comparison should be made by calculating the ratio of one to the other. If then a statistical table is to be really intelligible and useful, every column of absolute numbers should be accompanied by a column of ratios. This accordingly is done to a certain extent in the Census Reports, the publications of the Registrar General's office, and some other important statistical tables; but it is never done as much as would be desirable. The reason is obvious. Each ratio can only be obtained by a tedious long division sum, or by the use of logarithms. Many hours of tedious mental labour must be endured before a large statistical table can be reduced to its proper intelligible form. The result is that, in the absence of an office full of clerks, the labour is almost always shirked, and the reader of our statistical publications is left to extract their meaning as well as he can-which means very badly.

With the Arithmometer at hand, however, the work becomes rather amusement than labour, especially to those at all fond of ingenious and beautiful mechanism. The amount of time saved will vary with the character of the operation and the nature of the calculations; but about the saving of mental exertion there can be no possible doubt. The machine will also be of great use in effecting the reduction of numbers from one denomination to another, as from pounds to francs, dollars, rupees, &c.; tons to kilogrammes, yards to metres, &c., &c. It is requisite, however, that all numbers should be expressed in the purely decimal form, so that our absurd systems of money, weights, and measures, present obstacles to the easy use of the machine. When frequent alterations of any num

bers in a definite ratio have to be made, it will often be best to calculate at the outset a table of the multiples of that ratio; this can be done with the utmost facility by the machine, because each turn of the handle gives a fresh number for the table. A reduction table can thus be prepared as fast as the numbers can be written down, and all further labour of calculation is saved by reference to this table.

I should like to add, that if our science of statistics is to progress in the spirit of the times, frequent use must be made of the Method of Least Squares. This method is merely the method of means or averages employed in a more complete and elaborate way, to disentangle the probable values of several unknown quantities which happen to be involved together in our statistical data. The working of the process, as described in Merriman's "Elements of the Method of Least Squares" (Macmillan, 1877), in De Morgan's "Essay on Probabilities," and many other works on the same subject, can be carried on by mere rule of thumb; but it requires a great amount of multiplication. With Thomas' Arithmometer, however, the requisite calculations can be readily accomplished, and I conceive, therefore, that in this as well as in other cases, the fre

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quent use of the machine is indispensable as a condition of any distinct advance in statistical inquiry. Familiarity with the arithmetical machine would gradually lead to the undertaking of intricate numerical inquiries, which are practically impossible without its aid. The use of the machine as employed by actuaries has already been described by General Hannyngton in the "Journal of the Institute of Actuaries," vol. xvi, p. 244, and a very able paper "On the Arithmometer of M. Thomas (de Colmar) and its Applica"tion to the Construction of Life Contingency Tables," was printed by Mr. Peter Gray, in the journal of the same Society for 1874, and issued separately as a pamphlet. The operations therein described are, however, far more complicated than what the statist will usually need to perform.

The working of the machine is so easy, that it can be learnt by any person of ordinary intelligence in the course of an hour, and with a few little precautions, which are stated in the explanatory book of instructions delivered with the machine, there need be no fear of its getting out of order. It is said that machines are often worked daily for many years in succession, without any mishap or error occurring; but other operators find that certain springs are apt to break, and require replacement. The machine, though constructed only in Paris, can be inspected and purchased at a depôt in London. The smallest machine now made gives a product not exceeding twelve places of figures, which would be sufficient for most purposes; but the medium-sized machine, giving a product of sixteen places, is said to be more convenient in use, as there is greater scope and freedom of action. My own limited experience of the machine leads me to think that this may be so.

I have been induced to bring the Arithmometer under the notice of the Society, by the feeling that there must be many who are (as I was myself a few months ago) imperfectly acquainted with the value of the machine. Had I purchased a machine when I first saw it at the Paris Exposition of 1867, I should have been saved a great deal of mental fatigue during the eleven subsequent years, and I might have undertaken statistical inquiries which are beyond the power of a private unaided arithmetician. The conviction that this machine must prove no inconsiderable factor in the progress of statistical and social science, renders it desirable for those acquainted with its value, to endeavour to overcome the inertia, which, especially in this country, impedes the introduction of any new labour-saving invention. A machine which was in its essential features invented by the youthful genius of Pascal, in the year 1642, is only now coming into use. For two hundred and thirtysix years (236 years!) practical men have ignored what may prove one of the most practically useful, as it is certainly one of the most beautiful products of human reason.

