is biting at one cheese after the other, and looking at the balance as it alternately tips a little to either side. We have not thriven because water was poured into our milk, and we are to see if we can make a better drink by pouring the milk into the water. But yet something must be done to prevent persons from presenting the fruits of their labour to the first knave or fool who asks for them; and we must therefore have recourse to the slow, but sure, method of distributing a knowledge of the subject among the people. This may be done in two ways,-both good, but neither quite complete by itself. The general principles of the subject may be widely taught, but this is not enough; the number of those who can apply them rigorously to the questions which arise must be greatly increased. To this end, books of sound mathematical learning on the subject, accompanied by tables which will remove the necessity of much calculation, must be made attainable in a comparatively cheap form; and one of the sort is so made attainable in the work before us, which not only provides at least four times as much help to calculation as any of its predecessors, but will be sold at a price unprecedentedly low, for such a mass of table matter.

We see with great pleasure that in many different quarters popular essays on life assurance and annuities are making their appearance. We do not know whether these are good or bad, nor does it so much matter; all are friends in the present attempt. The worst of them will show many that there is more in the subject than can be mastered by taking a few shares in a new company, and will impress on the mind of its readers the necessity for some little inquiry. This much would be a great step gained, if it were universally gained. The smallest investigation, a mere pause till a friend or two could be consulted, would have saved many of those who were miserably taken in by the West Middlesex swindlers, from their disastrous fate. A very little knowledge of the subject, the mere idea that there is knowledge on the subject, would, if it were general, prevent the class of desperate rogues from combining for the mere purpose of plunder. But more must be done; for there are plenty of schemes, which, without being criminal, are most blameably rash. It is not a slight knowledge of principles which will serve to detect these; figures must be applied, and properly applied. It has been said that there is nothing which may not be proved by figures; this means that there is nothing which may not be shown from figures to those who only look at the figures. It was

the proverb of those who were always frightened by calculation, and silenced by a show of figures; so, in like manner, among savage nations, there is nothing which may not be gained by firearms, until they have got firearms for themselves. Arithmetical calculation is not now so much of a bugbear as it was; there are many who can not only endure, but practise it, and they must be shown how to do it.

As soon as the science of life contingencies had been successively licked into shape by Halley, De Moivre, Simpson, Dodson, and some others, and the exertions of Dr. Price had presented tables which appeared to represent the value of life in England, and the establishment of several assurance offices had given hopes of stability to its practical application, a succession of writers took upon themselves the charge of providing elementary instruction on the subject. Until the appearance of the work before us, the principal writings which were notorious as combining both instruction and tables, were the works of the late Mr. Morgan, of Mr. Baily, and of Mr. Milne. The well-known work of Dr. Price contained indeed the tabular matter connected with the Northampton tables, and a full account of their construction, but no elementary treatise on the application of the tables. Each of the preceding works had a merit of its own, and it will explain our remarks on the work particularly before us if we give a few words to each.

Mr. Morgan, a relation of Dr. Price, was introduced by the influence of the latter to the management of the affairs of the Equitable Society. He was of a cautious temperament, and, which is more to our present purpose, was a well-informed mathematician. He was the first who applied mathematical reasoning to the solution of the more complicated class of questions, on the assumption of a table of mortality, properly so called. The distinction is this: preceding writers had used what was called De Moivre's hypothesis; a mathematical assumption which superseded the use of a table of mortality, and assigning a simple (but not very accurate) law of life, made it possible to give tolerably simple formulæ for the most complicated problems. Mr. Morgan, in various papers published in the Transactions of the Royal Society, solved these problems in what is now called the usual method. Later writers have given conciseness and elegance to these solutions, but it was Mr. Morgan who led the way in this path. In his work on assurances (the second edition of which was published in 1821, the first having preceded it by more than forty years) Mr.

Morgan gives, 1. The principles of the subject and the solution of questions expressed in words at length, the algebraical part being reserved for the notes. The tables added

to the work are mostly those which Dr. Price had already given in his well-known treatise founded on the Northampton tables. This idea of expressing the formulæ by verbal rules was not a happy one. It was afterwards adopted by Baron Maseres (in 1783), in that most prolix of all possible works, his Treatise on the Principles of Life Annuities. The shortest way to comprehend this subject is to learn a very little of algebra; the time spent in doing without algebra must always be longer than that necessary to learn the requisite amount of it. Mr. Morgan's work, though the first elementary treatise, must not be called the first algebraical treatise, on the modern methods.

