CABINET CYCLOPAEDIA. 
 
LONDON : 
 
 Printed by A. SPOTTISWOODE, 
 New-Street-Square. 
 
THE 
 
 CABINET CYCLOPEDIA. 
 
 CONDUCTED BY THE 
 
 REV. DIONYSIUS LARDNER, LL.D. F.R.S. L.& E. 
 
 M.R.I.A. F.R.A.S. F.L.S. F.Z.S. Hon. F.C.P.S. &c. &c. 
 
 ASSISTED BY 
 
 EMINENT LITERARY AND SCIENTIFIC MEN. 
 
 AN 
 
 ESSAY ON PROBABILITIES, 
 
 AND ON THEIR APPLICATION TO 
 
 LIFE CONTINGENCIES AND INSURANCE OFFICES. 
 
 BY 
 
 AUGUSTUS DE MORGAN, 
 i> 
 
 OF TRINITY COLLEGE, CAMBRIDGE; 
 
 PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, AND 
 SECRETARY OF THE ROYAL ASTRONOMICAL SOCIETY. 
 
 LONDON: 
 
 PRINTED FOR 
 
 LONGMAN, ORME, BROWN, GREEN, & LONGMANS, 
 
 PATERNOSTER-ROW ; 
 
 AND JOHN TAYLOR, 
 
 UPPER COWER STREET. 
 
 1838. 
 
PREFACE. 
 
 IN order to explain the particular object of this Trea- 
 tise, it will be necessary to give a brief account of the 
 science on which it treats. 
 
 At the end of the seventeenth century, the theory of 
 probabilities was contained in a few isolated problems, 
 which had been solved by Pascal*, Huyghens, James 
 Bernoulli, and others. They consisted of questions re- 
 lating to the chances of different kinds of play, beyond 
 which it was then impossible to proceed : for the dif- 
 ficulty of a question of chances depending almost en- 
 tirely upon the number of combinations which may 
 arise, the actual and exact calculation of a result be- 
 comes exceedingly laborious when the possible cases are 
 numerous. A handful of dice, or even a single pack of 
 cards, may have its combinations exhausted by a mode- 
 rate degree of industry : but when a question involves 
 the chances of a thousand dice, or a thousand throws 
 with one die, though its correct principle of solution 
 would have been as clear to a mathematician of the six- 
 teenth century as if only half a dozen throws had been 
 considered ; yet the largeness of the numbers, and the 
 
 * Un probleme relatif aux jeux de hasard, propose & un austere janse- 
 niste par un homme du monde, a ete 1'origine du calcul des probabilites. 
 
 Poisson. 
 
 A 4 
 
 268730 
 
consequent length and tediousness of the necessary 
 operations,, would have formed as effectual a barrier to 
 the attainment of a result, as difficulty of principle, or 
 want of clear perception. 
 
 There was also another circumstance which stood in 
 the way of the first investigators, namely, the not hav- 
 ing considered, or, at least, not having discovered, the 
 method of reasoning from the happening of an event to 
 the probability of one or another cause. The questions 
 treated in the third chapter of this work could not 
 therefore be attempted by them. Given an hypothesis 
 presenting the necessity of one or another out of a 
 certain, and not very large, number of consequences, 
 they could determine the chance that any given one or 
 other of those consequences should arrive ; but given an 
 event as having happened, and which might have been 
 the consequence of either of several different causes, or 
 explicable by either of several different hypotheses, 
 they could not infer the probability with which the 
 happening of the event should cause the different hypo- 
 theses to be viewed. But, just as in natural philosophy 
 the selection of an hypothesis by means of observed 
 facts is always preliminary to any attempt at deductive 
 discovery ; so in the application of the notion of proba- 
 bility to the actual affairs of life, the process of reasoning 
 from observed events to their most probable antecedents 
 must go before the direct use of any such antecedent, 
 cause, hypothesis, or whatever it may be correctly 
 termed. These two obstacles, therefore, the mathema- 
 tical difficulty, and the want of an inverse method, pre- 
 vented the science from extending its views beyond 
 problems of that simple nature which games of chance 
 present. In the mean time, it was judged by its fruits; 
 
and that opinion of its character and tendency which is 
 not yet quite exploded, was fixed in the general mind. 
 
 Montmort, James Bernoulli, and perhaps others, had 
 made some slight attempts to overcome the mathema- 
 tical difficulty ; but De Moivre, one of the most pro- 
 found analysts of his day, was the first who made 
 decided progress in the removal of the necessity for 
 tedious operations. It was then very much the fashion, 
 and particularly in England, to publish results and con- 
 ceal methods ; by which we are left without the know- 
 ledge of the steps which led De Moivre to several of his 
 most brilliant results. These however exist, and when 
 we look at the intricate analysis by which Laplace ob- 
 tained the same, we feel that. we have lost some im- 
 portant links * in the chain of the history of discovery. 
 De Moivre, nevertheless, did not discover the inverse 
 method. This was first used by the Rev. T. Bayes, in 
 Phil. Trans, liii. 370. ; and the author, though now 
 almost forgotten, deserves the most honourable remem- 
 brance from all who treat the history of this science. 
 
 Laplace, armed with the mathematical aid given by 
 De Moivre, Stirling, Euler, and others, and being in 
 possession of the inverse principle already mentioned, 
 succeeded both in the application of this theory to more 
 useful species of questions, and in so far reducing the dif- 
 ficulties of calculation that very complicated problems 
 may be put, as to method of solution, within the reach 
 of an ordinary arithmetician. His contribution to the 
 science was a general method (the analytical beauty and 
 power of which would alone be sufficient to give him a 
 high rank among mathematicians) for the solution of 
 
 * The same may be said of several propositions given by Newton. 
 
air questions in the theory of chances which would 
 otherwise require large numbers of operations. The 
 instrument employed is a table (marked Table I. in the 
 Appendix to this work), upon the construction of which 
 the ultimate solution of every problem may be made to 
 depend. 
 
 To understand the demonstration of the method of 
 Laplace would require considerable mathematical know- 
 ledge ; but the manner of using his results may be de- 
 scribed to a person who possesses no more than a common 
 acquaintance with decimal fractions. To reduce this 
 method to rules, by which such an arithmetician may 
 have the use of it, has been one of my primary objects 
 in writing this treatise. I am not aware that such an 
 attempt has yet been made : if, therefore, the fourth, and 
 part of the fifth chapters of this work, should be found 
 difficult, let it be remembered that the attainment of such 
 results has hitherto been impossible, except to those who 
 have spent a large proportion of their lives in mathe- 
 matical studies. I shall not, in this place, make any 
 remark upon the utility of such knowledge. Those who 
 already admit that the theory of probabilities is a desir- 
 able study, must of course allow that persons who cannot 
 pay much attention to mathematics, are benefited by 
 the possession of rules which will enable them to obtain 
 at least the results of complicated problems ; and which 
 will, therefore, permit them to extend their inquiries 
 further than a few simple cases connected with gambling. 
 By those who do not make any such concession, it will 
 readily be seen, that the point in dispute may be argued 
 in a more appropriate place than with reference to the 
 question whether others, who hold a different opinion, 
 
should, or should not, be supplied with a certain arith- 
 metical method. 
 
 The first six chapters of this work (the fourth, and 
 part of the fifth exclusive) may be considered as a 
 treatise on the principles of the science, illustrated by 
 questions which do not require much numerical com- 
 putation. To this must be . added the first appendix, 
 on the ultimate results of play. Omitting the first pages 
 of the latter, the discussion on the noted game of rouge et 
 noir will, with the problems in page 108. &c,, serve to 
 show the real tendency of such diversion. I am informed 
 that this game is not played in England at any of the 
 clubs which are supposed to allow, of gambling : but it 
 was permitted in the Parisian salons until the very 
 recent suppression of those establishments; and the ac- 
 count given of it will show what has taken place in our 
 own day. The game of hazard is more used in this 
 country; but I have been prevented from giving it the 
 same consideration by the want of a clear account of the 
 manner in which it is played. Nothing can be more 
 unintelligible than the description given by the cele- 
 brated Hoyle. 
 
 The fourth chapter has been already alluded to : it 
 contains the method of using the tables at the end of 
 the work in the solution of complicated problems. The 
 seventh chapter, and the fourth appendix, contain the 
 application of the preceding principles to instruments of 
 observation in general. 
 
 The remainder of the work is devoted to the most 
 common application of this theory, the consideration of 
 life contingencies and pecuniary interests depending 
 upon them, together with the main principles of the 
 management of an insurance office. As this portion was 
 
not written for the sake of the offices, but of those who 
 deal with them, I have confined myself to such points 
 as I considered most requisite to he generally known. 
 Common as life insurance has now become, the present 
 amount of capital so invested is trifling compared with 
 what will be the case when its principles are better un- 
 derstood ; provided always that the offices continue to act 
 with prudence until that time arrives. At present, 
 while 'the public has little except results to judge by, the 
 failure of an office would cause a panic, and perhaps re- 
 tard for half a century the growth of one of the most 
 useful consequences of human association : but the time 
 will come when knowledge of the subject will be so 
 diffused, that even such an event as that supposed, if it 
 could then happen, would not produce the same result. 
 
 There are, however, one or two things to which I 
 should call the attention of those whose profession it is 
 to calculate life contingencies : 
 
 1. The notation for the expression of such contin- 
 gencies (pp. 197 204.). This notation was suggested 
 by that of Mr. Milne, from which it differs in what I 
 believe to be a closer representation of the analogies which 
 connect different species of contingencies. Thus, an 
 annuity to last a number of years certain does not differ 
 from a life annuity in any circumstance which requires 
 a difference of notation ; nor an insurance from an an- 
 nuity certain of one year deferred till a life drops. 
 Since writing the pages above referred to, I have learned 
 that I was not the first who considered an insurance in 
 that light. Some years ago the government granted 
 annuities for terms certain, to commence at the 
 death of an individual ; but refused to insure lives : the 
 consequence was, that, by a very obvious evasion, insur- 
 
ances were effected by buying annuities for one year 
 certain, to commence at the death of a person named. 
 This had the effect of putting an end to such annuities. 
 
 C 2. The form of the rule for computing the value of 
 fines, and its introduction into the method of calculating 
 the present value of a perpetual advowson (pp.231. 236. 
 and Appendix the Second). It will be found that the 
 rule of every writer on the subject is palpably wrong 
 in principle, with the exception of that of Mr. Milne. 
 
 3. The rule for the valuation of uniformly increasing 
 or decreasing annuities, given in the fifth appendix. A 
 simple application of the differential calculus is made a 
 striking instance of the position, that the labour of a 
 person of competent knowledge is seldom lost. The 
 annuities given by Mr. Morgan and Mr. Milne, are for 
 every rate of interest, from three to eight per cent.; 
 and perhaps those gentlemen may have had some doubts 
 as to the necessity of inserting the two last rates. It 
 now appears, however, that, in consequence of the extent 
 to which their tables are carried, the values of increasing 
 or decreasing annuities, can be calculated with great 
 accuracy for three and four per cent., and with sufficient 
 nearness for five per cent. ; and with very little trouble, 
 compared with that which it must have cost Mr. Morgan 
 to calculate the table referred to in page xxviii. of the 
 Appendix. 
 
 The rules, in page xxix. of the Appendix, contain a 
 point which, as no demonstration is given, may cause 
 some difficulty. In turning an annuity or insurance 
 which cannot be extinguished during the life of the 
 party into one which can, a direction to add is given 
 which will at first sight, perhaps, be supposed to be a 
 mistake, and that subtract should be written instead. But 
 
Xll PREFACE. 
 
 it must be remembered that an annuity of, say 3 a year, 
 diminishing by l every year, is equivalent, by the 
 first part of the rule, to an annuity of which the suc- 
 cessive payments are as follows : 
 
 3, 2, l, 0, (-l), (2), (3), &c 
 
 That is, the first part of the rule, when the annuity is 
 extinguished during the tabular life of the party, gives 
 the value of his interest upon the supposition that he 
 is to begin to pay as soon as he ceases to receive. If 
 then, this is not to be the case, the value of his interest 
 must be increased accordingly. 
 
 4. The method of the balance of annuities, or the 
 determination of complicated annuities by the addition 
 and substraction of simple ones. This has been done 
 before ; but it has not, to my knowledge, been carried to 
 the extent of making all the questions which commonly 
 occur deducible from the fundamental tables, without 
 the aid of any new series. It is desirable that the 
 beginner should be accustomed to deduction by reasoning, 
 without having recourse to the mechanism of algebra, 
 which, as a quaint editor of Euclid observed, " is the 
 paradise of the mind, where it may enjoy the fruits of 
 all its former labours, without the fatigue of thinking/' 
 Of no part of algebra is this more true, than of the 
 method by which complicated annuities are deduced 
 from simple ones, by the resolution of the series which 
 represent them into the simpler series of which they are 
 composed. The education of an actuary does not neces- 
 sarily imply the study of geometry ; and such processes, 
 for instance, as those by which are found the values 
 of a contingent insurance or a temporary insurance 
 (pp. 222. 226.), will serve, as far as they go, to ac- 
 
custom him to make those efforts of , mind, and to hear 
 that tension of thought, the necessity for which is the 
 distinction between a problem of geometry, and one of 
 ordinary algebra. 
 
 The considerations contained in this volume have, in 
 my opinion, a species of value which is not directly de- 
 rived from the use which may be made of them as an 
 aid to the solution of problems, whether pecuniary or 
 not. Those who prize the higher occupations of intel- 
 lect see with regret the tendency of our present social 
 system, both in England and America, with regard 
 to opinion upon the end and use of knowledge, and the 
 purpose of education. Of the thousands who, in each 
 year, take their station in the different parts of busy life, 
 by far the greater number have never known real mental 
 exertion ; and, in spite of the variety of subjects which 
 are crowding upon each other in the daily business of 
 our elementary schools, a low standard of utility is gain- 
 ing ground with the increase of the quantity of instruc- 
 tion, which deteriorates its quality. All information be- 
 gins to be tested by its professional value; and the know- 
 ledge which is to open the mind of fourteen years old is 
 decided upon by its fitness to manure the money- tree. 
 
 Such being the case, it is well when any subject can 
 be found which, while it bears at once upon questions 
 of business, admits, at the same time, the application of 
 strict reasoning ; and by its close relation to knowledge 
 of a more wide and liberal character, invites the student 
 to pursue from curiosity a path not very remote from 
 that which he entered from duty or necessity. Such a 
 subject is the theory of life annuities, which, while it 
 will attract many from its commercial utility, can hardly 
 fail to be the gate through which some will find their 
 
XIV PREFACE. 
 
 way to the general theory of probabilities, and, perhaps, 
 from thence to the pursuit of other branches of science. 
 There are strong instances in favour of such a suppo- 
 sition. Many persons in this country have begun by 
 the common studies of an accountant, have been led to 
 an elementary knowledge of algebra and to the use of 
 logarithms by seeing the value of such information in 
 their particular pursuit, and have ended by becoming, 
 in many cases well informed, and in some instances 
 eminent, mathematicians. 
 
 Nothing is of more importance, as a help in holding 
 out every bait by which students may be drawn to the 
 exact sciences, than the co-operation of the universities ; 
 which, though they do not possess much power of intro- 
 ducing subjects into general study, yet have great influ- 
 ence in the settlement of the manner in which those 
 things shall be learned, the advantages of which have 
 been, or may be, felt by the community at large. If 
 ever it should happen that a particular branch of know- 
 ledge becomes in request, it would be of much advan- 
 tage if those institutions would forthwith appropriate and 
 liberalise it; to do which nothing more would be neces- 
 sary than to promote the study of it among their aspir- 
 ants to distinction. The consequence would be, that it 
 would find a place in the elementary works which so 
 frequently appear; and not only a place, but its place; 
 that is, in proper connection with other branches of 
 learning, and treated by methods which would preserve 
 that connection. Those who begin to study it in their 
 younger days for professional purposes would be led to 
 the method which bore the sanction of the universities, 
 and not unfrequently to the pursuit of other subjects 
 immediately connected with it. 
 
The theory of insurance, with its kindred science of 
 annuities, deserves the attention of the academical 
 bodies. Stripped of its technical terms and its com. 
 mercial associations, it may be presented in a point of 
 view which will give it strong moral claims to notice. 
 Though based upon self-interest, yet it is the most en- 
 lightened and benevolent form which the projects of 
 self-interest ever took. It is, in fact, in a limited sense, 
 and a practicable method, the agreement of a community 
 to consider the goods of its individual members as com- 
 mon. It is an agreement that those whose fortune it 
 shall be to have more than average success, shall resign 
 the overplus in favour of those who have less. And 
 though, as yet, it has only been applied to the repara- 
 tion of the evils arising from storm, fire, premature 
 death, disease, and old age ; yet there is no placing a 
 limit to the extensions which its application might re- 
 ceive, if the public were fully aware of its principles, and 
 of the safety with which they may be put in practice. 
 
 It is of great importance at the present moment that 
 sound principles on the subject of insurance should be 
 widely and rapidly disseminated. Within the last twenty 
 years, many new institutions have been founded; and 
 during the busy portion of the London year, seldom a 
 month passes without the announcement of a novel plan. 
 Of many of these projects I am unable to speak, from 
 not having paid particular attention to them. But of 
 one thing I am certain, that the magnificent style in 
 which the prospectuses frequently indulge might often 
 remind their readers of the unparalleled benefits which 
 are promised by another description of traders, who 
 vie with each other in describing the rare qualities 
 of their several blackings. If there be in this country 
 
a person whose ambition it is to walk in the brightest 
 boots to the cheapest insurance office, he has my pity: 
 for, grant that he is ever able to settle where to send his 
 servant, and it remains as difficult a question to what 
 quarter he shall turn his own steps. The matter would 
 be of no great consequence if persons desiring to insure 
 could be told at once to throw aside every prospectus 
 which contains a puff: unfortunately this cannot be done, 
 as there are offices which may be in many circumstances 
 the most eligible, and which adopt this method of ad- 
 vertising their claims. If these pompous announce- 
 ments be intended to profess that every subscriber shall 
 receive more than he pays, their falsehood is as obvious 
 as their meaning; if not, their meaning is altogether 
 concealed. 
 
 Public ignorance of the principles of insurance is the 
 thing to which these advertisements appeal: when it 
 shall come to be clearly understood that in every office 
 some must pay more than they receive, in order that 
 others may receive more than they pay, such attempts 
 to persuade the public of a certainty of universal profit 
 will entirely cease. To forward this result, I have en- 
 deavoured, as much as possible, to free the chapters of 
 this work which relate to insurance offices from mathe- 
 matical details, and to make them accessible to all edu- 
 cated persons. Whether they act by producing convic- 
 tion, or opposition, a step is equally gained : nothing 
 but indifference can prevent the public from becoming 
 well acquainted with all that is essential for it to know on 
 a subject, of which, though some of the details may be 
 complicated, the first principles are singularly plain. 
 
 August 3. 1838. 
 
CONTENTS. 
 
 CHAPTER L 
 
 On the Notion of Probability and its Measurement ; on the Province 
 of Mathematics with regard to it, and Reply to Objections - Page 1 
 
 CHAPTER II. 
 
 On Direct Probabilities - - - - - - 30 
 
 . CHAPTER III. 
 On Inverse Probabilities - - 53 
 
 CHAPTER IV. 
 
 Use of the Tables at the end of this Work - - 69 
 
 CHAPTER V. 
 
 On the Risks of Loss or Gain - ----- 93 
 
 CHAPTER VI. 
 
 On common Notions with regard to Probability - 112 
 
 CHAPTER VII. 
 
 On Errors of Observation, and Risks of Mistake - - - 128 
 
 CHAPTER VIII. 
 On the Application of Probabilities to Life Contingencies - - 158 
 
 CHAPTER IX. 
 On Annuities and other Money Contingencies - - 181 
 
 CHAPTER X. 
 On the Value of Reversions and Insurances . - - 12 
 
XV111 CONTENTS. 
 
 CHAPTER XI. 
 
 On the Nature of the Contract of Insurance, and on the Risks of 
 Insurance Offices in general - - Page 237 
 
 CHAPTER XII. 
 
 On the Adjustment of the Interests of the different Members in 
 an;lnsurance Office - - - - - 267 
 
 CHAPTER XIII. 
 
 Miscellaneous Subjects connected with Insurance, &c. - - 294 
 
 APPENDIX. 
 
 APPENDIX THE FIRST. 
 
 On the ultimate Chances of Gain or Loss at Play, with a particular 
 Application to the Game of Rouge et Noir i 
 
 APPENDIX THE SECOND. 
 
 On the Rule for determining the Value of successive Lives, and of 
 Copyhold Estates - - - xv 
 
 APPENDIX THE THIRD. 
 On the Rule for determining the Probabilities of Survivorship - xxii 
 
 APPENDIX THE FOURTH. 
 On the average Result of a Number of Observations - - xxiv 
 
 APPENDIX THE FIFTH. ' 
 
 On the Method of calculating uniformly decreasing or increasing 
 Annuities - - xxvi 
 
 APPENDIX THE SIXTH. 
 
 On a Question connected with the Valuation of the Assets of an In- 
 surance Office -.-.- xxxi 
 
 Table I. . , - - - - xxxiv 
 
 Table II. - - - - xxxviii 
 
AN ESSAY 
 
 ON 
 
 PROBABILITIES. 
 
 CHAPTER I. 
 
 ON THE NOTION OP PROBABILITY AND ITS MEASURE- 
 MENT ; ON THE PROVINCE OP MATHEMATICS WITH 
 REGARD TO IT, AND REPLY TO OBJECTIONS. 
 
 WHEN the speculators of a former day were busily 
 employed in constructing celestial tables for the use of 
 prophets, or investigating the qualities of bodies for 
 the manufacture of gold, no one could guess that they 
 were accelerating the formation of sciences which should 
 themselves be among the most essential foundations of 
 navigation and commerce, and, through them, of civilis- 
 ation and government, peace and security, arts and liter- 
 ature. That good plants of such a species require the 
 warmth of mysticism and superstition in their early 
 growth is not a rule of absolute generality, for there are 
 cases in which cupidity and vacancy of mind will do 
 as well. Cards and dice were the early aliment of 
 the branch of knowledge before us ; but its utility is 
 now generally recognised in all the more delicate branches 
 of experimental science, in which it is consulted as the 
 guide of our erroneous senses, and the corrector of our 
 fallacious impressions. And more than this, it is the 
 source from whence we draw the means of equalising the 
 
2 ESSAY -ON PROBABILITIES. 
 
 accidents of life, and contains the principles on which 
 it is found practicable to induce many to join together, 
 and consent that all shall bear the average lot in life of 
 the whole. But the ill educated offspring of a vicious 
 parent is frequently fated to* bear the stigma of his de- 
 scent, long after his own conduct has created the good 
 opinion of those who know him. The science which I 
 endeavour, and I believe almost for the first time, to ren- 
 der practically accessible in its higher and more useful 
 parts to readers whose knowledge of mathematics ex- 
 tends no farther than common arithmetic, is still often 
 considered as foreign to the pursuits, and dangerous in the 
 conduct, of life. It is said to be necessary only to gam- 
 blers, and calculated to excite a passion for their worthless 
 and degrading pursuit. This refers to its practical and 
 moral consequences : with regard to its title to confidence, 
 it is often supposed to rest upon pure conventions of an 
 uncertain order, and to depend for the connection of 
 results with principles upon the higher branches of ma- 
 thematics; things understood by very few, and frequently 
 distrusted, if not by those who have reached them, by 
 those who have passed some way up the avenue which 
 leads to them. All these impressions must necessarily 
 be removed before the theory of probabilities can occupy 
 its proper place ; and it is, therefore, my preliminary 
 task to meet the arguments which arise out of them. 
 There is an indefinite dislike in many minds to all know- 
 ledge which they cannot reach ; it may tend to remove 
 this if I show that results, at least, are very easily at- 
 tained, and methods practised: but the notion that 
 asserted knowledge is not knowledge must be met by 
 preliminary reasoning, and imperfect as it must neces- 
 sarily be, considered as a view of the subject, it may 
 yet afford the means of dwelling on the first principles 
 to a greater extent than is usually done in formal treatises 
 on recognised subjects. 
 
 Human knowledge is, for the most part, obtained 
 under the condition that results shall be, at least, of that 
 degree of uncertainty which arises from the possibility of 
 
INTRODUCTORY EXPLANATIONS. 3 
 
 their being false. However improbable it may be, for in- 
 stance, that the barbarians did not overturn the Roman 
 empire, we do not recognise the same sort of sensible cer- 
 tainty in our moral certainty of the fact which we have in 
 our knowledge that fire burns, or that two straight lines 
 do not enclose space. And we perceive a difference in the 
 quality of our knowledge, when any alteration takes place 
 in our circumstances with respect to exterior objects. 
 That fire does burn is more certain than the account of 
 the fall of Rome : that fire yet to be lighted will burn 
 may or may not be more certain than the historical fact, 
 according to the temperament and knowledge of the in- 
 dividual. And thus we begin to recognise differences 
 even between our (so called) certainties ; and the com- 
 parative phrases of more and less certain are admissible 
 and intelligible. It is usual to begin the subject by 
 saying that our certainties are only very high degrees of 
 probability. This is not practically true at the outset ; 
 yet so far as deductions can be made numerically, 
 with respect to our impressions of assent or dissent, it 
 will be shown to be correct so to consider the subject. 
 We have a process to go through before we can arrive 
 at such a conclusion, as follows : When a child is 
 born, there is a certain degree of force, which we allow 
 to the assertion that he will die aged 50. To it we 
 answer that it may be, but that that particular age is 
 unlikely compared with all the rest, though, at first 
 sight, as likely as any other. If the assertion be made 
 of two children, that one or other will die aged 50, 
 we^ readily admit that our " it may be, but it is not 
 likely," is no longer the same assertion as it was before. 
 It is of the same sort, but not of the same strength : the 
 assertion is more probable , and wherever we have the 
 notion of more and less, we feel the possibility of an answer 
 to the question, ' f how much more or less ? " and which 
 we should produce if we knew how. First impressions 
 would induce us to suppose it twice as probable that the 
 assertion may be made of one or other of two children, 
 as of one alone ; and so on. Let this false measure (for 
 B 2 
 
4 ESSAY ON PROBABILITIES. 
 
 such it is) remain ; we are not here considering what is 
 the proper measure, but whether we can conceive the 
 possibility of a measure or not. Let the preceding me- 
 thod of measurement be admitted ; and let us ask how 
 we stand with regard to the same assertion, predicated 
 of one or other of a million of children born together. 
 The answer is, we feel quite certain, that many of them 
 will die at the age of 50. Supposing humanity to en- 
 dure 50 years, we feel as confident of the truth of the 
 assertion, as we do that Rome was taken by Alaric, or 
 that fire will burn. Without entering into the very 
 different sources through which conviction comes to us, 
 we put four propositions together : 
 
 The Roman em- 
 pire was over- 
 turned by north- 
 ern barbarians. 
 
 Two straight 
 lines cannot 
 enclose a 
 space. 
 
 Fire will 
 burn. 
 
 Of 1,000,000 of 
 children born,some 
 will die aged fifty, 
 if the race of man 
 last fifty years. 
 
 and, we ask, if you were to receive a certain advantage 
 upon naming a truth from among these four assertions, 
 what would guide your choice ? There is certainly a 
 little difference in the impressions of assent with which 
 we regard the four; but whether it be of any real 
 strength, we may test in this way : Supposing the 
 benefit in question to be 1000/., would you not let 
 another person choose for you, almost at his pleasure, 
 and certainly for a shilling ? 
 
 On this we remark, firstly, that by it we feel sensible 
 of our assent and dissent to propositions derived in very 
 different ways, being a sort of impression which is of 
 the same kind in all. To make this clearer, observe 
 the following: A merchant has freighted a ship, which 
 he expects (is not certain) will arrive at her port. Now 
 suppose a lottery, in which it is quite certain that every 
 ticket is marked with a letter, and that all the letters 
 enter in equal numbers. If I ask him, which is most 
 probable, that his ship will come into port, or that he 
 will draw no letter if he draw, he will answer, unques- 
 tionably, the first, for the second will certainly not hap- 
 
INTRODUCTORY EXPLANATIONS. 
 
 pen. If I ask, again, which is most probable, that his 
 ship will arrive, or that he will, if he draw, draw either 
 
 o, or 6, or c, or x, or y, or z, he will answer, the 
 
 second, for it is quite certain. Now suppose I write the 
 following series of assertions : 
 
 He will draw no letter (a drawing supposed). 
 
 He will draw a. 
 
 He will draw either a or b. 
 
 He will draw either a, or 6, or c. 
 
 He will draw either a or b or or y. 
 
 He will draw either a or b or or y or %. 
 
 and making him observe that there are, of their kind? 
 propositions of all degrees of probability, from that which 
 cannot be, to that which must be, I ask him to put the 
 assertion that his ship will arrive, in its proper place 
 among them. This he will perhaps not be able to do, 
 not because he feels that there is no proper place, but 
 because he does not know how to estimate the force of his 
 impressions in ordinary cases. If the voyage were from 
 London Bridge to Gravesend, he would (no steamers 
 being supposed) place it between the last and last but 
 one : if it were a trial of the north-west passage, he 
 would place it much nearer the beginning ; but he would 
 find difficulty in assigning, within a place or two, where 
 it should be. All this time he is attempting to compare 
 the magnitude of two very different kinds (as to the 
 sources whence they come) of assent or dissent ; and he 
 shows by the attempt that he believes them to be of the 
 same sort. He would never try to place the weight of 
 his ship in its proper position in a table of times of high 
 water* 
 
 We also see, secondly, that the impression called cer- 
 tainty is of the character of a very high degree of 
 probability. Out of 1,000,000 of children born, it is 
 certain some will die aged 50. But by gradual pro- 
 gression, our unassisted judgment makes us believe 
 that we may correctly say that it is 1,000,000 times as 
 B 3 
 
O ESSAY ON PROBABILITIES. 
 
 / 
 
 probable the assertion will be true of one or other out of 
 1, 000; 000 as of one alone. The method of measuring 
 is wrong, but that is here immaterial; suffice it 
 that, come how it may, the multiplication of the degree 
 of assent implied in ff there is a remote chance of it" 
 is found to give that which is conveyed in fc we are 
 quite sure of it." We have thus a sort of freezing and 
 boiling point of our scale of assent and dissent, namely, 
 absolute certainty against on the one hand, absolute 
 certainty for on the other hand, with every description 
 of intermediate state. 
 
 Thirdly, we have proposed two ascending scales of 
 assertions, in both of which first impressions would 
 make us suppose the probability of the second is double 
 that of the first, that of the third treble, and so on, as 
 follows : 
 
 A child born will die aged fifty. 
 Of two children born, one or 
 
 other will die aged fifty. 
 Of three children born, one or 
 
 other will die aged fifty. 
 &c. &c. &c. 
 
 a must be drawn, 
 a or b must be drawn. 
 
 a, or b, or c must be drawn. 
 &c. &c. &c. 
 
 Now it will hereafter be positively proved that our 
 notion is correct in the second case, but incorrect in the 
 first ; or at least that it cannot be correct in both. 
 Even then, if we should fail in assigning positive mea- 
 surements, we may succeed in drawing useful distinctions. 
 When we imagine two things to have a point of re- 
 semblance which they have not, it is worth while to in- 
 vestigate methods of correction, even though we cannot 
 assign how much the two properties differ which we 
 supposed were alike. 
 
 The quantities which we propose to compare are the 
 forces of the different impressions produced by different 
 circumstances. The phraseology of mechanics is here 
 extended : by force, we merely mean cause of action, 
 considered with reference to its magnitude, so that it is 
 more or less according as it produces greater or smaller 
 effect. It is one of the most essential points of the 
 
INTRODUCTORY EXPLANATIONS. 7 
 
 subject to draw the distinction we now explain. Pro- 
 bability is the feeling of the mind, not the inherent 
 property of a set of circumstances. It is frequently 
 referred to external objects, as if it accompanied them 
 independently of ourselves, in the same manner as we 
 imagine colour, form, &c. to abide by them. Thus we 
 hold it just to say, that a white ball may be shut up in 
 a box, and whether we allow light to shine on it or not, 
 it is still a white ball. And if we were to translate the 
 common notion, we should also say that in a lottery of 
 balls shut up in a box, each ball has its probability of 
 being drawn inseparably connected with it, just as much 
 as form, size, or colour. But this is evidently not the 
 case : two spectators, who stand by the drawer, may be 
 very differently affected with the notion of likelihood in 
 respect to any ball being drawn. Say that the question 
 is, whether a red or a green ball shall be drawn, and 
 suppose that A feels certain that all the balls are red, 
 
 B, that all are green, while C knows nothing whatever 
 about the matter. We have here, then, in reference to 
 the drawing of a red ball, absolute certainty for or 
 against, with absolute indifference, in three different 
 persons, coming under different previous impressions. 
 And thus we see that the real probabilities may be dif- 
 ferent to different persons. The abomination called 
 intolerance, in most cases in which it is accompanied by 
 sincerity, arises from inability to see this distinction. 
 A believes one opinion, B another, C has no opinion at 
 all. One of them, say A, proceeds either to burn B or 
 
 C, or to hang them, or imprison them, or incapacitate 
 them from public employments, or, at the least, to libel 
 them in the newspapers, according to what the feelings 
 of the age will allow ; and the pretext is, that B and C 
 are morally inexcusable for not believing what is true. 
 Now substituting* for what is true that which A be- 
 lieves to be true, he either cannot or will not see that it 
 
 '* The refusal of this substitution is what soldiers call the key of A's position: 
 he himself sees the absurdity of his own arguments the moment it is made ; 
 and he is therefore obliged to contend for a sort of absolute truth external 
 to himself, which B or C, he declares, might attain if they pleased. 
 B 4 
 
8 ESSAY ON PROBABILITIES. 
 
 depends upon the constitution of the minds of B and C 
 what shall be the result of discussion upon them. Let 
 it be granted that the intellectual constitution of A, B, 
 and C is precisely the same at a given moment, and 
 there is ground for declaring that any difference of 
 opinion upon the same arguments must be one of moral 
 character. Granting, then, that it were quite certain A is 
 right, he might be justified in using methods with B 
 and C which are reformative of moral character ; that 
 is to say, granting that state punishments are reform- 
 ative of immoral habits, as well as repressive of im- 
 moral acts, he would be justified in direct persecution. 
 But to any one who is able to see with the eyes of his 
 body that the same weight will stretch different strings 
 differently, and with those of his mind that the same 
 arguments will affect different minds differently by 
 difference not of moral but of intellectual construction 
 will also see that the only legitimate process of alteration 
 is that of the latter character, not of the former ; 
 namely, argument * and discussion. In the mean time, 
 we bring it forward as not the least of the advantages 
 of this study, that it has a tendency constantly to keep 
 before the mind considerations necessarily corrective of 
 one of the most fearful taints of our intellect. 
 
 Let us now consider what is the measure of proba- 
 bility. Any one thing is said to measure another when 
 the former grows with the growth of the latter, and 
 diminishes with its diminution. For instance, in the 
 tube of a thermometer, the height of the mercury above 
 freezing point (a line) measures the content of a cy- 
 linder ; not that a line is a solid, but twice as much 
 length belongs to twice as much content, and so on. 
 Again, the content of the cylinder measures the quantity 
 of expansion in a given quantity of mercury (and in 
 this case not only measures, but is). Thirdly, the 
 
 * It is frequently asserted, that opinions dangerous to the existence of 
 public order must not be promulgated. This is a question distinct from the 
 one in the text, so far as it is political. If we grant no morals except expe- 
 diency, (which, it appears to us, is necessary for the affirmation of the pre- 
 ceding,) the answer is, simply, that persecution is ineffective. 
 
INTRODUCTORY EXPLANATIONS. 9 
 
 quantity of expansion measures the quantity of heat 
 which produces it. 
 
 The exactness of mathematical reasoning depends 
 upon that of our knowledge of the circumstances em- 
 ployed. No theorem about triangles, for instance,, is 
 true of any approach to a triangle such as we make on 
 paper ; but only more and more nearly true, the more 
 nearly we make our lines lengths without breadths, and 
 straight. Similarly, we cannot apply any theory of pro- 
 babilities to the circumstances of life, with any greater 
 degree of exactness than the data will allow. But as in 
 geometry we invent exactness by supposing the utmost 
 limits of our conceptions attainable in practice, so in the 
 present case we begin by reasoning on circumstances de- 
 fined by ourselves, and require adherence to certain 
 axioms, as they are called, meaning propositions of the 
 highest order of evidence. 
 
 AXIOM 1. Let it be granted that the impression of 
 probability is one which admits perceptibly of the 
 gradations of more and less, according to the circum- 
 stances under which an event is to happen. 
 AXIOM 2. Let it be granted that when one out of a 
 certain number of events must happen, and these 
 events are entirely independent of one another, the 
 probability of one or other of a certain number of 
 events happening must be made up of the probabi- 
 lities of the several events happening. For instance, 
 in the lottery of letters, in which there are 26 inde- 
 pendent possible events, the probability of drawing 
 either a, b, c, or d is made up of the probabilities 
 of drawing a, of drawing b, of drawing c, and of 
 drawing d, put together. 
 
