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- The last professed treatise on the science, which I think worthy of notice*, is Mr. Baron Maseres's Principles of the Doctrine of Life Annuities (1783): wherein this celebrated author has explained the subject in so familiar a manner, as to be intelligible even to those who are unacquainted with the Doctrine of Chances, and who have made no great proficiency in mathematics. This treatise, however, (although consisting of more than 700 quarto pages) goe's no further, in the analysis of the subject, than the first two problems in the present work: but its value is greatly enhanced by containing a variety of new Tables of the value of Annuities on Single lives, and on two Joint lives of different ages, deduced from the probabilities of living as observed by M. De Parcieux amongst the Government Annuitants in France ; these being justly considered by the learned author as the most proper data whereon to found the value of Life-Annuities. There are, moreover, in that treatise several interesting observations on the best method of providing Annuities for Old Age, and on various subjects of finance and political economy; which render it particularly valuable to those who are desirous of information on these important questions, and will perpetuate the name and abilities of this truly public-spirited writer...

Soon after the publication of the fourth edition of Dr. Price's Observa. tions on Reversionary Payments, (which contained the valuable collection of Tables of Life-Annuities, deduced from the observations made at Northampton and in Sweden) Mr. Morgan was enabled to detect the inaccuracy of those rules which, not only Mr. Simpson and others had given for determining the value of contingent annuities and assurances, but also which he himself had deduced from the same principles, in his treatise above mentioned: and he immediately set about to correct them. His labours, on this subject; are contained in the several papers inserted by him in the Philosophical Transactions for 1788, 1789, 1791, 179+, and observed at Northampton : namely, one for Single lives, another for two Joint lives whose ages are equal, and another for two Joint lives whese difference of age is sixty years : the interest in each table being at 4 per cent. In this infant state of the science, every additional table contributed greatly to the means of information on this subject.

It may be here necessary to remark, that the fourth edition of Dr. Price's Obs. on Red. Pay. (which first contained the present valuable collection of Tables) did not appear till four years after the publication of Mr. Morgan's work above alluded to. So that, till within these thirty years, there existed only four tables of the value of Life- Annuitiés : viz: two founded on M. de Moivre's hypothesis, and two deduced from the

London observations; which are the Tables XLVII to L at the end of this work. As to the table formed by Dr. Halley, it was too contracted or any real use.

5*'I must not however onit mentioning the name of Dr. Waring: who, in a small pamphlet entitled On the Principles of translating Algebraic Quantities into Probable Relations, &c. (1792), bas devoted about thirty pages to the consideration of the subject of Annuities and Assurances. His style and manner, however, will not be much admired by those who have read the works of preceding writers on this subject. The formula in page 61, is taken from that pamphlet

1800. In the first volume, here alluded to, he has considered those cases only in which two lives are concerned : in the next two volumes, his object was to deduce the value of contingent assurances in all those cases where three lives are concerned, and which admit of a correct answer; and in the last two volumes he proposed to determine the value of contingent annuities and assurances in all the remaining cases of three lives.

•Whoever will take the pains to read over those papers with attention, must be struck with surprise and regret at the strange and confused manner which Mr. Morgan has pursued, in order to obtain the solution of the 'several problems under consideration. No one, at the present advanced state of the science (with so many models of simplicity and elegance before him), could expect to see any mathematical inquiries conducted in so loose, so obscure, so extraordinary a manner. The investigations are tediously and unnecessarily prolix, crowded with useless repetitions, and a variety of unmeaning quantities. All which might, indeed, be excused, if the resulting formula had been at once simple and correct : instead of which, we find the grossest errors committed, not only as to their form, but as to their accuracy. They are, for the most part, unnecessarily long ; abounding with useless quantities, (which render their numerical solution exceedingly intricate and difficult); and oftentimes at variance with the particulars mentioned in the investigation ; which, together with the erroneous manner in which they are printed, renders them of little or no use to the public. Most of his problems are investigated in two different ways, and are solved by the means of two distinct formulæ: but, notwithstanding the similarity of these methods is studiously kept from the observation of the reader, and although these double formulze are, in each problem, totally different in appearance; yet they will be found in all cases to be precisely the same disguised under different symbols! A curious and interesting branch of the science has been thus strangely distorted and enveloped in mystery; a depraved taste in mathematical reasoning has been introduced ; and, (what is, by far, of the greatest importance) many false solutions have probably resulted from too great a dependence on the general formulæ*' pp. xvi-xxiv.