Mr. Baily's Doctrine of Life Annuities and Assurances was published in 1810, and obtained a popularity among actuaries which not only soon put it out of print, but has, even up to the present time, made its second-hand price greatly exceed that at which it was published.* This celebrity it owed mostly to its being a clear and systematic algebraical treatise, most emphatically superior to that of Mr. Morgan in the notation employed; and in some measure to a larger range of tables. It was published at the period in which the defects of the Northampton tables began to be seen, and the want of others giving a longer duration to life began to be felt. The tables of Deparcieux, and the Swedish tables, then appearing for the first time in so extensive a form, were not demonstrated to be particularly applicable to English life; but they were valuable, in the absence of others, as giving longer life, it being known that the life of the Northampton was too short. Mr. Baily's work also introduced to the world the method of Mr. Barrett, of which we shall presently speak.

Mr. Milne published, in 1815, his Treatise on the Valuation of Annuities and Assurances, which has, since the disappearance of Mr. Baily's work as a book on sale, been the great authority in algebraical matters connected with life contingencies; and of late years has also taken that post with reference to its tables. In the mathematical part it combines a general account of the method of constructing tables of mortality, with a full investigation of the formula for their use, carried completely through all problems involving one,

* A French translation appeared some years ago.

two, or three lives. To the algebraical symmetry which Mr. Baily had introduced, is added the first general attempt to institute a distinct notation for the representation of life contingencies; the results of which, though somewhat cumbrous in appearance, and strange in their symbols, will in all probability lead to some general agreement on the subject; more than one writer having been led to the consideration of this branch of expression by the route traced out by Mr. Milne. The defect of the work to a learner, is too great a generalization at the commencement. With regard to the tables, Mr. Milne, by aid of observations on the value of life at Carlisle, presented a new table (the Carlisle), which subsequent verifications have shown to represent more accurately than any other the state of life among the middle classes in England.

Dr. Price and Mr. Morgan set the example of making the fundamental tables to consist in the values of annuities on one life for every age, and of two lives for every pair of ages in which the difference is five years, or any multiple of five years. Other differences of age must be supplied by interpolation, a process which gives results sufficiently accurate in most cases, but which sometimes does not admit of so much being said. In the construction of these tables, which is a work of tolerable labour, much valuable material is thrown away, and the results of it only presented. The want of the rejected part is very severely felt in many problems, which are tediously long when the answers are to be deduced from the tables of complete annuities. Mr. Barrett, of whom no more is known than that he linked his name inseparably to this subject, proposed a very ingenious modification of the tables, which, stopping short of the final results, leaves it in the power of the calculator to proceed as if he had before him all the raw material of the old tables in its easiest form. This method was first published by Mr. Baily, in an appendix to his own work,-which appendix was, we believe, subsequent to the treatise of Mr. Milne: it was thrown into a more convenient form, and extended in power, by Mr. G. Davies, and is now the most valuable instrument of which the actuary is possessed; and, what is more to our present purpose, it is the form in which knowledge of the modes of calculation will be diffused among arithmeticians in general.

At the appearance then of the work before us, the state of the science, as to points now under consideration, may be briefly summed up as follows: The superiority of the Carlisle Table was admitted;-the old methods, depending upon

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complete annuity tables, were the ones in general use ;—the practical correctness of results was somewhat lessened, and the trouble of obtaining them increased, by the incompleteness of the published tables of two lives;-many tables of simplification had been computed by actuaries for themselves, but none were published ;-Barrett's method was impeded by the want of published tables, though many private persons had them in manuscript for their own purposes;-the number of persons, not actuaries, who could work a simple question, was very small.

The treatise published under the superintendence of the Society for the Diffusion of Useful Knowledge began to be issued some years ago, and is not yet quite completed; but as the whole of the explanation has been some time before the public, and a mass of tables very much exceeding those of Mr. Milne has now appeared, a review of the whole will not be premature. The author is Mr. David Jones, actuary of the Universal Life Office. His preface (yet to appear) will, we hope, point out in detail the methods employed in the calculation of the several tables. The work consists of a treatise on interest; forty-three pages of tables to accompany this treatise (usually mixed up with the life tables); a treatise on life annuities, as to all cases in which not more than two lives are concerned; six hundred and thirty pages of lifetables, plus some yet to come. The treatise on life annuities, which belongs to this mass of tables, is only of one hundred and twenty-six pages. With regard to the treatise, independently of the tables, we have examined it (for a particular purpose) much more closely than we should have thought it necessary to do, previously to expressing a general opinion; and we can give our strongest testimony, both to the soundness of the writer's methods, and the accuracy of the printer. There is, however, no occasion to enlarge upon a question of fact, which must be established; for those who work questions of this kind must have the tables, and will therefore in most cases find out the merit of the work. The great defect is a badly chosen notation,-inferior, in our opinion, to Mr. Milne's. Its symbols are too small, and there is the defect of double indices; thus an annuity on joint lives of the ages m and n, to last k years, is denoted by

a

(m, n)