 The latter axiom may excite some discussion ; but we 
 must observe that it is the uniform practice of mankind 
 to act upon it, which is a sufficient justification ; for 
 what are we doing but endeavouring to represent that 
 which actually exists ? With regard to the value of each 
 chance, suppose that one of the letters is a prize of 26/., 
 and that the 26 letters have been bought. If I buy 
 
10 ESSAY ON PROBABILITIES. 
 
 up all the vested interests at less than II. a piece,, I am 
 certain to gain ; if at more, I am certain to lose. 1 /. 
 a piece is what I ought to give for each, if I buy all : 
 it is the universal practice to consider that II. a piece is 
 still the value, if I buy a part. To say this is in fact 
 to say that the force of the impression called certainty 
 should, in this case, be considered as made up of 26 
 equal parts, each of which is to be considered as the 
 representative of the impression of probability which a 
 right-minded man would derive from the possession of 
 one ticket. 
 
 On this I have to remark, 1. That so soon as any 
 notion receives the exactness of mathematical language, 
 though it be thereby not altered, objections are taken to 
 it. The reason is, that we frequently not only use ex- 
 pressions which can be rendered quite exact, but also 
 fairly act upon them as if they were exact, but not be- 
 cause we consider them exact. Why does the lottery 
 ticket of the preceding instance bear the character of 
 being exactly worth 11. ? Not as any consequence of 
 the accuracy of the preceding process, supposing it ac- 
 curate, but because we do not know why we should 
 exceed rather than fall short of it. It appears to me 
 that many of our conclusions are derived from this 
 principle, which is called in mathematics the want of 
 sufficient reason. A ball is equally struck in two dif- 
 ferent directions, the table being uniform throughout. 
 In what direction will it move ? In the direction which 
 is exactly between those of the blows. Why ? No posi- 
 tive reason is assignable (experiment being excluded) ; 
 but from the complete similarity of all circumstances on 
 one side and the other of the bisecting direction, it is 
 impossible to frame an argument for the ball going more 
 towards the direction of one blow, which cannot imme- 
 diately be made equally forcible in favour of the other. 
 The conclusion remains, then, balanced between an in- 
 finity of possible arguments, of which we can only see 
 that each has its counterpoise. Now whether we adopt 
 the above conclusion as to probability for its exactness 
 
INTRODUCTORY EXPLANATIONS. 11 
 
 or for its want of demonstrable inexactness one way 
 more than the other, it is still a principle of human 
 action, and as such is adopted. Many writers on pro- 
 bability speak of it as being a maxim which, if it were 
 not adopted, ought to be. Certainly, such an assertion 
 has some strong arguments in its favour ; but with me 
 they would not outweigh the importance I should attach 
 to exact deduction from the conceptions which actually 
 prevail. 
 
 Let the prospect of drawing any given letter be 
 of a degree of force represented by 1, all the several 
 prospects being equal. Then 2 is the chance of draw- 
 ing one or other out of any given pair ; and so on 
 up to 26, which is here the representative of certainty. 
 But if the lottery had 50 letters, the prospect of draw- 
 ing a given letter would no longer be represented by 
 1 ; or if so, the certainty of drawing one out of 50 
 in the second would be represented by 50, while the 
 certainty of drawing one out of 26 in the first is repre- 
 sented by 26. Now certainty, absolute certainty, should 
 have the same representation whatever contingencies it 
 may be supposed to be compounded of. If a man be 
 sure of 100/., it matters nothing whether his certainty 
 arise from the announcement of a prize in a lottery of 
 1000 tickets, or of a legacy to which 20 other people 
 were looking forward. To use a common phrase, a man 
 can but be certain ; and therefore it would be desirable 
 to use the same symbol for certainty in all cases. Let 
 this symbol be unity or 1 ; then in the first lottery the 
 chance of any given letter is represented by -r^p and in 
 the second by -$ 1 J. Similarly the chance of 1 out of 10 
 given letters in the first lottery is -J-g, and in the se- 
 
 Now I pause upon this result, which, in fact, con- 
 tains all the theory I shall be obliged to use ; grant 
 this, and you can be constrained, by demonstration, to 
 admit all the rest as simple logical consequences. A 
 writer on this subject, therefore, must take care not to 
 let an opponent of its principles choose his own ground of 
 
12 ESSAY ON PROBABILITIES. 
 
 attack, so as to wait until he can take advantage of the 
 length of a deduction, or of the mathematical character 
 of the steps. Do you admit, 1. That a certainty, if you 
 have it, of drawing a 10/. prize in a lottery, is precisely 
 the same thing whether there be 100 or 1000 tickets? 
 and 2. That if there be 3 white balls and 17 black in a 
 lottery, of which either white ball is to be a prize, you 
 are compelled to regard your chances of success and 
 failure with impressions of which it is reasonable to 
 suppose the force to be as 3 to 17; or to say, (C the 
 degree in which I fear failure is, to my degree of hope 
 of success, in the proportion of 17 to 3." If you say 
 this, it matters nothing whether you say it because you 
 feel the correctness of the proposition, or because you 
 feel a want of data to deny it in one way more than in 
 the opposite. Provided only that you do not deny it, 
 your occupation of opponent is gone ; for all that suc- 
 ceeds is merely a mathematical use of this mathematical 
 definition. In the words of the ritual, Speak now, or 
 ever after hold your tongue. 
 
 But it may be asked, with regard to the mathematical 
 part of this subject, What is the province of the science 
 of calculation ? Are we, because we reject the higher 
 mathematics, entirely without evidence; or can we ob- 
 tain any thing like conviction of the truth of our me- 
 thods ? Now it happens unluckily for objectors, that 
 the duty of mathematics in this science is very much 
 more simple in character than the same in astronomy, 
 mechanics, optics, music, or any other part of mathe- 
 matical physics. For in the whole of these sciences, 
 we have principles, as well as results, deduced by long 
 trains of mathematical reasoning ; whereas, in the 
 science before us, we ask nothing of mathematics but the 
 abbreviation of long numerical operations. For in- 
 stance : " If bodies move round another body, circu- 
 Jarly, and so that each body, in its own circle, describes 
 equal lengths in equal times, and if, moreover, the 
 squares of the times of revolution are in the same 
 proportion as the cubes of the distances, then it follows 
 that the cause of motion can be nothing but an attractive 
 
INTRODUCTORY EXPLANATIONS. lS 
 
 force directed towards the central body, which, for dif- 
 ferent distances, changes inversely as the square of the 
 distances." This is well known to be a fundamental 
 part of the system of astronomy which has enabled one 
 century to do more towards correct prediction of the 
 state of the heavens than the twenty centuries which 
 preceded it; and yet the apparatus of mathematics 
 which is required to establish this result, which is of the 
 nature of a principle, is enormous. But in the present 
 subject we shall establish all our principles without the 
 aid of any more mathematics than is contained in arith- 
 metic ; and when we draw upon the science, it shall be 
 for nothing but abbreviation of long processes. The 
 principle upon which mathematical abbreviation fre- 
 quently proceeds is this : that where the calculation of 
 a few results materially aids the production of a great 
 many more, it is advisable to calculate a multitude of 
 results, to arrange them in convenient tables of reference, 
 and to publish them ; so that by means of one person 
 taking a little more trouble than would otherwise fall to 
 his share, all others may be saved labour altogether. 
 Mathematical tables are frequently nothing but the re- 
 sult of labour performed once for all ; but it also some- 
 times happens, that the principle on which the labour 
 is performed can be exemplifie.d by a familiar case of it. 
 "We shall take that of logarithms as an instance. 
 
 Every table of logarithms is an extensive table of com- 
 pound interest. Not to embarrass ourselves with* frac- 
 tions, let us take a table of cent, per cent, compound 
 interest. We have then the following amounts of II. in 
 1, 2, 3, &c. years: 
 
 Yrs. 
 
 Am. 
 
 Yrs.; 
 
 Am. 
 
 Yrs. 
 
 Am. 
 
 Yrs. 
 
 Am. 
 
 
 
 1 
 
 7 
 
 128 
 
 14 
 
 16384 
 
 21 
 
 2097152 
 
 1 
 
 2 
 
 8 
 
 256 
 
 15 
 
 32768 
 
 22 
 
 4194304 
 
 2 
 
 4 
 
 9 
 
 512 
 
 16 
 
 65536 
 
 23 
 
 8388608 
 
 3 
 
 8 
 
 10 
 
 1024 
 
 17 
 
 131072 
 
 24 
 
 16777216 
 
 4 
 
 16 
 
 11 
 
 2048 
 
 18 
 
 262144 
 
 25 
 
 33554432 
 
 5 
 
 32 
 
 12 
 
 4096 
 
 19 
 
 524288 
 
 26 
 
 67108864 
 
 6 
 
 64 
 
 13 
 
 8192, 
 
 20 
 
 1048576 
 
 27 
 
 134217728 
 
14 ESSAY ON PROBABILITIES. 
 
 The property of this table is, that if we wish to 
 multiply together any two numbers called amounts, we 
 have only to add together the number of years they 
 belong to, and look opposite the sum in the table of 
 years. Thus, 11 and 12, added together, give 23; 
 2048 and 4096, multiplied together, give 8,388,608. 
 The reason is as follows: if 11. in 11 years yield 2048/., 
 and if this 2048/. be put out for 12 years more, then, 
 since II. in 12 years yields 4096/., 2048 times as much 
 will yield 2048X4096/. ; or the amount in 11 + 12 
 years is the product of the amounts in 11 and 12 years. 
 The only reason why the preceding table is not in the 
 common sense of the words a table of logarithms, is, that 
 its construction leaves out most of the numbers. We 
 can deal with 2048 and 4096, but there is nothing 
 between them. The remedy is, to construct a table of 
 compound interest, at such an excessively small interest, 
 that a year shall never add so much as a pound through- 
 out. Certain considerations, by which the table may be 
 shortened, but with which we have here nothing to do, 
 make it convenient to suppose such a rate of interest, that 
 11. shall increase to 101. in not less than 100,000 years, 
 at compound interest. Or we may suppose interest pay- 
 able 1 00,000 times a year, and say, let the whole yearly 
 interest be 1000 per cent, per annum. Taking the first 
 supposition, we have a part of a table of logarithms as 
 follows : 
 
 Am. 
 
 Yrs. 
 
 Am. 
 
 Yrs. 
 
 Am ! Yrs. 
 
 Am. 
 
 Yrs. 
 
 1000 
 
 300043 
 
 5232 
 
 371867 
 
 9997 
 
 399987 
 
 10001 
 
 400004 
 
 1001 
 
 300087 
 
 5233 
 
 371875 
 
 9998 
 
 399991 
 
 10002 
 
 400008 
 
 1002 
 
 300130 
 
 5234 
 
 371883 
 
 9999 
 
 399996 
 
 10003 
 
 400013 
 
 1003 
 
 300173 
 
 5235 
 
 371892 
 
 10000 
 
 400000 
 
 10004 
 
 400017 
 
 &c. 
 
 &c. 
 
 &c. 
 
 &c. 
 
 
 
 
 
 This is the light in which a common reader may view 
 a table of logarithms. Let 1 increase to 10 at compound 
 interest in 100,000 equal moments, then 1 will become 
 5234 in 371,883 such moments; and so on. We can 
 thus manage to put down every number, within certain 
 
INTRODUCTORY EXPLANATIONS. 15 
 
 limits, as an amount ; and thus, within those limits, we 
 reduce all questions of multiplication and division to 
 addition and subtraction, by reference to the tables. 
 
 We thus perceive a simple principle applied with 
 much labour, bnt such as is performed once for all. 
 The notion above elucidated was the first on which 
 logarithms were constructed; in time came more easy 
 methods. We now take another abbreviation which is 
 perpetually occurring in our subject. It is the multi- 
 plication of all the successive numbers from 1 up to 
 some high number ; that is, the continuation of the pro- 
 cess following. Let [10], for instance, represent the 
 product of all the numbers, from 1 up to 10, both in- 
 clusive, or let 
 
 {10] stand forlx2x3x4x5x6x7x8x9x 10 = 3628800 
 
 [1] is 1 ; [2] is 2; [3] is 6; [4] is 24; [5] is 120; [6] 
 is 720 ; [7] is 5040 ; [8] is 40,320 ; and so on. This 
 labour becomes absolutely unbearable when the numbers 
 become larger; thus, [30] contains 33 places of figures, 
 and [1000] contains 2568 figures. But, nevertheless, 
 we cannot deal with problems in which there are 1000 
 possible cases without knowing, either nearly or exactly 9 
 the value of [1000]. It will, however, be sufficient 
 to know this value very nearly ; within, say, a thou- 
 sandth part of the whole ; that is, as nearly as when, the 
 answer of a problem being 1 000, we find something be- 
 tween 999 and 1001. We now put before the reader 
 who can use logarithms a rule for this approximation, 
 with an example ; intending thereby to show the reader 
 who does not comprehend the process how mathematics 
 enter this subject in the abbreviation of tedious comput- 
 ations. 
 
 RULE. To find very nearly the value of [a given 
 number], from the logarithm of that number, subtract 
 4342945, and multiply the difference by the given 
 number, for a first step. Again, to the logarithm of the 
 given number add '7981799^ and take half the sum, for 
 a second step. Add together the results of the first and 
 
16 ESSAY ON PROBABILITIES. 
 
 second steps, and the sum is nearly the logarithm of the 
 product of all numbers up to the given number inclusive. 
 For still greater exactness, add to the final result its ali- 
 quot part, whose divisor is 12 times the given number. 
 
 EXAMPLE. What is [30] or 1 x 2 X 3 X x 29 
 
 X30? 
 
 log. 30 1-4771213 
 4342945 
 
 Subtract 1-0428268 2)2-2753012 Add. 
 
 30 
 
 . 1 '1376506 Second step. 
 
 Multiply 31-284804 First step. 
 
 1 -137651 Result of second step. 
 
 32-422455 log. of result. 
 
 The result has, therefore, 33 places of figures, of 
 which the first six are (nearly) 264,518 ; or, if this be 
 increased by its 360th (12 times 30) part, or about 
 735, the result is 265,253, followed by 27 ciphers; or 
 
 the approximate result is 
 
 265253000000000000000000000000000 
 The true result is 
 
 2652528598121 91058636308480000000 
 
 and the error is not so much of the whole, as one part 
 out of 500,000. 
 
 In this way, we are able to do with more than 
 sufficient nearness, and in a few minutes, what it 
 would take days to arrive at by the common method, 
 and with much greater risk of error. 
 
 If we wish to find the product of all the numbers, 
 say from 31 to 100, both inclusive, we find [100] and 
 30] approximately, and divide the first by the second. 
 We shall represent this by [31,100] : thus, 
 
 [7,15] stands for 7x8x9x10x11x12x13x14x15 
 
 But, though we can thus simply put the logarithmic 
 computer in possession of a great acquisition of power 
 we can get through much the greater part of our task 
 without such a process, by means of a table of which 
 
INTRODUCTORY EXPLANATION. 17 
 
 it is not possible to explain either the principle, or the 
 reason of its utility., to any but a mathematician. We 
 can only explain its mere construction, as follows : 
 
 Let A B be one (inch, for example) ; and take an inde- 
 finitely extended line A X, perpendicular to A B : from 
 A towards X let a curve be conceived to be described, 
 so that every ordinate N P shall be connected with its 
 abscissa A N, by the following law. Measure A N in 
 inches and parts of inches; and multiply the result by 
 itself; and the product by 4342945. Find the num- 
 ber to which this product is the common logarithm, 
 and divide ! 12 84 by the result. The quotient is the 
 fraction of an inch in N P, and in the table we find, 
 not N P, but the area A N P B expressed as a fraction of 
 a square inch. The curve itself is what is called an 
 asymptote to A X, continually approaching, but never 
 reaching, AX : and the whole area, AX being continued 
 for ever, is one square inch. To this table I shall have 
 continual occasion to refer : into it, in fact, is condensed 
 almost the whole use I shall have to make of the higher 
 mathematics. 
 
 I have thus drawn the distinction between the prin- 
 ciples of the subject, as derived from very obvious 
 results of self-knowledge, and the principles of mathe- 
 matics, applied merely to the abbreviation of the tedious 
 operations which large numbers require. I now pro- 
 ceed to the several assertions which have been made 
 upon the nature and tendency of the subject. 
 
 I. That it is not true. The whole weight of this 
 assertion, and of all arguments in its favour, falls 
 entirely upon the method of measurement in page 11., 
 and ultimately upon the second axiom, in page 9. 
 Again, as we are most unquestionably justified in say- 
 ing that it is more probable we shall draw one of the 
 c 
 
18 ESSAY ON PROBABILITIES. 
 
 two, a or by than that we shall draw a, the argument 
 must be directed against the method of measurement, 
 not against the possibility of a measure : for wherever 
 more or less are applicable terms, twice, thrice, &c. 
 must be also conceived to be possible, whether we can 
 ascertain how to find them or not. But no other 
 method of measurement has ever been proposed, nor, in 
 truth, have the assertors been aware that they could be 
 brought to such close quarters, but have generally ob- 
 jected to the theory as a whole, without any particular 
 knowledge of its parts. It will be time enough to 
 refute their notion, when they begin to be so particular 
 that refutation becomes possible. 
 
 II. That it is not practical. By this it is either 
 meant, (for practical is one of the words employed in 
 shifting an argument, which are sometimes so con- 
 venient), that it has not been reduced to practical 
 form, or else that it is not capable of being so reduced ; 
 or perhaps that it is not useful. The working results 
 hitherto obtained may be divided into : 1 . The method 
 of obtaining probabilities. 2. The method of estimating 
 the probability of more or less departure from the 
 results indicated by the main branch of the theory as 
 most probable. The first has been frequently made 
 practical ; the second not hitherto, except to mathema- 
 ticians. That the whole can be made practical, I hope 
 to establish by the contents of this work. To the asser- 
 tion that it is not useful, we oppose : 1 . The unanimous 
 opinion of astronomers, (meaning thereby persons capa- 
 ble of applying the subject to astronomy) that the 
 exactness of our present knowledge is very much owing 
 to the application of it, and their uniformly continuing 
 to use it in the deduction of results from the necessary 
 discordances of observations. 2. The extent to which 
 it has been applied in the very choicest view of 
 the word practical, (which frequently means money- 
 making) in concerns which now employ many millions 
 sterling. 3. The light in which it is regarded by a 
 very large majority of those who have studied it, as a 
 
INTRODUCTORY EXPLANATION. 19 
 
 corrector of false impressions, and indicator of just and 
 necessary, though not always perceptible, distinctions. 
 
 4. The beauty of the study itself, considered merely as 
 a speculation, and as a method of exercising certain 
 powers of mind, which might otherwise lie useless. 
 
 5. The necessity of informing the public as to the real 
 nature of the occupation called gambling, and of the 
 class of men who live by it ; the latter being persons 
 who are using knowledge of these principles success- 
 fully, to the daily loss and ruin of those who are not 
 aware of what constitutes unequal play. If such argu- 
 ments be not sufficient to counterbalance a simple asser- 
 tion, to the extent of making it worth while to decide 
 the question by an examination of the subject itself, we 
 may safely dispute the utility of any branch of know- 
 ledge. 
 
 III. That it has a tendency to promote gambling. 
 Those who make this objection generally use the common 
 signification of the term gambling ; and the motives for 
 this pursuit are, in their view, either the pleasure of 
 suspense, acting as a stimulus to a mind weary of its 
 own vacancy, or the desire of gain. On the first notion, 
 the assertion is self-destructive ; it amounts to saying 
 that knowledge which diminishes suspense, by giving a 
 better view of the circumstances, has a tendency to 
 promote gambling, by affording the pleasure arising 
 from suspense. So far as the theory of probabilities 
 bears upon gaming in general, its tendency is to convert 
 games of chance into something more resembling games 
 of skill. Now games of skill are seldom made the ve- 
 hicles of very high play. So far, then, the tendency of 
 our study is to substitute the satisfaction of mental 
 exercise for the pernicious enjoyment of an immoral 
 stimulus. With regard to the desire, of gain, we may 
 safely admit that those who are already actuated by this 
 motive in an undue degree, will sometimes be led to 
 gamble by knowing how to do so properly ; and just in 
 the same manner some of them will be led to make 
 forgery the means of increasing their store, from knowing 
 
20 ESSAY ON PROBABILITIES. 
 
 how to write. But the fear that those who seek 
 a livelihood by what is commonly called gambling, 
 which always means cards, dice, or horse-racing, &c., 
 would be much increased in number, if at all, by such 
 a pursuit as the mathematical appreciation of proba- 
 bilities, seems to me grounded upon a want of know- 
 ledge of human nature. Putting out of view the 
 tendency of all serious thought to lead the mind to a 
 perception of its own resources, and to furnish methods 
 of employing time ; and not even considering that the 
 demand for this baneful excitement is controlled by the 
 opinion of society, and lessened by the amount of edu- 
 cation: there still remain the means of showing that 
 the baknce is in favour of a study of the theory of 
 probabilities, even as a preventive of this very gambling 
 which it is said to provoke. Nemo repente fuit tur- 
 pissimus : and, it will be one of our objects to show, 
 that the person who lives by gaming, deserves the 
 strongest form of the adjective. No one ever said to 
 himself, I have not played hitherto, but I will begin 
 henceforward to make it my trade. A young man who 
 is ruined by play in the first instance, or who, at least, 
 has begun by courting as an amusement what he ends 
 by requiring as an occupation, is the subject of which 
 a gambler is made. Now, suppose that ah 1 those who 
 have been ruined by play had been trained to under- 
 stand the true nature of their pursuit. Let it be 
 granted that some of them are so fond of acquisition, 
 that it is only necessary to point out a plausible method 
 to insure their following it : yet we must grant, on the 
 other side, that there will be some who can be per- 
 suaded that when they play against a bank or a 
 gamester, they are almost certain of playing on very 
 unequal terms, which is never what they contemplate 
 and intend. The only question is, which of the two 
 numbers will be the greatest ; 1 . Of those who be- 
 come gamesters prepense, or, 2. Of those who either 
 take a total or a partial warning; -the latter in a 
 degree sufficient to insure a fair chance for them- 
 
INTRODUCTORY EXPLANATION. 21 
 
 selves. The thoughtlessness of youth will be urged 
 against my opinion, that the latter number would be very 
 much the greatest. I reply, that, comparatively speak- 
 ing, and with respect to maturer years, young men 
 are thoughtless; but, absolutely speaking, they are not 
 so with respect to dangers of which they know the 
 risks. The ih 1 success of others does not deter them, be- 
 cause they attribute it to fortune ; and, because they have 
 superstitions hanging about them with respect to luck 
 which are tolerably prevalent in all classes. They think 
 they are trying their luck, as the phrase is; but if 
 they could be convinced that it is not their luck 
 which they are trying, but only a fraction of it, 
 their opponent having the rest in his pocket, they 
 would show themselves in this, as in other matters, 
 averse to risks in which it is more than an even 
 chance against them. They come to the consideration 
 of the subject fraught with wrong notions, which have 
 been carefully instilled as preventives. The character 
 of a gambler is represented as dishonest, in the com- 
 mon sense of the word. That is to say, the term 
 gambler is confounded with that of sharper, meaning a 
 person who would mark a card, or load a die. They 
 find the falsehood of this notion in their commerce 
 with the world : gamblers show themselves in the face 
 of day who really appear to be, and are, men of ho- 
 nour in the common sense of the word, and who would, 
 scorn any under-handed proceeding, under ordinary 
 temptation at least. What then becomes of the pre- 
 vious warning ? It is proved to be false in an essential 
 part ; and is therefore lost altogether. Add to this 
 that the principle of the occupation is misrepresented : 
 admonition is given against trying fortune, instead of 
 proof that fortune is not tried. A proposition is ad- 
 vanced which is an absurdity : equal play is supposed, 
 and yet it is maintained that the luck will generally be 
 against the inexperienced. Skill is considered as only 
 adding to the chances against the unskilful, instead of 
 creating a certainty, and arguments drawn from a single 
 c 3 
 
22 ESSAY ON PROBABILITIES. 
 
 game,, which are really good, are applied to collections 
 of many games, with regard to which they are not ap- 
 plicable. I will leave it to any one to say, whether the 
 considerations pointed out in the succeeding pages have 
 the tendency to promote the pursuit of fair gaming as 
 a means of profit. 
 
 With regard to gambling as a stimulus, it must be 
 observed, that the passion has every where subsided with 
 the increase of education and occupation. If the his- 
 torians who write for schoolboys could spare a little space 
 among their interminable accounts of kings, treaties, 
 battles, to insert some account of the manners of the 
 several ages of Europe, it would be matter of surprise 
 that the universal rage for games of chance, should have 
 left any time for the (so called) great actions which fill 
 the books. The wars of the middle ages would be 
 looked upon as belonging only to one particular class of 
 the stimuli by which the universal vacancy was sought 
 to be filled up. From the old Germans, who played 
 away, to one another, their wives, their children, and 
 lastly themselves, down to the time * of the French re- 
 volution, the continent of Europe (and during a part 
 of the time, Great Britain, though in a less degree,) 
 gives, comparatively to ourselves, the notion of succes- 
 sive races devoted to gambling throughout the upper 
 class, the only one upon whose occupations we get fre- 
 quent details. 
 
 IV. That the basis of it is an irreligious principle. 
 There is a word in our language with which I shall 
 not confuse this subject, both on account of the dis- 
 honourable use which is frequently made of it, as an 
 imputation thrown by one sect upon another, and of the 
 variety of significations attached to it. I shall use the 
 word anti-deism to signify the opinion that there does 
 not exist a Creator, who made and sustains the universe. 
 The charge is, that a theory of probabilities (called 
 chances) is necessarily anti-deistical, because it refers 
 
 * Quanri, avant la revolution francaiae, les etats de certaines provinces 
 etaient assembles, on y jouait un jeu terrible, et tel que 1'endroit oii il se 
 tt'nait; dans la ci-devant province de Bre'tagne, s'appellait Veitfer Diet. 
 0es Jeux (Encyc. Meth.) 1792. 
 
INTRODUCTORY EXPLANATION. gg 
 
 events to chance. Various modifications of this asser- 
 tion present themselves, but they may all be referred 
 either to that just made, or to a tendency argument of 
 the same character. All the sciences have had to 
 encounter this aspersion, each in its turn ; but it is to be 
 remarked, that philosophy and philosophers have always 
 been charged with the worst thing going. The believers 
 in sorcery never failed to attribute an intimate connection 
 with infernal spirits to all who investigated nature in 
 any form : the believers in anti-deism follow in their 
 steps. There is in the proposition above mentioned, a 
 shifting of the meaning of terms : it has been customary 
 to designate anti-deism as the opinion that the world was 
 made by chance, meaning, without any law or purpose 
 existing ; but the word chance *, in the acceptation of 
 probability, refers to events of which the law or purpose 
 is not visible. Thus a great part of the application of 
 this subject has been destroyed by successive discoveries. 
 When the observatory at Greenwich was founded, the 
 chance errors of observation were large in the fixed stars. 
 Nothing could be said but that there was a deviation 
 which appeared of one sort in one observation, and of 
 another in another, without visible law or order. Brad- 
 ley's discoveries removed much of this, that is, pointed 
 out law where law was not seen to exist before. Im- 
 provement of instruments and methods of observation 
 has still more distinguished the error into parts with a 
 visible, and parts with an invisible, cause. As an 
 answer to the species of argument employed, nothing 
 more is necessary : those who can, may consider this 
 science as not bearing on religion, either in one way 
 or the other, so far as anything in the preceding argu- 
 ment is concerned, or in the explanation which is no 
 more than necessary for an answer. But there is a view 
 of the subject, and that one most indispensable, which 
 
 * Generally speaking, the abstract singular term chance has the anti- 
 delstic meaning, while the plural chances is used for the several possibilities 
 of an event happening. Thus Hume says : '* Though there be no such 
 
 thing as chance in the world there is certainly a probability which 
 
 arises from a superiority of chances." 
 
 c 4 
 
24 ESSAY ON PROBABILITIES. 
 
 better deserves to be made a fundamental principle, than 
 an incidental answer to a futile objection. The past 
 contains our grounds of expectation for the future : 
 Why ? Because we cannot help supposing that there 
 were causes which produced the past, and which con- 
 tinue to act. If there be any one to whom this is not 
 a truth, he cannot proceed with us one step. Suppose 
 that 100 drawings out of an urn all give white balls, the 
 presumption is very strong that the 101st will give a 
 white ball also. But if there neither be, nor ever were, 
 some reason why the balls so drawn should be white 
 rather than black, that is, if the event be pure chance, 
 then the 100 drawings afford no presumption whatever 
 that the 101st will be either white or black. So far 
 then as we have yet gone we have the following positive 
 and negative conclusion : 
 
 The theory of probabilities 
 absolutely requires, in its fun- 
 damental principles, the rejec- 
 tion of the notion that pure 
 chance can produce any two 
 events alike ; that is, it pre- 
 sumes causation and order of 
 some kind or other, that is, 
 providence of some kind or 
 other. 
 
 The theory of probabilities, 
 so far as considerations of ab- 
 solute necessity are concerned, 
 neither denies nor asserts, in 
 whole or in part, any thing 
 whatsoever respecting the mo- 
 ral or intellectual character of 
 the providence which it re- 
 quires to be granted. 
 
 From the preceding we may be certain that no con- 
 clusion in any way leading to natural religion, however 
 faint, is tacitly assumed in the premises. If there- 
 fore such a conclusion should follow legitimately, it 
 stands upon a basis of absolute security. This is not 
 often the case in arguments drawn from nature in gene- 
 ral, on account of the mixture of considerations with 
 which the mind is affected by them. When we speak 
 of the vastness, the regularity, and the permanency of 
 the solar system in general, the very immensity of the 
 argument would prevent the mind from being aware whe- 
 ther there was or was not either an appeal to constitutional 
 feeling, distinct from reason, or even an assumption of 
 the question in the manner of deducing it. The cele- 
 
INTRODUCTORY EXPLANATION. 25 
 
 brated work of Paley may be considered as a treatment 
 of the following syllogism : ' f If there be contrivance, 
 there was design; but there is contrivance, there- 
 fore there was design," the minor of which is proved by 
 appeal to observation. But the author refers the 
 opponent to the beauty and ingenuity of the methods in 
 which the contrivance is brought about : to the general 
 effect on our notions of what is done, compared with what 
 we could do. It may very often be discovered what the 
 real tenor of an argument is, by observing what would 
 refute it : now imagine an individual possessed with the 
 notion that he could execute * better contrivances, and 
 Paley 's argument must (to him) be imagined to be ineffec- 
 tive, t It appears to me that the result of the treatise in 
 question is this : f{ If there be a contriver, he must be 
 one of infinite power and intellect." But the argument of 
 contrivance against chance cannot, from the complication 
 and non-numerical character of the instances, be illustrated 
 by any reference to what might have been if chance had 
 prevailed. Taking, for example, the chamois, as the 
 result of a contrivance for the support of animal life on 
 frozen mountains, we have no method of comparing the 
 chamois of design with any notion that we can form, 
 and call the chamois of chance. But where a consider- 
 ation is pure number, we then have other ideas, of the 
 homogeneity of which with that in question, we feel 
 assured : and we can absolutely try the question with 
 chance in precisely the same manner as we try it in the 
 common affairs of life. Let us assume, as we must, 
 that a number produced by chance alone (in the anti- 
 deistical sense of the word,) might as well have been 
 any other as what it is. And further, let us require 
 before we grant intelligence and contrivance, not merely 
 the presence of an adaptation which would have been 
 unlikely from chance alone, but two such phenomena, 
 
 * Such as is said actually to have struck Alfonso of Portugal, when the 
 Ptolemaic theory of the heavens was explained to him. 
 
 f The fault of "most treatises on Natural Theology is to draw the reader's 
 attention from the mere design, to the complication and ingenuity of the 
 design. The Bridgwater Treatises have a consistent title, and it is worthy 
 of remark, that this was the doing of the testator himself, 
 
26 ESSAY ON PROBABILITIES. 
 
 perfectly distinct from each other considered as phe- 
 nomena, each of which might have existed without 
 the other, and both tending to the same object, which 
 would have been defeated by the absence of either. 
 Let it also be granted, to fix our ideas, that we admit as 
 proved, a proposition which has a hundred million to one 
 in its favour. 
 
 This being premised, and laying it down as our object 
 to show that the necessary results of the theory of pro- 
 babilities lead to the conclusion that the existence of 
 contrivance is made at least as certain, by means of it, 
 as any other result which can come from it, we proceed 
 to state a consequence : The action of the planets upon 
 each other, and that of the sun upon all (the most cer- 
 tain law of the universe), would not produce a perma- 
 nent * system unless certain other conditions were fulfilled 
 which do not necessarily follow from the law of attrac- 
 tion. The latter might have existed without the former, 
 or the former without the latter, for any thing that we 
 know to the contrary.t Two of these conditions are, 
 that the orbital motions must all be in the same direc- 
 tion, and also, that the inclinations of the planes of these 
 orbits must not be considerable. Granting a planetary sys- 
 tem which is what ours is, in every respect except either 
 of these two, and it is mathematically shown that such 
 a system must go to ruin : its planets could not preserve 
 their distances from the sun. Neither of these phe- 
 nomena can be shown to depend necessarily on the 
 other, or on any law which regulates the system in 
 general. For any thing we know to the contrary, then, 
 they are distinct and independent circumstances of the 
 organisation of the whole. Now let us see what are 
 the phenomena in question : 
 
 * Permanent, not liable so to change as to destroy the organisation of 
 the parts. If the earth could ever approach so near to the sun that all the 
 water should be vaporised, the permanency of the system would be de- 
 stroyed, so far as our planet is concerned 
 
 t The only way in which we can guess any two things to be independent 
 It must be remembered, as a result of the theory, that, ot things perfectly 
 unknown, the probability of their coming to act, when known, against an 
 argument, is counterbalanced by the equal probability of the future dis- 
 covery being on the other side. 
 
INTRODUCTORY EXPLANATION. 27 
 
 I. All the eleven planets yet discovered move in one 
 direction round the sun. 
 
 II. Taking one of them (the earth) as a standard, the 
 sum of all the angles made by the planes of the orbits of 
 the remaining ten with the plane of the earth's orbit, is 
 less than a right angle, whereas it might by possibility 
 have been ten right angles. 
 
 Now it will hereafter be shewn that causes are likely 
 or unlikely, just in the same proportion that it is likely 
 or unlikely that observed events should follow from them. 
 The most probable cause is that from which the observed 
 event could most easily have arisen. Taking it then as 
 certain that the preceding phenomena would have fol- 
 lowed from design, if such had existed, seeing that they 
 are absolutely necessary, ceteris manentibus, to the main- 
 tenance of a system which that design, if it exist, actu- 
 ally has organised, we proceed to inquire what prospect 
 there would have been of such a concurrence of circum- 
 stances, if a etate of pure chance had been the only ante- 
 cedent. With regard to the sameness of the directions 
 of motion, either of which might have been from west 
 to east, or from east to west, the case is precisely similar 
 to the following : There is a lottery containing 
 black and white balls, from each drawing of which it is 
 as likely a black ball shall arise as a white one ; what 
 is the chance of drawing eleven balls, all white. Answer, 
 2047 to 1 against it. With regard to the other question, 
 our position is this. There is a lottery containing an 
 infinite number of counters marked with all possible dif- 
 ferent angles less than a right angle, in such manner that 
 any angle is as likely to be drawn as another; so that in 
 10 drawings the sum of the angles drawn may be any 
 thing under 10 right angles. What is the chance of 10 
 drawings giving collectively less than 1 right angle ? 
 Answer, ] 0,000,000 to 1 against it. Now what is the 
 chance of both of these events coming together ? An- 
 swer, more than 20,000,000,000 to 1 against it. It is 
 consequently of the same degree of probability that there 
 has been something at work which is not chance, in the 
 
28 ESSAY ON PROBABILITIES. 
 
 formation of the solar system. And the preceding does 
 not involve a line of argument addressed to our percep- 
 tions of beauty or utility, but one which is applied 
 every day, numerically or not, to the common business 
 of life. 
 
 Now whether what precedes amounts to the means 
 of producing rational conviction, it is not necessary 
 for me to stop and inquire. The question is, how 
 do the results of this theory affect those moral and 
 intellectual considerations which it has been stated to 
 have a tendency to overthrow ? It matters nothing to 
 my present purpose how much of the preceding a reader 
 will admit ; for the point, considered with reference 
 to the objection before us, is this : Does the preceding 
 deduction weaken the probability of the 'existence of an 
 intelligent creative power ? for if not, the objection is 
 overthrown, and whatever strength is conceded to belong 
 to the reply is so much addition to other arguments 
 in favour of the same conclusion. Let us suppose a 
 reader so much biassed against the higher parts of 
 mathematics that he does not feel any confidence in 
 the united work of all ages and countries amounting 
 to more than a millionth of certainty. There still re- 
 mains 20,000 to 1 in favour of the conclusion above 
 stated, after weakening the preceding by introducing a 
 probability that all the exact sciences may be wrong, 
 such as his state of mind requires. With respect to the 
 bearings of the theory, we may now add the follow- 
 ing to the statement in page 24. 
 
 Applying the first principles of the theory of proba- 
 bilities, by means of mathematics, to the phenomena of 
 the universe, it is a necessary conclusion that the ex- 
 istence of something which combines together different 
 and independent arrangements to produce an end which 
 could not, ceteris manentibus, be produced without 
 them, must be added to the notion of a Providence, in- 
 telligent or not, which is required in the first prin- 
 ciples. 
 
 With regard to the existence of a revelation from the 
 
INTRODUCTORY EXPLANATION. 29 
 
 Supreme Being, this theory leaves the question exactly 
 where it found it ; and the same of all questions of his- 
 torical evidence. If we were to assume fictitious data, 
 we might, as in all other sciences of inference, produce a 
 consequence which should be as true as the premises, 
 standing or falling with them. The science itself is 
 the deduct'on of the probability in a complicated case 
 from the probability in a known and simple case. But 
 where is the known and simple case in the historical 
 question ? In valuing testimony, no theory of the 
 method in which conflicting evidence should be com- 
 bined will help us to the original value of the several 
 parts of it, any more than an investigation of the 
 method of solving an equation will help us to a know- 
 ledge of the particular equations which apply in any 
 given case. 
 