After giving this copious account of the labours of English authors, Mr. Baily glances slightly at the performances of foreigners. This part of his history we think raiher defective; yet our author has done enough to prove that there is still ample room, or rather indeed an absolute necessity for a new treatise, in which the principles of this departinent of science shall be methodically unfolded, and their practical

5* The Philosophical Transactions not being within the reach of every person, Mr. Morgan has inserted his formula, for the solution of the several problems here alluded to, in the last edition of Dr. Price's Obs. on Rev. Pay. note (P). But, the errors of the original are multiplied in the copy: and Mr. Morgan, if he studies his own reputation as a mathe. matician, had better expunge them altogether in future, than suffer thema to remain in their present worse than useless state,

utility evinced by an application to a due variety of cases. For Dr. Price's work, though of much intrinsic merit, is rather a collection of insulated essays than a connected treatise ; Maseres's, though admirable as far as it goes, is very limited in its extent; and the results of Mr. Morgan's inquiries, even supposing them to be always accurate and unobjectionable, are thrown into so many different repositories of information, that it would be unreasonable to expect they should all fall into the hands of any one reader, not a professed collector of papers on this particular topic.

We received Mr. Baily's work, therefore, with considerable satisfaction, and are willing to acknowledge that we have read inuch of it with pleasure and advantage. He has considered the subject with great attention ; and our mathematical read. ers, we presume, will be gratified if we permit such an author to speak for himself at greater length, than we are in the habit of allowing to mathematical writers in general. The comparative novelty and importance of the subject seem indeed to demand this.

The following,' says Mr. B. is the outline of my plan. •The first chapter contains a few elementary principles of the Laws of Chance ; some remarks on the Probabilities of Life, with an account of the several Tables of Observations made at different parts of the world ; and an explanation of the general method adopted to express those probabilities in all cases. This preliminary chapter will prevent much unnecessary repetition in the course of the work.

The second chapter shews the method of determining the Value of Annuities on any Single or Joint lives ; on the Longest of any number of lives; &c. &c. The second corollary to the first problem is of considerable importance in enabling us to dednce, in a very easy and expeditious manner, the value of annuities on any single or joint lives, from real observations. For, it should be particularly observed that, tables of such values being once formed, the solutions to the subsequent problems become extremely easy; since the formulæ are expressed in terms denoting the value of such annuities,

• The third chapter contains the four necessary problems for the solution of all cases of absolute Reversionary Annuities; and at the end of that chapter I have selected all the possible cases of two and three lives, in order that they may be more easily referred to. The formulæ there given will be found of considerable utility also in enabling us to determine the value of the Fines that ought to be paid for the Renewal of Leuses held on two or three lives.

• The fourth chạptes comprehends various cases of annuities depending on Survivorships between two and three lives. These cases might have been considerably augmented, but without any real benefit : since the most frequent ones are there inserted : and any other which may arise) is easily solved by the same method of proceeding.

The fifth chapter relates to such cases of Contingent Reversionary Annuities as could not, for want of some previous information, be inserted in the two preceding chapters : and I believe that the method of solution, which I have there adopted, will come nearer to the correct value than any that has hitherto been published.

• The sixth chapter treats of Assurances : a subject of great importance and extensive utility at the present day. A full explanation of the doctrine is given in the two problems inserted in that chapter.

The seventh chapter contains the method of determining the value of annuities on Successive lives ; the value of Fines in Copyhold estates held on lives; the value of Presentations, Advowsons, and things of a like kind. It likewise enables us to determine the value of the Fines that ought to be paid for Renewing or Exchanging any lives held on a lease originally granted for three lives and afterwards for a number of years cortuin: a practice pursued by several corporations in this country.

• The eighth chapter is devoted to an investigation of the value of Con. 'tingent Assurances ; wherein I have considered every possible case in . which not more than three lives are concerned. In this branch of the science I flatter myself that I have made considerable improvements. 'I have divested the subject of all extraneous matter; have not introduced more cases than were absolutely necessary; have exposed the singular formulæ given by Mr. Morgan (the only person who has preceded me in these inquiries); and have, for the most part, introduced more correct expressions for the value of the several cases there alluded to.

