science and of its applications. It may be conceived, therefore, that propositions which are apparently as general and certain as those we have discussed to-day may be analysed in the same manner, and shown to be really statements about the apparatus of thought.

In my second lecture I endeavoured to explain the difference between a discrete and a continuous aggregate. In a row of marbles, which is a discrete aggregate, we can find between any two marbles only a finite number of others, and sometimes none at all. But if two points are taken on a line, the hypothesis of continuity supposes that there is no end to the number of intermediate points that we can find. Precisely the same difference holds good between number and continuous quantity. The several marbles, beginning at any one of them, may be numbered one, two, three, &c.; and the number attached to each marble will be the number of marbles from the starting-point to that marble inclusive. If the points on a line are regarded as forming a continuous aggregate, then lengths measured along the line from an arbitrary point on it are called continuous quantities. So also, if the instants of time are regarded as forming a continuous aggregate-that is, if we suppose that between any two instants there is no end to the number of intermediate ones that might be found-then intervals or lengths of time will be continuous quantities. And just as we may attach our numbers one by one to the marbles which form a discrete aggregate, so we may attach continuous quantities (or shortly quantities) one by one to the points which form a continuous aggregate. Thus to the point p will be at

A

Р

B

tached the quantity or length Ap. And we see thus that between any two quantities there may be found an infinite number of intermediate quantities, while between two numbers there can only be found a finite number of intermediate numbers, and sometimes none at all. That is to say, continuous quantities form a continuous aggregate, while numbers form a discrete aggregate. Thus, the science of quantity is a totally different thing from the science of number.

Notwithstanding that this difference was clearly perceived by the ancients, attempts have constantly been made by the moderns to treat the two sciences as one, and to found the science of quantity upon the science of number. The method is to treat rational fractions as a necessary extension of numerical division, and then to deal with incommensurable quantities by way of continual approximation. In the science of number, while five-sevenths of fourteen has a meaning, namely, ten, five-sevenths of twelve is nonsense. Let us then treat it as if it were sense, and see what comes of it. A repetition of this process with every impossible operation that occurs is supposed to lead in time to continuous quantities. The results of such attempts are the substitution of algebra for the fifth book of Euclid, or some equivalent doctrine of continuous ratios, and the substitution of the

I

differential calculus for the method of fluxions. For my own part, believe this method to be logically false and educationally mischievous. For reasons too long to give here, I do not believe that the provisional use of unmeaning arithmetical symbols can ever lead to the science of quantity; and I feel sure that the attempt to found it on such abstractions obscures its true physical nature. The science of number is founded on the hypothesis of the distinctness of things; the science of quantity is founded on the totally different hypothesis of continuity. Nevertheless, the relations between the two sciences are very close and extensive. The scale of numbers is used, as we shall see, in forming the mental apparatus of the scale of quantities, and the fundamental conception of equality of ratios is so defined that it can be reasoned about in the terms of arithmetic.2 The operations of addition and subtraction of quantities are closely analogous to the operations of the same name performed on numbers, and follow the same laws. The composition of ratios includes numerical multiplication as a particular case, and combines in the same way with addition and subtraction. So close and far-reaching is this analogy that the processes and results of the two sciences are expressed in the same language, verbal and symbolical, while no confusion is produced by this ambiguity of meaning, except in the minds of those who try to make familiarity with language do duty for knowledge of things.

Just as in operations of counting there is a comparison of some aggregate of discrete things with a scale of numbers carried about with us as a standard, so in operations of measuring, real or ideal, there is comparison of some piece of a continuous thing with a scale of quantities. We may best understand this scale by the example of time. To indicate exactly the time elapsed from the beginning of the century to some particular instant of to-day, it is necessary and sufficient to name the date and point to the hands of a clock which was going right and was stopped at that instant. This is equivalent to saying that the whole quantity of time consists, first, of a certain number of hours, specified by comparison with the scale of numbers already constructed, and, secondly, of a certain part of an hour, which being a continuous quantity can only be adequately specified by another continuous quantity representing it on some definite scale. In the present case this is conveniently taken to be the arc of a circle described by the point of the minute-hand. On the scale in which that whole circumference represents an hour, this arc represents the portion of an hour which remains to be added. With the help of the scale of numbers, then, any assigned continuous quantity will serve as a standard by which the whole scale of quantities may be represented. And when we assert that any theorem, e.g., the binomial theorem, is

* Defining a fraction as the ratio of two numbers, Euclid's definition of proportion is equivalent to the following:-Two quantity-ratios are equal if every fraction is either less than both, equal to both, or greater than both of them.

true of all quantities whatever, whether of length, of time, of weight, of intensity, we really assert two things: first, this theorem is true on the standard; secondly, relations of the measures of quantities on the standard are relations of the quantities themselves. The first is (in regard to the kind of quantity) a particular statement; the second is involved in the meaning of the words 'quantity' and 6 measurement.'