 The two great theoretical questions before us are : 
 I. What is the measure of probability ? II. What is 
 the way of using it ? The necessary preliminary to 
 application is, the result of the measurement in a case 
 to which the method of measuring can be applied, and 
 has been applied. The mistakes which have arisen from 
 confounding these considerations are numerous. For in- 
 stance, tell me how many times per cent, a given man will 
 be wrong in his judgment, and I can tell you exactly, 
 positively, and mathematically, how much more likely a 
 unanimous jury (not starved) is to have arrived at a true 
 decision, than another in which the voices are 8 to 4. 
 But that does not put me one step nearer to ascertaining 
 what is the per centage of erroneous conclusions in 
 the judgments of a single individual. The miscon- 
 ceptions just alluded to are equally prevalent with regard 
 to all the sciences ; a person who studies astronomy is 
 frequently asked what the moon is made of. 
 
 Much of the objection, religious or not, made against 
 probability in general, is connected with the notion 
 already mentioned, (page 7.) that it is a fundamental 
 quality of events, external to ourselves, which is under 
 consideration : on which the rational feeling must be, 
 
SO ESSAY ON PROBABILITIES. 
 
 that there is no such thing. The term probability is 
 as difficult to explain as gravitation : and the method of 
 proceeding is the same with regard to the properties of 
 both. We cannot tell what they are, in simpler terms, 
 but we know them by, or rather trace and define them 
 by, their manifestations. In both, we first see a com- 
 pound result, depending upon the patient as well as the 
 agent. In the case of the mental phenomena, we can- 
 not decompose the effect produced, still less ascend a 
 step, and find any of the laws which regulate the hu- 
 man disposition to doubt or expect. I shall conclude by 
 again reminding the reader, that the impression produced 
 by circumstances upon his own mind is the thing in 
 uestion ; and that nothing can be more liable to cause 
 confusion than a lurking notion that the results of theory 
 are anything more, before the event arrives, than a re- 
 presentation of the relative force of his own impressions, 
 as they should be if unassisted reason could follow the le- 
 gitimate consequences of some simple and universally 
 admitted principles. 
 
 I proceed in the next chapter to develope the leading 
 rules of the science. 
 
 CHAP. II. 
 
 ON DIRECT PROBABILITIES. 
 
 WE now proceed on the supposition that the probability 
 of an event is measured by the fraction which the 
 number of favourable cases is of all that can happen. 
 Thus, if there be 20 white balls and 27 black (20 + 27 
 or 47 in all), the probability of drawing a white ball is 
 measured by -J-Q-, and that of drawing a black ball by ~^L. 
 We shall say that these probabilities are |^ and |^. 
 Nothing is more common than to substitute a measure 
 
ON DIRECT PROBABILITIES. 31 
 
 for the thing itself: thus we talk of a temperature of 
 60, when, in fact, each 1 only means a certain length 
 on the tube of the thermometer. It will illustrate this 
 to take a case in which we do not confound the thing 
 and its measure; in the barometer, for instance, we 
 never say that the air's weight is 30 inches, but that the 
 height of the barometer is 30 inches. 
 
 The technical words, probability and improbability, 
 must now be considered as meaning the same thing in 
 different degrees. If there be only one white ball out 
 of a thousand, we usually say that to draw the white 
 ball is possible, but not probable. We now speak of it 
 as having a small probability, namely TO^T) ', we might 
 say it has a great improbability, namely -f-^^j but this 
 phrase is not customary. The moment we obtain either 
 numerical measures, or distinctions which are not verbal, 
 the distinctions which are verbal frequently become su- 
 perfluous and inconvenient. Thus in the art of book- 
 keeping, profit and loss never appear as separate words, 
 but only as part of a complex term prqfit-and-loss, 
 meaning, one or the other, according to the side of the 
 account on which the item is found. To a mercantile 
 reader, we should say that probability means probability- 
 and-improbability the first or second, according as its 
 measure is greater or less than -J-. When the number 
 of favourable and unfavourable cases is the same, say 50 
 of each, the probability of the event is T 5 ^ or \ ; and in 
 this case we say in common life that there is a balance 
 of probabilities, or that the event has an even chance. 
 
 By the word chance with the article (a chance) we 
 mean one single way in which an event may happen, as 
 when we say that every white ball adds a chance to the 
 prospect of drawing a white ball. In the first instance 
 above, the chances of white and black are as 20 to 27. 
 It is also usual to say that the odds are here 27 to 20 in 
 favour of black against white. 
 
 Questions on probability are twofold in character : 
 1 , Where we know the previous circumstances and re- 
 quire the probability of an event. 2. Where we know 
 
32 ESSAY ON PROBABILITIES. 
 
 the event which has happened, and require the proba- 
 bility which results therefrom to any particular set of 
 circumstances under which it might have happened. 
 The first I call direct, and the second inverse, questions. 
 
 We must begin with direct questions,, though the in- 
 verse precede the direct in practice. For, as we know 
 the whole range of possible cases in hardly one instance, 
 we cannot proceed with points which have reference to 
 matters of life until we know what presumptions arise 
 with respect to the whole, from observation of a part. 
 
 In direct questions of probabilities, the event may 
 either be simple, that is, depending on one indivisible 
 event ; or compound, consisting of several events which 
 may happen together. Thus, suppose four events, either 
 of which may happen, and call them A, B, C, D. 
 Knowing the circumstances of each, I may ask the fol- 
 lowing questions, every one of which states an event 
 simple or compound. 
 
 1. What is the chance of A. 2. What is the chance 
 that one shall happen and only one. 3. What is the 
 chance that one or more will happen. 4. What is the 
 chance that one at least will happen, and one at least 
 will not. 5. What is the chance that a given pair, and 
 no others, will arrive. 6. What is the chance that a 
 given pair at least will arrive. 7. What is the chance 
 that some pair or other will arrive, but only a pair. 
 8. What is the chance that a pair at least will happen, 
 &c. &c. All these are most evidently distinct questions 
 when they are clearly proposed; but it is almost as 
 evident that they are very liable to be confounded. 
 
 The mathematical definitions and theorems which 
 will be necessary for our purpose, are the following : 
 
 1. A permutation means a number of cases selected 
 out of all possible cases, in some particular order ; so 
 that different arrangements of the same things make dif- 
 ferent permutations. Thus if all the possible cases be 
 A, B, C, and D, we have the following 
 
 Permutations of one out of four, A, B, C, and D. 
 
ON DIRECT PROBABILITIES. 33 
 
 Permutations of two out of four, AB, BA, AC, CA, AD, 
 DA, BC, CB, BD, DB, CD, DC. 
 
 Permutations of three out of four, ABC, ABD,BAC,BAD, 
 ACB, ACD, CAB, CAD, ADB, ADC, DAB, DAC, 
 BCA, BCD, CBA, CBD, BDA, BDC, DBA, DBC, 
 CDA, CDB, DCA, DCB, 
 
 Permutations of four out of four (different arrangements of 
 four), ABCD, ABDC, BACD, BADC, ACBD, ACDB, 
 CABD, CADB, ADBC, ADCB, DABC, DACB, BCAD, 
 BCDA, CBAD, CBDA, BDAC, BDCA, DBAC, DBCA, 
 CDAB, CDBA, DCAB, DCBA, 
 
 To find the number of permutations of one number 
 (m) out of another (n), begin with the whole number 
 (n) and write down as many numbers, reckoning down- 
 wards . as there are units in the number (m) which are to 
 be in each permutation : then multiply all together. 
 For instance : How many permutations are there of 4 
 out of 12. The answer is. 
 
 12 x 11 x 10 x 9 or 11880 
 
 The following table will furnish examples, or serve 
 for reference. 
 
 1 
 
 10 j 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 L 
 
 4 
 
 3 1 2 1 1 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 3 
 
 2 
 2 
 
 1 
 
 2 
 
 90 
 
 72 
 
 56 
 
 42 
 
 30 
 
 20 
 
 12 
 
 6 
 
 
 3 
 
 720 
 
 504 
 
 336 
 
 210 
 
 120 
 
 60 
 
 24 
 
 6 
 
 
 
 4 
 
 5040 
 
 3024 
 
 1680 
 
 840 
 
 360 
 
 120 
 
 24 
 
 i 
 
 
 
 5 
 
 30240 
 
 15120 
 
 6720 
 
 2520 
 
 720 
 
 120 
 
 
 
 
 6 
 7 
 
 151200 
 
 60480 
 
 20160 
 
 5040 
 
 720 
 
 
 
 
 
 604800 
 
 181440 
 
 40320 
 
 504O 
 
 
 
 
 
 
 8 
 
 1814400 
 
 362880 
 
 40320 
 
 
 
 
 
 
 
 
 
 9 
 10 
 
 3628800 
 
 362880 
 
 
 
 
 
 
 
 
 3628800 
 
 
 
 
 1 
 
 
 
 
 
34 ESSAY ON PROBABILITIES. 
 
 Thus the number of permutations of 6 out of 9 is 
 opposite to 6 under 9> or 60480. 
 
 2. By a combination is meant the same thing as a 
 permutation, except only that arrangement is no part 
 of the idea. Thus, of all the permutations of three out 
 of four in A, B, C, D, the following, A B C, B A C, 
 ACB, CAB, BCA, and CB A, are all the same 
 combination. Thus we have 
 
 a permutation : a selection in a certain order. 
 
 a combination : a selection without reference to order. 
 
 To find the number of combinations, divide the 
 number of permutations by the product of ail the 
 numbers up to the number in each combination. Thus, 
 how many combinations can be made of 4 out of 
 12 ? The answer is found by 
 
 dividing 12x11x10x9"! ,. , . 
 
 by 1* 2x 3 x 4 j whlch S' Tes 495 ' 
 
 The process may always be shortened by striking 
 common factors from the dividend and divisor. Thus, 
 if we wish to know the number of different hands which 
 can be held at whist (combinations of 13 out of 52) we 
 must 
 
 divide 52. 51. 50. 49. 48. 47. 46. 45. 44. 43. 42. 41. 40. 
 by 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 
 
 The shortest way is to decompose every number in 
 both into its factors, when it will appear that all the 
 factors of the divisor are found" among those of the di- 
 vidend; as follows: 
 
 IS. 2.2. 17.3.5.5.2.7.7. 2,2.3.2.2. 47. 2.23. 5.3.3. 11.2.2. 
 
 43. 2.7.3. 41. 2.2.2.5. 
 
 1. 2. 3. 2.2. 5. 3.2. 7. 2.2.2. 3.3. 5.2. 11. 2.2.3. 13. ; 
 
 and the factors which must be multiplied to give the 
 result are 
 
 17. 7. 47. 23. 5. 43. 2.7. 41. 2.2.2.5. 
 or 17. 47. 23. 43. 41. 35. 560 giving 635013559600. 
 
 When the number in each combination is more than 
 half of the whole number, the rule may be shortened. 
 
ON DIRECT PROBABILITIES. 35 
 
 Thus, if I ask how many combinations of 21 can be taken 
 out of 25, I do in effect ask how many combinations 
 of 4 may be taken. For there are just as many ways 
 of taking 2 1 as there are of leaving 4. 
 
 The following table corresponds to the one preced- 
 ing : 
 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 1 1 
 
 1 
 
 1 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 2 
 
 45 
 
 86 
 
 28 
 
 21 
 
 15 
 
 10 
 
 6 
 
 3 
 
 1 
 
 
 3 
 
 120 
 
 84 
 
 56 
 
 35 
 
 20 
 
 10 
 
 4 
 
 1 
 
 
 
 4 
 
 210 
 
 126 
 
 70 
 
 35 
 
 15 
 
 5 
 
 1 
 
 
 
 
 5 
 
 252 
 
 126 
 
 56 
 
 21 
 
 6 
 
 1 
 
 
 
 
 
 6 
 
 210 
 
 84 
 
 28 
 
 7 
 
 1 
 
 
 
 
 
 
 7 
 
 120 
 
 36 
 
 8 
 
 1 
 
 
 
 
 
 
 
 8 
 
 45 
 
 9 
 
 1 
 
 
 
 
 
 
 
 
 9 
 10 
 
 10 
 1 
 
 1 
 
 _ 
 
 
 
 ~ 
 
 
 
 
 
 
 
 
 
 
 Thus the combinations of 5 out of 9 are 126 in 
 number. 
 
 3. When the same event may be repeated in a per- 
 mutation,, the number of permutations is the product 
 of as many numbers, all equal to the whole number of 
 possible events, as there are of events in each permutation. 
 Thus, of four events, A, B, C, D, the number of per- 
 mutations of two, with repetition, is 4 x 4 or 1 6 ; with- 
 out repetition 4 x 3 or 12. The additional 4 in the 
 first case, are A A, B B, C C, and D D. The number 
 of permutations of three, with repetition, is 4 x 4 x 4, 
 or 64 without repetition, 4x3x2 or 24. Of the addi- 
 tional 40, 10 are where A only is repeated; namely, A A B, 
 A A C, A AD, A B A, A C A, A D A, B A A, C A A, 
 D 2 
 
36 
 
 ESSAY ON PROBABILITIES. 
 
 D A A, that is nine where A is repeated twice, and 
 A A A, one, where A is repeated three times. Ten 
 more are those in which B is repeated twice, &c. 
 
 4. The number of combinations in which there are 
 repetitions must he determined without rule, in every 
 particular case. Suppose I wish to know how many com- 
 binations, including repetitions, can be made of four 
 out of seven. Let the seven be A, B, C, D, E, F, G. 
 
 I. The number, without repetition, is 35. 
 
 II. Those in which A is repeated twice, and no other, 
 are evidently as many as the number of pairs in B, C, 
 D, E, F, and G, that is, J 5. Thus we have A A B C, 
 A A C D, &c. There are as many in which B is re- 
 peated twice, &c., so that 7 X 15 or 105, is the num- 
 ber in which one only is repeated twice. 
 
 III. The number in which A is repeated three times 
 is evidently 6, and the number in which one or other is 
 repeated three times only, is 6 X 7 or 42. 
 
 IV. The number in which one is repeated four 
 times, is 7. 
 
 V. The number in which two are repeated twice, is 
 as many as the number of pairs in 7j or 21. 
 
 Consequently, the whole number is made up of 35 
 105, 42, 7, and 21, or it is 210. 
 
 We have not so much to do with combinations allow- 
 ing repetition, as with permutations of the same kind. 
 
 To avoid the perpetual occurrence of long phrases 
 for simple ideas, I shall use the following abbrevia- 
 tions: By P [4,20 } is meant the number of per- 
 mutations of 4 out of 20, without repetition : by P P 
 { 4,20 ] the same with repetition. By C [ 4,20 ] 
 is meant the number of combinations of 4 out of 20, 
 without repetition : by C C [ 4,20 j the same with 
 repetition. Again, by [4,20], as in page 15., is meant 
 the product of all numbers from 4 to 20, both inclu- 
 sive ; and, by 7 10 , as in algebra, is meant ten sevens 
 multiplied together. Thus, a reference to the preceding 
 rules will show the following : 
 
ON DIRECT PROBABILITIES. 37 
 
 P |5, 12? is [12,8] , PP/5, 12J is 12' 
 
 c|4, 25 1- is [25, 22] divided by [1, 4] 
 
 5. T there be an event composed of several others in 
 succession, of which the first may happen in, say 10 
 different ways, the second in 14, and the third (let there 
 be but three) in 6, then the compound event must be 
 one out of 10 x 14 X 6, which is the number of dif- 
 ferent ways in which it may happen. When all the 
 component events may happen in the same number of 
 ways, this reduces itself, both in principle and rule, to 
 the case of permutation with repetition. 
 
 I now proceed to some problems requiring nothing 
 but a knowledge of the measure of probability and the 
 preceding rules. Any event of known circumstances 
 may be familiarly represented by a lottery of balls of 
 different colours. Thus, suppose ten Russian ships, 
 twelve French, and fourteen English, are expected in 
 port, or may have arrived. Let one be as likely to ar- 
 rive as another, and suppose it known that two have 
 arrived, but not of which nation they are. Let a cer- 
 tain advantage accrue to A, if they should happen to be 
 Russian and French, of which he is desirous of selling 
 his chance immediately. The question is precisely the 
 following : Let there be a lottery of green, white, and 
 red balls, 10, 12, and 14 in number. Let two have 
 been drawn, and suppose a certain advantage to arise to 
 A, if they be green and white. What should be given 
 to A, certain, in consideration of his chance ? 
 
 Here we must first consider the number of ways in 
 which two can be drawn. All the balls are 36 in num- 
 ber, and the whole number of pairs is 36 X 35 -=- 2 
 or 630. Of these a green ball (1 out of 10) may be 
 paired with a white ball (1 out of 12) in 120 different 
 ways. Consequently, the probability of his gaining the 
 advantage is f g or - ? \. The probability against it is 
 therefore ~Hr : or out of every 2 1 possible events, 4 are 
 in ,favour of, and 17 against, the advantage. It is there- 
 D S 
 
38 ESSAY ON PROBABILITIES. 
 
 17 to 4-, or 4| to 1, against the advantage. If the con- 
 tingent gain were 2 11. the value of the chance would be 
 4/., as will hereafter be more fully shown. 
 
 Of all the pairs which can be drawn, there are 
 
 10 x 1 2 = 1 20 green and white. 10 x 9 -r- 2 = 45 both green. 
 12 x 14=168 white and red. 12 x ll-s-2 = 66 both white. 
 14 x 10= 140 red and green. 14 x 13 -f- 2 = 91 both red. 
 
 The sum of these is 630,, as it should be. The reader 
 should explain to himself the reason of the difference 
 of process. The least probable case is both green, 
 the most probable white and red. Nothing is more 
 common than the idea that the event most likely to 
 happen, which is compounded of two or more events, is 
 a repetition of the event which is individually the most 
 probable. This is true of repetitions : red being most 
 probable, it is more probable that red should be repeat- 
 ed than that white should be repeated, or that green 
 should be repeated ; but that two white ones should be 
 drawn is not so probable as that a white and red should 
 be drawn. The drawing, whatever it Tnay be, is con- 
 sidered independently of succession and this makes an 
 important difference. If the balls were drawn succes- 
 sively, both red is more probable than white followed 
 by red, or than red followed by white, but not more 
 probable than the chance of one or other, which is the 
 preceding case. 
 
 I here also take occasion to notice the common 
 error, that because an event is more probable than any 
 other, it is the one to be looked for. The question 
 ought to be, Is that event more probable than some one 
 or other out of all the other events which may happen ? 
 If ten persons engage in a competition, with an equal 
 chance of success, and if two of them, A and B, enter 
 into partnership, it is now more probable that the firm 
 of A and B will win than that C will win, or that D 
 will win, &c. But the chances against the firm are 
 still 8 to 2 or 4 to 1. If a hundred halfpence be 
 tossed up into the air, the result which is more probable 
 than any other, is 50 heads and 50 tails. But com- 
 mon sense will tell us that the chances of this result are 
 
ON DIRECT PROBABILITIES. 39 
 
 very small, out of the whole, and calculation would con- 
 firm it. 
 
 Before laying down any more specific rules, I shall 
 take an instance of a somewhat complicated deduction, 
 in order to show that we do not really require any 
 other principle than is contained in the measure of 
 probability. Suppose A and B to play at whist against 
 C and D. A begins, holding ace, king, and queen of 
 trumps, and these trumps only : what right has he to ex- 
 pect that, hy playing them out in succession, he will force 
 all the trumps of the other party ? That is to say, ten 
 trumps being distributed among B, C, and D, and any 
 single card having the same chance of belonging to either 
 of the three, what is the probability that neither C nor 
 D holds more than three trumps ? 
 
 Firstly, as to the number of tenures, which are per- 
 fectly distinct. To find the number of ways in which any 
 number of distinct objects can be divided among any 
 number of persons, use the following RULE : 
 
 Multiply together numbers equal to the number of 
 persons as often as there are things to be divided among 
 them. Thus, to find in how many different ways ten 
 distinct cards can be divided among three persons, 
 find 
 
 3x3x3, &c. (ten threes) or 3 10 which is 59049. 
 
 The question now is, how many of these 59049 
 ways favour the supposition that neither C nor D holds 
 more than three out of ten. Both together they cannot 
 hold more than six : if, then, we pick out any given 
 six of the ten trumps, that set may be divided 
 among C and D in 2 6 or 64 ways. But of these, there 
 are two ways in which and 6 may be held, twelve 
 ways for 1 and 5, and thirty ways for 2 and 4, none 
 of which must be included. It must be observed, that 
 in the last sentence, the six distinct ways into which 
 6 may be divided into parcels of 1 and 5 must be 
 doubled, because * each gives two of our cases, ac- 
 
 * It is important to observe that no duplication must take place on this 
 account, if the two numbers be the same; for instance, in dividing 6 into 
 parcels of 3. 
 
 D 4 
 
40 ESSAY ON PROBABILITIES* 
 
 cording as C holds 1 and D holds 5, or C holds 5 and 
 D holds 1 : and so of the other cases. Consequently, 
 out of a given set of six, there are 64 all hut 44, that 
 is, 20 ways, in which, when they happen, A could force 
 all the adversary's trumps. But every set of six yields 
 20 such cases : hence the ways in which C and D can 
 together hold six trumps, are 20 times C (6,10 j or 
 20 X 210, or 4200. Similarly, a given set of 5 trumps 
 may be held by C and D in 2 5 or 32 ways, of which 2 
 and 10 must be excluded. Hence 20 x C ( 5,10 } 
 or 20 x 252, or 5040 sets are the number favourable 
 to the event in this case. Four trumps can be held 
 in 2 4 or 16 ways, of which 2 must be excluded, or 14 
 X C {4,10] that is, 2940, is the number of favour- 
 able cases. On the supposition that C and D together 
 hold only 3, or 2, or 1 trump, no exclusions are neces- 
 sary, and the number of cases are 2 3 x C {3,10J 
 or 960, and 2 2 X C { 2,10 j or 180, and 2 X 
 C [l,!0] or 20; and there is one case in which C 
 and D hold no trumps. All the favourable cases are, 
 therefore, in number 
 
 4200 + 5040 + 2940 + 960+180+20+1 or 13341. 
 The chance of A being able to force all the adversary's 
 trumps is -j-gjJ-J-g, or nearly 3 1 / to 1 against it. 
 
 Given a fraction less than unity, and which has high 
 numbers in its terms, required a set of fractions which 
 shall be very nearly equal to it, and each of which shall 
 be nearer than any other fraction of the same order 
 of simplicity. Required, also, a near estimate of the 
 error committed in each case. 
 
 RULE. First perform the process for finding the 
 greatest common measure of the numerator and deno- 
 minator. 
 
ON DIRECT PROBABILITIES. 
 
 1 13341)59049(4=4 
 
 53364 
 
 2 5685)13341(2X4+1=M 
 
 11370 
 
 2x2+1=5 1971)5685(2x9+4=22 
 
 39*2 
 
 5x1+2=7 1743)1971(1 x 2+9=31 
 
 1743 
 
 7x7+5=54. 228)1743(7x314-22=239 
 
 1596 
 
 54x1+7=61 147)228(1x239+31=270 
 
 147 
 
 81)147(1 
 81 
 
 66 &c. 
 
 Opposite to the first quotient write 1, and proceed tc 
 form columns out of the several quotients, in the fol- 
 lowing manner : 
 
 1st Quotient = 1st Denomina- 
 tor. 
 
 2d Qu. x 1st Qu. + 1 = 2d 
 Den. 
 
 3d Qu. x2d Den. + 1st Den. 
 = 3d Den. 
 
 4th Qu. x 3d Den. + 2d Den. 
 
 1 = 1 st Numerator. 
 2nd Qu. = 2d Num. 
 
 2d Num. x 3d Qu. + 1st Num. 
 
 = 3d Num. 
 3d Num. x 4th Qu. + 2d Num. 
 
 = 4th Num. 
 
 4th Num. x 5th Qu. + 3d Num. 
 = 5th Num. 
 
 Any numerator multiplied by 
 the next quotient, and pro- 
 duct increased by preceding 
 numerator, gives succeeding 
 numerator. 
 
 = 4th Den. 
 
 5th Qu. x 4th Den. + 3d Den. 
 = 5th Den. 
 
 Any denominator multiplied 
 by the next quotient, and 
 product increased by pre- 
 ceding denominator, gives 
 succeeding denominator. 
 
 Thus, 1, -|, -%, J_, -^-g, -f^ } &c. &c., is a set of 
 fractions which approach nearer and nearer to -Jilo^-g- 
 The first is always too great, the second too small, the 
 third too great, the fourth too small : every odd one too 
 great, every even one too small. 
 
 Test of correctness. Take any two successive numer- 
 ators and denominators, multiplied crosswise ; they give 
 products which differ by unity. 
 
42 * ESSAY ON PROBABILITIES. 
 
 54 61 54x270 = 14580 
 
 239 270 239 x 61=14579 
 
 1 
 
 Estimation of the error. The error of J is less than 
 3 V (4 X 9 = 36) ; that of f is less than T J^- (9 X 
 22 = 198) ; that of -^ is less than ^ (22 X 31 = 
 682), &c. 
 
 any Numerator . 1 
 
 The error of : = : is less than 
 
 its Denominator That Den. x the next. 
 
 I now take the following problem : A die is thrown 
 time after time; in how many times have we an 
 even chance of throwing an ace. The common error 
 attached to this problem is, that since there are six 
 faces, it is most likely all will have come up in six 
 throws. 
 
 In the first throw there are six events, five of which 
 are unfavourable. In the first two throws, considered 
 as giving one event, there are 6x6 or 3 6 possible 
 cases, for every possible case of the first throw may com- 
 bine with any case of the second. But of these 36 
 throws, any one of the five unfavourables of the first 
 throw may combine with any one of the second throw, 
 and there are therefore 5 x 5, or 25 unfavourable com- 
 pound events. Hence the following table : 
 
 One throw gives 6 cases ; 5 unfavourable 
 
 Two throws give 36 cases ; 25 unfavourable 
 
 Three 216 ; 125 (odds still 
 
 against.) 
 
 Four 1296 ; 625 (odds 
 
 turned in favour.) 
 
 Five 7776 ; 3125 &c. 
 
 Answer : There is not quite an even chance of doing it 
 in three throws, but more than an even chance of doing 
 it in four. 
 
 RULE. When the odds against success in one trial 
 are n to 1, then T 7 () of n. (or the nearest whole num- 
 ber to it) is about the number of trials in which there 
 is an even chance of one success: more correctly -J--J of 
 n. Thus, if it be 144 to 1 against a single attempt, 
 
ON DIRECT PROBABILITIES. 
 
 there is about an even chance of one success in 100 
 throws. 
 
 A table of the number of trials (odds being n. to 1) 
 in which there are various odds of succeeding once : 
 
 1 to 1 
 
 69 
 
 13 to 1 
 
 264 
 
 30 to 1 
 
 343 
 
 1000 to i 
 
 691 
 
 
 
 2 to 1 
 
 110 
 
 14 to 
 
 271 
 
 40 to 1 
 
 371 
 
 2000 to 1 
 
 760 
 
 
 
 3 tol 
 
 139 
 
 15 to 
 
 277 
 
 50 to 
 
 393 
 
 3000 to 1 
 
 8D1 
 
 
 
 4tol 
 
 161 
 
 16 to 
 
 283 
 
 60 to 
 
 411 
 
 4000 to 1 
 
 829 
 
 
 
 Stol 
 
 179 
 
 17 to 
 
 2X9 
 
 70 to 
 
 426 
 
 5000 tol 
 
 852 
 
 
 
 6 tol 
 
 195 
 
 18 to 
 
 294 
 
 80 to 
 
 439 
 
 6000 to 1 
 
 870 
 
 
 
 7 to I 
 
 208 
 
 19 to 
 
 300 
 
 90 to 
 
 451 
 
 7000 to 1 
 
 885 
 
 
 
 8 to 1 
 
 220 
 
 20 to 
 
 304 
 
 100 to 1 
 
 462 
 
 8000 tol 
 
 899 
 
 
 
 9 to 1 
 
 230 
 
 21 tol 
 
 309 
 
 110 tol 
 
 471 
 
 9000 tol 
 
 911 
 
 
 
 lOtol 
 
 240 
 
 22 to 1 
 
 314 
 
 120 to 1 
 
 48j) 
 
 10,000 to 1 
 
 921 
 
 
 
 11 tol 
 
 248 
 
 23 tol 
 
 318 
 
 130 to 1 
 
 488 
 
 
 
 
 
 12 to 1 
 
 256 
 
 24 to \ 
 
 322 
 
 140 to 1 
 
 495 
 
 
 
 
 
 The method of using this table is as follows : Sup- 
 pose the odds to be 20 to 1 against success in a single 
 trial : required in how many throws it is 100 to 1 there 
 shall be one success (or more). Look in the table 
 opposite to " 100 to I/' and we see 4-62 ; multiply 20 
 by 462 and divide ly 100 (which is always to be done), 
 this gives 92 fa, or 92 is about the number of throws 
 required. 
 
 I now proceed to the method of compounding the 
 probabilities of single events, so as to find those of com- 
 pound events : that is, the way of methodising the re- 
 sults of actual inspection. 
 
 Let there be two events, one of which may happen in 7 
 ways, and may not happen in 5; and the other of which 
 may happen in 4 ways, and may not happen in 9 ' 
 whence the probabilities of the two events (A and B) 
 are T V and -*-%. Now, the compound event may hap- 
 pen in 12 X 13 different ways, for any case of the first 
 (12 in all) may come up with any case of the second 
 (13 in all). By similar reason, the compound case in 
 which both A and B happen may come up in 7 X 4 
 different ways ; hence the probability of A and B both 
 happening, is 
 
 -jlil which is the product of 1 and _ 
 RULE. When events are perfectly independent, so 
 
44 ESSAY ON PROBABILITIES. 
 
 that the happening of one has nothing to do with that 
 of the other, the probability that both will happen is 
 the product of the probabilities that each will happen. 
 
 This rule applies to any number of independent 
 events. 
 
 It is indifferent whether the events are to happen to- 
 gether, or one after the other : thus the chances of a 
 compound drawing out of two lotteries, one drawing 
 out of each, are the same whether two persons draw 
 simultaneously, or one person first draws out of one, and 
 then out of the other. 
 
 EXAMPLE. Let there be three lotteries, as follows : 
 f 6 white! f 7 white "I f 8 white! 
 \ 5 black J \ 2 black J \ 1 black J 
 
 What is the chance of drawing from the three, white, 
 black, and white ? The probabilities of these events are, 
 _6_^ ^ an( j _&_ } the product of which is ^V^, or about 1 7 
 to 1 against the compound event. 
 
 The probability that A will happen, and B will not, 
 is the product of the chance for A, and that against B. 
 The probability that one will happen, and one only, is 
 the sum of the probabilities 1. that A will happen and 
 B will not ; 2. That A will not happen and B will 
 happen. The probability that one or both will hap- 
 pen is the remainder when the probability that neither 
 will happen is subtracted from unity. The following 
 are evident results of the measure of probability : 
 
 1. When either P, Q, or R must happen, the sum of 
 their probabilities must be unity, which is always the 
 representative of certainty. Thus, if a lottery contain 7 
 white, 5 black, and 3 red, either white, black, or red 
 must be drawn, and -fa, -j 5 3 , and -fa, together make 1, 
 and the same if the events be more or less than three in 
 number. 
 
 2. When of two events, each excludes the other, 
 the probability that one or other of them will happen is 
 the sum of their probabilities. For to say that each 
 excludes the other, is to say that they are connected 
 events, or different possible cases of the same set. Thus, 
 
ON DIRECT PROBABILITIES. 45 
 
 in the preceding lottery, if I draw white I cannot draw 
 black, and vice versa, there being one drawing only 
 supposed. Hence the probability of drawing black or 
 white is T T ^ -f- T 5 - or i| . But suppose there had been 
 two lotteries, as follows : 
 
 (7 white, 8 red) (5 black, 10 red), 
 
 and I draw in one without knowing in which I am 
 drawing. The individual probabilities of white in the 
 two, if the lottery be known, are -$ and -f^ as before ; 
 but the question, what is the chance of drawing white 
 or black, and not red, is a different one from the pre- 
 ceding. If I draw without knowing in which lottery I 
 am drawing, my position is the same as if one lottery 
 had been thrown into the other, and I had drawn from 
 both in one, giving 
 
 (7 white, 5 black, 18 red). 
 
 This union, which will readily be admitted to make no 
 alteration in the probabilities, is not admissible unless 
 the total number of balls in both be the same. For, in 
 the mixture, the 8 red balls of the one and the 1 of 
 the other, are considered as severally of equal proba- 
 bility. But this they are not, unless the number of 
 possible cases in the two were the same. Suppose, for 
 instance, I had the following lotteries : 
 
 (10 white, 10 red) (4 black, 5 red) ; 
 
 the probability of each ball of the first is 1 TT , but of 
 each of the second -^. Before I can draw a white ball, 
 two events must happen. 1 . I must happen to select 
 the first lottery. 2. I must happen to draw a white 
 ball rather than a red. The chances of these are -i and 
 |-{{- or l ; consequently, \ X \ or -*- is my chance of a 
 white ball. But if I mix the two lotteries, that chance 
 will be I", which is more than J. The reason is, that I 
 have mixed together balls which had unequal chances of 
 being drawn, and treated them as if they were to have 
 equal chances. But it is evident, from the measure of 
 probability, that I do not alter the chances for an event 
 
40 ESSAY ON PROBABILITIES. 
 
 or the general chances which any set of circumstances 
 affords, if I multiply the chances for an event any num- 
 ber of times, provided I multiply the chances against it 
 as often. Reduce the preceding lotteries to common num- 
 bers of balls, by putting 20 balls for one into the second, 
 and 9 for one into the first. We then have 
 (90 white, 90 red) (80 black, 100 red). 
 Now mix the two, and we have 
 
 (90 white, 80 black, 190 red) : 
 
 and the chance of a white is -$-$ 9 or -i, that is, the pro- 
 blem is not altered by the mixture. And the same may 
 be shown in any other case. 
 
 Let us now suppose that the chance of A is and 
 that of B . The chances against A and B are there- 
 fore J- and -|. 
 
 The possible cases are The probability of which is 
 That A and B shall both happen x f or ^ 
 
 That A shall happen, and not B | x | or 5 \ 
 
 That B shall happen, and not A $ x f or 5 5 T 
 
 That neither shall happen. x f or 5 2 T 
 
 One of these cases must happen, and the sum of the 
 chances is -T or 1, each compound event being ex. 
 elusive of the others. Again, we find 
 
 (~Both or neither is 
 One or other, but only one , 
 One or other, or both 
 
 B or both } B or neither 5 \ 
 j A or neither, or both i| 
 ^B or neither, or both || 
 
 The latter set of events does not consist of those which 
 are mutually exclusive. Any thing which happens falls 
 under six of them, that is, six of them must happen. 
 Thus, A, and not B, actually arriving, would secure any 
 gain which depended upon either of the following : 
 
 One or other, but only one A, or both 
 
 One or other, or both A, or neither 
 
 One or neither A, or neither, or both, 
 
ON DIBECT PROBABILITIES. 47 
 
 If we add together all the probabilities of the preceding, 
 we find l -ff } or 6. This is the indication that every 
 possible event enters in six different ways into the 
 contingencies whose probabilities united amount to 
 six. 
 
 EXAMPLE. Twelve halfpence, A x A 2 . . . A x 2 , are 
 thrown up, required the probability of all the cases 
 which can happen, and which we shall symbolise thus : 
 (HsTg) means that there are three heads and nine 
 tails. The chance of H or T in any one piece is -A; 
 consequently, the chance of the several pieces A 19 A , 
 &c., yielding each any particular letter is 1 x 4- X~-J 
 X (twelve factors), or T0 ^. Now, the 4096 pos- 
 sible cases are thus classified : 
 
 T n ...... H n 
 
 TT >y TT 
 
 H 5 T 7 H 7 
 
 H 6 T 6 happens in 924 cases. 
 
 It appears, then, that the most likely individual re- 
 sult is He T 6 , against which, however, it is about 31 
 to 1 . But if we ask for the probability either of this 
 or of a single variation on one side or the other, that is, 
 of the following event 
 
 Either H 5 T 7 , or H 6 T , or H 7 T 5 
 
 wefind792 X 24-924or2508 t (m rethanhalfof4 96) 
 cases in which the event arrives. That is, the chances 
 are in favour of one or other of these arriving. As the 
 number of events increases, a given degree of nearness 
 to the most probable event becomes more and more 
 likely. To illustrate this : suppose 24 halfpence thrown 
 up, the notation remaining as before. The total number of 
 cases is now 2 24 or 1 6777216: calculating the number 
 
 T happen in 
 
 
 1 case each. 
 1 2 cases each 
 
 T 2 C 
 
 2,12 
 
 or 66 
 
 T, .. ,.. C 
 
 
 
 3, 12 
 
 or220 
 
 T, .. .. C 
 
 4,12 
 
 or 495 
 
 T 5 C 
 
 5,12 
 
 or792 
 
48 ESSAY ON PROBABILITIES. 
 
 of combinations of 1,2,3., &c. out of 24, we have the 
 following summary : 
 
 H T 24 and H 24 T happen in 1 case each. 
 
 H 1 
 
 ? 
 