PROFESSOR JEVONS'S DESCRIPTION of the CALCULATING MACHINE.

Dr. FARR said the Society was indebted to Professor Jevons for bringing forward the machine and explaining its uses. It was of great use in the Registrar-General's office in determining the ratios and the percentage of deaths, births, marriages, and so on, and this was done on a very large scale. The clerks were allowed to work by logarithms or by arithmetic, but they invariably preferred the machine. Undoubtedly in order to make it of universal application, there should be a decimal system of calculation in weights, measures, and money; but at present he recommended its use for calculations on a large scale.

Dr. BALFOUR said that for the last five or six years, when he was at the head of the Army Medical Department, they could make all their calculations as quickly as they could by the machine, but he thought it minimised the chance of error. In working out logarithms there was a source of error, which was avoided by using the machine. All the time it was used in the office under his charge he never had any cause to complain of it except that the springs that were worked on the pegs were apt to go wrong. He was indebted to General Hannyngton, who pointed out how this could be remedied, and after he had done so he had never found the machine to go wrong once. On one occasion the clerk of the Department said that the machine had gone wrong, but it was afterwards found that the error was on the part of the clerk, and not on the part of the machine. In working out ratios it minimised the chance of error, which was a consideration of great importance.

Mr. WALFORD said he had been familiar with the machine some years ago. He had seen some sixty calculating machines, but he thought on the whole that the one exhibited was the most available for general purposes. He did not think it was a safe thing to use, except by persons familiar with it, because if it was not set with great care errors would arise, and if one number was wrong, the whole would be wrong. He would minimise that difficulty by having two machines to commence at decennial points, and work the one with the other-not, however, by the same operator-and if the results were both the same, they might be pretty well sure that they were correct. It would save a great amount of labour, money, and thought, and simply required care.

Mr, A, H. BAILEY, President of the Institute of Actuaries, said that the machine grew upon people the more they used it. There was a variety of purposes to which it could be applied. It was very useful in the distributing of a bankrupt's estate, for instance, and one of its great uses would be to introduce a decimal system of arithmetic; but even with our present system of weights and mea

sures, and of pounds, shillings, and pence, the machine could be used. It should, however, be understood that it was entirely a decimal machine.

The PRESIDENT, in moving a vote of thanks to Professor Jevons, said that he had shown that it was an extremely valuable instrument, and he thought it might be looked forward to as one of the principal instruments in Government offices. Although it might not save time, it would save a good deal of mental labour, and that was one of the greatest considerations in a Government department.

ADDRESS of the PRESIDENT of SECTION F of the BRITISH ASSOCIA TION, at the FORTY-EIGHTH MEETING, held at DUBLIN, in August, 1878. By PROFESSOR INGRAM, LL.D., F.T.C.D.

HAD I been called upon at any other time to preside over this Seetion, I should have followed the example of most of my predecessors, in selecting as the subject of the discourse which it is usual to deliver from this chair, some one of the special economic questions of the day, which my knowledge might have enabled me most adequately, or, let me rather say, least inadequately to treat. But I have felt that the matter with which I should deal has been prac tically determined for me beforehand. An important crisis in the history of our Section has taken place. Its claim to form a part of the British Association has been disputed. Some of the cultivators of the older branches of research but half recognise the right of Political Economy and Statistics to citizenship in the commonwealth of science; and it is not obscurely intimated on their part that these studies would do well to relinquish pretensions which cannot be sustained, and proceed, with or without shame, to take the lower room to which alone they are entitled.

How far this sentiment is entertained by those who would be recognised as the best representatives of the mathematical, physicochemical and biological sciences, I am unable to say. But it is natural to suppose that no one clothed with an official character in the Association, could have assumed towards us such an attitude as I have described, unless supported by a considerable weight of opinion amongst those within the body who are regarded as competent judges. Still more-and this is what lends a peculiar gravity to the incident-such a step could scarcely have been taken if the general mass of the intelligent public entertained strong convictions as to the genuinely scientific character of political economy, as it is usually professed and understood amongst us. It is, in fact, well known that there is a good deal of scepticism current on this question. There may be seen in various quarters evidences sometimes of contemptuous rejection of its claims, sometimes of uneasy distrust as to their validity. And even amongst those who admit its services in the past, there is a disposition to regard it as essentially effete, and as having no scientific or practical future before it.

When some of our leading economists met not long ago to celebrate the centenary of the publication of the "Wealth of