiples of some particular magnitude. If they do, this magnitude is presumably the lost ancient measure, or some fraction of it. Of course, all this may be the mere result of coincidence; whether it is so or not must be decided on principles of probability, by seeing how many independent magnitudes have been examined for the purpose.

It is obvious, therefore, that one tacit condition always underlies the doctrine of taking an average-viz.: that every intermediate value is not indeed equally likely, but, at any rate, equally possible. When this is worked out, we shall soon see that a reason can be given for the advice, which depends partly upon observation and partly upon mathematical deduction. Let the reader recur to what was said, some pages back, about the way in which the measurements of objects (in common with many other things) cluster about their mean or central point. The central values are the commonest indeed, but all are possible, and will, in the long-run, get duly represented. Now, suppose that three of these measures are set before us, and suppose also (what is always the case in practice) that we do not know where the ultimate mean is (in fact, so far from knowing it, the position of this mean or true value is just the very thing we want to ascertain), what are we to do with our three discordant results-our three different values of the angle or distance which we want to measure? Shall we toss up to decide? Shall we just take the intermediate one of three, or shall we take them all three into account? If the latter, in which way are they to be combined, out of all the different sorts of combination that might be proposed? It is here that the mathematician steps in, for we are now in his province. He is able to assure us that, by taking the mean or average of the three results, we are not only adopting the simplest plan, but the best. No method is absolutely good, for, as we have so often repeated, no method can entirely avoid error; but the one in question has the merit that the errors to be feared from it are, on the average, fewer and smaller than those to be feared from the use of any other.

We have said just above that it is not in every case that an average can be taken. Revert once more to the instance of the target. Suppose that several shots having been fired at a point on a wall, the point aimed at had been afterwards removed, so that nothing was left to indicate what spot had been aimed at, except in so far as this could be determined from the shotmarks themselves. This case is a very fair analogy to a large class of instrumental observations. For simplicity, let only three shot-marks be assigned. No one can here give the sum

mary recommendation, Take the average; for an average, in the conventional sense, does not exist. The next suggestion might perhaps be: Take a point that shall be just equally distant from all three as the most likely one. In some cases this rule would answer fairly well, but it would break down utterly if the three points were nearly in a straight line, as in that case the selected point would be very far from all three; and, when they were quite in a straight line, it would be infinitely far off, which is absurd. The suggestion of the mathematicians sounds a little complex, but is really the simplest that could be chosen for working purposes.' It is to select, as the most likely spot to have been aimed at, that point which shall make the sum of the squares of its distances from the three shot-marks the least. possible. Thence is derived the name of Least Squares. We cannot attempt any proof of this result, of course, but must merely remark that, in our judgment, its justification is of the same kind as that which was offered for the simpler case of taking the average. We start with the assumption that the shot-marks would tend to group themselves uniformly round the spot aimed at (rejecting or allowing for any permanent disturbing causes, such as constant wind, etc.), an assumption which reason and experience justify. We then determine by mathematics what mode of combining any three or more of these elements, and thus deducing a "probable" centre for the shots, will give a result which will, in the long-run, diverge least often and least widely from the truth. The mathematicians seem to have decided that this is the case with the result given by the method of Least Squares, whence its justification.

We must say something now in answer to the question, suggested some way back, how the chances of any particular events are practically to be determined. There are two ways of doing this, broadly corresponding to what may be termed artificial and natural modes of producing events. In the former we determine à priori-that is, by a consideration of the shape of the object, or the mode in which it is worked, in what proportions the events which it produces will ultimately distribute them

It is also the simplest in the sense that it is doing nothing more than choosing the centre of gravity of the three points, regarding these as of equal weight.

selves. In the latter, we determine these facts à posteriorithat is, by a direct appeal to statistics-to inform us what they actually are. If, for instance, we want to know what is the chance of getting just three trumps in a hand in a game at whist, we should never dream of taking notes of what had occurred in former deals. A serious objection to such a course would be the time which it would demand, for millions of millions of years would be required to exhaust all the possible combinations only once. What we do, of course, is to calculate the number of possible combinations, and then compare these with the number which will answer our purpose. There is plenty of justification for such a plan, one simple argument being this: Why should a particular card come out oftener than another? Presumably from some difference in its shape or nature; either because it is larger or thicker, or more or less slippery, etc. The makers, by taking care to reduce such differences to an amount absolutely indistinguishable, insure for us perfect fairness as between card and card. And the experience of every player, at least as regards the commoner sorts of occurrences, confirms us in the belief that the events do really happen as we expect that they should. We may say, therefore, that, however true it may be that the ultimate justification of the premises depends upon observation and experience, our calculations in such cases are rested entirely on what may be called à priori considerations.

The above remarks apply to cards, dice, roulette tables, and generally to every application of probability in which the constitution and behavior of the objects in question are under our own control, or can from any other reason be accurately determined. From the nature of the case they apply to nearly all games of chance; and, for that matter, to not much else. But when we come to examine natural objects and processes, we are, as a rule, quite unable to say beforehand what these things will do. The objects themselves are often far too iregular, and their modes of action far too complicated and obscure, for us ever to be able to penetrate into their nature and calcu late their behavior. Who, by looking at a man and studying the state of society in which he lives, would be able to guess how long a duration of life he might expect to attain? We

cannot even distinguish beforehand, except very roughly, between healthy and unhealthy modes of life; and cannot in the least determine how much advantage the one will have over the other. In these matters, therefore, insurance societies, and others interested in the results, have to rely entirely on the data offered by experience. Many people would imagine that the life of an English agricultural laborer is a "good" one, for his wages are no longer really low, and he is always in the fresh air. But the fact is that the Accidental Insurance societies class him among their "hazardous" lives. We presume that the chances of being kicked by a cart-horse, gored by a bull, crippled by rheumatism and so forth, more than counterbalance the small tradesman's worry and anxiety, and the clerk's cramped position and unwholesome air.

Of these two methods, the first, or à priori method, where practicable, is far the best. It assigns at once in the limit, in the form of a fraction, that ultimate average towards which experience slowly and gradually gropes, knowing that it will never reach it. The chance, or fraction so assigned, is subject to one kind of error only; the inability, viz., to secure any certain justification of it within the comparatively short space of time which can generally be afforded. This we have already explained, and have pointed out that an ultimate tendency can never be proved, or even illustrated, within a finite range of examples. But then this same limitation of experience tells with twofold force against the results obtained by the à posteriori method. It not only, as above, invalidates the experimental proof of the result, but it also invalidates the process of obtaining it. When we say that the chance of ace with a die is one sixth, we are at least sure that this fraction is right— that is, that it really represents the limit. But when we say that the chance of a man aged 32 living 27 years is, say, we are not sure of this fraction. It is itself obtained from a too limited experience, which being unable certainly to verify the chances if they are given to us, is equally unable to take the prior step of obtaining them for us. That is to say, we cannot tell for certain what is the real chance of any given. man's life; partly because we cannot get any but a limited range of statistics to yield it, and partly because the social and