 HTT 
 23 L l 
 
 . .. H 99 T 9 
 
 24 cases each, 
 276 
 
 -"2 
 
 T 
 
 ..H O ; T" . 
 
 2024 
 
 H 
 
 T 
 
 :: H 2 2 T :: 
 
 10626 
 
 H 
 
 *.- 
 
 . Hol\ , 
 
 42504 
 
 B 
 
 i" 
 
 H! T 
 
 .. 134596 
 
 H 
 
 T'" 
 
 
 . 346104 
 
 H ' 
 
 T 
 
 ::::. 2" T! ::"::::;: 
 
 .. 735471 
 
 H 
 
 T" 
 
 H T 
 
 * 1307504 
 
 H 9 
 
 T , . 
 
 : s;!T 9 : 
 
 .. 1961256 
 
 5.' 
 
 T 
 
 J. 1 < 
 
 
 .. 2496144 .. 
 
 H 12 T 12 happens in 2704156 cases. 
 The odds are now about 6 to 1 against the even di- 
 vision of the pieces into heads and tails. But let us 
 consider the same degree of departure from the most 
 probable case as we took before. An alteration in one 
 piece out of twelve answers to that of 2 pieces out of 
 24. Now the number of cases in which either HIO 
 
 arrives, and the odds of one or other of them, is com- 
 puted as follows : 
 
 H 10 T 14 or H 14 T 10 3922512 16777216 whole No. of cases 
 H n T 13 or H 13 T n 4992288 11618956 No. favourable 
 H 6 T 6 2704156 - 
 
 - 5158260 No. unfavourable 
 11618956 
 
 or it is now more than 2 to 1 in favour of the heads 
 lying between 10 and 14, both inclusive. 
 
 I now put together the principles on which we have 
 hitherto gone, adding two more, the first of which 
 (Principle III.) is obviously a consequence of the pre- 
 ceding, and the second of which will be presently ex- 
 plained. 
 
 Principle I. When all the ways in which an event 
 may happen are equally probable, the chance of its 
 happening is the number of ways in which it may 
 happen, divided by all the number of ways in which it 
 may happen and fail. 
 
ON DIRECT PROBABILITIES. 49 
 
 Principle II. The probability of any number of in- 
 dependent events all happening together, is the product 
 of their several probabilities. 
 
 Principle III. The probability of two events arriving 
 together being known ; and also,, that of one of them : the 
 probability of the other is found by dividing the first 
 mentioned probability by the second. 
 
 Principle IV. When an event may happen in several 
 ways, whether equally probable or not, the probability 
 of the event is the sum of the probabilities of its hap- 
 pening in the several different ways. 
 
 The best way of illustrating the last principle is by 
 beginning with one of the numerous errors into which 
 we may fall, either in proceeding towards it, or applying 
 it. Let there be an event which may happen in two 
 different ways, for each of which there is an even chance. 
 Then, according to the principle, it is certain ( J- -f -J- = 1 ) 
 that the event must happen. In this there is nothing 
 inconsistent. Let there be a lottery containing ten white 
 balls, and let them be sub-divided into two sets of five 
 by a mark. Then there are two ways of drawing a 
 white ball, for each of which there is an even chance ; 
 namely, I may choose a ball of one mark, or of the other. 
 But it is evidently certain that in this case a white ball 
 must be drawn. Now, suppose an event can happen in 
 three different ways, for each of which there is an even 
 chance. Then the event k more than certain (-J- + A 
 H- -J- = 1 1) ; which is absurd. But the absurdity is in 
 the supposition : an event can only have an even chance 
 of happening in one particular way, when that way 
 involves half of the total number of individual cases of 
 the event ; and it is impossible that three different ways 
 of arriving can each contain half of the whole number 
 of possible cases. Consequently, when we have made a 
 calculation of the probabilities which different ways of 
 arriving give to an event, there has certainly been an 
 error if the sum of the probabilities exceed unity. 
 
 But when we throw three half -pence into the air, are 
 there not three different ways of throwing head, for each 
 
50 ESSAY ON PROBABILITIES. 
 
 of which there is an even chance ? If by H we here 
 mean a single H, the three events by which we propose 
 to attain it are not H, H, H, but H T T, T H T, and 
 TTH, the probability of each of which is 4, and, by the 
 application of the principle, ^ -f -^ + -^ or -^ is the 
 probability of the event, one single head. If by* 
 throwing a head we mean one head or more, the ways 
 under which it may be brought about are one head 
 only (involving the cases H T T, T H T, T T H) two 
 heads only (involving H H T, H T H, and T H H) 
 and three heads (involving H H H.) The probabilities 
 of these are -J , , and -1, or -J- is the chance required. 
 
 Another error to which we are liable, is the wrong 
 estimation of the probability of the different cases. For 
 instance, a person is to go on until he throws H. and is to 
 win if he do it in less than five throws. There are then 
 five possible cases ; namely, H, T H, T T H, T T T H, 
 T T T T. In four of these cases he wins ; in the fifth he 
 loses. His chance of winning then appears to be 4. 
 But this supposes the five cases to be equally likely, 
 which is not true. Their several probabilities are \, , 
 , -jW and -^6 : not -$> -k> & c -> as supposed. Consequently 
 the sum of the probabilities which the several winning 
 cases actually have is -ff, or it is 15 to 1 that he wins. 
 If we put together all the cases, which four throws pre- 
 sent, thus, 
 
 1. HHHH 5. HTHH 9. THHH 13. TTHH 
 
 2. HHHT 6. HTHT 10. THHT 14. TTHT 
 
 3. HHTH 7. HTTH 11. THTH 15. TTTH 
 
 4. HHTT 8. HTTT 12. THTT 16. TTTT; 
 
 all those which begin with H are 8 in number, those 
 which begin with T H are 4, with T T H, 2, and with 
 TTTH, one. Hence 8 -f 4 f 2 -f 1, or 15, is the 
 number of winning cases. But the argument against 
 us, is this : most of the preceding cases are impossible, 
 for the condition is, that the play shall stop as soon as 
 H occurs : so that, in fact, the only possible cases are, 
 H, T H, T T H, T T T H, T T T T. Let it be so ; 
 we must then represent the several events as follows : 
 
ON DIRECT PROBABILITIES. 51 
 
 1st event ; certainly happens,, and gives either H or T. 
 Probability of H, |. 
 
 2nd event ; does not certainly happen, but is con- 
 tingent upon the first throw being T ; and gives either 
 H or T. Probability \ that the throw is made, that 
 if it be made it gives H ; probability that the throw is 
 made, and that, being made, it gives H, -J- x -J- -J. 
 
 3rd event; contingent upon the two first throws 
 giving T and T, of which the chance is | . Probability 
 of winning at this throw, | X -J- = -g. 
 
 4th event ; contingent upon the three first throws 
 giving T, T, T. Probability of winning at this throw, 
 
 4 * i = /* 
 
 It is obvious enough, when stated, that every con- 
 tingency must enter into the consideration of a question ; 
 so that if some of the circumstances depend upon the 
 manner in which preceding contingencies arrive, this 
 circumstance itself influences the method of proceeding. 
 If we wish to avoid the necessity of considering a con- 
 tingency the trial of which is itself contingent, that is, if 
 we wish to make a contingency certain, we must in- 
 troduce all the new events which such change of con- 
 tingency into certainty brings with it. The whole 
 problem is exactly the same as if we made the four 
 throws certain, and made the gain dependent upon one 
 head or more being thrown : but we revert again to the 
 former state of the question, if we agree to mark the first 
 H as the winner. In this point of view, the distinction 
 between the two is evidently iin material. 
 
 EXAMPLE. There are seven lotteries, as follows (W 
 means white, B black) : 
 
 L(2W ' 3B) 
 
 and the conditions of drawing are the following. I. i's 
 drawn, and then II. or III., according as I. gives W or B. 
 If II. be drawn, then IV. or V. is to be drawn, according 
 as II. gives W or B. But if III. be drawn, then VI. or 
 
 E 2 
 
33 ESSAY ON PROBABILITIES. 
 
 VII. is to be drawn, according as III. gives W or B. 
 What are the probabilities of the several possible 
 drawings ? 
 
 If from I. we draw W (of which the chance is ), 
 we proceed to II. If we still draw W (chance, -J), we 
 proceed to IV. And here the chance of W is -J-. 
 Hence, 
 
 the chance of WWW is f x x \ = 2 3 5 
 
 Computing all the other chances in the same way, we 
 get the following : 
 
 1. WWW | f i = A 
 
 2. WWB | i - ft 
 
 3. WBW f = ft 
 4.WBB f | J-A 
 
 5. BWW f J J = A 
 
 6. BWB | M | 
 
 7. BBW f j = ft 
 
 8. BBB | \ I = fa 
 
 The sum of all these is equal to unity, as it should 
 be, since one or other of these cases must happen. And 
 by reducing all to the common denominator 5. 5. 4. 6, 
 or 600, we have the following chances : 
 
 '7. BBWfif 
 
 1. WWW, 
 
 2. WWB , 
 
 3. \VBW 
 
 5. BWW 
 
 6. BWB 
 
 That all shall he white, 7 to 1 against (nearly). 
 
 Two ^vhite and one black, 3 to 1 against (very nearly) 
 
 Two black and one white, an even chance (nearly) 
 
 All black, 10 to 1 against (nearly). 
 
 From what has been said in this chapter, no great 
 difficulty will be found in ordinary questions. The cir- 
 cumstances are supposed to be fully known, and the 
 probabilities will be found, of the strength which it 
 follows they must have, to those who admit the axioms 
 on which the measure of probability is founded. 
 
ON INVERSE PROBABILITIES. 53 
 
 CHAPTER III. 
 
 ON INVERSE PROBABILITIES. 
 
 IN the preceding chapter, we have calculated the 
 chances of an event, knowing the circumstances under 
 which it is to happen or fail. We are now to place 
 ourselves in an inverted position : we know the event, 
 and ask what is the probability which results from the 
 event in favour of any set of circumstances under which 
 the same might have happened. This problem is fre- 
 quently enunciated as follows: An event has hap- 
 pened, such as might have arisen from different causes : 
 what is the probability that any one specified cause did 
 produce the event, to the exclusion of the other causes ? 
 By a cause, is to be understood simply a state of things 
 antecedent to the happening of an event, without the 
 introduction of any notion of agency, physical or 
 moral. 
 
 In order that we may secure a problem of sufficient 
 simplicity, we must limit the number of possible ante- 
 cedent states. Let us suppose that there is an urn, of 
 which we know that it contains balls, three in number, 
 and either white or black, all cases being equally pro- 
 bable : that is, before any drawing takes place, all we 
 can say is, that we are going to draw out of one of the 
 following, having no reason for supposing one in pre- 
 ference to another 
 
 A B C DEF 
 
 !.() II. ( o ) III. ( o o ) IV. ( o o o) 
 
 A drawing takes place, and a white ball is produced, 
 consequently I. is immediately excluded ; for from it 
 the observed event could not have been produced. This 
 much is certain ; but we are also tempted to say that 
 E 3 
 
O* ESSAY ON PROBABILITIES. 
 
 II. is rendered unlikely, because, from such an ante- 
 cedent state of things, a black ball would have been 
 more likely than a white one. On the same prin- 
 ciple III. is more likely than II., and IV. the most 
 likely of all. We have then to decide the relative pro- 
 babilities of II., III., and IV. 
 
 Before the drawing took place, the probability of each 
 set of circumstances was J ; and, the lottery being 
 given, the probability of any one ball in it was i. 
 Thus the chance of III. being the lottery, and the 
 second white ball being drawn from it, was x -^, or 
 -^ . The same of other balls : so that, in fact, our 
 primitive position was that of having to draw from 12 
 balls, 6 white and 6 black, all equally probable. But 
 the observed event changed that position ; a white ball 
 was drawn : was it a given ball (namely, the white ball 
 in II.), or was it one of two given balls (those in III.), 
 or was it one of three (those in IV.) ? There are six 
 cases in question, namely, A, B, C, D, E, F, and one 
 of them happened, we do not know which. We have 
 used all the knowledge we have (namely, that a white 
 ball was drawn,) in excluding the black balls. 
 
 Hence the chance that A was drawn, or that 
 
 II. was the lottery - is -J- : 
 
 That either B or C was drawn, or that III. 
 
 was the lottery - - is |- : 
 
 That either D, E, or F was drawn, or that 
 
 IV. was the lottery - - is -|-. 
 
 In the preceding instance, owing to the number of balls 
 being the same in every lottery, the antecedent proba- 
 bility of each ball was the same. Previous to deducing 
 a rule, I take an instance in which this is not the case. 
 
 PROBLEM. A white ball has been drawn, and from 
 one or other of the two following urns : 
 
 (2 white, 5 black) (3 white, 1 black). 
 What are the probabilities in favour of each urn ? 
 
 The case is not now that of a lottery of 5 white and 
 6 black balls ; for the chance of our going to the first 
 urn (which is-l), and thence drawing a given white 
 
ON INVERSE PROBABILITIES. 55 
 
 ball (chance -0, is \ x -y or -^ ; while our chance of 
 going to the second urn (which is -<), and thence draw- 
 ing a given white ball (chance J), is 1 X J or -J. But 
 since we do not alter the chance of producing a white 
 ball from either urn, if we double, or treble, &c. the 
 number of white balls, provided we at the sanle time 
 double, or treble, &c. the number of black balls, let us 
 put four times as many balls into the first, and seven 
 times as many into the second, as there are already. 
 Thus we have : 
 
 (8 white, 20 black) (21 white, 7 black). 
 
 There are now 28 balls in each : every individual ball 
 has the antecedent probability -J- x --% ', and since our 
 knowledge of the event (a white ball was drawn) ex- 
 cludes the black balls, the question is simply this : 
 Out of 29 possible, and equally probable cases, was 
 the event which did happen one out of a certain 8, or 
 one out of the remaining 21 ? The chances of these 
 are 7% and -g-J- ; consequently it is 21 to 8 that the 
 second lottery was that which was drawn, and not the 
 first. 
 
 On looking at the resulting chance for the first urn, 
 namely, -j^, or the (21 -f- 8)th part of 8, we see that/ 
 8 and 21 are in proportion to the two chances for a 
 white ball being drawn, when we know that we are 
 drawing from the first urn, or from the second. For 
 these chances are ^ and ^, which, reduced to a common 
 denominator, are -^ and -|i, which are in the propor- 
 tion of 8 to 21. The same reasoning may be applied 
 to any other cases, and the result is as follows : 
 
 Principle V. When an event has happened, and 
 the state of things under which it happened must have 
 been one out of the set A, B, C, D, &c., take the 
 different states for granted, one after the other, and 
 ascertain the probability that, such state existing, the 
 event which did happen would have happened. Divide 
 the probability thus deduced from A by the sum of the 
 probabilities deduced from all, and the result is the 
 E 4 
 
56 ESSAY ON PROBABILITIES. 
 
 probability that A was the state which produced the 
 event : and similarly for the rest. [Or, reduce the re- 
 sults of the first part of the rule to a common deno- 
 minator^ and use the numerators only in the second 
 part of the rule.] 
 
 EXAMPLE I. There is a lottery which is one or 
 other of the two following : 
 
 (3 white, 7 black) (all white). 
 
 A ball is drawn, and restored; this takes place five 
 times, and the result is always a white ball. What are 
 the chances for each lottery ? 
 
 Upon the supposition that the first lottery was that 
 in question, the chance of the observed event is the 
 product of -^ tV io> fo> and TO> or Ti >VbW When 
 the second is the lottery, the observed event is certain, 
 and its probability is 1 or -ViyaoSo- Consequently, the 
 probability for the second lottery is -JSJJ-g^-j, or the 
 second has the odds 100000 to 243, or more than 411 
 to 1 in its favour. 
 
 EXAMPLE II. Two witnesses, on each of whom it 
 is 3 to 1 that he speaks truth, unite in affirming that 
 an event did happen, which of itself is equally likely 
 to have happened or not to have happened. What is 
 the probability that the event did happen ? 
 
 The fact observed is the agreement of the two wit- 
 nesses in asserting the event: the two possible ante- 
 cedents (equally likely) are, 1. The event did happen. 
 2. It did not happen. If it did happen, the probability 
 that both witnesses should state its happening is that 
 of their both telling the truth, which is f x J, or ^. 
 If it did not happen, then the probability that both 
 witnesses should assert its happening is that of their 
 both speaking falsely, which is x J, or -*$. Conse- 
 quently, the probability that the event did happen is 
 the (9 -f l)th part of 9, or T ; that is, it is 9 to 1 
 in favour of the event having happened. 
 
 EXAMPLE III. There are two urns, having certainly 
 3 and 2 white balls; and in one or other, but which 
 
ON INVERSE PROBABILITIES. 57 
 
 is not known, is a black- ball. A ball is drawn and 
 replaced ; and this process is repeated, but whether out 
 of the same urn as before is not known. Both draw- 
 ings give a white ball : what is the probability of the 
 several cases from which this result might have hap- 
 pened ? 
 
 Since the black ball is as likely to be in one as in 
 the other, the antecedent state of things is (so far as a 
 single drawing is concerned,,) the same as if there were 
 four urns, as follows : 
 
 I. (3 white) 
 
 II. (3 white, 1 black) 
 IV. (2 white, 1 black). 
 
 III. (2 white) 
 
 There are 16 possible cases, PP [2, 4,] numbered in 
 the first columns following, described in the second, 
 and having the probability which each would give 
 to the observed event (both drawings white) registered 
 in the third, together with the numerator, when all the 
 fractions are reduced to a common denominator 1 44. 
 
 1 
 
 I. I. 
 
 1, 144 
 
 5 
 
 II. I. 
 
 |, 108 
 
 2 
 
 I. II. 
 
 |,108 
 
 6 
 
 II. II. 
 
 TT 81 
 
 3 
 
 I. III. 
 
 1, 144 
 
 7 
 
 II. III. 
 
 |, 108 
 
 4 
 
 I. IV. 
 
 4, 96 
 
 8 
 
 H. IV. 
 
 Ta" 7 w 
 
 9 
 
 III. I. 
 
 1, 144 
 
 13 
 
 IV. I. 
 
 f , 9 
 
 10 
 
 III. II. 
 
 1 108 
 
 14 
 
 IV. II. 
 
 iV> 7 
 
 11 
 
 III. III. 
 
 1, 144 
 
 15 
 
 IV. III. 
 
 |, 9 
 
 12 
 
 III. IV. 
 
 4, 96 
 
 16 
 
 IV. IV. 
 
 ' 6 
 
 That is to say, if (case 8), II. and IV. were the urns 
 of the first and second drawings, the chance of the 
 observed event is \ or T ^. But, it must be remem- 
 bered, that we do not suppose the black ball may have 
 been removed from one urn into the other before the 
 second drawing takes place. Most of the preceding 
 cases are, therefore, to be rejected; in fact, I. can 
 combine with nothing but I. or IV., and II. with 
 nothing but II. or III. Reject, therefore, cases 2, 3, 
 5, 8, 9j 12, 14, 15, and the sum of the numerators in 
 the rest is 841. Hence the probability (for instance,) 
 
58 ESSAY ON PROBABILITIES. 
 
 1. That the black ball is with the three white ones. 
 
 2. That the first drawing is from the lottery which has 
 the black ball, and the second from the other, is (case 7) 
 -J-g-f- . To find the total probability that the black ball 
 is with the three white ones, we must add the proba- 
 bilities of all the cases (as to drawings) which can take 
 place under this arrangement, namely -f-^ y -J-J-^ -|Jv 
 J-4T> g ivin S iti- Consequently, from the observed 
 event, it is slightly more probable that the black ball is 
 with the three white ones, than with the two. 
 
 The principle which we have illustrated, though a 
 mathematical consequence of those which precede, is 
 nevertheless received in common life upon its own 
 evidence. When an event happens, we immediately 
 look to that cause or antecedent which such event most 
 often follows. When it rains, we suspect the barometer 
 must have fallen ; because, when the barometer falls, 
 it usually rains. 
 
 Our next step is to inquire, what is the probability 
 which an event gives to its several possible antece- 
 dents, upon the supposition that they are not all equally 
 likely beforehand ; as in the following instance. 
 
 PROBLEM. A white ball is drawn, and from one or 
 other of the following urns : 
 
 (3 white, 4 black) (2 white, 7 black) : 
 but before the drawing was made, it was three to one 
 that the drawer should go to the first urn, and not to 
 the second. What is the chance that it was the first 
 urn from which the drawing was made ? 
 
 We may immediately reduce the preceding to the 
 case where all the antecedent circumstances are equally 
 probable, by introducing urns enough of the first kind 
 to make it 3 to 1 that the drawing is made from one 
 or other of them. Let us suppose the urns to be as 
 follows : 
 
 (3 white, 4 black) (3 white, 4 black) (3 white, 4 black) 
 (2 white, 7 black): 
 
 these urns being equally probable, the hypothesis of 
 
ON INVERSE PROBABILITIES. 59 
 
 the problem exists. If we number the urns A 1? A , 
 A 3 , B, the chances which they severally give to the 
 observed event are , -y> T> an ^ "f> ^ e numerators of 
 which, reduced to a common denominator, are 27? 27 5 
 27? and 14. Consequently, the probability that A 1 was 
 chosen, is ^i; and the same for A 2 and A 3 . There- 
 fore, the chance that one or other of the three, A 1? A 2 _, 
 and A,,, was chosen, is -|J-; which is the probability of 
 the ball having been drawn from the urn (3 white, 
 4 black,) in the first statement of the problem. 
 
 The rule to which the preceding reasoning conducts 
 us is as follows : When the different states under which 
 an event may have happened are not equally likely to 
 have existed, then having found the probability which 
 each state would give to the observed event, multiply 
 each by the probability of the state itself before using 
 the rule in page 55. The following is another example. 
 
 An event has happened, the possible preceding states 
 of which are represented by A, B, and C. The chances 
 of the existence of these different states (independently 
 of all knowledge of the observed event) are, say, -|, -J, 
 and -J : the probabilities that the observed event would 
 have happened are -f 19 -fa, and T 2 T , if A, B, or C were 
 certainly existing. Form the three products 
 
 T> i U-TX f A f Probability that the event would! 
 Probability of A x | happeni / A were known to exist, j &c - 
 
 there are 
 
 T * 7T> T X TP and T * TT; 
 
 the numerators of which (the denominators being com- 
 mon) are 20, 12, and 4 Then the probability that 
 A was the state under which the event happened, is 
 20 divided by 20 + 12 -f 4, or -||-; those of B and C are 
 
 7T and IT" 
 
 Let us now suppose, that having only a first event 
 by which to judge of the preceding state of things, we 
 ask what is the probability of a second event yet to 
 come. For instance, an urn contains two balls, but 
 whether white or black is not known ; the first draw- 
 
60 
 
 ESSAY ON PROBABILITIES. 
 
 ing gives a white ball, and the ball is replaced. What 
 is the chance that a second drawing shall give a black 
 ball ? 
 
 The preceding states under which the first event may 
 have happened, are 
 
 (2 white) (1 white, 1 black); 
 
 and 1 and % are the chances of a white ball, if one or 
 other state were absolutely known to exist. Hence., by 
 the last principle, -| and -^ are the chances which the 
 observed e*vent gives to the two states; that is, it is 
 two to one that both balls were white. Now, the black 
 ball can only appear at the second drawing, upon the 
 supposition of the second state existing ; and this sup- 
 position being made, the chance of a black ball at the 
 second drawing is -J. Hence, page 43., the second 
 event depending upon two contingencies, of which the 
 chances are ^ and -J, its chance is J, or it is five to 
 one against the second drawing being black. But let 
 us now ask what is the chance of a white ball at the 
 second drawing ? Either of the preceding states admit 
 of such an event, and, in fact, the event proposed a 
 white ball at the second drawing means 
 
 One or other of these C (2 white) and white drawn. 
 
 two combinations ( (1 white, 1 black) and white drawn. 
 
 In the first combination, the first contingency (the 
 chance of which is -J) ensures the second: so that 
 A- x 1 is the chance of a white ball being drawn, and 
 of (2 white) being the lottery from which it is drawn. 
 In the second combination, the chances of the two con- 
 tingencies are -] and 1, whence J- is the chance of a 
 white ball being drawn, and being drawn from (1 
 white, 1 black). But the event proposed happens if 
 either of these cases occur ; therefore, - 4. l, or | , is 
 the chance of a white ball at the second drawing, as 
 might have been inferred from the probability already 
 obtained for a black ball. By such reasoning as the 
 preceding, the following principle is established : 
 
 Principle VI. Having given an observed event A, 
 
ON INVERSE PROBABILITIES. O'l 
 
 to find the probability which it affords to the suppo- 
 sition that a coming event shall be B, find the proba- 
 bility which A gives to every possible preceding state ; 
 multiply each probability thus obtained by the chance 
 which B would have from that state, and add the 
 results together. 
 
 PROBLEM. There is a, lottery of 10 balls, each one 
 white or black, but which is not known : drawings are 
 made, after each of which the hall is replaced. The 
 first five drawings are white ; what chance is there 
 that the next two drawings shall be white ? 
 
 Let S (20) denote the sum of all numbers up to 20 ; 
 S (20 2 ) the sum of the squares of all numbers up to 
 20 2 or 400 ; and so on. The possible preceding states 
 are 
 
 (1 W, <>B) (2 W,8 B.) ---- (10 W,OB); 
 and the probabilities of W five times running from 
 each, are 
 
 TO iV ' TV * To * To To" ' To ' T o" * Fo To> & C> 
 
 iS iS> or 1* the 
 
 . . -- 
 
 certain, if the last state existed. The numerators of 
 these products (the common denominator being 10 5 ,) 
 are I 5 , 2 5 , - 10 5 : whence, page 55., the proba- 
 bilities of the several states are 
 
 is & ......... 1Q5 
 
 S 105' S 105* S 105 
 
 By the same reasoning, the probabilities of the proposed 
 events (two more white balls,) are 
 
 12 22 ^ ^ 1Q2 
 
 10 2 10 2 102 ; 
 
 the different preceding states being successively sup- 
 posed to exist; whence the actual chance which the 
 observed event gives to the proposed is 
 
 _I!_x.il+_^Lx^l + ...... * 105 .' 2 
 
 S 105 102 S 105 102 S 105 102 ; 
 S 107 
 
 which is 
 
OX ESSAY ON PROBABILITIES. 
 
 By precisely the same reasoning, if there had been 
 1000 balls in the lottery, and if 157 had been drawn 
 white, the probability that 27 more drawings would 
 have given white balls, would have been 
 
 S 1000^4 
 
 (157 + 27 = 184) 100027 S 10001*7 
 
 The difficulty of calculating S 1000 184 is insuperable: 
 but a mathematical theorem which we shall proceed to 
 explain, makes it very easy to find a near approxima- 
 tion to the preceding result, and the nearer the greater 
 the numbers in question. 
 
 Take the sums of the powers of the different num- 
 bers, as follows : 
 
 First powers 1+2+ 3+ 4 + 5+ 6 + 
 
 Squares 1+4+ 9+ 16+ 25+ 36 + 
 
 Cubes 1+ 8+ 27+ 64+ 125+ 216 + 
 
 Fourth powers 1 + 16+ 81+ 256+ 625 + 1296 + 
 
 Fifth powers 1 +32+243 + 1024 + 3125 + 7776 + 
 
 and examine the sum of any number of terms in any 
 line, as compared with the term immediately below the 
 last in the sum j thus : 
 
 1+2 + 3 + 4 1 + 16 + 81+256 + 625 
 
 16 3125 
 
 Form fractions with such sums as numerators, and their 
 compared terms as denominators, and observe how 
 much each fraction, so formed, differs from the fraction 
 written in the last column, as foUows : 
 ]+2_3_l 1 l+2+3_6_l 1 
 
 ~4 4~2 4 9 9~ 2 6 
 
 ilj^l+i J^li * , and so on, 
 16 16 2 8 25 25 2 10 
 
 whence 
 
 S (any number) JL __ _J 
 
 square of that number 2 twice that number 
 
 Hence it follows, that when the number is large, the 
 preceding fraction is very nearly one half, or 1 -f- 2 -j- 
 
 3 -f up to a large number, is very nearly one half 
 
 the square of that number. 
 
Again: 
 
 ON INVERSE PROBABILITIES. 63 
 
 1+4 5 1 7 l' + 4 + 9 1 5 
 
 8 8 3 24 27 3 27 
 
 S42 30 1 13 S52 1 
 
 
 64 or 43 64 3 9(5 125 or 5 3 3 75 
 
 ^03 1000 3 3000 
 
 In this way, it appears that the sum of all the 
 squares of numbers is nearly one third of the cube of 
 the last number, and that the greater the number of 
 squares taken, the greater the proximity in question. 
 This proposition is general, namely, that the sum of the 
 nth powers of numbers is nearly the (n -f l)th part 
 of the (n + l)th power of the last of the numbers : 
 thus, the sum of all the 13th powers I 13 + 2 13 -f 
 up to 1000 13 , is very nearly the 14th part of 1000 14 , 
 This proposition, never absolutely true, may be made as 
 near the truth as we please, by taking the number of 
 terms sufficiently great; and the error made by the 
 substitution, is nearly such a fraction of the whole as 
 has one more than the index of the power for its nu- 
 merator, and twice the number stopped at for its 
 denominator. Thus, if the tenth powers of all num- 
 bers were summed up to 1 0,000 10 , the substitute for 
 this sum given by the theorem, namely, -fa of 10,000 1] , 
 would be too small by about 
 
 10+1 or ll of the whole. 
 
 2 x 10,000 20,000 
 
 PROBLEM. A lottery contains 10,000 balls, each of 
 which may be white or black. A ball is drawn and 
 then replaced, and 100 such drawings give nothing but 
 white balls : what is the chance that the five next 
 drawings shall all be white ? 
 
 S 10,000*00+ s 
 
 This chance, by what precedes, is 
 
 10,0005 S 10,CKX)ioo 
 
 But S 10 3 OOOioo+5 = J_ x 1 o,000i <5 very nearly. 
 106 
 
ESSAY ON PROBABILITIES. 
 
 S 1 0,000100 = _ x 1 0,000*01 
 
 101 
 
 Consequently, the chance is TO T , or , nearly. 
 
 -j-I-j- of 10,000106 106 
 
 If the number of balls had been a million, instead 
 of 10,000, the preceding odds, namely, 101 to 5, 
 would still more nearly have represented the chance 
 that after 100 drawings, all white, the next five should 
 be white also. If the number of balls had been abso- 
 lutely unlimited, the preceding odds will correctly ex- 
 press that same chance. But a lottery with an unlimited 
 number of balls, each of which may be either white or 
 black, is a lottery which may be anything whatever. 
 For instance : [3 W, 7 B] is equivalent to an unlimited 
 lottery, in which for every 7 black balls there are 3 
 white ones. Again, an unlimited lottery in which any 
 number of balls may be black, and the rest white, is 
 one in which the chance of drawing a white ball may 
 be any whatever, and is absolutely unknown. The draw- 
 ing of a white ball from such a lottery may be likened 
 to the occurrence of an event, about the preceding 
 chances of which we are in total ignorance. The pre- 
 ceding process furnishes us with the following theorem. 
 When an event which may, for any thing we can see 
 to the contrary beforehand, happen in either of two 
 different ways, happens one way m times in succession, 
 it is m -f 1 to n that it shall happen n times more in 
 the same way, if it happen n times more at all. Thus, 
 suppose a person on the bank of a river, not knowing 
 in what country he is, and not having the smallest 
 reason to know whether the vessels which come up the 
 river carry flags or not : the first ten ships which 
 come up all carry flags (m = 10) ; then it is 10 -f 1 
 to 3, or 1 1 to 3 that the next three ships shall carry 
 them, and 10 4- 1 to 1, or 11 to 1 that the next ship 
 shall carry a flag. And it is always m -f 1 to 1, that 
 an event which has occurred in one out of two possible 
 ways m times in succession, shall happen the same way 
 on the next occasion. 
 
ON INVERSE PROBABILITIES. 65 
 
 The preceding affords some view of the way in 
 which chances are obtained, in cases where the ante- 
 cedent probability of the events stated may be any 
 whatever. The following are conclusions upon the same 
 subject, obtained by a more complicated reasoning of 
 the same kind. 
 
 If an event, each repetition of which may be either 
 A or B, have happened m-\-n times, and if A have 
 occurred m times, and B n times ; then it is m -\- 1 
 to n + 1 that the next event shall produce A, and 
 not B. And in the same case the chance that out of 
 p -f- q events to come, p shall produce A, and q shall pro- 
 duce B, is (see pages 15 and 16, for explanation of [ ]). 
 \p+q\ 
 
 [f] x M x [m+n+2> m+n+p+q+ 1]' 
 and 
 
 . 
 [m + n+2, m + n+p + 1] [m + n+2, m+n+q + l] 
 
 are the chances that in p new events, all shall give A 
 and that in q new events all shall give B. 
 
 EXAMPLE. In a lottery containing an unlimited 
 number of balls, in which the proportion of black and 
 white is absolutely unknown, six drawings give four 
 white and two black ; what are the chances that four 
 drawings more shall give all white, or one only black, 
 or two only black, &c. 
 
 Let us first take the case of three white and one black : 
 here m = 4, n = 2, p = 3, q =. 1. 
 
 [p + q] = 1.2.3.4, [m + l,m+p] = 5.6.7, 
 [n + l,n + q] =3[p] = 1.2.3. [q] = 1, 
 [m + w + 2, m + n+p + q + l~\ = 8. 9. 10.11. 
 
 Chance required is 1-2.3.4. x 5.6.7. * 3. _ 7 
 1.2.3, x 1. x 8.9.10.11. 22 
 
 That two shall be white and two black (m = 4, n = % 3 
 p = 2, q = 2), the chance is 
 
 1 2.3.4. x5.6. x3.4. 3 
 
 > or 
 
 1.2. x 1.2. x 8.9.10.11. 11 
 That one shall be white and three black (p = J. 
 g = 3), the chance is 
 
66 ESSAY ON PROBABILITIES. 
 
 1.2.3.4x5x3.4.5 Qr 5 
 
 1x1.2.3x8.9.10.11 33 
 
 That all shall be white, or all black (p = 4, q = 4, 
 second and third formulae), the chances are 
 
 Jli,or .land 3 - 4 ' 5 - 6 , or 1 
 8.9.10.11 33 8.9.10.11 22 
 
 and the verification of the whole is 
 73571 
 
 J. 4 1 -I = 1 
 
 22 11 33 33 22 
 
 which must be, since one or other of the cases con- 
 sidered must happen. 
 
 When it is known beforehand that either A or B 
 must happen, and out of m + n times A has hap- 
 pened m times, and B n times, then (page 65.) it is 
 m -f 1 to n -f- 1 that A will happen the next time. 
 But suppose we have no reason, except what we gather 
 from the observed event, to know that A or B must 
 happen ; that is, suppose C or D, or E, &c. might have 
 happened: then the next event may be either A or 
 B, or a new species, of which it can be found that the 
 respective probabilities are proportional to m + 1, 
 w-j-1, and 1 ; so that though the odds remain w-fl 
 1 to n + 1 for A rather than B, yet it is now m + 1 
 to n + 2 for A against either B or the new event. 
 Thus, suppose a game at which one party or the other 
 must win, and suppose that out of 20 games A has 
 won 1 3 and B 7 : and this is all we know of the game 
 or of the players. Then, it is 134-1 to 7-fl> or 
 14 to 8, or 7 to 4, that A shall win the 21st game. 
 But suppose that it is possible to have a drawn game ; 
 then there is some chance that the 21st may be a 
 drawn game, though but a small one, as might be 
 inferred from such a thing never happening in 20 
 trials. The 21st game may be either A's or B's, or 
 drawn: of which the chances are as 15 + 1, 7+1, 
 and 1 ; or as 1 4, 8, and 1 . Consequently, though in 
 the preceding case it was 14 out of 22 in favour of A's 
 
ON INVERSE PROBABILITIES. 6j 
 
 winning, it is now 14 out of 23, and 1 chance out of 
 23 remains for the next game being drawn. 
 
 When a number of different events have happened, 
 A, B, C, &c., write down each number increased by 1, 
 and the results will express the several relative proba- 
 bilities, on the supposition that no events can happen 
 except those which have happened. But if new events 
 may happen, write down 1 for the relative probability 
 of such an occurrence at the next trial. Thus, if out 
 of a box, and in 100 drawings, there have appeared 49 
 white balls, 37 red, and 14 black ; then if it be known 
 that nothing but white, red, or black can appear, con- 
 sider the chances of these to be as 50, 38, and 15 ; that 
 is, -j 5 ^ is the chance of drawing a white ball at the 
 101st trial. But if another sort of ball may appear., 
 then the chances of the four cases being as 50, 38, 15, 
 and 1, it follows that -ffy is the chance of a white ball 
 at the 101st trial. 
 