But the most familiar and perhaps the most natural form of the scale of quantities is that in which it is supposed to be marked off on a straight line, starting from an arbitrarily assumed point which is called the origin. If we make the four assumptions of Euclidian or parabolic geometry, the position of every point in space may be specified by three quantities marked off on three straight lines at right angles to each other, their common point of intersection being taken as origin, and the direction in which each of the quantities is measured being also assigned. Namely, these three quantities are the distances from the origin to the feet of perpendiculars let fall from the point to be specified on the three straight lines respectively. In all space of three dimensions the position of a point may be specified in general by a set of three quantities; but two or more points may belong to the same set of quantities, or two or more sets may specify the same point; and there may be exceptional sets specifying not one point but all the points on a curve or surface, and exceptional points belonging to an infinite number of sets of quantities subject to some condition. There are three kinds of space of three dimensions in which this specification is unique, one point for one set of quantities, one set of quantities for every point, and without any exceptional cases. These three are the hypothetical space of Euclid, with no curvature; the space of Lobatchewsky, with constant negative curvature; and the space I described at the end of my second lecture, with constant positive curvature. In only one of these, the space of Euclid, are the three quantities specifying a point actual distances of the point from three planes. In this alone we have a simple and direct representation of the scale of quantities. Now, if we remember that the scale of quantities is a mental apparatus depending only on the first of our four assumptions about space, we may see in this distinctive property of Euclidian space a probable origin for the curious opinion that it has some à priori probability or even certainty, as the true character of the universe we inhabit, over and above the observation that within the limits of experience that universe does approximately conform to its rules. It has even been maintained that if our space has curvature, it must be contained in a space of more dimensions and no curvature. I can think of no grounds for such an opinion except the property of flat spaces which I have just mentioned.

W. K. CLIFFORD.

IS INSANITY INCREASING?

THE Lancet has recently directed attention to the bearing of certain figures collected, and computations made, by me in 1875-6-7, in the course of an inquiry instituted by that journal, upon the question, 'Is Insanity increasing?' The statistics then compiled were derived from special sources of information, very carefully tested, compared, and counter-checked, so that while, in the nature of things, there may be some mistakes, it is not probable any serious errors have vitiated the results obtained. I had no pet theory to uphold, and the question upon which it is now thought these calculations may help to throw light was not one in which I was, at that moment, particularly interested; they are therefore free from the influence of conscious bias. The statistician will know that the last-mentioned circumstance must sensibly enhance the value of any evidence the figures are found to supply. It is always easy, and the most honest worker is apt to make convenient figures fit into the niches of a preconstructed theory. As the issue raised is economic rather than medical, and may play an important part in the deliberations believed to be impending on the subject of Lunacy Law Reform, I offer no apology for submitting them to fuller consideration in the pages of the Nineteenth Century, without further preface than the remark that if the lines of argument adopted should appear arbitrary, and the conclusions stated seem to be too hastily reached, I can assure the reader they have not been presented without much deliberation. For the reasons that have compelled me to the position taken up with regard to this deeply practical question, I must refer him to the volumes wherein the grounds of my judgment have been more fully set out.1

The materials at my disposal may be indicated as follows: 1. The returns made to the Commissioners and embodied in the valuable, though by no means exhaustive, tables appended to their annual reports. The present series, issued by the Board, was preceded by the reports of the Metropolitan Commissioners, which appeared at uncertain intervals. These were included in my survey. 2. With considerable difficulty I collected the reports and returns made by the medical officers and The Care and Cure of the Insane. (Hardwicke and Bogue, 1877.)

divers authorities of the asylums in Middlesex, Surrey, and several other counties, and extracted from these trustworthy sources particulars wholly wanting in the Commissioners' Reports. 3. I have consulted the statistical papers and works published during the century, together with the statements made to Boards of Magistracy. The asylums most closely studied were those of Middlesex, Surrey, and the City of London. Colney Hatch supplied the largest number of facts; Hanwell the longest period of inquiry. Colney Hatch had a history of 24 years, and a record of 12,539 cases; Hanwell a career of 44 years, with 10,887 cases; Wandsworth Asylum, with a period of 34 years, gave 7,583 cases; the City of London, 850 cases, in 9 years; and Brookwood, in 8 years, 2,069 cases. This provided me with a gross total of 33,928 cases, which, after careful sifting, I found to represent, as nearly as could be determined, 25,850 patients. I venture to think this total, the largest population as yet studied for conditions of age,' and 'length of residence,' in an asylum-both matters of high moment-may be considered to supply a reasonably broad basis for the investigation prosecuted and detailed. What I have first to urge will, however, relate to the figures in the Commissioners' Reports.

The facts with which we have to deal are, briefly, these:—In round numbers rather under 27,000 persons were classed as insane in 1846, whereas in 1875 the total so described was not far short of 64,000. There has in fact been an apparent increase of 137.0 per cent. in about thirty years. Meanwhile the population enumerated in 1841 was 15,914,148, and the census of 1871 gave 22,712,266, an increment of only 42-72. Thus the increase of insanity in three comparable decades would seem to be 94.28 per cent. greater than that of the population. This is a sufficiently alarming statement. The question I propose to discuss is: has the formidable rise in numbers returned been due to a progressive increase of mental disease, or is it not in part, perhaps mainly, the result of an accumulation of patients? I use the word patients advisedly, because, owing to the loose manner in which the returns have been made, it is impossible to draw the very obvious and necessary distinction, in all statistics of disease, between cases and patients, the same individuals being counted again and again, owing to the neglect to eliminate transfers' from one asylum to another, and even readmissions' to the same institution after relapse. How important this last discrimination would be will appear from the fact that, examining the total returns for the public asylums of Middlesex, Surrey, and the City of London, I find the relapsed cases readmitted, that is, those which returned to the same asylums from which they had been discharged as 'recovered,' amounted to not less than 26 per cent. on the total of the so-called cures; and making the same calculation for all the county and borough asylums in England and Wales for