The three remaining chapters complete the analysis of the science, and relate to such subjects as could not properly be introduced into either of the preceding ones. The ninth is confined to an explanation of the cele. brated hypothesis of M. De Moivre : wherein its great utility and cond venience, in many obvious cases, is defended against the recent attacks of Dr. Price and Mr. Morgan. The tenth treats of the method of determining the value of life-annuitics payable Halfyearly, Quarterly, &c: also of the value of life-annuities Secured by Land : and of the value of Assurances of sums of money payable immediately on the extinction of any given lives. The eleventh shows the method of finding in Annual Paye ments the value of any Assurance or of any deferred Annuity : problems 'which will be found of very extensive use in practice.

The twelfth chapter contains a variety of very useful Questions connected with the subject : to which are added the Rules for the solution of the same ; and a numerous collection of Examples. These are thrown together into one chapter for two obvious reasons : in the first place, by being separated from the body of the work, they do not interrupt the analytical investigations; and secondly, they may be used together with the tables which follow) by such persons as are not a quainted with mathematics. Consequently, the present work will be accommodated to the use of both classes of readers; and' (although some repetition is un. avoidably occasioned thereby) may be thus rendered doubly valuable.

•The chirleenth chapter shows the direct application of the sixth, thir. teenth and eighteenth questions, in the preceding chapter, to some of the most useful and important concerns of life : namely, to the method of forming the best Schemes for providing annuities for the benefit of Old Age, and for Widows. These, observations are brought together under one

head, in order that they might not interrupt the regular arrangement of the questions : and because it gives me, thereby, an opportunity of enJarging more fully on this very interesting subject.

The last chapter contains an account of the various Societies or Compa. nies that have been formed in this metropolis for the purpose of granting Annuities, Assurances and Endowments; together with a review of their several plans and constitutions. By this statement the world may be enabled to judge of their comparative merits, and will easily decide on their respective claims to public patronage. This additional chapter cannot be considered foreign to the nature of the present work. For, after explaining the principles of the science, and after entering at large into the value of such subjects, it becomes me not only to show where a person might apply who is desirous of negociating for such things, but likewise to point out those offices at which he may do it with the greatest security and advantage to himself. And I am the more induced to do this, as I have lately observed several mean attempts to mislead and deceive the pub. lic. Uninterested in, and unconnected with, any of these Societies, I have been anxious only to give a clear and unprejudiced account, deduced from their own plans and proposals, as submitted by them to the public. At the end of this chapter I have also inserted a brief account of the recent plan, adopted by Government, for converting the 3 per cent stocks into Life. Annuities. The propriety and advantage of a measure of this kind, I had suggested in my Doctrine of Interest and Annuities.

• Respecting the Tables of Observations, and of Life-Annuities deduced therefrom, which are inserted at the end of this work, I can only observe that they are a collection of all that have hitherto been published, in this country, on the subject : and that they are here carefully reprinted from the authorities mentioned in the note in page 41. The necessity of other ta. bles I have often had occasion to mention ; but I fear it will be long before I shall be able to congratulate the public on this point. It consequently becomes the more necessary to have all those, which do exist, brought into one view, in order that they may be more conveniently consulted. The Tables of the value of Assurances for Single and Joint Lives, according to the observations made by M. De Parcieux and in Sweden, have been calculated expressly for this work; and will be oftentimes found extremely useful.' pp. xxvii-xxxiv.

Such is the author's own account of the plan and substance of his performance; and we are happy to add that he has performed his task with ability. His arrangement is logical; his demonstrations as perspicuous as the nature of the subject will allow: the new notation by which he simplifies his theoretical processes is ingenious; and the numerous practical rules which he has deduced from his investigations are plain and free from ambiguity. He has carefully guarded against a loose and unscientific use of terms; and has so contrived the subdivisions of his work, as to keep himself tolerably free from necdless repetitions. The tables which are in number 59, and occupy 100 pages, are neatly, and (as far as we may venture to speak from a pretty cautious inspection) accurately printed. On the whole,

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