 In judging of future events by those which have 
 passed, we must be extremely cautious always to pre- 
 serve the same method of considering the event pro- 
 posed. If, for instance, in ] 00 trials, A has appeared 
 49 times, B 37 times, and C 14 times, we know that 
 there is one chance out of 104 that the 101st drawing 
 shall give neither A, B, nor C, but something else. 
 What the new character may possibly be is left un- 
 known ; it may be another letter, or it may be a num- 
 ber, a picture, or a blank. Are we to understand that 
 all these are equally probable ? Common sense tells us 
 the contrary ; experience makes us feel it much more 
 likely that the letter D should appear at the 101st trial,, 
 than any stated number or picture. But we have 
 now changed the question, and, dropping the distinction 
 between A, B, and C, have considered them merely as 
 letters. Having drawn a letter 100 times running, we 
 are to infer (page 64.) that it is 101 to 1 in favour 
 of our drawing a letter at the 101st trial; or that |g-J 
 is the chance of this. But it is already 103 to 1 that 
 the next drawing shall be, not merely a letter, but one 
 F 2 
 
68 ESSAY ON PROBABILITIES. 
 
 of the letters A, B, and C : that is, to all appearance, 
 we have this strange result ; the chances of drawing 
 one of the three, A, B or C, are greater than those of 
 drawing one of the set, A or B, or C or D, &c. &c. up 
 to Z. This paradox will afford me a good opportunity 
 of again inculcating the maxim, that the probability of 
 an event is the presumption drawn from certain obvious 
 principles, as to what the state of our minds ought 
 to be with regard to belief in the happening of that 
 event, as influenced by our knowledge of previous 
 events. Consequently, if John know that A, B, and 
 C have been drawn 49, 37, and 14 times, and nothing 
 more, he has reasonable ground, with his knowledge, 
 for assenting to the proposition that the 101st trial 
 shall give either A, B, or C, as to a proposition 
 which has ^-g-J of probability, or 103 to 1 in its 
 favour. But if Thomas only know that 100 draw- 
 ings have all given letters, then he, with his know- 
 ledge, has no ground of inference with respect to A, B, 
 and C, in particular, but may reasonably assent to the 
 proposition, " the 101st drawing will also give a letter," 
 as having a probability of j-g 4, or 101 to 1 in its 
 favour. But the paradox in question requires that 
 John should make believe he knows no more than 
 Thomas, and then be surprised that a discordance 
 should arise from his using that knowledge in recon- 
 sidering a result, which he has suppressed in the me- 
 thod of attaining it. 
 
 It very rarely happens that we meet with a case in 
 which we can so distinctly specify the antecedent cir- 
 cumstances which influence our assent or dissent, as to 
 enable us to apply direct calculation to the determin- 
 ation of the rational probability of an event to come. 
 And in most of the practicable cases, the largeness of 
 the numbers employed would check our progress, if it 
 were not for the approximative methods of the higher 
 mathematics, and the tables at the end of this work. I 
 now proceed to the method of using those tables. 
 
USE OF TABLES. 
 
 CHAPTER IV. 
 
 USE OF THE TABLES AT THE END OF THIS WORK. 
 
 I HAVE endeavoured to accumulate in this chapter 
 a considerable part of the uninteresting details of com- 
 putation which accompany the solution of complicated 
 problems. It is at the readers' pleasure to omit the ' 
 whole of it, referring to it afterwards in cases where 
 its assistance may be necessary. 
 
 In Table I. we see (I.) a column headed t^ con- 
 taining the series -00, -01 '99, 1-00, 2, or 
 
 every hundredth of a unit from to 2 (I I.) a column 
 headed H, deduced in the manner pointed out in page 
 17, (HI.) columns headed A and A 2 , which are 
 only the differences of the numbers in column H 
 (marked A), and the differences of those differences, 
 (marked A"). The following is an extract from the 
 table : 
 
 t 
 
 H 
 
 A 
 
 A2 
 
 47 
 
 49374 52 
 
 900 46 
 
 8 61 
 
 48 
 
 50274 98 
 
 891 85 
 
 
 49 
 
 51166 83 
 
 
 
 
 
 
 
 The columns A and A 2 must be made up to 7 places 
 of decimals * by means of ciphers : thus,, 90046 means 
 0090046'; and 86l means -0000861. The formation 
 of A and A is then as follows : 
 
 5116683 -0090046 
 
 5027498 -0089185 
 
 Subt. -0090046 
 
 0089185 
 
 0000861 
 
 * A little practice will show how to dispense with the decimal points 
 altogether till the end of the process. 
 F 3 
 
70 ESSAY ON PROBABILITIES. 
 
 Since it is very seldom necessary to use more than five 
 places of the table marked t, the sixth and seventh places, 
 and those which arise from them in the differences, are 
 separated from the rest by a blank space. The sixth 
 and seventh places are allowed to remain, on account of 
 the use which will hereafter be made of the differences 
 derived from them. 
 
 In Table II. we find to five places only a column 
 marked K, and another marked A, containing the dif- 
 ferences of the former column. This is a modification 
 of the former table, the reason of which will hereafter 
 appear. In the meanwhile, however, observe that we 
 can directly find the value of H and K by these tables 
 only, when t is *QO or -01, &c. ; that is, when t is a 
 given decimal of two places. But supposing it required 
 to find H when t lies between two of the values in the 
 table; suppose, for instance, we ask what is H when 
 t = -47694 ? The method is as follows : 
 
 QUESTION. What is the value of H (Table I.) 
 when t = '47694'., correct to five places of decimals ? 
 
 RULE. 
 
 Take out of Table I. the 
 value of H answering to the 
 two first decimal places and 
 the whole number preceding 
 them, if there be one. Retain 
 only five places of decimals. 
 
 Take the figures of the first 
 difference (as far as the blank 
 space), and multiply them by 
 the remaining figures in the 
 value of t, and cut away as 
 many places from the result as 
 there were remaining figures. 
 
 Add the figures in the last 
 result to the right hand of the 
 first, and the sum is the an- 
 swer required. 
 
 ;MrLIFJCATIOX. 
 
 Opposite to *47 we find 
 49375 * 
 
 Three figures remaining, 594. 
 
 900x694 = 624600. 
 Cut away three figures, 
 625.* 
 
 49375' 
 $& 
 
 50000 
 When t= -47694, H= -50000. 
 
 * Whenever decimal figures are rejected, if the first rejected be five or 
 upwards, the last retained is increased by a unit 
 
USE OF TABLES. 
 
 71 
 
 As another example, suppose the value of t to be 
 1-51209. 
 
 When t=l'51, H = . 96728 
 24 
 
 When t = 1-5 1209, H= '96752 
 
 209 
 
 1026 
 228 
 
 23,826 
 
 We shall now take the inverse problem, and sup- 
 posing H to be intermediate between two values in the 
 table, require the value of t. For instance, let H = 
 93972. 
 
 RULE. 
 
 Find in the table the value 
 of H next below the given 
 value ; note the corresponding 
 value of t, and subtract the 
 nearest value of H from the 
 given value. 
 
 Annex three ciphers to the 
 difference just found, and di- 
 vide by the figures of the dif- 
 ference in the table which 
 come before the blank space, 
 rejecting fractions, and taking 
 the nearest whole number. 
 
 The quotient cannot have 
 more than three places : if 
 three, annex to the value of t 
 already found; if less than 
 three, place ciphers at the be- 
 ginning to make up the defi- 
 ciency, and annex. 
 
 EXEMPLIFICATION. 
 
 H -93972, 
 next below 1 . 93807t = 
 
 > *Q 
 
 in the table J" 
 
 165 
 Tabular diff. 195. 
 
 195)165000(846 
 1560 
 
 900 
 780 
 
 1200 
 1170 
 
 30 
 t=l'32 846 
 
 F 4 
 
72 USE OF TABLES AT END OF WORK. 
 
 Let H = '97169 ; required the value of t? 
 
 H = -97169 
 Nearest below = -97 162 t=l\55 
 
 101)7000(69 t= 1-55069 Answer. 
 
 606 
 
 940 
 909 
 
 31 
 
 The Table II. is used in exactly the same way, except 
 towards the end,, from t = 3*40 upwards ; in which 
 case the cipher at the end of t must be neglected, and only 
 one decimal place taken out of the table in the value 
 of t. For instance, to find the value of t answering to 
 K = -98222. 
 
 K= -98222 
 Next below -98176 t=3-5 
 
 306)46000(150 t = 3'5150 
 306 
 
 1540 
 1530 
 
 100 
 
 In the first table, there is another result which will 
 frequently be wanted, and which I shall call H'. It 
 arises from adding half the second difference to the first 
 difference*, if the value of t be in the tables, and 
 making five decimal places. But if the value of t be 
 not in the tables, then H' must be formed for the 
 values of t immediately above and below ; and by means 
 of the first and the difference of the two, H' must be 
 found in exactly the same manner as H is found in 
 the first of the preceding rules, page 70., remembering 
 to subtract at the last step instead of adding, if the second 
 H' thus previously determined be less than the first. 
 
 * Meaning the whole differences ; not the parts which precede the blank 
 space, as in the preceding rules. 
 
USE OF TABLES. 
 
 EXAMPLE 1. When t is 1-56. what is H'> 
 A= 9745 
 
 *A2 = 151 
 
 / H'= -09896 
 
 EXAMPLE 2. When t z= 1-23412, what is H' ? 
 t=l-23 H'= -24852 
 
 ,_ 
 t=l-24 H'=- 24245 ' 6 7 
 
 412 -24852 
 
 607 250 
 
 2884 Subt. -24602:=H' when t= 1-23412 
 
 2472 
 
 250,084 
 
 We have already seen that when two events, A and 
 B, one of which must happen at every trial, have 
 severally happened m times and n times in m + n 
 trials, it is m -f 1 to n -f 1 that A shall happen 
 at the next trial. But m + 1 to n -f 1 is very 
 nearly m to n, when m and n are considerable num- 
 bers: for instance, 248 to 11? is very nearly 247 to 
 116. That is, when a great many trials have been 
 made, the numbers of times which A and B have hap- 
 pened express very nearly the odds (relative proba- 
 bilities) for A against B ; or, inverted, for B against A. 
 Let us convert the problem, and supposing that we 
 know beforehand the chances of A and B, are we to 
 suppose that in a great many trials A and B will happen 
 in proportion to their respective probabilities ? Common 
 sense tells us that such will always be nearly the case, 
 but that the odds are great against an. exact result. 
 Suppose 3000 drawings to be made from a lottery con- 
 taining two As and one B. We must then, it seems 
 clear, draw A twice as often as B, in the long run. 
 Our reason convinces us thus. Let one of the As be 
 distinguished from the other by an accent, so that we 
 have A, A', and B. If the urn be well shaken before 
 each drawing, it is impossible to believe that, in the 
 
7* ESSAY ON PROBABILITIES. 
 
 whole result of 3000 trials, we shall have drawn the 
 three in very unequal numbers ; so that, destroying the 
 distinction between A and A' 3 we feel secure of drawing 
 A twice as often as B ; and it is obviously two to one 
 in favour of A at each trial. The following phrases 
 seem to common sense to mean the same thing. 
 
 It is two to one that A 
 shall happen, and not B. 
 
 It is an even chance for 
 head or tail. 
 
 It is more than a hundred 
 to one, that a ship at sea will 
 not be lost. 
 
 In the long run, A will 
 happen twice as often as B. 
 
 In a large number of tosses, 
 the heads and tails will occur 
 in nearly equal numbers. 
 
 Of all the ships which sail, 
 the number which is not lost 
 exceeds that which is lost more 
 than a hundred times. 
 
 I now proceed to some problems, which will exhibit 
 the method of applying the tables, and will illustrate 
 and confirm the preceding notions. 
 
 PROBLEM I. The odds for A against B being a to b, 
 to find the chance that in n times a -f- b trials, A shall 
 happen exactly n X a times, and B n X b times. 
 
 RULE. Divide the II' belonging to t = (page 72) 
 by the square root of the following : 8 times the product 
 of n, a, and 6, divided by a + b. 
 
 Suppose, for instance, a die is thrown 6000 times ; 
 what is the chance that exactly 1000 of the throws shall 
 give an ace ? Here it is 1 to 5 that an ace shall be 
 thrown in any one trial, and 6000 is 1000 times 1-f 5. 
 Hence a = 1, b = 5, n = 1000 : 8 times the product 
 of n, a, and b is 40,000, the sixth part of which is 6667 
 (sufficiently near), and the square root of this is 81*65. 
 Again, when.* = 0, we have in Table I. 
 
 A = 112833, A2 = H ; whence H' is 1-12844 
 
 and 1-12814 divided by 81-65 gives -014 very nearly. 
 This is very near the real probability that 6000 throws 
 with a die shall give exactly 1000 aces: for such an 
 event there are only 14 chances out of a thousand ; and 
 
USE OF TABLES. 75 
 
 it is 1000 14 to 14, or about 70i to 1, against the 
 event. This result is rather above what we should have 
 expected ; we might have imagined it to be more 
 than 71 to 1 against 6000 throws giving exactly 1000 
 aces. 
 
 As another example, let us find the probability that, 
 out of 200 tosses with a halfpenny, there shall be ex- 
 actly 100 heads and 100 tails. Here a = 1, b = 1, 
 n = 100, and 8nab* is 800, which divided by a + b, 
 or 2, gives 400, the square root of which is 20. And 
 IT, when t (or 1-12844) divided by 20, gives '056. 
 It is therefore about 944 to 56', or 17 to 1, against the 
 proposed event ; and (page 42.) we must repeat 200 
 throws 1 2 times to have an even chance of the equality 
 of heads and tails happening once. 
 
 Generally speaking, the rules in this chapter are very 
 accurate only when the number of trials is considerable. 
 Suppose only 12 tosses; required the chances of 6 
 heads and 6 tails. Here o = 1, 6=1, n = 6, 
 Snab = 48, which divided by 2 gives 24, whose square 
 root is 4'9 very nearly. And 1*12844 divided by 4*9 
 gives -23, or 77 to 23, that is 3^ to 1 against the 
 event. That is (page 47)> this rule is not very inaccu- 
 rate, even when the number of trials is as low as 12. 
 
 We shall call the event whose chance is sought in the 
 preceding problem, the probable mean; understanding 
 by that term the event which is more likely to happen 
 than any other. Thus, when 12 halfpence are thrown 
 up, 6 heads and 6 tails is the probable mean, being the 
 event which is more likely than any other, though not 
 in itself more likely than not. When 6000 throws are 
 made with a die, the probable mean is 1000 aces, 1000 
 deuces, &c. 
 
 PROBLEM II. The odds for A against B being a to &, 
 required the chance that in n times a + b trials, the 
 As shall fall short of the probable mean by a given 
 
 * Juxtaposition of numbers, in algebra, stands for their product. 
 
76 ESSAY ON PROBABILITIES. 
 
 number I : I being small, compared with the whole number 
 of As in the probable mean. 
 
 RULE. Divide twice / by the square root obtained in 
 the last example ; and find the value of H'from Table I., 
 taking the preceding quotient for t. DivideH', so found, 
 -by the square root just used, and the quotient is the 
 answer required. 
 
 N. B. This rule also applies when the number of A s 
 is to exceed the probable mean by /. 
 
 EXAMPLE I. In 6000 throws with a die, what is 
 the chance that the aces shall fall short of (or exceed) 
 1000 by exactly 50 ? Here a = 1, b = 5, n = 1 000, 
 and the square root is 81*65, as before. And twice /, 
 or 100, divided by 81'65, gives 1-22: to which the 
 value of H' is 25162 -f- | of 6ll, with five decimal 
 places, or '25468. This last, divided by 81 '65, gives 
 0031 ; so that it is about 997 to 3, or 332 to 1, against 
 the proposed event : and the 6000 throws must be re- 
 peated 232 times to give an even chance of succeeding 
 once. 
 
 EXAMPLE II. What is the chance that in 200 tosses, 
 there shall be exactly 95 heads ? Here a = 1, b = 1, 
 n = 100, I =. 5, and the square root, as before, is 20. 
 And twice I, or 10, divided by 20, gives "50, which 
 being t, the value of H' is '87882, which divided by 
 20 gives -044 very nearly. It is, then, 956 to 44, or 
 about 22 to 1, against the proposed event. 
 
 EXAMPLE III. In 12 tosses, what is the chance of 
 exactly 7 heads ? Here a = 1, 6=1, n = 6, /=- 1, 
 the square root, as before, is 4*9., and 2 divided by 4'9 
 is '41 nearly ; which being t, H' is '95384, which 
 divided by 4*9 gives -194. It is therefore 806 to 194, 
 or 4-,-^j- to 1, against the proposed event. In page 47- 
 it is 3304 to 792 against this event, or 4^! nearly. Hence 
 the incorrectness of our rule is very small. 
 
 PROBLEM III. The odds for A against B being 
 a to b y required the chance that in n times a -f b throws, 
 the number of As shall not differ from the probable 
 mean by more than /. 
 
USE OF TABLES. 77 
 
 RULE. Divide one more than twice I by the square 
 root already mentioned, and the quotient being made 
 t, the value of H in Table I. is the probability re- 
 quired. 
 
 EXAMPLE I. In 6000 throws with a die, what is 
 the chance that the number of aces shall not differ from 
 1000 by more than 50; that is, shall lie between 950 
 and 1050, both inclusive. Here a = 1, 6 5, 
 n = 1 000, 1 50, and the square root as before is 8 1 *65. 
 Divide 2 /+ 1. or 101, by 81'65, which gives 1*237, 
 which being t, H is '91977. Hence it is 920 to 80 in 
 favour of the proposed event, or about 11^ to 1 . 
 
 EXAMPLE II. In 200 tosses, what is the chance that 
 the number of heads shall lie between 97 and 103, both 
 inclusive ? Here a = 1, 6 = 1, n = 100, I = 3, and 
 the square root, as before, is 20. And 2 / + 1, or 7, 
 divided by 20, gives '35, which being t, H is -379. 
 Hence it is 621 to 379. or about 31 to 19, against the 
 proposed event. 
 
 EXAMPLE III. In 12 tosses, what is the chance of 
 the heads being either 5, 6, or 7 in number ? Here 
 a =. 1, 6 = 1, n =. 6, / = 1, and the square root, as 
 before, is 4'9- And 2 I -{- 1, or 3, divided by 4*9, gives 
 6l, which being t, H is *6ll7- Hence it is about b'l 2 
 to 388, or 153 to 97, in favour of the proposed event, 
 In page 47 the chance of this event is 
 
 792 + 924 + 792 2508 
 
 or or '612 
 
 4096 4096 
 
 PROBLEM IV. The odds for A against B being a to 6, 
 and n times a -j- b trials being to be made, for what 
 number is there a given probability H that the As shall 
 not differ from the probable mean by more than that 
 number ? 
 
 RULE. Find in Table I. the value of t answering to 
 that of H (page 71), multiply it by the square root 
 already described, subtract 1, and divide by 2 : the 
 
78 ESSAY ON PROBABILITIES. 
 
 quotient, or its nearest whole number, is the answer 
 required. 
 
 EXAMPLE. In 6000 throws with a die, within 
 what limits is it two to one that the aces shall be con- 
 tained ? The square root is 81 '65, and H is -| or '66667, 
 to which the value of t is -68409, found as follows 
 
 (page 71): - 
 
 H= -66667 
 Nearest below -66378 t=68 
 
 Tab. Diff. 706) 289000(409 t= -68409 say -6841 
 2824 81-65 
 
 6600 34205 
 
 6354 41046 
 
 . 6841 
 
 246 54728 
 
 2)54-86 
 
 Answer. It is a little more than 2 to 1, that the 
 aces shall lie between 1000 - 28 and 1000 + 28, and 
 a little less than 2 to 1 that they shall lie between 
 1000 27 and 1000 + 27. 
 
 But the most convenient way of solving this problem 
 is by first finding for what degree of departure from the 
 probable mean there is an even chance. In this case, 
 since H = -5 (page 70), t is =. '476936, which the 
 method in page 41, will show to be very nearly |^. It 
 will be worth while to re- state the whole process. 
 
 The odds for A against B being a to 6, and the pro - 
 posed number of trials being n times a -f fe, required 
 the limits of departure from the probable mean na, 
 within which it is an even chance that the number of 
 As shall be contained. 
 
 RULE. Multiply together 8,71,0, and 6, and divide by 
 
USE OF TABLES. 79 
 
 a + 6 : extract the square root of the quotient., and 
 multiply it by 3 1 : subtract 65, and divide by 1 30 : the 
 nearest whole number is the answer required. Thus in 
 the preceding instance, where the square root is 81*65, 
 multiply this by 31, which gives 2531 '15 ; and 65 less 
 is. 2466-15, which divided by 130 gives 18'97- Hence 
 it is very little less than an even chance that the aces 
 in 6000 throws shall be between 1000 +19 and 
 1000 19, or 1019 and 981. 
 
 Having found the limits of departure for which there 
 is an even chance, we can now use Table II. as follows. 
 The values of t in Table II. are the proportions of 
 various departures (each increased by -5) to that depar- 
 ture which has an even chance, as just ascertained, and 
 also increased by *5 : the values of K are the proba- 
 bilities of the departures answering to those of t. Having 
 then ascertained 18 '97 to be the departure for which 
 there is an even chance, suppose I ask what is that limit 
 of departure within which it is two to one that the 
 aces shall be contained. Two to one gives J for the 
 chance, or -66667 : I look into Table II., and find that 
 when K is -66667, t is 1-43433, found as follows:- 
 
 K= -66667 t= 1-43 
 Next below -66521 
 
 Tab. Biff. 337) 146000(433 t= 1-43433 
 1348 
 
 1120 
 1011 
 
 1090 
 1011 
 
 79 
 
 This is the proportion which the departure in ques- 
 tion, increased by *5, bears to 18*97 increased by -5 or 
 19-47. Multiply 1-43433 fcy 19'47, giving 27*93; 
 from which subtract -5, giving 27*43 for the limit of 
 departure, the same as in page 78. 
 
80 ESSAY ON PROBABILITIES. 
 
 Suppose the question to be that of page 77) namely, 
 what is the probability that the number of aces in 6000 
 throws shall lie within 50, one way or the other, of the 
 probable mean 1000? Now,, 18'97 -f- '5 is 19'47, and 
 50 + -5 is 50-5, and 50-5 divided by 19'47 gives 
 2-594, which being t (in Table II.), K is '91981, ex- 
 tremely near to the result in page 77- 
 
 There are, therefore, two distinct methods of treating 
 these problems, connected with the two tables : and this 
 is a great advantage, since it is a very strong presump- 
 tion of a correct answer, when the results of the tables 
 agree. The problems III. and IV. .being of great import- 
 ance, I shall now recapitulate their details, with the ad. 
 dition of some new phraseology. Let the instance be 
 6000 throws of a die, and the event A the arrival of an 
 ace, and B the arrival of some other face. The most 
 probable number of aces is 1000, though the arrival of 
 that exact number is not probable in the common sense 
 of the word. There will then most likely be a de- 
 parture from the number 1000 in the number of aces 
 thrown ; of which departure we are now entitled to 
 say, that it is very improbable it should be considerable. 
 Let the term neutral departure mean that degree of de- 
 parture for which it is just an even chance that the 
 actual event shall be contained within its limits : in 
 the present instance it is 18'97- We may explain the 
 fraction as follows : suppose a person to receive 100/. 
 for every unit by which the number of aces falls short 
 of or exceeds 1000. Then, supposing him to try this 
 stake a great many times, he will in the long run re- 
 ceive less than 1897/. at a trial, as often as he receives 
 more. But his receipts will oftener exceed than fall 
 short of 1800/. ; while they will oftener fall short of 
 than exceed 1900/. Roughly speaking, there is here the 
 same probability that the aces shall not lie between 
 1000-19 and 1000+ 19 (both inclusive), and that 
 they shall lie between these numbers. 
 
 In all these problems there is a square root to be 
 found, which we call the square root, as there is no other. 
 
USE OF TABLES. 81 
 
 The odds are a to >, for A against B, and n (a + 6) 
 trials are contemplated. Though we have only instanced 
 whole values of n, yet it may be a fraction : thus, if the 
 odds are 3 to 2, and 96 trials are contemplated, n (3 -f 2) 
 must be 9&? or n must be 191. In this case, the pro- 
 bable mean is that A shall happen 57|? and B 38 J- ; by 
 which it must be understood^ that a person who should 
 repeat Q6 throws a great many times, receiving 1 L for 
 every A, would, in the long run, gain on the average 
 57f /. per trial of 96 throws. 
 
 The square root in question, represented algebrai- 
 cally, is 
 
 or the square root of the product of S, n, a, and 6 di- 
 vided by a -f- & I now subjoin the two principal pro- 
 blems, with the two rules in parallel columns. 
 
 PROBLEM. What is the chance that the number of 
 As in w(a-f-&) trials shall lie between na -f / and 
 na I, both inclusive ? or what is the chance that the 
 departure from the probable mean shall not exceed / ? 
 
 BY TABLE I. BY TABLE II. 
 
 Find the square root, and 
 divide one more than twice I 
 by it ; call the result t, and 
 find H answering to t in the 
 table. (Use the rule in p. 70. 
 if necessary.) This H is the 
 probability required. 
 
 Find the square root, and 
 multiply it by 31 ; then divide 
 by 130. To I add -5, and 
 divide by the preceding 
 quotient ; call the result t, 
 and find the value of K 
 answering to t : this is the 
 probability required. 
 
 N. B. The neutral departure 
 
 j is -5 less than the quotient first 
 
 I found. 
 
 PROBLEM. What is that degree of departure within 
 which it is p to q that the number of As in n (a + b) 
 trials shall lie ? 
 
ESSAY ON PROBABILITIES. 
 
 BY TABLE I. 
 
 Divide p by p + g, and call- 
 ing the result II, find the cor- 
 responding value of t in the 
 table. Multiply it by the 
 square root, subtract 1 and 
 divide by 2 ; the quotient 
 being called Z, it is then p to q 
 that the As in n (a + b) trials 
 shall be contained between 
 nal and na + l, both inclu- 
 sive. 
 
 BY TABLE II. 
 
 Find the square root ; mul- 
 tiply by 31, and divide by 
 130. Divide p by p+q, and 
 calling the quotient K, find the 
 corresponding value of t in 
 the table ; multiply t by the 
 pieceding quotient, subtract '5 
 from the product, and I being 
 the remainder, it is then p to q 
 that the As in n (a-t-fc) trials 
 shall be contained between 
 nal and na + 1, both inclusive. 
 
 I shall conclude this problem with an example of each 
 case, worked by both methods, without explanation. 
 There is a lottery containing 3 white and 2 black balls : 
 what is the chance that in 50,000 drawings the num- 
 ber of white balls shall be between 30,000 + 100 and 
 30,000-100? 
 
 fl-3 6 = 2, n=10,000, 7=100 
 8 x 3 x 2 x 10,000 
 
 = 96,000 
 
 V 96,000 =309 -84 
 
 309 -84) 201 ( -64873 = t 
 H= -64109 
 
 309-84 
 31 
 
 130) 9605 -04(73 -885 
 
 73-885)100'5(l-3602 
 K= -64109 
 
 This question shows how nearly a great many trials 
 may be expected to agree with the probable mean : in 
 50,000 trials, it is nearly two to one against the 
 number of white balls differing from 30,000 by more 
 than a hundred. 
 
 In 1 00,000 tosses, between what limits is it 99 to 1 
 that the heads shall be contained ? 
 
 l, 6=1, n = 50,000,^ = 99, j=l 
 100)99('99 = H 
 t^l-8215 
 
 8 x 1 x 1 x 50,000 
 
 -=200,000 
 
 447-21 
 31 
 
 130)13863-51(106-64 
 100)99(-99-=K 
 
USE OF TABLES. 83 
 
 V 200, 000 = 447 '21 
 
 1-8215 106 ' 64 
 
 447-21 
 
 814-6 
 1 
 
 2)813-6 
 
 t = 3 821 
 
 406 '97 = 
 
 406 -8 = 1 
 
 Answer. Between 50,000 407 and 50,000 + 407. 
 
 Now for the inverse method attached to the preceding 
 problem. If I be totally unacquainted with the na- 
 ture of the events A and B, except only that one or 
 other, and not both, must happen every time, it is then 
 clear that, as the matter stands, it is to me 1 to 1 for A 
 against B, with a very great chance that, if I were better 
 informed, I should form a different opinion. At the same 
 time (page 10), I choose 1 to 1 as my rule of action, 
 because, though coming events may not justify my pre- 
 diction, I know of nothing to warrant my assuming 
 that the odds are in favour of A, rather than in favour of 
 B. A trial takes place, and A happens ; it becomes im- 
 mediately most safe to assume that the odds for A against 
 B are 2 to 1, but still that safety is not very decided. 
 But if 1000 trials be made, and if A have happened 
 520 and B 480 times, I can then confidently say, that 
 the odds for A against B are very nearly, if not exactly 
 520 + 1 to 480 + 1, which is nearly 520 to 480. The 
 notion then formed has a strong presumption that it is 
 nearly correct. 
 
 PROBLEM. In a -f- b trials A has happened a times 
 and B b times : from which, if a and b be considerable 
 numbers, it is safe to infer that it is a to & nearly for 
 A against B. What is the presumption that the odds 
 for A against B really lie between ak to b -f k and 
 a-f A; to&-.fc? 
 
 G 2 
 
ESSAY ON PROBABILITIES. 
 
 RULE. (TABLE L) 
 Divide twice the product of 
 a and b by their sum, and ex- 
 tract the square root of the 
 quotient, by which divide k. 
 Then the last quotient being 
 't, the H of the table is 
 probability required. 
 
 the 
 
 RULE. (TABLE II.) 
 
 Having found the square 
 root, and divided k by it, as 
 opposite, from seven times the 
 quotient, take the hundredth 
 part of the quotient, and take 
 
 three tenths of the remainder. 
 Make the result t, and K in 
 the table is the probability re- 
 quired. 
 
 Suppose that in a thousand trials, A has happened 
 exactly 6*00 times, and B 400 times ; what is the pre- 
 sumption that the odds for A against B lie between 570 
 to 430 and 630 to 370 ? 
 
 600,6 = 400, = 30. 
 
 2 x 600 x 400 = 480,000 
 
 480,000 
 1000 
 
 = 480 
 
 V480 =21-91 
 SO 
 
 H = 94713 
 
 1-369 
 7 
 
 9-583 
 014 
 
 9-569 
 3 
 
 28-707 t = 2- 871 
 K= -94719 
 
 Answer. About 95 to 5, or 19 to 1 in favour of the 
 odds being between the limits specified. N 
 
 In the preceding problem, A and B have happened a 
 and b times; whence the most likely of all individual cases 
 is, that the odds for A against B are a to b; or, in other 
 words, the result which has the strongest presumption 
 in its favour is, that 
 
 Probability of A was 
 
 a + b. 
 
 Probability of B was r 
 
 a + b' 
 
 Now we have found, in the preceding problem, the 
 presumption that 
 
 Probability of A lies between and J 
 
 a + b a+b 
 
USE OF TABLES. 85 
 
 or, which is the same thing, that 
 
 Probability of B lies between -- and 
 
 a + b a+b 
 
 For since it is our hypothesis, that either A or B must 
 happen at every trial, whatever presumption there is that 
 the chance of A is #, there is the same presumption that 
 the chance of B is 1 #. 
 
 But we might ask the following questions : A and B 
 having happened a and b times in a + b trials, what are 
 the values of the following presumptions ? 
 
 1. That the probability of A lies between and ; 
 
 a + b a + b 
 
 or its equivalent, that the probability of B lies between 
 b-k b 
 
 ; an <* T 
 
 a + b a + b* 
 
 2. That the probability of A lies between and , 
 
 a+b a+b 
 
 or that the probability of B lies between and - 
 
 a+b a+b 
 
 To solve these by the help of the following rule, 
 remember that, if a be greater than 6, it is more likely 
 that the chance of A falls short of a -4- (a -f 6) than ex- 
 ceeds it : and if a be less than b, then it is more likely 
 that the chance of A exceeds a-f-(a-|~6) than falls short 
 of it. 
 
 RULE. First find the result of the preceding pro- 
 blem, and find from Table I. the H' (p. 72} belong- 
 ing to the value of t. Subtract this from the H / 
 derived from in the table (which is 1-12844); mul- 
 tiply by the difference between b and a, and divide by 
 the product of the square root used in the preceding 
 problem and three times the whole number of trials : 
 call the result V. To one half of the result of the 
 preceding problem add V ; and from it subtract V : and 
 call these results ^H + V and |H-V. 
 
 Then, if B have happened most times, JH-f V is the 
 G 3 
 
86 ESSA5T ON PROBABILITIES. 
 
 presumption that the chance of A lies between - and 
 
 a-j-6 
 a+k 
 
 or that the odds for A against B lie hetween a to b 
 
 and a + k to b k. But in this case, |H V is 
 the presumption that the chance of A lies between 
 a k a 
 
 - - and - or that the odds for A against B lie 
 a + b a + b' 
 
 between a k to b + k and a to b. 
 
 But if A have happened most times,, make -JH -f V 
 and |H V change places in the preceding paragraph, 
 every thing else remaining the same. 
 
 EXAMPLE. In a thousand trials, A has happened 
 600 times, and B 400 times. What is the pre- 
 sumption," 1. that the odds for A against B lie 
 between 600 to 400 and 630 to 370 ; 2. that the 
 odds for A against B lie between 570 to 430 and 600 
 to 400 ? 
 
 From the preceding problem t=l'369., say = 1*37; 
 the square root is 21'91, and H is '9471. 
 
 t = l-S7A17036 1-12844 6-a = 200 
 
 JA2 232 -17268 
 
 - - 3000x21-91=65730 
 H'= -17268 -95576 
 
 200 
 
 65730)191 -152(-0029 =V 
 
 4765 =JH + V 
 4707=JH-V 
 
 Hence (since A has happened most times), it is "4707 
 to *5293, or about 47 to 53, that the odds for A against 
 B lie between 600 to 400 and 630 to 370. And it is 
 4765 to -5235, or about 48 to 52, that the odds for A 
 against B lie between 570 to 430 and 600 to 400. 
 
 The problems which the preceding part of this 
 chapter has enabled us to solve, are the determination 
 of the chance which exists (under known circumstances) 
 for the happening of an event a number of times which 
 
USE OF TABLES. 87 
 
 lies between certain limits, and its converse. The 
 latter problem contains a consideration of some diffi- 
 culty, namely,, the probability of a probability, or, as we 
 have called it, the presumption of a probability. To 
 make this idea more clear, remember that any state of 
 probability may be immediately made the expression of 
 the result of a set of circumstances, which being in- 
 troduced into the question, the difficulty disappears. 
 Thus, suppose a large number of urns containing 
 various proportions of black and white balls. Let 
 there be 100 urns, and let one of them only contain 
 equal numbers of black and white balls. If then 
 I lay my hand upon one of these urns with the intention 
 of drawing, it is, before the drawing, 99 to 1 against 
 my having placed my hand upon an urn from which, 
 in the long run, equal numbers of both sorts of balls 
 will be produced : the presumption that black and 
 white balls have an even chance is only T -J^ ; the pre- 
 sumption that the probability of a white ball is -J, is 
 
 TFo* 
 
 In speaking of compound probabilities, writers have 
 
 employed six words synonymously probability, chance, 
 presumption, possibility, facility, and expectation. Re- 
 jecting only the word possibility, as indicating a thing 
 of which there cannot be different degrees., the five 
 remaining terms have their advantages, each one 
 pointing out a peculiar and useful view of the main 
 idea. Thus the word presumption refers distinctly 
 to an act of the mind, or a state of the mind, while in 
 the word probability we feel disposed rather to think of 
 the external arrangements on the knowledge of which 
 the strength of our presumption ought to depend, than 
 of the presumption itself. When, therefore, having 
 observed an event, we want to know how strongly we 
 are to suppose that the observed event was preceded by 
 a given arrangement of circumstances, the term pre- 
 sumption of probability is very appropriate. The word 
 facility applies particularly to the notion which we 
 form when we see one event happen more often than 
 
88 ESSAY ON PROBABILITIES. 
 
 another, namely, that it is easier to produce the first 
 than the second. In our problems,, however, the 
 facility is not that arising from art, but from previous 
 (it may be accidental) distribution of means. The 
 word expectation will lie applied throughout this work 
 to that state of things for the production of which 
 there is an even chance. If (p. 79)^ 6000 throws be 
 made with a die, it is an even chance that the number 
 of aces lies between 981 and 1019 : the odds are 
 against any smaller amount of departure on both sides 
 of the probable mean, and against any greater amount ; 
 this is then our expectation of the number of aces. 
 
 When one of two possible events happens oftener than 
 the other, it being understood that one, and only one^ 
 can happen each time, we are led to suppose that the 
 excess of one event is the consequence of some arrange- 
 ment which would, had we known it, have made us 
 count that event more probable than the other. If A 
 or B must happen, and if in a thousand trials the As 
 outnumber the Es very much, we feel perfectly cer- 
 tain that such must have been the case. The theory 
 of probabilities confirms this impression, as will appear 
 by the solution of the following 
 
 PROBLEM. In a-\- b trials, the number of As was 
 , and that of Bs was b. If a exceed b considerably *, 
 required the presumption that there was at the outset 
 a greater probability of drawing A than of drawing B, 
 in any one single trial ? 
 
 RULE. Divide the difference of a and b by the 
 square root of twice their sum, and let the result be t. 
 Find (page 72) the H' corresponding to t : multiply 
 the result by the sum of a and b, and divide by the 
 product of 8, ^ and the square root of the product of 
 a and b. The result subtracted from unity gives the 
 answer required. Suppose, for instance, that out of 
 50 trials A occurs 32 times, and B 18 times. Then, 
 
 * In order that the result may be very correct, a must exceed b so much 
 that the excess of a above b, multiplied by itself, may considerably exceed 
 the sum of a and 6. 
 
USE OP TABLES. 89 
 
 50x2=100, VIo61=10 S2 ~ 18 _i-4o_ 
 H'=-15891 15891x50 = 7-9455 
 
 ^32x 18 = 24, 8 x 24 x 1-40=268-8 
 
 7-9 divided by 268-8 is ^ nearly: 1 -^ is r ?|i 
 
 Hence it is about 26l to 8 that A was more probable 
 than B. 
 
