A TREATISE ON FRIENDLY SOCIETIES, IN WHICH THE DOCTRINE OF INTEREST OF MONEY, AND THE DOCTRINE TO THE AFFAIRS OF SUCH SOCIETIES: WITH NUMEROUS TABLES; AND AN APPENDIX, CONTAINING THE ACTS OF PARLIAMENT RELATING BY CHARLES ANSELL, Esq., F.R.S., ACTUARY TO THE ATLAS ASSURANCE COMPANY. PUBLISHED UNDER THE SUPERINTENDENCE OF THE SOCIETY FOR THE COMMITTEE. Chairman-The Right Hon. LORD BROUGHAM, F.R.S., Mem. Nat. Inst. of France. W. Allen, Esq., F. R. & R.A.S. Sir C. Bell, F.R.S., L. & E. C. Hay Cameron, Esq. J. Bonham Carter, Esq., M.P. Rt. Rev. the Bishop of Chichester, D.D. William Coulson, Esq. Wm. Crawford, Esq. J. Fred. Daniell, Esq., F.R.S. Rt. Hon. Lord Denman, F.R.A.S. C. L. Eastlake, Esq., R.A. Right Hon. Visc. Ebrington, M.P. Sir H. Ellis, Prin. Libr. Brit. T.F. Ellis, Esq.,M.A., F.R.A.S. Rt. Hon. Sir J. C. Hobhouse, David Jardine, Esq., M.A. Henry B. Ker, Esq. F.R.S. The Rt. Hon. the Earl of Kerry, M.P. Thos. Hewitt Key, Esq., M.A. Geo. C. Lewis, Esq., M.A. James Loch, Esq.,M.P., F.G.S. George Long, Esq., M.A. J. W.Lubbock, Esq. F.R., R.A. & L.S.S. Henry Malden, Esq., M.A. W. H. Ord, Esq., M.P. The Rt. Hon. Sir H. Parnell, Bart., M.P. Dr. Roget, Sec. R.S., F.R.A.S. J. Abel Smith, Esq., M.P. John Wrottesley, Esq. M.A., THOMAS COATES, Esq., Secretary, No. 59, Lincoln's Inn Fields. BL London: Printed by W. CLOWES, Duke-street, Lambeth. PREFACE. THE parts of the following small work which the author considers to deserve most attention, are, 1st, those which relate to the actual experience of a large number of Friendly Societies as to the quantity of sickness, as well as the rate of mortality, among the members-now for the first time published, as regards such Societies in England; and 2d, the mode of applying that experience in determining the requisite contributions to provide for an allowance in sickness to these objects the Author has directed the chief part of his attention. He is not aware that he has been preceded in the publication of any accredited data as respects the sickness experience among Friendly Societies in England; or in the application of the doctrine of interest and probability to that part of the subject: and since there are few who act as pioneers on such occasions but leave much room for improvement, he is quite prepared to learn that his work is defective and capable of amendment. If it be objected by any that some of the theorems might have been extended to cases which he has not introduced, he can only say he has, throughout, been actuated by a desire to render the Treatise short; and that it was found, in almost every instance, more difficult to compress than it would have been to dilate. Care has been taken to introduce nothing but what may be very easily understood by persons acquainted with the most simple principles of algebra; and from this cause, some of the demonstrations may, by the experienced mathematician, be objected to as unnecessarily prolix. The notation adopted was fixed on for the particular purpose to which it is applied: because it generally keeps in view the several elements which enter into the demonstrations or theorems; and, notwithstanding it may sometimes appear cumbrous to the eye, it will probably be less diffi cult for the mind to retain, than one which may be shorter in its written expression. As it is not intended that the work should be confined to the reading of those only who have made some progress in a knowledge of algebra, it has been deemed proper to give rules, in words at length, by which all the arithmetical operations, required in computing tables for the use of Friendly Societies, can be performed without any reference whatever to algebraical formulæ. Most of the values and tables introduced are expressed in decimal fractions, and since the work may be consulted by some persons not very familiar with that mode of expression, it is thought well to print the rule and tables given in this and the three following pages; the application of which rule and tables will, it is hoped, be sufficiently plain to obviate any difficulty that might, without them, arise in making use of the several tables inserted in the body of the work. Rule for converting Decimal Fractions of £1 into Equivalent Values, expressed in Shillings, Pence, and Farthings. 1st. Multiply the decimal fraction by 20, and in the product separate by a comma as many figures on the right hand as there were figures in the decimal fraction: the figure to the left hand of the comma will be the number of complete shillings in the fraction. 2nd. Then multiply those figures remaining to the right hand of the comma by 12, and, as before, separate, by a comma, from the second product as many figures on the right hand of this second comma as were contained in the original fraction: the figures to the left hand of this second comma will be the number of complete pence in the fraction. 3rd. Again multiply the figures to the right of the comma, in this second product, by 4; and separate as many places of figures from the right hand of this third product as the original fraction contained: the figure to the left of this last comma will be the number of complete farthings in the fraction, 4th. If the first figure on the right hand of the last comma be 5 or more, an additional farthing should be added to the sum obtained by the foregoing operation. EXAMPLE. What is the value, in money, of the decimal fraction ⚫7623? ⚫7623 15,2460 15 shillings. 12 2,9520 2 pence. 4 3,8080 3 farthings. And as 8, the first figure to the right hand of the last comma, is more than 5, one farthing must be added to the money-value found, and the total money-value sought will be as under .— As a result of multiplying any decimal fraction of a pound sterling by 20-there will always, consistently with the foregoing rule, be twice as many shillings produced in the money-value as there are units in the first decimal figure; and an additional shilling will be contained in the fraction, if the second figure thereof amount to 5: so that we can, from inspection, immediately see the exact number of complete shillings contained in any decimal fraction presented to us. Thus, the decimal fraction •7000, we should at once know to be fourteen shillings; because 7000×20= 14,000, or fourteen shillings, by the preceding rule. Again, ⚫7500 × 20 = 15,000, or fifteen shillings. This is, in the first case, the same as doubling, for the shillings, the first figure of the fraction; and in the latter instance, it is the same as doubling the first figure and adding 1, because the second figure amounts to 5. If there be any remainder over the 5 in the second figure, or if the second figure fall short of 5, the Table A, on the next page, will show the money-value, to the nearest farthing, of that part of the fraction which may remain above |