 ADDITIONAL RULE. When a and b are nearly equal, 
 find t, as in the last rule; find H (not H') correspond- 
 ing to t, add I, and divide by 2 : the result is the pro- 
 bability required. 
 
 The additional rule belongs to the more important 
 case of the two, namely, that in which A has not hap- 
 pened so much oftener than B as to justify an imme- 
 diate conclusion that it was the more probable event of 
 the two. Suppose, for instance, that A has occurred 
 10,100 times out of 20,000 trials, and B 9,900 times: 
 then t = 200 divided by 200, or 1 ; to which H is 
 843, and this increased by 1, and the result divided by 
 2, gives -922. It is, therefore, about 11 J to 1 that A 
 was the more probable. 
 
 The preceding solution can be applied to various 
 species of observations ; of which we shall see more 
 hereafter. The following may be considered as closely 
 connected with it. If we make two different sets of 
 trials, in circumstances which we suppose to be the 
 same, it will generally happen that the As will not bear 
 the same proportion to the Bs in both sets. If r for 
 instance, we find 1000 As arrive in 2000 trials, the 
 odds are very much against the arrival of exactly 5000 
 As in a new set of 10,000 trials, though the expectation 
 is that something near that number of As will arrive. 
 Suppose that the first and second sets of trials give 1st, 
 50 As, 30 Bs; 2nd, 112 As, 6l Bs. 
 
 In the second set the As bear a larger proportion to 
 the whole than in the first : and our present question is 
 what presumption thence arises that there is some dif- 
 ference of circumstances between the two sets, which 
 
90 ESSAY ON PROBABILITIES. 
 
 gives A a greater facility in the second than in the first, 
 or a greater probability of being drawn at any one trial ? 
 Or, if in a first set of a -f b trials, A happen a times 
 and B happen b times ; and if in a second set of a' -f b' 
 trials,, A happen a times, and B happen &' times ; and if 
 a' be a larger proportion of a x -f b' than a is of a 4- b ; 
 required the presumption that there was a greater chance 
 of drawing A at a single trial in the second set than in 
 the first ? 
 
 RULE. Divide the cube of the sum of a and b by 
 twice their product : do the same with a' and 6 X : mul- 
 tiply the two results together, and add them together : 
 divide the product by the sum, and extract the square 
 root of the quotient. 
 
 Divide a by of + &', and a by a -f b, and subtract 
 the less result from the greater. Multiply the difference 
 by the square root previously found, and let the product 
 be t. Then the H corresponding to t, increased by 1, 
 and divided by 2, is the presumption required. 
 
 In the example a = 50, b = 30, a = 1 12, &' = 6l. 
 
 80 x 80 x 80 = 512000 2 X 50 X 30 = 3000, 51200 = 1707 
 
 3000 
 
 173 x 173 X 173 = 5177717 2 X 112 X 61 = 13664 j 5177717 = 378-9 
 
 13664 
 1707 X 378-9 = 64678-23, 170'7 + 378*9 =549 '6 
 
 64678-23 = 117.7, v ? H77 = 10-85 
 549-6 
 tyjfjj= -6474 -6250 = 0224 
 
 0224 X 10-85 = -24304 = t, H = -2689 
 
 J (H -f- 1) = '635, the probability required ; and it is 
 therefore about 16 to 9 in favour of the excess of As 
 at the second set of trials not being accidental fluctua- 
 tion, but arising from some new circumstance or dif- 
 ferent arrangement of the old ones. 
 
 If, in 1000 trials, A should happen 520 times, and 
 B 480 times, there is strong presumption that in any fu- 
 ture number of trials the whole number will be divided 
 among As and Bs nearly in the proportion of 520 to 
 
USE OF TABLES. 91 
 
 480. But this is not the same set of circumstances as 
 that of the problem in page 77- We are there sup- 
 posed to know exactly in what proportion As and Bs are 
 contained in an urn ; and with this positive knowledge 
 we can ascertain the probability of drawing any given 
 number of As in a given number of trials. In the 
 present instance we do not know the contents of the 
 urn, but only the result of a certain number of drawings, 
 from which we can draw presumptions, as in page 53. 
 about the whole contents. The determination of chances 
 relative to a new set of trials depends upon two risks in 
 the latter case, and upon one only in the former. The 
 latter problem is therefore more complicated in its prin- 
 ciples though not so in its results. 
 
 Let us suppose two different persons, John and 
 Thomas, thus situated with respect to the contents of an 
 urn: John knows that there are as many As as Bs; 
 Thomas has observed a hundred successive drawings, of 
 which (so let it have happened) fifty have given A, and 
 as many have given B. That which John knows is 
 rendered not improbable to Thomas by the result of 
 the trials, while the same result would have been thought 
 not unlikely beforehand by John. But there is this 
 difference between their degrees of knowledge, that John 
 has the certainty of a fact (the equality of As and Bs), 
 of which Thomas can only say that the fact, or some- 
 thing near it, is extremely probable. No one could 
 argue with John against any particular venture in such 
 a lottery upon the ground of the possibility of the As 
 much exceeding the Bs; while with Thomas it might 
 be urged as possible, though not probable, that the 
 former might exceed the latter a hundred-fold. Again, 
 suppose John and Thomas, having equal fortunes, are 
 disposed to venture as far as produce woultl warrant, 
 upon the results of a hundred (to Thomas a second 
 hundred) trials. It is obvious to common sense that 
 Thomas must not venture so much as John ; for he 
 runs a larger risk, seeing that he assumes as an average 
 result what possibly may have been a rare occurrence. 
 
92 ESSAY ON PROBABILITIES. 
 
 The following is the rule pointed out by the theory of 
 probabilities: The expectation of fluctuation should 
 be greater to a person who proposes to try q new in- 
 stances, upon the assumption that p preceding instances 
 have fairly represented the long run, than it should be 
 to another person, who knows in what proportions the 
 As and Bs really exist ; and greater in the proportion of 
 the square root of p augmented by q to the square root 
 of p. Thus, if in the preceding case, John and Thomas 
 propose to embark in a matter which depends on 300 
 more trials, the proportion of the square root of 100 -f 300 
 to that of 100, being that of 2 to 1, it follows that, 
 whatever reason John may have to guard against the 
 possibility of 300 drawings giving x more than 150 As, 
 Thomas has as much reason to guard against 2 x more 
 than the same number. 
 
 PBOBLEM (to be compared with that in page 77- ) 
 When a + b trials have happened to give a As and b 
 Bs, required the chance that in n times a -j- b new. 
 throws, the number of As shall not differ from na by 
 more than /. 
 
 RULE. Divide one more than twice I by a square 
 root to be immediately mentioned, and the quotient being 
 made t, the value of H in Table I. is the probability re- 
 quired. 
 
 The square root in the 
 former problem was that of the 
 product of 8, n t a, and b 
 divided by a + b. 
 
 The square root in the 
 present problem, is that of 
 the product of 8, w, n+l t a t 
 and b divided by a -f- b. 
 
 The additional rule in page 81. may also be applied 
 verbatim, the square root *now meaning the second 
 square root above given ; and the inverse rule (p. 82) 
 may be applied in exactly the same way. 
 
 EXAMPLE. In 600 drawings A occurred 100 times, 
 and B 500 times ; what presumption thence arises that 
 in 6000 more drawings A would occur somewhere be- 
 tween 1000 50, and 1000 + 50, or 950 and 1050 
 inclusive ? (See page 77 for the corresponding pfo- 
 blem.) 
 
ON THE RISKS OP LOSS OR GAIN. 9 
 
 (a + 6) = 73333 
 
 _ 101 
 
 V73333 = 270-8; 27+1 = 101; 75^ = ' 373=t 
 
 H = -4 very nearly, or about two to three against the 
 proposed event. 
 
 1 Having thus shown the use of the tahles at the end 
 of this work, in the solution of complicated questions, I 
 now proceed to the application of the theory to questions 
 involving loss and gain. 
 
 CHAPTER V. 
 
 ON THE RISKS OP LOSS OR GAIN. 
 
 THE proverb which advises us to throw a sprat to catch 
 a whale, shows that mankind consider a chance of a 
 gain to be a benefit for which it is worth while to give 
 up a proportionate certainty. The principle on which 
 depends the determination of the amount which it is 
 safe to hazard, must vary with the circumstances of the 
 person who runs the risk. A man should not hazard 
 his all on any terms ; but in ventures the loss of one of 
 which would not be felt, we may suppose the venturer 
 able to make a large number of the same kind; in 
 which case, the common notions of mankind, reinforced 
 by the results of theory, tell us that the sum risked 
 must be only such a proportion of the possible gain 
 as the mathematical probability of gaining it is of 
 unity. For instance : suppose I am to receive a shilling 
 if a die, yet to be thrown, give an ace ; in the long 
 run, an ace will occur one time out of six, or I shall 
 lose five times for every time which I gain. I must 
 therefore make one gain compensate the outlay of six 
 Dentures, or one sixth of a shilling is what I may give 
 
94 ESSAY ON PROBABILITIES. 
 
 for the prospect, one time with another. But ^ is the 
 probability of throwing the ace. 
 
 PRINCIPLE. Multiply the sum to be gained by the 
 fraction which expresses the chance of gaining it, and 
 the result is the greatest sum which should be given for 
 the chance. 
 
 PROBLEM. I am to gain by the throw of a die as 
 many pounds as there are spots on the face which is 
 thrown, with this exception, that I am to lose as many 
 pounds as there are spots on both the face in question 
 and on that of another die which is thrown at the same 
 time, if the two give doublets. What should I give 
 for the chance in question ? 
 
 Call these dice P and Q : then the chance of an ace 
 from P, and something not an ace from Q, is the pro- 
 duct of i and |> or ^ ; and the same for any given 
 throw from P, and not the same from Q. Consequently, 
 the chance of winning, independently of the chance of 
 losing, is worth 
 
 3 5 g x 1 + 3 5 5 x 2 + g * g x 3 + 3 3 g x 4 + 3 5 g x 5 + 3 5 g x 6, 
 
 or - 3 5 6 - x 21, or 2-|-J. But if the chance of winning 
 must be paid for, the chance of losing must be paid. 
 Now the chance of throwing any one pair of doublets 
 is the product of J and ^, or ^ : whence the sum I 
 must receive if I pay 2^ (or not pay out of the 2-\-$, 
 which is the same thing) is 
 
 3S X 2 + 35 x 4 + 38 x 6 + & X 8 + J a X 10 + 3 'g X 12, 
 
 or 1 J. Consequently, I must pay only If. If I were 
 to stand a million of such hazards, paying l-| for each, 
 I might expect my ultimate gain or loss to be extremely 
 small, compared with the whole sum risked. If I had 
 besides a very small profit upon each, I might be sure 
 of winning. 
 
 We have in the last chapter considered the amount of 
 fluctuation for which there is a given probability : we 
 will now look at the percentage of fluctuation, that i^, 
 
ON THE RISKS OF LOSS OB GAIN. 95 
 
 at the proportion which the fluctuation bears to the whole. 
 It is for want of consideration on this point that many 
 notions prevail which are utterly at variance both with 
 the daily evidence of facts, and the results of exact in- 
 vestigation. 
 
 Name any sum of money as a probable gain or loss 
 of a commercial transaction, and we immediately pro- 
 ceed to compare it with the whole capital by which it is 
 to be borne, if lost. A hundred pounds is nothing in 
 one case, because it is only a shilling per cent, on the 
 outlay; in another, the same sum is important, because 
 it is ten pounds per cent. The importance of a gain 
 or loss, then, depends upon the relative, and not on the 
 absolute, value of the sum in question. Similarly, in a 
 large number of transactions of the same sort, the 
 number of them which may go against previous calcu- 
 lation is only important as compared with the whole 
 number in question. The following is the correct prin- 
 ciple, namely, that the percentage of fluctuation for 
 which there is a given chance, varies inversely as the 
 square root of the whole number of trials. Suppose, 
 for instance, I find that on a certain hundred risks it 
 is an even chance that one out of twenty goes against 
 calculations made by the preceding methods ; then if I 
 take four times as many trials, it is an even chance that 
 one out of forty only will disappoint the calculations ; 
 if nine times as many, one out of sixty, and so on. 
 Thus it appears that the chances may be made to be 
 against any named percentage of fluctuation, however 
 small, by sufficiently multiplying the number of risks. 
 The probable amount of fluctuation increases, but not 
 so fast as the number of risks, in such manner that the 
 proportion of the probable fluctuation to the whole 
 diminishes. 
 
 Gambling, according to the common notion of the 
 word, means the habit of risking considerable sums in 
 hazardous transactions. The gambler, properly so called, 
 at cards and dice, and the jobber at the Royal Exchange, 
 are called by the same name. Moral considerations 
 
96 ESSAY ON PROBABILITIES. 
 
 must induce us to regard with an evil eye the person, 
 whatever be the name of his occupation, who thrives 
 by the actual losses of others, in transactions which are 
 not commercially beneficial to the community. I am not 
 prepared to say that stock-jobbers deserve to be included 
 in this class ; it may be that the acquisition or sale of 
 bond fide investments may by their means be rendered 
 more easy to be obtained, and that they themselves may 
 be a useful circulating medium between those who 
 really wish to buy and sell for their own purposes. 
 However this may be, the character of a gambler 
 has no claim, when he is skilful, to the sort of 
 respect which we pay to those who risk. The man of 
 cards and dice, if he be cool and stick to his principles, 
 secures a certainty to himself, and throws all the hazard 
 upon his opponents : taking care never to risk too large 
 a proportion of his means, and thus always enabling 
 himself to take the benefit of the long run, he manages 
 to play upon terms slightly unequal, and, of course, in 
 his own favour. He runs a greater risk in games of 
 mixed skill and chance than in games of pure chance 
 alone. The latter he does not play at, unless he know 
 the chances to be in his own favour ; while in the for- 
 mer it is next to impossible that he should always be 
 more than a match for every opponent. The keeper of a 
 gambling house has the surest game imaginable : the play 
 is so managed that there shall be some chances more for 
 the bank, as it is called, than for those who play against 
 it ; care is taken to provide a sum sufficient to stand a 
 considerable succession of losses ; and the preceding 
 principles will enable us to show that no individual re- 
 sources can stand against a large fund thus used. The 
 following problems will illustrate this. 
 
 PROBLEM. An indefinite number of successive ha- 
 zards are tried of the following kind: One of the 
 events, A, B, C, &c. must happen at every trial, and 
 each event brings with it a specified gain or loss. 
 What may be expected to be the result of continuing to 
 play a very great number of trials ; or what, in the long 
 
ON THE BISKS OF LOSS OR GAIN. 97 
 
 run, may be expected to be the average produce or loss 
 per game ? 
 
 RULE. Multiply each' gain or loss by the probability 
 of the event on which it depends ; compare the total 
 result of the gains with that of the losses: the balance 
 is the average required, and is known by the name of 
 the mathematical expectation. When the balance is 
 nothing, then the play is equal. 
 
 For example, a person plays against a bank in the fol- 
 lowing manner : Equal stakes of one shilling each are 
 laid down, and if head be thrown twice successively he 
 wins, or if tails be thrown twice he loses. But if 
 head and tail be thrown, then another throw is to be 
 made, by which, if it be head, the player only recovers 
 his stake, but wins nothing: while, if it be tail, he 
 loses his stake. In such problems, the stake is a mere 
 evidence of solvency, with which the mathematical 
 question has nothing to do. Let us suppose, then, that 
 both parties keep their money until it is called for by the 
 result. The events on which the player gains or loses 
 are as follows (H stands for head, T for tail) : 
 
 H H gains him a shilling. 
 
 TT, HTT, THT lose him a shilling. 
 
 HTH and THH give no result. 
 
 The chance of HH is 1, and the same of TT; while 
 for each of the set HTT, THT, HTH, and THH, 
 the chance is -*. Hence the value of the prospect of 
 gain is J of one shilling ; that of loss \ -j- -J + \> or 2 f 
 a shilling : consequently the balance against the player 
 is ^ of a shilling each time, and this he would certainly 
 lose in the long run ; that is to say, the number of times 
 he loses would eventually so far exceed the number of 
 times he wins as to make the playjcost him threepence 
 per game. 
 
 Every risk of loss must then be compensated by an 
 equal chance of gain, when the play is equal. But it 
 does not follow that equal play means prudent play ; 
 
98 ESSAY ON PROBABILITIES. 
 
 and for the following reason. Prudence requires that 
 no one should expose himself to great risks of loss, and 
 does not accept it as an excuse that there was a remote 
 chance of enormous gain, or that another had the same 
 chance of the same gain or loss. Again, the mathe- 
 matical expectation is derived from the result which, as 
 can be shown, will be produced in the long run : con. 
 sequently no one can prudently play, even with the play 
 in his favour, unless he continue the occupation through 
 such a number of trials that he may reasonably expect 
 an average of all sorts of fortune. And though I have 
 hitherto appeared to speak only of games of chance, yet 
 precisely the same considerations apply to mercantile 
 speculations, and to every species of affair in which no 
 absolute certainty exists. If any possible event enter 
 into the play which, from the nature of the game, cannot 
 often occur, and if a stake be made upon the arrival of 
 that event, proportionate to some enormous benefit which 
 it is agreed that event shall secure, then prudence re- 
 quires that the game shall be very often repeated, or, if 
 that cannot be done, that it shall not be played at all. 
 There is a celebrated case known by the name of the 
 Petersburgh problem, which is one of the most in- 
 structive lessons in this subject, both on account of its 
 paradoxical appearance, and also because very eminent 
 writers have considered it as a sort of stumbling-block, 
 and have endeavoured to evade the conclusion. Con- 
 dor cet and others have taken a proper view of the subject; 
 while among those who have considered the problem as 
 an anomaly, we may instance D'Alembert. 
 
 If p, g, and r be three fractions whose sum is unity, 
 it follows that we may suppose three events, one of which 
 must happen, and neither of the others, and of which the 
 chances are p, q, and r : and similarly of more fractions 
 than three, whose sum is unity ; or of any number of 
 fractions, however great, provided their sum be unity. 
 Let us, then, take an infinite number of fractions whose 
 sum is unity : either of the following series will do. 
 
nv THE RISKS OF LOSS OR GUN. 99 
 
 1 l i ft ^& C ' I ft * T<?M TOoV&C. 
 
 i i ft & fr &* r J # A % & c - 
 
 Let the game be as follows : The events which may 
 happen at every trial are Ej, of which the chance is J ; 
 E ; of which the chance is | ; E 3 , of which the chance 
 is -J ; and so on, ad infinitum. And one of these must 
 occur. The bank engages to give 2/. if E should turn 
 up, 4/. for E 2 , 8/. for E 3 , l6l. for E 4 , and so on, 
 ad infinitum. What should the player give to the bank 
 for one trial ? Write the several possible gains in a row, 
 and underneath each the chance of its being won, as 
 follows : 
 
 2 4 8 16 32 64 128, &c. 
 i i * * & A ik &c. 
 Multiply each gain by the chance of gaining it, and 
 each result is 1 ; consequently the mathematical expect- 
 ation of the player is unity repeated an infinite number 
 of times, or an infinite amount. No sum, then, how- 
 ever great, can compensate the bank for its risk. The 
 Petersburgh problem realises the preceding supposition as 
 follows : A halfpenny is tossed up until a head arrives, 
 which is the event in question. If this happen at the 
 first toss, the player receives 2/. ; if not till the second, 
 4/. ; if not till the third, 8/., and so on. Now, H 
 standing for head and T for tail, the chance of H is ; 
 of TH, |; of TTH, $; of TTTH, ^ and so on. 
 But can it be believed, that if I am only to throw until 
 head arrives, and to receive 2/., or 4/., or 8L, &c. ac- 
 cording as this happens at the first, second, third, &c. 
 throw can it be believed, you will say, that this pros- 
 pect is even worth 100/. ; and is it not altogether mon- 
 strous to say that an infinite amount of money ought 
 to be given for it ? 
 
 Firstly, I will advert to a large number of trials 
 which was actually made. Buffon tried 2048 experi- 
 ments, or sets of tosses, the results of which were as 
 follows: In 106l, H appeared at the first toss; in 
 H 2 
 
100 ESSAY ON PROBABILITIES. 
 
 494, at the second ; in 232, at the third ; in 1 37, at 
 the fourth ; in 56, at the fifth ; in 29, at the sixth ; in 
 25, at the seventh ; in 8, at the eighth ; and in 6, at 
 the ninth. Let us, then, compute the amount which he 
 would have received if he had bond fide played all these 
 games on the preceding terms. 
 1061 x 2 = 2122 
 
 494 x 4 = 1976 
 
 232 x 8 = 1856 
 
 137 x 16 = 2192 
 
 56 x 32 = 1792 
 
 29 x 64 = 1856 
 
 25 x 128 = 3200 
 
 8 x 256 = 2048 
 
 6 x 512 = 3072 
 
 The 2048 games would have 
 given 20,114Z. or nearly 
 10Z. per game, one game 
 with another. 
 
 No person would stake at this game for a single trial, 
 upon the prospect of head being deferred till the ninth 
 throw. Nevertheless, in this instance, it appeared that 
 out of 2048 trials, such a rare occurrence happened often 
 enough to realise more than any other, with one excep- 
 tion. If Buffon had tried a thousand times as many 
 games, the results would not only have given more, but 
 more per game. A larger net would have caught, not 
 only more fish_, but more varieties of fish ; and in two 
 millions of sets, we might have expected to have seen 
 cases in which head did not appear till the twentieth 
 throw. Let us turn back to page 43, and inquire, by 
 the rule there given, in how many trials it is 10,000 to 
 1 that head will be deferred till the twentieth throw. 
 Out of 2 20 , or 1,048,576 cases, representing the number 
 of different arrangements which may happen in 20 
 throws, the arrangement in question is but one ; it is 
 then 1,048,575 to 1 against its arrival in any one given 
 trial. Look in the Table opposite to " 10,000 to 1," 
 and we find 921: multiply 1,048,575 by 921, and 
 divide by 100, which gives 9)657,375. It is then 
 more than 10,000 to 1 that head is deferred till the 
 twentieth throw somewhere in ten millions of trials, and 
 more than an even chance that it is found to occur in 
 seventy thousand trials. Thus the reader may readily 
 
ON THE RISKS OF I-OSS ' Cl$ $3ATN. <X 
 
 conceive that with unlimited license of proceeding in 
 this play, the player might continue until he had realised 
 not only any given sum, but any given sum per game : 
 a result which is indicated hy the application of our rule, 
 when it tells us that the mathematical expectation of 
 the player upon a single game is infinite. 
 
 The result of all which precedes shows us that great 
 risks should not be run, unless for sums so small that 
 the venturer can afford to repeat them often enough 
 to secure an average. But it should seem as if we were 
 thus told either not to gamble at all, or else to play in- 
 cessantly. With a little reservation, this is true ; the 
 stake must be lowered, and more games played, instead 
 of risking a large fraction of the whole upon one game. 
 It is better to buy the sixteenth of sixteen different 
 tickets than to stake all upon one ticket ; and this even 
 though it should be better than either not to buy at all. 
 It is more prudent to play twenty games, staking one 
 shilling upon each, than to stake a sovereign upon one 
 game. Lay a proper proportion of the whole capital 
 upon any hazard, and stipulate for as many trials as you 
 please, and it will follow that with any mathema- 
 tical advantage, however trifling, in your favour, you 
 must come off a winner. The mistake committed by 
 those who attempt to gamble with professional men, is 
 twofold : firstly, they set out upon unequal terms ; se- 
 condly, if the terms were equal, their stakes would be 
 too large a proportion of their means. That the terms 
 are unequal may readily be supposed, and will presently 
 appear. No bank or individual gamester can play on 
 fair terms, without losing as much as he wins in the 
 long run. But even in such a case, the player of 
 superior fortune has a great advantage over his anta- 
 gonist, unless the stake be very small. If A with 
 twenty guineas engage B with forty, all other things 
 being equal, and if they are to play on until one or 
 other has lost all, it is obviously much more likely that 
 A shall lose his money before B, than the converse. 
 If the play be unequally in B's favour, as well as the 
 IT 3 
 
lf/2 ESSAY ON PROBABILITIES. 
 
 largeness of the fund, then it is still more against A in 
 any given succession of games. The truth is, that to a 
 young man who is determined to gamble, whether at 
 one of the private receptacles in London, or the (till 
 lately) recognised saloons of Paris, it is of little con- 
 sequence whether his stakes be high or low, except in 
 this particular, that a longer process of ruination will 
 give him more chances of seeing his error. The play is 
 against him in both cases, and sooner or later he must 
 be ruined. Nor if his means be ever so great, could he 
 'make use of them, against the banks in Paris, at least. 
 Those who conducted the play at the Palais Royal 
 were perfectly aware of the necessity of not staking too 
 much, and limited not only the amount of each stake, 
 but also the number of persons whom they would en- 
 gage at once. The consequence was, that though they 
 played with perfect fairness (inasmuch as the inequality 
 which existed in their favour was known to, and recog- 
 nised by, their opponents), they gained large returns 
 upon their capital, besides paying a considerable duty 
 to the government. 
 
 A gambler (meaning a bold venturer, which the term 
 commonly implies) ceases to be such when he makes 
 his stakes bear a proper proportion to his capital, and 
 takes no hazards which are unduly against him. If, 
 then, a government should attempt to discourage the 
 acquisition of great losses and gains, by limiting the 
 number of hazards which an individual should be allowed 
 to take, it might defeat its own object ; and this is the 
 case with our law, as it stands at present. In order to 
 prevent individuals from gambling in life-insurance, the 
 legislature has declared that A shall not insure the life 
 of B, unless he have what is called an insurable interest 
 in that life ; that is, unless A have some pecuniary in- 
 terest in B's continuing to live. The insurance offices, 
 for the most part, have virtually, and very wisely, 
 refused to live under this law, by paying all fair claims 
 without questions asked. But supposing that the law 
 were enforced, its effect would be as follows. It is 
 
ON THE RISKS OP LOSS OR GAIN. 103 
 
 tolerably easy to create a bond fide insurable interest on 
 a few lives, while it would be difficult and attended with 
 danger of detection to do the same with many lives. 
 Under the system,, then, proposed by the law, it would 
 be easy to gamble, but not easy to carry speculation to the 
 extent which would make it cease to be gambling. Allow 
 the venturer to extend his traffic, and he will soon begin 
 to feel the average, not to his gain but to his loss. For 
 the mathematical advantage is in favour of the insurance 
 offices, which are sure to gain in the long run. If, then, 
 the law had been intended to save the gambler from 
 certain loss, and to make it a real toss up, it would have 
 been rational, considered as means ; but if, as I ima- 
 gine, it was meant to hinder immoral gain, a more futile 
 contrivance can hardly be conceived. 
 
 It must be remembered that, in the long run, events 
 will happen in proportion to the chances of their happen- 
 ing in a single trial. We see this result in Buffon's 
 trial of the Petersburgh problem, for which I write 
 down the numbers of cases as they did arise, and un- 
 derneath as they would have arisen, one time with an- 
 other, if a great many series of 2048 trials each had 
 been made. 
 
 2048 1061 494 232 137 56 29 25 8 6 
 2048 1024 512 256 128 64 32 16 8 4. 
 
 This rule, however, will only apply when so many 
 cases are taken as will produce a great many of every 
 event to which reference is made. For instance, in 
 2048 trials, one time with another, we can only expect 
 the deferment of head till the seventh throw 16 times; 
 the result gave 25 times, or 56 per cent, more than the 
 probable average. But the occurrence of head at the 
 first throw, which, one set with another, would have 
 occurred 1024 times, did really occur 1061 times, or 
 3 per cent, too many times. If we remember that in 
 the long run, and on 2048 trials, we might expect two 
 H 4 
 
1> ESSAY ON PROBABILITIES. 
 
 sets in which head should not appear till the tenth throw, 
 and one in which no such thing should take place till 
 the eleventh, and if we calculate the total amount 
 which would have been realised had the average case 
 occurred, we shall find it to be 1 1 per game. In the 
 experiment in question, it would have produced < 10. In 
 precisely the same way, sets each consisting of 2" games 
 would have realised n per game (2048 is the eleventh 
 power of 2). 
 
 I now come to the estimation of the chances of 
 fluctuation in loss or gain, meaning by fluctuation 
 any departure from that general average to which the 
 results of more and more trials will continually approach. 
 It has been assumed in what precedes, that the propor- 
 tions which the fluctuation will bear to the whole will 
 diminish without limit as the number of speculations 
 increase. The following problems are easy deductions 
 from those in the last chapter. 
 
 PROBLEM. It is known to be a to b for A against B. 
 A is an event which brings a loss or gain of g pounds ; 
 B is another event which brings a loss or gain of h 
 pounds. What is the general average of such trials ; 
 and what is the chance that in n times a -f b trials, the 
 result as to loss or gain shall differ from the general 
 average by not more than v pounds. 
 
 RULE. Find g times a, and h times 6, and if g and 
 h be both gains or both losses, take their sum ; but it 
 one be a gain and the other a loss, take the difference, 
 counting it gain or loss, according as the term which 
 contained the gain or the loss was the greater. Multiply 
 the result by n, wjiich gives the most probable total result 
 (call this M). The general average is the n ( a -f- 6)th 
 part of this ; or, more simply, the (a + &)th part of 
 the balance of g times a and h times b. Take the differ- 
 ence of g and h, if of the same name, or their sum, it 
 of opposite names, and by it divide v. Take one more 
 than twice the quotient. Having found this result, 
 divide it by a square root immediately to be described, 
 and let the quotient be t. Then the value of H in 
 
ON THE RISKS OF LOSS OR GAIN. 105 
 
 Table I. is the probability that the resulting gain or loss 
 shall lie between M -f v and M v pounds. 
 
 If it be absolutely known that the chances are as a 
 to b for A against B, then, as in page 81, the square root 
 ' is that of the product of 8, n } a, and b, divided by a -j- b. 
 But if all that is known be that in a -f- b previous spe- 
 lations, a gave A, and b gave B, then, as in page 92, 
 the square root is that of the product 8, n, n -J- 1, a, 
 and b } divided by a -t- 6. 
 
 EXAMPLE. It has been observed, that of 100 specu- 
 lations, 70 yielded a profit of 20 each, and the re- 
 mainder a loss of 25 each. What is the probability 
 that in 1 50 more such speculations the total result shall 
 not differ by more than 100 from its most probable 
 amount ? 
 
 a = 70, b = 30, n = 11, g ^ 20 gained, h = 25 lost, 
 v = 100. 
 
 g times d = 1400 gain"! -, rt ^ . 
 
 h times b = 750 toss j ? = <H general average of gam. 
 
 650 gain x l = 975 probable total. 
 
 ' n llfiQ _ 0- 
 
 - J 
 
 _ -. 9'9999 v 9 J. 1 
 
 h = 25 loss J 45 - J ^^ , J m* * J + 
 
 Add 45. 
 8 x n x n +Ixax6 = 8xlx2lx70x 30 = 63,000. 
 
 6 -^- = 63 > Visa = 25-JOO,^! 4 = -217 = t 
 Table L, if t = -217, H = -241. 
 
 Hence it is more than 3 to 1 agaiijst the result lying 
 within the given limits. 
 
 PROBLEM. All things remaining as in the last pro- 
 blem, what is the amount of departure from the pro- 
 bable total for being within which there is the given 
 odds p to q 9 
 
 RULE. Turn p divided by p -f- q into a decimal 
 fraction, and find it in the column H of Table I., 
 taking out the corresponding value of t. Multiply t by 
 
106 ESSAY ON PROBABILITIES. 
 
 the square root above mentioned, and having subtracted 
 1, divide the remainder by 2. Multiply the quotient 
 by the difference or sum of g and h, according as they 
 are of the same or different names, and the product is 
 the answer required. 
 
 EXAMPLE. In the preceding example, within what 
 departure from .975 is it 10 to 1 that the result shall 
 be contained? 
 
 p= 10,?= l,p + (p + q) ='9091, -9091 x 25'1 = 22-8184. 
 (22-8184 1) = 10-9092 10-9092 x 45 = 490'9/. 
 
 It is, then, 10 to 1 that the balance of 150 speculations 
 shall lie between 975 + 491 and 975 491 pounds, or 
 1466/. and 484/. Even such a case shows the effect of 
 multitude in diminishing risks. The possible extremes 
 of the problem (or the result of the problem itself, if 
 we supposed one speculation instead of 150) are a gain 
 of 3000/., and a loss of 37501. 
 
 I will now add an example which will tend to show 
 the ultimate effect of gambling against a bank with a 
 slight mathematical advantage in its favour. Suppose 
 the game is such, that at each trial it is 30 to 29 * that 
 the bank shall win, the stake on both sides being one 
 sovereign. Here a -f- b is 59, and making n a whole 
 number for convenience of calculation, let n = 50, or 
 let 2950 games be tried. The bank has each time a 
 mathematical advantage (page 97) of ^ --, or 
 of a sovereign, and will, in the long run, realise 50 
 upon 2950 games. What are the chances in favour of 
 the individual fluctuation of this one set of 2950 games 
 leaving the bank without any profit, and with more or 
 less loss ? To apply the preceding rule, we must first 
 ask what are the chances that the departure from the 
 probable total of 50/. shall not exceed 507. ; that is, 
 that the bank shall realise between 01. and 100/. Here 
 we have 
 
 * This supposition is more in favour of the player than is often the 
 case. 
 
ON THE BISKS OF LOSS OB GAIN. 107 
 
 a = 30, b = 29, n = 50, g-\ gained h = l lost v = 50. 
 
 1 gain x 50 = 50 probable total. 
 
 Add 2. 
 
 In this case the probability is absolutely given. The 
 square root is therefore the first one mentioned. 
 
 8xwxox6 = 8x50x30x29 = 348,000 
 
 348000 / - 51 
 
 ^ = 5898-3, V 5898-3 = 76-800, ^ = -664 = t. 
 
 Table L, if t = '664, H = '652. 
 Consequently T 6 6 5 ^ is the chance that the bank shall not 
 
 lose,, but shall gain something less than ^ 100; and 
 consequently the chance that the bank shall either lose, 
 or gain more than ,=100 is -jo 4 oV ^ n account of the 
 nearness of 30 and 29, the results last mentioned are 
 nearly equally probable, and it is near enough for our 
 present purpose, to say that -jftffo is the chance of the 
 bank gaining more than 100, the supposition being 
 against us ; for it is more likely that the bank should 
 gain more than 1 00 than that it should lose. Hence 
 it follows, that the chance of the bank gaining on 2950 
 games is more than -j^efo* or aDout fi ve t one ' K sucn ^ 
 be the case with a bank much less unfairly constituted 
 than is often the case, against a player who can not only 
 command 2^50, but who has the prudence to deter- 
 mine that he will only play 2950 games, at a stake of 
 , 1 for each game, what must be the chances against 
 those who risk a larger proportion of their means at 
 more unequal play, with a determination to win that is, 
 to go on till they are ruined ? 
 
 The inequality of means is an important consider- 
 ation in calculating the chances of two antagonist game- 
 sters. If two persons, with equal means and equal 
 chances, play for equal stakes, it is an even chance 
 
108 ESSAY ON PROBABILITIES. 
 
 whether A shall ruin B, or B shall ruin A ; but that 
 one or other will ultimately be ruined is certain. Sup- 
 pose each party to have a hundred guineas, the stake 
 being one guinea, and suppose two millions of games 
 are to be played. The most probable individual case is 
 that each shall win a million of games ; but if the 
 fluctuation amount to 100 in favour of either, the other 
 is ruined. Now, page 81, the probability that the 
 number of games won by A shall lie between a million 
 -f- 100 and a million 100 is ! 12, which is therefore the 
 chance that neither player shall be ruined. Consequently, 
 it is about nine to one that one player or other is ruined 
 or more than ruined in two millions of games. And 
 the chance is even greater than this : for the preceding 
 method of treating the problem supposes the players 
 not to balance their account till two million of games 
 have been actually played, so that one player or the 
 other may have been repeatedly playing on credit. The 
 same rule may be easily applied to any inequality of play, 
 the fortunes of the players being equal ; and the result 
 is, I. that ultimate ruin to one or other player is certain ; 
 2. that, if the stake be a sufficiently small fraction of 
 the player's income, the number of games which must 
 be played to render probable the ruin of either may be 
 made as large as we please. There are but two con- 
 ditions under which gambling can be prudently followed 
 as an amusement small stakes and equal play. In 
 games of pure chance it is possible to obtain the latter, 
 and almost impossible in games of mixed skill and 
 chance. Unfortunately, the stimulus of -gambling, a 
 combination of suspense and hope of large gain, cannot 
 be obtained upon any terms which prudence would 
 sanction. 
 
 When two players, of unequal fortunes, play together 
 for the same stake, however equal the play may be, the 
 larger fortune has an unfair advantage. To estimate 
 the amount of the disadvantage, proceed as follows. 
 
 PROBLEM. Two players, A and B, having funds of 
 m and n times their stake, play a game, at which it is a 
 to b that A wins, or b to a that B wins. What is the 
 
ON THE RISKS OP LOSS OB GAIN. 109 
 
 chance that in the long run B will rum A, and that A 
 will ruin B ? 
 
 RULE 1. If a and b be equal, it is m to n that A 
 will mm B, and n to m that B will ruin A. 
 
 RULE II. If a and b be unequal, let M represent 
 the difference of the rath powers of a and of 6, multiplied 
 by the nth power of a, and let N represent the difference 
 of the nth powers of a and of b, multiplied by the mth 
 power of b. Then it is M to N that A ruins B, or 
 N to M that B ruins A. 
 
 N. B. If m and n be considerable, it is almost im- 
 practicable to apply the above rule without the aid of 
 logarithms. VThen b is many times a, take the nth 
 power of a for M, and the nth power of b for N. 
 
 RULE III. When m and n are equal, it is as the 
 mth power of b to the twth power of a that B will 
 ruin A. 
 
 RULE IV. If the means of both players be unlimited, 
 then it is certain the player who has odds in his favour 
 on a single game will, in time, gain any sum, however 
 great ; if the means of the stronger player be unlimited, 
 then it is certain he must at last ruin the weaker. 
 
 RULE V. If the means of the weaker player be 
 unlimited, and those of the stronger limited, the chance 
 that the latter will in time win any sum, however great, 
 from the former, is as follows. Let B be the stronger 
 player (that is, b greater than a), and let him begin 
 with n times his stake, while A has unlimited means : 
 then it is the excess of the nth power of b over that of 
 a to the nth power of a that B gains any sum, in the 
 long run, from A, and vice versa for A ruining B. 
 
 EXAMPLE. A has 51. and B 31., and they play at a 
 game for which it is three to two that B shall win any 
 one game. What are the chances for the ultimate 
 success of each player ? 
 
 a = 2, 6 = 3, m = 5, n = 3 
 
 M or (3 5 2 5 ) x 2 3 is (243 - 32) x 8 or 1688 
 N or (3 3 2 3 ) x 3 5 is (27 - 8) x 243 or 4617 : 
 It is therefore 4617 to 1688 that B shall ruin A. 
 
 A gaming bank must be considered as a player of 
 
110 ESSAY ON PROBABILITIES. 
 
 limited means, playing against all who choose to enter., 
 that is, playing against unlimited means. It is, 
 therefore, essential to its existence that some mathe- 
 matical advantage should be allowed, even more than 
 is necessary to reproduce the expenses of its ma- 
 nagement. What I have hitherto said on the subject 
 refers to the relation between the bank and the indi- 
 vidual player against it . but considering the former as 
 the antagonist of all who choose to play, it absolutely 
 requires the protection of a mathematical advantage. 
 But having this advantage, it must, in the long run, 
 ruin its individual opponents ; so that bankruptcy to 
 itself, or degradation and suicide to its customers, are 
 the initial conditions of its existence. But since the 
 banks flourish, it is plain that whatever advantage is 
 necessary to their continuance, is really obtained by 
 them ; and I shall now inquire how much this advantage 
 must be in several cases. 
 
 EXAMILE (Rule V.) It is 30 to 29 for the bank 
 upon each game, and the bank stakes the tenth part of 
 its means at every game. What are the chances of its 
 perpetual continuance ? (b = 30, a = 29, n = 10). 
 
 SO 10 = 590,490,000,000,000 
 29 io _. 420,707,233,300,201 
 
 169,782,766,699,799. 
 
 Answer. About 170 to 421, or such a bank would 
 not be likely to last; that is, in the long run, only 170 
 out of 421 such banks would avoid ruin. 
 
 PROBLEM. What is the mathematical advantage 
 which a bank must have, in order that its permanent 
 continuance may have /c to 1 in its favour ; the sup- 
 position being that the bank stakes the nth part of its 
 means at every game? 
 
 RULE. The odds in favour of the bank, on a single 
 game, must be the rith root of 1 -f k to 1. Thus if, 
 judging by the experience of the Parisian* banks, we say 
 
 * These banks were open to the public and to the municipal police. Of 
 the gaming-houses in London, those who know them must speak. The 
 
ON THE BISKS OF LOSS OR GAIN. Ill 
 
 that the permanency of each had 100 to 1 in its favour, 
 we are entitled to conclude that (the tenth root of 101 
 being 1*59) the games played were each not less than 
 3 to 2 in favour of the bank, if they staked the 10th 
 of their resources at each throw, or 11 to 10 if they 
 stake one fiftieth. 
 
 PROBLEM. The odds in favour of the bank being b 
 to a, required the greatest proportion of the fund which 
 may be staked at one game, in order to insure the chance 
 A; to 1 for the permanency of the establishment. 
 
 RULE. Divide the logarithm of one more than A: by 
 the excess of the logarithm of b over that of a : the 
 quotient is the denominator of the fraction required, 1 
 being the numerator. Suppose, for instance, that the 
 odds for the bank, on single games, are 30 to 29, then, 
 if 99 to 1 be required to be the chance that the bank 
 shall continue to exist, divide the logarithm of 99 + 1 
 (which is 2) by the excess of the logarithm of 30 (or 
 1-47712) over the logarithm of 29 (or 1-46240), and 
 2 -7- "01472 gives a fraction more than 135 ; whence 
 the 135th part of the capital is the highest which 
 should be staked. 
 
 A merchant, who engages in speculations which must 
 produce a fixed loss or a fixed gain, and who offers td 
 deal with any one in such a manner, is precisely in the 
 position of a bank such as is above described. The reason 
 why neither party need be ruined in this instance is 
 that the produce of the earth and sea is an unlimited fund, 
 upon which the merchant draws. Trade by itself would 
 tend to ruin the many, and accumulate all the stakes in 
 the hands of a few, and the theory of probabilities would 
 enable us to foretell that continual approximation to- 
 wards the extremes of wealth and poverty which com- 
 mercial countries always present. We have seen that 
 the poorer player must, to maintain his ground, have a 
 
 protection and encouragement which legal regulation of gambling-houses 
 would appear to give to gambling in general, is a good reason for the state 
 of our own law : otherwise, there can be no doubt that much particular evil 
 would be prevented. by allowing a regulated system. 
 
112 ESSAY ON PROBABILITIES. 
 
 mathematical advantage in his favour. Now, it is the 
 nature of free trade that whatever mathematical ad- 
 vantage can be gained at all, is more accessible to the 
 rich speculator than to the poor one. Consequently, 
 the richer player, for that reason, can make himself the 
 stronger player. If, then, a certain number of persons 
 were to play upon a fixed total of stakes, equally divided 
 at the commencement, with the condition that every 
 stake won should enable the winner to make his next 
 throw with somewhat more (no matter how little) of 
 mathematical advantage than he had before, it is certain 
 that, in the long run, the whole of the stakes would be 
 in the hands of some one of the players. But, in the 
 actual state of things, there is always an accession of 
 new stakes and new players, so that the original players 
 are contending against an unlimited fund. If the con- 
 tinual augmentation of stakes and players be not suffi- 
 cient to counterbalance the tendency to extremes, a wise 
 government would throw the burthen of taxation more 
 upon the rich and less upon the poor. The mathe- 
 matical advantage of wealth would be taxed, as well as 
 its power of procuring luxuries. Such a result never 
 can be expected until the public mind is better informed 
 upon the subject of which this work treats. 
 
 CHAPTER VI 
 
 ON COMMON NOTIONS WITH REGARD TO PROBABILITY. 
 
 THOSE who have not considered this subject with par- 
 ticular attention, seldom fail to think that there must be 
 more or less of fallacy in the attempt to connect its prin- 
 ciples with its results. Some, indeed, of the latter are 
 strange and new, and are used as arguments against the 
 validity of the theory. I propose in this chapter to 
 
ON COMMON NOTIONS OF PROBABILITY. 113 
 
 turn those which precede to account, in examining 
 opinions of various kinds, whether on this subject at 
 large, or on particular cases of its application. 
 
 The doctrine of probabilities seems to some to assume 
 a sort of power of prophesying,, or of predicting the 
 run of events; to others, it appears that unless such a 
 power of prophesying be attained, the theory can be of 
 no use. Both notions are correct in one sense and in- 
 correct in another : there is prophecy, but not of par- 
 ticular events, and derived, not from inspiration, but 
 from observation. The astronomer predicts and all 
 the world knows that his predictions daily come true. 
 His means of prophecy are aided by deduction from 
 certain notions of which, be the cause what it may, we 
 are as certain as of our own existence. From his very 
 distinct (and therefore often called intuitive) perception, 
 that two straight lines cannot inclose a space, and various 
 other axioms of arithmetic and geometry, he is able to 
 make his observations tell him more as to the future 
 motions of our system than his unassisted perceptions 
 of the past could ever have accomplished. He is a 
 dealer in probabilities of a very high order. But be- 
 fore his prediction appears^ it is necessary that he 
 should consider much more doubtful questions of pro- 
 bability. The minute errors of observation, coupled with 
 the various trifling effects which result from yet un- 
 discovered causes, oblige him to have recourse to the 
 principles which we have explained in the preceding 
 chapters. 
 
 Again, there is no prophecy of particular events in 
 the theory of probabilities, of which it is the very es- 
 sence that there should be more or less tendency to 
 falsehood in every one of its assertions. No result is 
 announced, except as having a certain chance in its 
 favour, which implies also a certain chance against it. 
 
 With regard to the second class of assertions, namely, 
 that unless the theory of probabilities enable us to 
 predict, it can be of no use it may be said that, for 
 the purpose contemplated, it is of no use. Theory would 
 
114} ESSAY ON PROBABILITIES. 
 
 never enable us to tell what face of a die will be turned 
 up in any one instance, nor would the maxims of our 
 science be worth putting into practice with respect to 
 an event which is to happen only a few times. If a 
 man were determined to run six hazards, and never to 
 gamble afterwards, say if he were determined to wager 
 twice upon a pair of dice giving doublets, I should 
 think it perfectly immaterial whether he accepted an 
 even wager, namely 5 to 1, or not. For though, in the 
 long run, only one throw out of six will give doublets, 
 yet the probability that six throws will give such a pair 
 once at least is not very great. It is as a provider of 
 general rules of conduct that the science is valuable ; 
 the adherence to rules being desirable on precisely the 
 same principles as those which obtain in morals or legis- 
 lation, no maxim of which will be found to meet every 
 case which will occur. 
 
 It is an assumption of this theory that nothing ever 
 did happen, or ever will happen, without some particular 
 reason why it should have been precisely what it was, 
 and not any thing else. Conceive it possible that a ball 
 which is white might have been black, without the 
 alteration of any action or circumstance which took 
 place in time previous to the moment at which the 
 ball is shown, and the foundations of the theory of 
 probabilities have ceased to exist in the mind which has 
 formed that conception. There is no one but will admit, 
 that out of a box, which contains nothing but two black 
 balls, nothing but black balls can be drawn ; and that 
 out of a box which contains only two white balls, 
 no black balls can be drawn. The difficulty lies in a 
 clear perception of the remaining assertion ; namely, 
 that when the box contains one white ball and one black 
 ball, a very large number of drawings will give as 
 many white as black nearly, and the more nearly the 
 greater the number. This proposition might be proved 
 in three ways : firstly, by actual experiment ; secondly, 
 by showing that out of all the possible cases which can 
 happen, those in which black and white are equal, or 
 
ON COMMON NOTIONS OP PROBABILITY. 115 
 
 nearly equal, much exceed in number all the rest put 
 together ; thirdly, by showing that there can be no 
 possible reason for an excess of white, which does not 
 equally, by express condition of the question, apply in fa- 
 vour of an excess of black. The last is more unanswer- 
 able than convincing ; the second really shows that the 
 event which we propose and treat as one event, namely, 
 " as many white as black, or nearly so," is, in fact, a 
 collection of a large number of events, much exceeding 
 in number all the rest which can happen. It is as if, 
 having a million of possible cases, I separated nine 
 hundred thousand from the rest, called each of them A, 
 and each of the rest B, and then asserted that A would 
 happen more often than B. But, nevertheless, I sus- 
 pect that to the first mode of demonstration, actual 
 experiment, most persons owe that degree of con- 
 fidence in the theory, which (often without knowing 
 it) they exhibit in the affairs of life ; and I derive 
 such a suspicion from observing that every result of the 
 theory of probabilities which is not of a nature to admit 
 of e very-day confirmation, or which would escape an 
 inattentive observer, is looked upon with distrust. In 
 no case is this more obvious than in the prevailing 
 notions with regard to luck. 
 
 It is observed that some people always have luck at 
 cards. The order of things seems disturbed at their 
 caprice ; if they sit opposite to the dealer at whist, then 
 there is always an undue proportion of trumps among 
 the cards which come second, sixth, tenth, &c., up to 
 the fiftieth; while, when they become before the dealer 
 Hocus Pocus (for to no other spirit, ancient or 
 modern, can the agency be attributed) puts all the 
 good cards, third, seventh, eleventh, &c. The fact is 
 stated as a sort of mystery, and we hear of people 
 who are always lucky at cards and never at dice, or 
 vice versa. The statement implies that the parties who 
 make it believe there is something in luck an asser- 
 tion which I do not think of questioning; for, as I 
 i 2 
 
116 ESSAY ON PROBABILITIES. 
 
 shall proceed to show, it would be the most improbable 
 thing imaginable that there should be no such lucky 
 people. 
 
 Firstly, every question of probabilities stands in 
 precisely the same relation to our faculties, whether \ve 
 suppose a moral government of the universe, or none at 
 all, provided that we have no reason to suppose we know 
 any thing of the plan of that government in the par- 
 ticular case in question. If I am before an urn which 
 contains a black and a white ball, which is all I know, 
 I am then disposed to say the chances are even. The 
 ball which I am to draw is undetermined (by me), and 
 that which we call chance appears to exist. But suppose 
 1 draw the ball, and without looking at it hold it in my 
 hand. That which we call chance has ceased to exist 
 the ball is actually determined, and I am clearly and 
 physically placed in the same position as I should have 
 had before the drawing, if a superintending power, 
 capable of guiding my thoughts and actions without my 
 perceiving it, had predetermined which I should draw. 
 But my position with respect to knowledge of the ball is 
 not in any way changed, either by the predetermination 
 of the superintending power before the drawing or by 
 my own act of drawing, as long as 1 do not know what 
 I am to draw or have drawn. It tells me nothing, if I 
 hear that the drawing is settled, unless it be in a manner 
 by which I can form some guess as to the nature of the 
 settlement. Consequently we must not, unless some 
 reason be shown for it, consider the question of the 
 luck of individuals in any other light, with reference to 
 calculation, than that in which it would have been 
 placed by the supposition that, ah 1 imaginable species of 
 fortune being described on the tickets of a lottery, each 
 individual had one drawing made for him at birth, 
 which should describe his future successes and reverses. 
 To create an analogous question, within reasonable 
 numerical limits, let us suppose a thousand individuals, 
 each of whom is to play two thousand deals at 
 
ON COMMON NOTIONS OF PROBABILITY. 117 
 
 whist, with a given suit as trumps.* Let there be a 
 lottery, containing an enormous number of books, in 
 each of which 2000 deals are described, and let the 
 books be so many in number that among them is one 
 containing every possible set of 2000 deals which can 
 be imagined, the four hands in each deal being described, 
 and that allotted to the drawer of the book being 
 marked as such. Let each individual draw one of these 
 books, replacing it before the next drawer arrives : 
 these individuals are then precisely in the same situation 
 with regard to us, if their hands are to be dealt to them 
 according to the directions laid down in their books, as 
 if the distribution were made by accident (as we call it). 
 Now the question is, which is most likely, that the luck 
 of these individuals shall all be nearly the same, or that 
 some of them shall have a marked predominance over 
 others ? To take one simple question : consider only 
 the chance of gaining the ace of trumps. Excluding 
 the dealer's advantage, to simplify the question, the 
 chance of any one individual gaining the ace of trumps 
 at any given deal is ^. Considering 2000 deals, he 
 has a very good chance of gaining it 500 times, or a 
 few more or less. But the probabilities are much in 
 favour of several of these 1000 individuals having a 
 very different lot from the average. Frame a set of cir- 
 cumstances in this respect against which it shall be 
 twenty to one, and (page 43) it is a hundred to one that 
 this fate (or a better) shall be found to be that of some 
 one or other out of any 92 individuals taken at hazard 
 from among the thousand. And when to the chance of 
 holding the ace of trumps we add the various others 
 which constitute a good hand at the game, we thereby 
 much increase the probability of large fluctuations, one 
 way or the other; and though itis certain that uniformity 
 will be found in the average lot of a large number of per- 
 
 * This does not alter the question ; since the substitution of four possible 
 different sorts of trumps would only multiply every possible case of good 
 and bad fortune four times. 
 
118 ESSAY ON PROBABILITIES. 
 
 sons, yet the larger the number, the greater will be the 
 extremes of fluctuation. 
 
 Now it must be noticed, that this variation is the 
 thing observed not on one side only, but on both. 
 For every one who is lucky at cards, there is another who 
 is unlucky. It would be, indeed, such a sort of mystery 
 as that which I am endeavouring to explain, if the ex- 
 ceptions to common luck were all on one side, or if there 
 were no such thing at all as uncommon luck, or only in 
 very few instances. This latter would be the same sort 
 of phenomenon as we should see if a halfpenny gave 
 head and tail alternately through an enormous recurrence 
 of throws. The event observed is precisely that which 
 might have been expected beforehand. If by thinking 
 mysteriously of the fluctuations of luck which are 
 observed in comparing the fortunes of individuals, 
 any reader should mean to imply that the alternative, 
 namely, slight individual departure from the average, 
 would not have been mysterious, he is in a singular 
 error. The state of things which he would regard with 
 no wonder would be an apparent interference with the 
 material world on the part of its governor, without the 
 intermediate agency of any second causes ; that is, 
 something resembling a miracle. For though the phi- 
 losopher, in such a case, would suspect an intermediate 
 cause, and endeavour to discover it, this consideration 
 does not enter in to the view of people in general. When 
 the world wonders, whether at one side or another of a 
 question of probabilities, it is at the want of any ap- 
 parent physical or moral reason : on which account they 
 refer It to the Creator in a manner different from that 
 in which they refer what they call usual occurrences. 
 
 The law of individual cases is, that there shall be 
 marked differences ; of the masses, that there shall be 
 great approach to uniformity. There are a hundred 
 years in which, and hundreds of diseases by which, 
 any individual who is born may die : a lottery, which 
 should contain one ticket for every disorder, repeated as 
 often as there are years of age in which it has been 
 
ON COMMON NOTIONS OF PROBABILITY. 11Q 
 
 fatal, would present at least 20,000 chances. Before an 
 individual is born, it is, say 2 0,000 to 1 against his dying 
 at a given age of a given malady ; and yet, even with 
 such imperfect observations as exist at present, it begins 
 to be seen that uniformity is the law of large masses 
 compared with each other. I will illustrate this by 
 some cases. Few things appear more varied than the 
 distribution of maladies, that of criminal acts, and that 
 of the sex of children in different families. I have 
 taken purposely a case of evil, physical and moral, and 
 one which is neither. The experience of any one 
 individual might lead him to say that it is no uncommon 
 thing for three or four times as many persons to die of 
 consumption in one period of five years as in the 
 previous period ; but the experience of one large city 
 will show that such is not the case. The bills of mor- 
 tality in London showed the following results; the 
 upper line denoting the last year of the five in question, 
 and the lower line the average number in every thousand 
 deaths which were caused by consumption, or what was 
 called such.* 
 
 1732 3737 1742 1747 1752 1757 
 135 163 165 180 ]87 197. 
 
 Here is nothing like enormous fluctuation. The 
 gradual increase of the number shows an increasing 
 tendency to the complaints then described under the 
 head consumption, but cannot be called fluctuation, being 
 itself regular. 
 
 The number of murders committed in the whole 
 extent of any one country might be supposed liable to 
 very large yearly fluctuation, and still more the compara- 
 tive numbers committed with different classes of weapons. 
 A few years ago, extreme derision would have followed 
 the assertion that the sword and pistol would be felo- 
 
 * The known loose manner in which these Tables were put together 
 does not affect the argument, further than to favour its conclusion. The 
 chances of error in the description increase the probability of fluctuation 
 in the results. 
 
 i 4 
 
120 
 
 ESSAY ON PROBABILITIES. 
 
 niously used in different years by nearly the same num- 
 bers. Let us look at the following Table, extracted 
 from the Essai de Physique Sociale of M. Quetelet. The 
 country referred to is France. 
 
 
 
 
 
 
 i 
 
 YEARS. 
 
 1826 
 
 1827 
 
 1828 
 
 1829 
 
 1830 ! 1831 
 
 Total number of murders 
 brought to justice. 
 
 241 
 
 234 
 
 227 
 
 231 
 
 207 
 
 266 
 
 Fire-arms 
 
 56 
 
 64 
 
 60 
 
 61 
 
 57 
 
 88 
 
 Sharp weapons of 
 
 
 
 
 
 
 
 war 
 
 15 
 
 7 
 
 8 
 
 7 
 
 12 
 
 30 
 
 Knives - - 
 
 39 
 
 40 
 
 34 
 
 46 
 
 44 
 
 34 
 
 Clubs or sticks 
 
 23 
 
 28 
 
 31 
 
 24 
 
 12 
 
 21 
 
 Stones 
 
 20 
 
 20 
 
 21 
 
 21 
 
 11 
 
 9 
 
 Sharp instruments 
 
 
 
 
 
 
 
 not above de- 
 
 
 
 
 
 
 
 scribed 
 
 35 
 
 40 
 
 42 
 
 45 
 
 46 
 
 49 
 
 Striking or kicking 
 
 28 
 
 12 
 
 21 
 
 23 
 
 17 
 
 26 
 
 Other modes, and 
 
 
 
 
 
 
 
 unknown 
 
 25 
 
 23 
 
 10 
 
 4 
 
 8 
 
 9 
 
 I now compare the number of male uid female bap- 
 tisms registered in England in ]821, and the nine fol- 
 lowing years. For these successive years it is found 
 that for 1000 girls baptised there were 1048, 1047, 
 1047, 1041, 1049, 1046, 1047, 1043, 1043, and 1034 
 boys. 
 
 Such cases as the preceding tend to establish the law 
 in question ; namely, that different large masses of facts, 
 collected under the same circumstances, will present 
 nearly the same averages. I now proceed to another 
 point, 
 
 When two circumstances happen to change together, 
 it is frequently presumed that they are connected with 
 each other, when, in truth, there is no reason for any 
 such supposition. In order to justify any notion of 
 
ON COMMON NOTIONS OF PROBABILITY. 121 
 
 necessary connection it is necessary that the two cir- 
 cumstances should always happen together, and that one 
 should never happen without the other. If it should 
 .only be observed that one is very frequently accompanied 
 by the other, we must then inquire into the probability 
 that either may happen without the other. If two 
 events are almost always happening, then it is evident 
 that very frequent coincidence is no evidence of connec- 
 tion., so long as exceptions tell us that there is no neces- 
 sary connection. And even if" we always observe A to 
 be immediately followed by B, it does not immediately 
 follow that they are necessarily connected. We must 
 remember that the phenomenon is this our perception 
 of A is immediately followed by our perception of B. 
 This may arise in different ways, as follows. 
 
 1. A may make B necessarily follow it ; that is, the 
 common deduction from the phenomena may be true. 
 
 2. Our perception of A may make B follow. This 
 is the case with regard to all effects produced upon our 
 own minds by A. 
 
 3. A itself may make our perception of B follow. 
 The latter may be always happening, but it may require 
 the arrival of A to make us see it. 
 
 4. Our perception of A may make our perception of 
 B follow ; that is, B may be always happening, 'but it 
 may need both the arrival of A and our knowledge of 
 that arrival to make us see B ; or the circumstances 
 which favour our perception of A may be the same, or 
 have necessary connection with, those which do the same 
 forB. 
 
 5. It may be B which is the antecedent event ; but 
 our perception of A, the consequence of B^ may be ne- 
 cessary to our perception of B itself. 
 
 We sometimes, for example, note what takes place 
 from the time A happens, and compare it with previous 
 events, merely because the arrival of A suggests the 
 comparison. In many instances we do this correctly, 
 and merely for convenience ; but whenever the events 
 
122 ESSAY ON PROBABILITIES. 
 
 which are compared with A are such as to exhibit what 
 we call accidental fluctuations, we are apt to imagine 
 that the difference between the two sequences is a 
 consequence of A. In the mysterious subject of luck,, 
 already alluded to, this tendency to error produces 
 superstition. There are many who imagine that the 
 change of seats,, or a new pack of cards, changes the 
 luck. They imagine it, because they observe that 
 the luck is not the same before the change as after it, 
 which, for the most part, is true. But it is equally 
 true that, for the most part, no number of games exhi- 
 bits the same fortune as those which precede it ; that 
 is, this change of luck is always taking place, but is 
 usually only perceived when the introduction of some 
 novel circumstance affords a point to date from, on one 
 side and the other. The growth of the superstition is 
 this : An individual who has been unlucky during 
 several games, happens to begin to win after the intro- 
 duction of new cards. His fortune changes, as most 
 probably it will do ; for if the chances be even, and 
 three games have previously been lost, it is seven to one 
 against the next three games resembling them, and an 
 even chance that he shall win the next game. If he 
 win the next, or, indeed, if he do not go on losing, he 
 notes the circumstance, and the next time a run of ill 
 luck occurs, he takes particular care to repeat the expe- 
 riment. In this way he soon furnishes himself with a 
 tolerable number of facts in support of his theory. 
 The exceptions are forgotten ; for it is the character of 
 negative events to lay less firmly hold of the mind* 
 than positive ones. Thus the theory of the change of 
 the weather with that of the moon receives more con- 
 firmation from one fact in its favour than of doubt from 
 two against it. This last notion is another case in 
 point. The weather of any three days, in by far the 
 most instances, differs from that of the preceding three 
 
 * The lucky hit of a prophet of the weather, in foretelling the coldest 
 day of January, 1838, did more to establish his infallibility than weeks of 
 succeeding mistakes could destroy. 
 
ON COMMON NOTIONS OP PROBABILITY. 123 
 
 days. When once the notion is obtained that a change 
 of weather will follow that of the moon, the epoch is 
 watched, and the change which is in most instances 
 observed, is admitted as evidence. If any one would 
 carefully note for a considerable time the weather pre- 
 ceding and following prorogations of parliament, he 
 would perhaps astonish the world with a result which 
 no one has yet dreamt of. These cases fall under the 
 third head above described. 
 
 It was frequently supposed, a few years ago, that 
 comets produced hot weather. An examination of the 
 number of comets discovered in years of different 
 average temperature gave it as a result that there were 
 more comets in hot summers than in cold ones. But 
 since hot summers are generally fine, with clear skies, 
 and cold summers cloudy and rainy, it is obvious that 
 the former are more favourable to the discovery of 
 comets than the latter. The fact, then, from which the 
 inference was drawn amounted to this, that the years of 
 heat are those in which we see most comets. With 
 what we know of the matter, there is no more reason 
 to suppose that comets bring heat than that heat brings 
 comets. We must, in all instances of presumed con- 
 nection, look closely at these two distinct things the 
 happening of an event, and our perception of it ; other- 
 wise, we shall always be liable to suppose that an event 
 may produce the first, when it produces only the second. 
 
 There is, however, a class of events which does not 
 appear capable of any of the preceding explanations ; 
 namely, the occurrence of what are called runs of luck 
 at play, and repetitions of similar events in common 
 life, which give rise to such proverbs as this that it 
 never rains but it pours, and misfortunes never come 
 alone. We shall first attempt to destroy the extraor- 
 dinary character of these occurrences, and shall then 
 show that any other order of things would be indeed 
 extraordinary. 
 
 Let there be any event of an unusual character, say 
 
124 ESSAY ON PROBABILITIES. 
 
 the drawing of a black ball out of a lottery of twenty 
 balls in which only one is black. Surely, any one would 
 say, we shall never draw the black ball fifty times 
 running. I answer, continue drawing as long as I 
 direct, and you certainly shall, that is, if you will admit 
 that an event must some time or other arrive, which has 
 ten thousand to one in its favour. Supposing the 
 ball to be replaced immediately after drawing, and the 
 lottery to be shaken, so that every ball has its fair chance,, 
 I will calculate the number of drawings in which it is 
 much more than 10,000 to 1 that a black ball shall be 
 drawn five times running, and it will be evident that by 
 the same principles the number might be calculated 
 which would give as great odds for a run of 50, or 
 5000, or any number, however great. 
 
 The chance of drawing the black ball at one trial is 
 -Q, and through five successive trials it is -^ x -}Q X 
 
 so- X -2V x sV r -3Y oo (Too- !t is > therefore, 3,199,999 
 to 1, say 3,200,000 to 1, against five successive black 
 balls. In page 43, look in the Table opposite to 10,000 
 to 1, and we see 921. Let every succession of five 
 drawings be called a set, then it is 3,200,000 to 1 against 
 any given set being all black. Multiply 3,200,000 by 
 921, and divide by 100, which gives 29,472,000, and 
 this is the number of sets in which it is 10,000 to 1 
 that one whole set, at least, shall be black. Much 
 greater are the odds that there shall be a run of five 
 somewhere in all the drawings which make up these 
 sets ; for this latter event would arrive, not only when 
 all of a set are black, but when the last of one set, and 
 four of the next, or the two last of one set and three of 
 the next, &c. are black. 
 
 Let two players, with equal chances, play a large 
 number of games in succession. It is 63 to 1 against 
 a run of six games in one given manner, and 62 to 
 2, or 31 to 1 against a run of six games for one or 
 the other. Now, 31 x 921-^-100 285*51, or it is, 
 as before, more than 10,000 to 1 that in 286 sets of 
 six, a run of six shall occur for one or the other. This 
 
ON COMMON NOTIONS OF PROBABILITY. 125 
 
 gives 1716 games, and yet a sturdy whist player, who 
 plays on an average a dozen deals an evening for 200 
 evenings in the year, or 2400 deals per annum, will look 
 grave when he relates that he had bad cards for six 
 deals together, and will assure you that there is some* 
 thing in luck. 
 
 But at the same time, the number of trials which 
 makes a run of the unlikely event extremely probable, 
 will give the same probability to a much larger run in 
 favour of the more probable event. To find to what 
 extent this goes, use the following 
 
 RULE. If it be a to & for A against B, in any single 
 trial, then subtract the logarithms of a and b separately 
 from that of a -{- 6; take any two whole numbers 
 which are very nearly in proportion to these differences, 
 and to each add 1 ; the sums show the runs of the 
 two events, which have the same probability (be it small 
 or great) of happening in a very large number of 
 throws. Thus, suppose it is 10 to 3 for A against B. 
 
 Log, 13 1-11394 log. 13 1-11394 
 Log. 10 1 -00000 log. 3 -47712 
 
 0-11394 0-63682 : : 11 ; 64 nearly. 
 
 Consequently, whatever chance there is for a run of 12 
 B's, there is as much for a run of 65 A's. 
 
 In the last mentioned lottery, when a = 19, 6=1, 
 we have log 20 log 19 = -02228, and log 20 log 1 
 = 1*30103, which differences are 1 to 65 nearly, and 
 as 6 to 390. Consequently there is as much reason to 
 expect a run of 391 white balls as of 7 black balls. 
 
 No person can take a rational view of probabilities 
 until he ceases to recoil from the supposition that an 
 event is never to happen because the odds are very much 
 against his choosing, out of a large number of trials, 
 the one in which it is to happen. The best way to force 
 the mind upon the consideration is to return to the first 
 
126 ESSAY ON PROBABILITIES. 
 
 principles upon which the method of judging is founded. 
 You find it difficult to imagine, that out of twenty balls, 
 one only of which is black, you shall draw the black 
 ball five times running. But yet in 30,000,000 sets of 
 five drawings each, it is asserted that you are what is 
 called " almost sure" of drawing the black ball through- 
 out the whole of one set. Waiving the question of 
 probabilities, I will now state what it is of which ma- 
 thematical demonstration makes us quite sure. Let 
 vol. i. be a book which describes 30,000,000 of sets ; 
 vol. ii. another, which describes 30,000,000 more, differ- 
 ing from the preceding in some, many, or all, its sets, 
 and so on until every possible collection of 30,000,000 
 of sets is described in one volume or another. Now it is 
 quite certain that out of the innumerable volumes which 
 will thus be produced, the volumes which somewhere or 
 other describe a set all black will outnumber those which 
 do not describe such a set many thousand times, 10,000 
 at least. Suppose the black sets when they exist, to be in 
 a frontispiece; the question then is, having an enormous 
 library, with books at the rate of 10,000 with a frontis- 
 piece, for one which has none, and taking down a book 
 blindfold, which do you suppose to be most likely, that 
 you shall draw a frontispiece, or none at all ? Un- 
 questionably you answer that you are almost mathema- 
 tically certain of not drawing the latter. But this is 
 (page 124) an exaggerated statement of the case of the 
 chance of a run of five black balls in 30,000,000 of sets. 
 But it is said that no person ever does arrive at such 
 extremely improbable cases as the one just cited. That 
 a given individual should never throw an ace twelve 
 times running on a single die, is by far the most likely ; 
 indeed, so remote are the chances of such an event in 
 any twelve trials (more than 2000,000,000 to 1 againstit), 
 that it is unlikely the experience of any given country, 
 in any given century, should furnish it. But let us 
 stop for a moment, and ask ourselves to what this argu- 
 ment applies. A person who rarely touches dice will 
 hardly believe that doublets sometimes occur three times 
 
ON COMMON NOTIONS OP PROBABILITY. 127 
 
 running ; one who handles 'them frequently knows that 
 such is sometimes the fact. Every very practised user 
 of those implements has seen still rarer sequences. Now 
 suppose that a society of persons had thrown the dice so 
 often as to secure a run of six aces observed and recorded, 
 the preceding argument would still he used against 
 twelve. And if another society had practised long 
 enough to see twelve aces following each other, they 
 might still employ the same method of doubting as to 
 a run of twenty-four, and so on, ad infinitum. The 
 power of imagining cases which contain long combin- 
 ations so much exceeds that of exhibiting and arranging 
 them, that it is easy to assign a telegraph which should 
 make a separate signal for every grain of sand in a globe 
 as large as the visible universe, upon the hypothesis of 
 the most space-penetrating astronomer. The fallacy of 
 the preceding objection lies in supposing events in 
 number beyond our experience, composed entirely of 
 sequences such as fall within our experience. It makes 
 the past necessarily contain the whole, as to the quality 
 of its components ; and judges by samples. Now the 
 least cautious buyer of grain requires to examine a 
 handful before he judges of a bushel, and a bushel be- 
 fore he judges of a load. But relatively to such enor- 
 mous numbers of combinations as are frequently pro- 
 posed, our experience does not deserve the title of a 
 handful as compared with a bushel, or even of a single 
 grain. 
 
 Further to illustrate this point, let us turn back to 
 page 64, where we see that when an event has happened 
 m times running, it is m -f 1 to n that it shall happen 
 n times more. This proceeds upon the supposition that 
 the chances of the event were entirely unknown before 
 its happening, and the presumptions drawn are therefore 
 entirely derived from experience. When an event has 
 happened 1000 times one way, it is 1001 to 1 that it 
 happens in the same way next time. But it is only 
 1001 to 1000 that it happens 1000 times more the 
 same way, and only 1001 to 1,000,000 that it happens 
 
128 ESSAY ON PROBABILITIES. 
 
 1,000,000 of times more in the same way. Hence ex- 
 perience can never, on sound principles, be held as fore- 
 telling all that is to come. The order of things,, the 
 laws of nature, and all those phrases by which we try to 
 make the past command the future, will be understood 
 by a person who admits the principles of which I treat 
 as of limited application, not giving the highest degree of 
 probability to more than a definite and limited continu- 
 ance of those things which appear to us most stable. No 
 finite experience whatsoever can justify us in saying that 
 the future shall coincide with the past in all time to come y 
 or that there is any probability for such a conclusion. 
 
 CHAPTER VII. 
 
 ON ERRORS OF OBSERVATION, AND RISKS OF MISTAKE. 
 
 IN every measurement, as well as in unassisted esti- 
 mation, the observer is liable to error ; the only dif- 
 ference being that the mistakes of careful instrumental 
 measurement are likely to be less than those of esti- 
 mation. That which we call estimation means guess 
 formed by a person whose previous habits and ex- 
 perience are such as to make it very likely that he can 
 tell nearly true that which would require instruments 
 to obtain with great approach to accuracy. To illustrate 
 this distinction, imagine a certain small length, say 
 about twelve inches, to be presented to a large number 
 of persons, who are required to write on separate bits 
 of paper how many inches and tenths of inches it 
 appears to them to contain. If these persons had been 
 used to estimate lengths by the unassisted eye, it would 
 be extremely probable, 1. That the average of their 
 guesses would be very near the truth. 2. That their 
 widest limits of error would be small. If their habits 
 
ON ERRORS OP OBSERVATION. 129 
 
 have not been accurate, it is still reasonably probable 
 that the average of their guesses would be nearly true ; 
 the limits of error would certainly be larger. It is the 
 object of the present chapter to show how the theory of 
 probabilities must be applied to the detection of the 
 most probable result, when various observations are 
 discordant with each other. 
 
 Error, as used in this part of the subject, merely 
 means discordance of which the cause is unknown. In 
 the different branches of physics, in their application to 
 the arts, &c. c., that which we signify by the pre- 
 ceding word may arise from various causes, the chief of 
 which are here enumerated. 
 
 1 . From some law of nature not known to the 
 observer. Thus before the discovery of the aberration 
 of light, all the small yearly changes which the places 
 of stars receive from that cause, only appeared in the 
 form of embarrassing differences between the observ- 
 ations of different months. Those who used astro- 
 nomical instruments might suspect the existence of 
 some unknown motion in the heavenly bodies; they 
 might think it extremely improbable that their improve- 
 ments in the art of observing should permit purely 
 casual discrepancies of so large an amount as those 
 which occurred : but still, so long as no account of the 
 magnitude of these possible results of law could be 
 given, those who observed could in no manner dis- 
 tinguish them from the imperfections of the instru- 
 ments, or of the human senses. But had it been 
 shown that these discrepancies were always the same at 
 the same time of the year, for any one star, they would 
 then have ceased to be errors, and would have become 
 the objects of prediction, as soon as one year's observ- 
 ations had been registered. The physical cause might 
 or might not have been subsequently discovered, without 
 altering the state of the question : the certainty of a 
 phenomenon is all that is required to remove it from 
 the domain of probability. 
 
 2. From the personal constitution of the observer ; 
 
 K 
 
ISO ESSAY ON PROBABILITIES. 
 
 by which is meant, not that general facility of misper- 
 ception which is common to all the human race, but 
 that particular habit or temperament which causes some 
 to differ from persons in general in their method of 
 perceiving. Thus it will frequently happen that when 
 two observers note the time of a phenomenon, by the 
 same watch,, one will always see the event, or imagine 
 he sees it, before the other does the same. This 
 personal error, wbich is seldom large, is beginning to 
 receive the attention of observers. It is not perceptible 
 as long as the natural data of a science remain imper- 
 fectly known, being mixed up and lost in errors of 
 greater magnitude ; but it produces discoverable effects 
 so soon as the science approaches towards accuracy. 
 
 3. From fixed sources of error peculiar either to 
 the species of apparatus employed, or to the individual 
 instrument with which the observations are made. 
 This answers precisely to the personal error of the 
 observer in its effects : it matters nothing whether the 
 clock be one second too fast, or the observer, in the 
 result of the observation. 
 
 4. From the imperfection of human senses and in- 
 struments. To note a measurable phenomenon without 
 any error at all, would require sight and touch by 
 which every magnitude, however small, could be per- 
 ceived and correctly estimated. Such senses belong to 
 no one, and the degree of approach towards perfection 
 not only varies with the observer, but is different at 
 different times with the same observer. Many errors 
 to which instruments are subject ought in strictness to 
 be classed under the first head ; if, for instance,, an 
 astronomical circle gradually change its form, or 
 undergo daily expansion and contraction by variations 
 of temperature, the diversity of results which such a 
 piece of brass will shew are certainly subject to laws, 
 and might be predicted, if we possessed sufficient 
 knowledge of the constitution of the metal, and the 
 laws which regulate the effect of pressure, temperature, 
 moisture, &c. upon it. But so long as such laws are 
 
ON ERRORS OP OBSERVATION. 131 
 
 unknown, and the variations do not follow any dis- 
 tinguishable rule, their effect upon general results 
 differs in nothing perceptible from that of the observer's 
 own errors, with which they are mixed up in the par- 
 ticular results of observation. 
 
 Before any trials are made, that is, before any thing 
 is known of the character of the observer, of the instru- 
 ment he uses, or the perceptibility of the phenomenon, 
 we can have no reason to suppose that any one observ- 
 ation is more likely to exceed the truth than to fall 
 short of it. When any observation is greater than the 
 reality, the error is called positive ; when less, negative. 
 The hypothesis, therefore, of an equal presumption for 
 positive and negative errors, is one with which we must 
 commence ; and it follows from the supposition that 
 the average is the most probable result of a number of 
 discordant observations. The sum of all the observations 
 will be affected by the balance of all the errors, but 
 will be without error itself if the amount of the positive 
 errors be equal to that of the negative ones. This last 
 supposition, though not probable in itself, is never- 
 theless more probable than any other, and the odds are 
 very much in favour of its being nearly true. Now 
 whatever may be the error of the sum of observations, 
 say 100 in number, the average, or the hundredth part 
 of the sum, contains only the hundredth part of that 
 error; and the presumption that such an average is 
 very near indeed to the truth, greatly exceeds the pro- 
 bability in favour of any one of the observations. 
 
 But before we proceed further, it becomes necessary 
 to ask what laws of error can be absolutely determined, 
 and shown in the nature of things to exist? And 
 first, what do we mean by a law of error ? Let A B 
 be a length to be measured * or estimated, subject to 
 error of observers and instruments ; and let the greatest 
 possible errors be B K (negative), and B L (positive) ; 
 so that the result of measurement may be any thing 
 
 * The second figure is an enlargement of part of the first. 
 K 2 
 
132 
 
 ESSAY ON PROBABILITIES. 
 
 between A K and A L. Let K B and B L be equal, 
 and suppose positive and negative errors equally likely. 
 
 The probability of any one measurement giving 
 exactly any predicted result, say A N, must be in- 
 appreciably small ; since A N is only one out of an 
 infinite number of possible cases. But take any point 
 V, however near to N, and the chance that the result 
 
 ETV I. 
 
 of an observation shall lie between AN and AV is 
 capable of being imagined to be finite, though small. 
 'Whatever the law of errors may be, let a curve KCL 
 be so drawn that the chance of something between 
 AN and AV shall be the proportion which the area 
 NPQV is of the whole area KCL. Or if we call the 
 whole area 1, the area RPNW, for example, is that 
 fraction which expresses the chance of a result lying 
 between AW and AV. The symmetry of the curve 
 on the two sides of CB is an expression of the hypo- 
 thesis of positive and negative errors being equally 
 likely : and the approach of C K and C L towards the 
 axis is equally an expression of the supposition that 
 large errors are not so likely as small ones. If we wish 
 
ON ERRORS OP OBSERVATION. 133 
 
 to express that errors of all magnitudes are very nearly 
 equally likely, we draw such a curve as the upper 
 dotted line : while, if we wish to express that the pre- 
 sumptions are very strong for each measure being very 
 nearly true/ we describe the lower dotted line. We 
 can thus figure to the eye a representation of the law of 
 errors, be it what it may ; and the description of a law of 
 error is that of a curve. 
 
 Generally speaking, it is impossible to commit an 
 error of more than a certain magnitude : but this cir- 
 cumstance is one which embarrasses the mathematical 
 treatment of the subject exceedingly. It is practically 
 the same thing to consider any error, however great, as 
 possible, but errors of more than a certain magnitude 
 as extremely improbable. If, for instance, a case should 
 arise in which an error of more than an inch is impos- 
 sible, let it be agreed to consider such an error as not 
 impossible, but so improbable that there shall not be 
 an even chance of its happening once in a million of 
 times. The curve which exhibits the law of error 
 must then be of the following kind, never meeting the 
 axis, but continually approaching towards it, so that the 
 whole of its area from and after L, is incomparably 
 small by the side of the area C B L. 
 
 The curve here drawn is something like that in 
 p. 17., and we might suspect from the utility of the 
 latter and of the tables derived from it, that it should 
 play an important part in the present subject. This 
 we shall presently see : in the mean while I proceed to 
 explain the sense in which I use the terms average 
 balance, average error, mean risk of error and probable 
 error, to which I direct the reader's particular at- 
 K 3 
 
134 ESSAY ON PROBABILITIES. 
 
 tention. When we consider the errors of different 
 kinds as balancing each other, it follows that positive 
 and negative errors being equally likely, the balance of 
 all the errors will be trivial in a very great number of 
 observations. The average of all the errors will be 
 extremely small : and, in the long run, nothing. In 
 this instance then, it is said that the average balance of 
 error is nothing. But if positive and negative errors 
 were not equally likely, the more probable class would, 
 in the long run, predominate, and the average balance 
 of error would be of a definite magnitude, positive when 
 positive error is more likely than negative, and vice 
 versa. 
 
 The preceding has nothing to do with the average 
 of the absolute magnitudes of errors, considered without 
 reference to the distinction of positive and negative. For 
 example, whether the greatest error may be a mile or an 
 inch, it is equally true that the long run will establish 
 a compensation, when positive and negative errors are 
 equally likely. But in the former case, the average 
 magnitude of the errors which occur will, cceteris paribus, 
 as much exceed that in the latter, as a mile does an 
 inch. This average magnitude of errors, independently 
 of sign, is an important element of the whole question, 
 because a tolerably probable estimation of its value can 
 be found from the observations. Suppose, for instance, 
 that fifteen observations or estimations gave the follow- 
 ing results.* 
 
 722 1311 967 1309 
 
 933 1089 1344 858 
 1033 972 1250 1029 
 
 917 1294 744 
 
 The average of these is 1051, and assuming this as 
 the true result, the errors are, 
 
 329 260 84 258 
 118 38 293 193 
 18 79 199 22 
 134 243 307 
 
 * These are not numbers written at hazard, but actual results of 
 estimation, on a subject which it is not here necessary to explain. . 
 
ON ERRORS OF OBSERVATION. . 135 
 
 The average of the errors is 172 ; subject of course to 
 the supposition that the average of the observations is 
 nearly true. But the error of the last assumption 
 must be considerable before it can much affect the 
 present result. 
 
 The average of all the errors, taken without reference 
 to sign, will, in the present hypothesis, be the same 
 thing in the long run as the average of positive error, 
 or the average of negative error. The reason is, that 
 the number of positive and negative errors will in 'the 
 long run be equal, and also their sums. If, in a very 
 large number of observations, there be s positive and 
 s negative errors, and if the sum of each set be S, it 
 is evident that the sth part of S (which is the average 
 positive error, or the average negative error), is the 
 same as the 2sth part of 2S (which is the average 
 error without reference to sign). But it is customary 
 to prefer to the average error another function of the 
 errors, which may be called the mean risk of error, and 
 which differs from the average in the following manner. 
 Every error, positive or negative, is an increase or 
 diminution of the final result ; just as every game 
 won or lost at gambling is an increase or diminution 
 of the stock of the player. On precisely the same 
 principles as those explained in chapter V, I may con- 
 sider an even chance of an error 2 as a thing to be 
 compounded for by a certainty of an error 1. If, in a 
 large number of observations 2s, there will come a sum 
 of positive errors equal to S, and the same of negative 
 errors, and if, as in taking the average, the sum of the 
 errors be the only material point, it may be considered 
 that every observation will have either a positive error or 
 a negative error, of the value of the sth part of S. The 
 results of such a supposition will, in the long run, and so 
 far as the sum of errors is concerned, represent the actual 
 case under consideration. If, therefore, a person could 
 compound for positive errors alone, leaving the negative 
 ones to chance, he must suppose every observation to 
 have the half of S - s, or S -r- 2s of positive error, 
 
 K 4 
 
136 ESSAY ON PROBABILITIES. 
 
 combined with such a negative error as chance may 
 yield. That is, he must suppose all the negative 
 errors altered by the introduction of such an additional 
 positive error, and each of the positive errors increased 
 or reduced to S -5- 2s. And the same if he would 
 compound for negative errors only : while, to compound 
 for both, he must suppose every observation affected 
 both by a positive error of S -f- 2s and by a negative 
 error of the same amount. This latter case supposes 
 every observation to be correct, which is the result in 
 the long run. The use of this consideration is, to keep 
 before the mind the average effect of positive error, not 
 upon those observations which have positive error, but 
 upon all the observations; and the same for negative 
 errors. Suppose I have an instrument which makes 
 positive and negative errors in equal numbers and to 
 equal amounts, in the long run ; and suppose that it is 
 in my power totally to destroy positive error, leaving 
 the chance of negative error as before. What is the 
 effect upon the whole system of errors, positive and 
 negative, one with another ? What must I do with all 
 the errors to reproduce the same amount of absolute 
 error as before ? I must affect every observation with 
 one half of the average amount of positive error, or one 
 half the magnitude which the positive errors have, one 
 with another. Or look at it in this way ; if I have to 
 pay a shilling for every unit of positive error, for how 
 much should another person take the risk off my hands, 
 that is, for how much per observation, whether its 
 error be positive or negative. If the average positive 
 error were 1, I should have in the long run to pay at 
 the rate of one shilling for every two observations, against 
 which I should insure at the rate of sixpence per 
 observation. 
 
 The mean risk of positive error, then, is the average 
 positive error, when the errors of this kind are equally 
 distributed over all the observations : and the same for 
 negative error. When positive and negative error are 
 
ON ERRORS OP OBSERVATION. 137 
 
 equally likely, each is one half the average error, con- 
 sidered without reference to sign. 
 
 By the probable error I mean that amount of error 
 wjiich is such, that there is an even chance for exceed- 
 ing or falling short of it. Thus if it be 1 to 1 that 
 the error shall lie between and 10, and of course 
 the same that it shall exceed 10, then 10 is called the 
 probable error. For any number greater than 10, the 
 chances are (no matter how little) in favour of the 
 error being within that number ; for any thing less 
 than 10, the chances are against the error falling 
 within that amount. 
 
 PROBLEM. The number of observations being n, 
 and positive and negative errors being equally likely, 
 required the probability that the average of the n 
 observations lies within a given quantity k of the truth ; 
 or, M being the average, that the truth lies within 
 M + k and M k. 
 
 RULE. (By Table I.) Take the average of the 
 observations, find all the errors upon the supposition 
 that the average is the true result, add together the 
 squares of the errors, and divide the square of the 
 number of observations by twice the sum of the squares 
 of the errors. Call the result * the weight of the 
 average. Multiply k by the square root of the weight ; 
 let the result be t ; then the H answering to t in table 
 I. is the probability required. 
 
 RULE. (By Table II.) Find the weight as in the 
 last rule, and divide 62 by 130 times the square root 
 of the weight. The result is the probable error of the 
 average. Divide k by the probable error, and let the 
 quotient be t ; then the K answering to t in table II. is 
 the probability required. 
 
 EXAMPLE. In the preceding instance, what is the 
 probability that the average 1051 lies within 50 of the 
 truth. The squares of the errors are, 108241, 13924, 
 324, 17956, 67600, 1444, 6241, 59049, 7056, 
 
 * The reason of the appellation will be afterwards explained. 
 
138 ESSAY ON PROBABILITIES. 
 
 8584-9, 39601, 94249, 66564, 37249, 484, the sum 
 of which is 605831; and twice this is 1211662. 
 Divide 225, the square of 15, by 1211662, which gives 
 0001856953, the weight of the average. The square 
 root of the weight is -013627, which multiplied by 50 
 gives *6814, the value of t: that of H is then -66. 
 So that it is more than 3 to 2 that the true result of 
 the preceding very discordant observations lies between 
 1001 and 1101 
 
 To use the second table, multiply -013627 by 130 
 which gives 1 '771 51, by which divide 62, giving 35 
 very nearly. This is the probable error, so that it is 
 an even chance the result lies between 1051 35 and 
 1051 + 35, or 1016 and 1086. Divide 50 by 35 
 giving 1-43 the value of t ; for which in table II., K 
 is -66, as before. 
 
 I now proceed to explain the meaning of the term 
 weight, as used above. When an observer has made 
 various observations, one or more of which he thinks 
 superior to the rest, as to the favourableness of the 
 circumstances under which they were made, it follows 
 that the good observations should tell more in the 
 formation of the most probable result than the indif- 
 ferent ones. If for example, a remarkably good trial 
 give 10 and an indifferent one 11, it is not reasonable 
 to say that 10^ is the most probable result. If the 
 first observation be remarkably good, it may seem not 
 unfair to give it the force of four observations, or to 
 let the number 10 have the weight which would result 
 from four observations giving 10, 10, 10, 10, a fifth 
 giving 11. On this supposition the average is the 
 fifth part of 51, or 10, instead of 10 J. This was called 
 giving the observations 10 and 11 weights of 4 and 1, 
 and the method of finding an average is this : multiply 
 every observation by its weight and divide the sum of 
 the products by the sum of the weights. Such a method 
 was adopted before the theory of probabilities was ap- 
 plied to the subject, as a direction of common sense. 
 When that theory came into use, it was found that the 
 
ON ERRORS OF OBSERVATION. 139 
 
 square of the number of observations divided by twice 
 the sum of the squares of the reputed errors (the 
 average being reputed correct) ought to stand in the 
 place of the weight in the preceding rule, whenever 
 different averages are to be combined together to form 
 one general average. If, for instance, one average of 
 100 observations gave 10 and another of 50 gave 11, 
 and if the squares of 100 and 50 respectively divided 
 by twice the sums of the squares of the errors gave 1*5 
 and 1*1, the most probable average of these averages 
 would not be 10^, but the product of 10 and 1*5 
 increased by that of 11 and 1*1, and divided by the 
 sum of 1-5 and 1*1, which gives 10-4. Hence the 
 term weight is now applied to the quotient above de- 
 scribed. 
 
 When the law of error is of the kind figured in 
 p. 17., the mean risk of either sort of error, the probable 
 error, and the weight of a single observation, are con- 
 nected by very simple relations, as follows: 
 
 1. The mean risk is 200 divided by 709 times the 
 square root of the weight ; more nearly, "2820953 
 divided by the square root of the weight. 
 
 2. The probable error is 62 divided by 130 times 
 the square root of the weight ; more nearly, -476936 
 divided by the square root of the weight. 
 
 3. The weight is 113 divided by 1420 times the 
 square of the mean risk. 
 
 4. The probable error is 1 T 7 , T of the mean risk; 
 more nearly 1 -690694 of the mean risk. 
 
 5. The weight is 5 divided by 22 times the square 
 of the probable error ; more nearly, '227468 divided 
 by the square of the probable error. 
 
 6. The mean risk is -^4f of the probable error ; more 
 nearly, '591473 multiplied by the probable error. 
 
 We can thus find the remaining two, when either 
 of the three is given. Of the three, I apprehend that 
 the probable error refers to the most instructive notion ; 
 but the mean risk and the specific weight enter more 
 usefully into formulae .of .calculation. The average 
 
140 ESSAY ON PROBABILITIES. 
 
 error, being twice the mean risk, is readily determined 
 when wanted. 
 
 Many persons confuse the average and the probable 
 error in their own minds : that is,, they imagine it to 
 be as likely that any error should exceed the average 
 as fall short of it. That such cannot be the case is 
 evident from the following consideration. 
 
 The average error depends upon the magnitude of 
 the error, as well as upon the proportions in which 
 errors of different magnitudes enter ; the probable error 
 depends only upon the latter. If, then, small errors 
 enter in larger numbers than great ones, the probable 
 error is rendered less than it would otherwise be. In 
 determining the probable error, the error 100 entering 
 once, counts no more than the error 1 entering once. 
 But in the average error, the error 100, entering once, 
 counts 100 times as much as the error 1 entering once. 
 Consequently the former must be less than the latter. 
 But whether the probable error exceed or fall short of 
 the mean risk (half the average error), must depend on 
 the law of error. In the present case the former con- 
 siderably exceeds the latter. 
 
 It may be asked whether the preceding results are 
 always strictly true. Granting that the probability of 
 an error diminishes with its magnitude, and that 
 positive and negative errors are equally likely (which 
 are the only hypotheses of the preceding question), does 
 it necessarily follow, whatever may be the law of the 
 diminution of probability, that the mean risk of error 
 will, in the long run, be -^ of that error which is 
 as often exceeded as not ? The absolute answer to this 
 is, that the assertion is not strictly true, except upon 
 further suppositions as to the law of error. The 
 manner in which this inconsistency is explained, depends 
 upon whether the person asking the question be sup- 
 posed to be an inquirer seeking methods of disciplining 
 his judgment, or an experimental philosopher requiring 
 only a sufficient practical rule for the treatment of a 
 set of observations. 
 
ON ERRORS OP OBSERVATION. 141 
 
 To one who is looking for sound principles, I observe 
 that he does not want in this matter the exposition of 
 the consequences of any one law of facility of error, 
 but an account of the general character of those laws 
 to which common sense and daily experience assure 
 him that his faculties and means of observation are 
 subject. The facts of which he stands assured are, 
 that the probability of error does not diminish very 
 rapidly at first, but that as the error we consider grows 
 larger, its probability does diminish very rapidly, and 
 becomes insensibly small for errors of a certain magni- 
 tude and upwards. No curve of comparison, drawn in 
 the manner described in p. 132., will be a true repre- 
 sentation of what we know on this subject, unless it 
 have the general form, of which the following varieties 
 are instances. Now though the preceding results are 
 not strictly true for every curve which has such a form, 
 yet there is a class of curves of this form, some variety 
 or other of which will approach tolerably close to any 
 line which can be drawn to resemble one of those in the 
 
 figure. In every one of this standard class of curves, 
 all the preceding relations are strictly true, and there- 
 fore are nearly true for all the curves which resemble 
 any one of the standard class. Thus though the 
 average error may not always be -}4f of the probable 
 error, yet the former is always some fraction of the 
 latter, not differing very greatly from -J-if. There 
 is another reason for the adoption of this law of error 
 as a standard, for which the reader may consult the 
 fourth appendix to this work. 
 
142 ESSAY ON PROBABILITIES. 
 
 The experimenter, looking for a method of treating 
 observations which shall produce trustworthy results., 
 well knows that it matters nothing whether a method 
 be true or false, if demonstration can be given 
 that the consequences of the method are true. That 
 falsehood necessarily produces falsehood is a fallacy, 
 pardonable enough in everything but mathematics. 
 True reasoning on true hypotheses must necessarily 
 produce true results ; false reasoning, or false principles, 
 or both, may, and most probably will, lead to false con- 
 sequences, but may lead to the direct reverse. In 
 every part of knowledge, except mathematics, error 
 must be carefully avoided, because there is no method 
 of distinguishing between the cases in which it leads to 
 truth, and the contrary cases. But in the exact 
 sciences, the knowledge of the consequences of false- 
 hood and of those of truth are equally exact : and it is 
 possible to introduce an erroneous addition to the con- 
 ditions of a problem, to trace the consequences of such 
 error, and to annihilate them at any part of the process. 
 It is possible also to substitute for truth an erroneous 
 supposition, in such manner that the effect of successive 
 lapses of this kind shall be compensatory of each other, 
 or so that the more often the error is repeated, the nearer 
 is the result to the truth. The preceding case affords an 
 instance : let the law of error be what it may (provided 
 only that positive and negative errors are equally likely 
 and that of two errors the larger is always the less 
 probable), and let a moderately large number of observa- 
 tions be in question, and it follows that the results of the 
 real law, and those of the preceding supposition, are 
 nearly identical. Let the number of observations be 
 still larger, and the resemblance is still nearer, and so 
 on without limit. And this is true, even when the 
 law of error, as regards a single observation, or two or 
 three observations, varies to a large amount from that 
 which is used above. Consequently, for a tolerable 
 number of observations, it is absolutely indifferent 
 whether the real law of error be known, or whether 
 
ON ERRORS OF OBSERVATION. 143 
 
 the nearest variety of the class under consideration be 
 substituted for it. 
 
 Having then asserted, as a result of investigation, the 
 existence of a standard law of facility of error which 
 not only represents or resembles the impressions which 
 unassisted reason would form a priori, but the results of 
 which are more than sufficient mathematical approxi- 
 mations to truth, whatever (with some easily admissible 
 limitations) the law of error may be, I proceed to 
 describe it more particularly, calling it in future the 
 standard law of facility of error. The sole datum 
 necessary for its specific application, is either of the 
 three, the weight of an observation, the mean risk of 
 error, and the probable error, any two of which may be 
 deduced from the third by the rules in p. 137- 
 
 PROBLEM. Given either of the three data, required 
 (A) the chance that the error of any one observation 
 shall lie between e positive and e negative., or that the 
 observation shall give something between e too much, 
 and e too little (B) the chance that the preceding 
 shall not happen (C) the chance that the error shall 
 be positive, but not exceeding e ; or that it shall be 
 negative, not exceeding e. 
 
 RULE. Multiply e by the square root of the weight, 
 and let the product be t ; then (A) is the H corres- 
 ponding to t in table I., and (B) is the remainder of (A) 
 subtracted from unity ; each of the chances called (C) 
 is one half of (A). 
 
 EXAMPLE. The mean risk of error is 10 ; required the 
 chance of the error lying between -f 15 and 15 ; that 
 is, between 15 too much, and 15 too little. Since the 
 square root of the weight is 200 divided by 709 times 
 the mean risk of error 10, or T 2 %, and since 15 times this 
 result is -^-g-g- or *42 ; the probability required IB '45 ; 
 or 11 to 9 against the event. The probability of a 
 positive error less than 15 is "225, and the same for a 
 negative error within the same limits. 
 
 When all the individual observations are made under 
 the same circumstances, so as to have the same weight, 
 
144 ESSAY ON PROBABILITIES. 
 
 the common average is the most probable truth, and its 
 weight is as many times the weight of one observation as 
 there are observations in all : or, the average of n 
 observations each of which has the weight w, is entitled 
 to the same confidence as one observation made under 
 circumstances which give it a weight nw. Of this we 
 have an example in p. 137-:, and I now give another, 
 detached from the method there given of finding the 
 weight. 
 
 EXAMPLE. The weight of each observation being 
 18, what is the probability that the average of 50 
 observations lies within '05 of the truth. The weight 
 of this average is 18 X 50 or 900, the square root 
 of which is 30. Now -05 X 30 is 1*5, which, being t, 
 H is '966, so that the probability required is about 23 to 
 1 in favour of the event. The mean risk of error of such 
 an average is -fg-g- divided by 30, or ^-ff-^ less than 
 01 ; by which it is meant that if repeated sets of 50 
 observations each were made, the errors of these sets, 
 neglecting their signs, would not average so much as 
 01 x 2 or -02 each. 
 
 Let us now suppose that positive and negative errors 
 are not equally likely. Hitherto, the absolute truth 
 has been the most likely ; that is, though the pro- 
 bability of any one observation giving mathematical 
 truth was infinitely small, yet so was that of any given 
 error being exactly attained, and the infinitely small 
 probability of the first case was greater than that of 
 the second.* Let us now suppose that errors equal to 
 or near to P are more probable than any other, and so 
 that, x being the truth, any observation is equally likely 
 to exceed or to fall short of (not #) but x + P. This 
 is equivalent to describing a curve of the following 
 figure in place of that in p. 132. Here A B is to be 
 
 * To compare the proportions of these indefinitely small probabilities, 
 say of absolute truth and of either the error +e, or that of e exactly, 
 take H' from table I., corresponding to 0, and to e multiplied by the square 
 root of c. Thus, c being 1, the probabilities of an error and an error 2 
 are as I'l to -02, or as 55 to 1. 
 
ON ERRORS OF OBSERVATION, 145 
 
 measured, BK is P, and the chance of any error 
 falling between and B N is such a fraction of unity as 
 the area BPN is of the whole area of the curve. The 
 case before us is precisely that of an observer with a 
 personal error equal to P., in addition to casualties. If 
 we imagine a very large number of observations, made 
 under circumstances equally favourable to positive and 
 negative error, and if the error P be added to or taken 
 away from each of the results, according as P is a 
 positive or a negative error, we shall then have such a 
 succession of results as might be looked for on the 
 present hypothesis. For instance, let the quantity P 
 be 10, then whatever may be the prospect of having 
 the error 1, when positive and negative errors are 
 equally likely, we have the same chance for the error 
 11, in the present case. 
 
 For simplification, I shall adopt the algebraical 
 method of signifying positive and negative errors. 
 Thus -f 3 means 3 too much, or 3 added to the truth ; 
 4 means 4 too little. Thus + 4 5 is 1 ; mean- 
 ing that four too much from one cause, and 5 too little 
 from another, gives on the whole 1 too little. Again, 
 we consider 2 less than 1, since the effect of the 
 former is to lessen the result more than would be done 
 by the latter. To show that the supposition now before 
 us is equivalent to that of an observer with a personal 
 error, or an instrument with an individual error, imagine 
 an instrument wrongly graduated, so that for every 
 reading we ought to read 10 less : thus for 125 we 
 ought to read 115. In other respects let the instru- 
 ment be equally likely to give positive and negative 
 departures from truth. If then an observation give 
 176, we know immediately that the truth is 176" 10 
 
146 ESSAY ON PROBABILITIES. 
 
 + a casual error, the effect of which disappears in a 
 large number of observations. The whole error then 
 is 10 + the casual error, 10 -f x and 10 # being equally 
 likely. This is precisely the hypothesis in question, 
 in the case in which P = 10. 
 
 The average of observations, in the case before us, 
 does not necessarily give an approximation to the truth. 
 Calling the quantity P a fixed error, meaning by a fixed 
 error one which is as likely to be exceeded as not, we 
 see, in the following theorem, a justification of the term : 
 the most probable result of a large number of observations 
 is the truth, increased or diminished by the fixed error, 
 according as it is positive or negative. This error may 
 either be fixed in the instrument, fixed in the observer, 
 or in different degrees in both. 
 
 Let the phenomenon to be observed be perfectly 
 unknown, except by what the instrument tells us. 
 Then it is totally impossible to discover the amount of 
 this error ; which, nevertheless, must be assumed to 
 exist, unless the contrary be shewn. For example, an 
 instrument wrongly graduated throughout could never 
 tell the truth, either in individual or average results. 
 But it is obvious that such an apparatus, incapable as 
 it is of telling absolute truths, might nevertheless detect 
 such results as are obtained by measuring the differences 
 of other results. Thus a clock which goes truly, but 
 is set too fast or too slow, will serve to find the time 
 elapsed between two events, though it will not show the 
 real time of either. Instruments which, on account of 
 some permanent error affecting all their results, can 
 only be used to determine differences, are called dif- 
 ferential instruments. 
 
 All instruments, as well as observers, are subject 
 more or less to this species of error ; how then is it 
 possible to depend upon the results of any observations ? 
 The answer to this question will require some detail. 
 Since perfect exactness cannot be attained, either on 
 the part of the instrument, or of the observer, we can 
 only call either good, when positive errors are as likely 
 
ON ERRORS OP OBSERVATION. 14?7 
 
 as negative. The average of a large number of observ- 
 ations will in such a case be extremely near the truth, 
 and provided this condition can be fulfilled, the absolute 
 amount of the tendency to error is comparatively un- 
 important. Let two observers, A and B, have instru- 
 ments the average error of the first of which is double 
 of that of the second. A given number of observations 
 made by A is not as likely to be within a given amount 
 of the truth in the average as the same number made 
 by B. But the former may more than make the 
 difference, by -taking a larger number of observations. 
 The rule is, that the square roots of the numbers of 
 observations must be in proportion to the average errors 
 of the instruments. That is, if A's instrument have an 
 average error double of that of B, he must make four 
 times B's number of observations before he can place 
 the same reliance upon his own observations which he 
 ought to do upon those of B. And the same is true 
 if for average error we read mean risk, or probable 
 error. But if the weights of the observations be known, 
 the numbers of observations (and not their square roots), 
 must be inversely as the weights. 
 
 When, however, there is a fixed error in the instru- 
 ment, independently of casual errors, or of such as are 
 as likely to be positive as negative, there are two 
 modes of proceeding. The average of the observations 
 will now be too great or too small, according as the fixed 
 error is positive or negative. 
 
 1. If the truth can be found by any other means, 
 in any one instance, a large number of observations, 
 such as would be made if the truth in that one in- 
 stance were the object of inquiry, will serve to detect 
 the fixed error, with a high degree of probability that 
 the result shall be correct. If a result should be 29 
 and the average of 100 observations give 28, then it 
 must be presumed that instead of errors -j- x and x 
 being equally likely \-\-oc and - 1 an are equally 
 likely, or there is a fixed error of 1. If A be the 
 true result, and if P be the fixed error, then A -f- P is 
 L 2 
 
148 ESSAY ON PROBABILITIES. 
 
 the result which the instrument would give, in the long 
 run. In p. 137. is shewn the method of determining 
 the chance that in s observations, the average should lie 
 within e of the final average, that is, within e of A -f P. 
 This ascertains the chance that the final average just 
 mentioned lies between A + P -f e and A +- P e. Let 
 R be the result shown by the instrument, the truth A 
 being otherwise known. Then if R lies between 
 A -f- P e and A -f- P -f- e, it follows that P lies between 
 R A 4- e and R A e, and the chance of the first 
 is that of the second. 
 
 EXAMPLE. The truth being known to be 30, and 
 the average of 20 observations giving 31, what is the 
 chance that there is in the instrument a fixed error 
 lying between 1 + 1 and 1 J, or between -J and J. 
 
 The weight of the observations must first be found, 
 which is done by summing the squares of the errors, 
 taking the average given by the instrument as true, 
 precisely in the manner used in p. 137. Suppose this 
 weight to be 10; then e or -, multiplied by the square 
 root of 10, or 3'l62, is '79, to which in table I., the 
 value of H is "74. It is therefore about 3 to 1 that 
 the instrument, in the long run, would give a result 
 between 31 -f- and 31 -J- ; that is, that there is a 
 fixed error in the instrument lying between 1 + -J- and 
 
 !-i- 
 
 This first method, then, of ascertaining the fixed 
 
 error of a set of observations, supposes that there are 
 cases in which the result is known beforehand, so that 
 the instrument may first be read by the aid of phe- 
 nomena, instead of phenomena by the instrument. The 
 first observation is that of the error of the latter, found 
 by comparing its indications with the known truth ; 
 the second, the observation of unknown phenomena, 
 follows : accurate results being obtained, not by altering 
 the instrument, but by applying the correction to the 
 observations which the preceding class of observations 
 has rendered necessary. 
 
 A reader unused to astronomical works, on opening 
 
ON ERRORS OP OBSERVATION. 14p 
 
 a book on the practical part of the science, might imagine 
 that no part of the subject pretended even to ordinary 
 accuracy. Nothing appears to be done which is unaf- 
 fected by serious error ; and it seems as if a little more 
 attention to the fabrication of instruments would render 
 nine tenths of what has been written altogether useless. 
 This appearance is the victory of the head over the 
 hands ; the means of detecting the errors of instru- 
 ments are much more powerful than those of correcting 
 them. It is also the victory of astronomy over the 
 other physical sciences, on our knowledge of which the 
 manufacture of utensils depends : we know more of 
 the laws which regulate the changes of the heavens 
 than we do of those on which the stability and fluctu- 
 ation of instruments depend. Nor does the semblance 
 above mentioned entirely spring from unavoidable 
 error : for it is frequently the most convenient plan to 
 allow an error to subsist which might be corrected at 
 once, but which may be more easily corrected at 
 another stage of the process. It is also sometimes 
 useful to allow an error to remain of a larger magnitude 
 than is physically necessary, if by that means another 
 risk of error may be avoided. For instance, if re- 
 quisite correction be either an addition or a subtraction, 
 sometimes one and sometimes the other, the most 
 practised calculator will be very liable to confound the 
 two. This may be remedied by allowing to the instru- 
 ment a fixed error, either additive or subtractive, of 
 such a magnitude that casual fluctuations will never 
 alter its name. The correction, therefore, will always 
 require the same process, and the risk of error arising 
 from taking the wrong method will be avoided. 
 
 2. The next plan of eliminating the fixed errors 
 of an instrument is by giving it such a construction 
 that an observation can be made in two different ways, 
 in which the fixed error must necessarily have dif- 
 ferent signs, and must be of the same amount in both 
 cases. This is in reality a method of making the 
 positive and negative errors of the same amount in 
 L 3 
 
150 ESSAY ON PROBABILITIES. 
 
 the long run. Suppose, for example, that (as in the 
 transit-instrument) the correctness of the observations 
 depends (partly) upon the line of sight of a telescope heing 
 always exactly perpendicular to the axis upon which 
 the telescope turns. Such exact perpendicularity is 
 a mathematical fiction, which was never yet realised : 
 the telescope will incline more or less to the right 
 or to the left. But if the telescope he fixed to its 
 axis, and if the axis itself rest on pivots, from which 
 it can be taken off and the position of the instrument 
 reversed, it is obvious that such a reversal of the ends 
 of the axis will alter the error of the instrument, 
 throwing the line of sight as much to the left as it 
 was before to the right, or vice versa. The average of 
 a large number of observations will now present no 
 signs of fixed error, arising from this cause at least : 
 provided that the numbers of observations made in the 
 two different positions be equal. The chance of the 
 average of s observations in each position lying within 
 a given degree of nearness to the truth is precisely that 
 of twice s observations made with no fixed error, and 
 the same tendency to positive and negative casualties 
 as before. When the result has been obtained by 
 combination of the different sets, the fixed error of 
 the instrument may be ascertained by comparing the 
 combined average with the separate averages of each 
 set. If the observations be numerous, and the reversal 
 of the method of observing introduce no new errors, 
 then the combined average will be an arithmetical 
 mean (or nearly so) between the other averages, and 
 the difference between the former and either of the 
 latter will be the fixed error required. 
 
 Independently of one or other of these two methods, 
 the only result directly furnished by the observations 
 (except the average affected by the fixed error) is 
 their weight, which is obtained precisely as in p. 137. 
 It will be seen that either of the preceding methods 
 introduces entirely new elements ; in the first we have 
 previous truths for comparison with subsequent ob- 
 
ON ERRORS OF OBSERVATION. 151 
 
 serrations ; in the second, an adaptation of the instru- 
 ment which reproduces an equal likelihood of positive 
 and negative error, by making the fixed error itself as 
 often positive as negative. What we have called a 
 fixed error is in fact a part of the phenomenon, styled 
 an error because it is not a part of the result we wish 
 to observe. The errors which a simple application 
 of our theory removes are those of which no account 
 whatever can be given, and of which nothing can be 
 previously known. Such is not the case when positive 
 error is more likely than negative, or vice versa: for 
 this very circumstance is itself a phenomenon, which 
 must arise from some unvarying cause. 
 
 Having stated that it is indifferent, in a mathematical 
 point of view, whether the law of the facility of error 
 above explained be true or not, because any law what- 
 soever which falls within the widest permission of 
 common sense leads to the same results as above ex- 
 plained, when the number of observations is consider- 
 able I will now point out, in some simple cases, 
 how different laws of error are to be reduced to the 
 preceding. 
 
 1. Let all errors, positive and negative, between 4- E 
 and E be equally probable, and all others impossible. 
 Treat large numbers of such observations as in pages 
 187. and 144.j on the supposition that the weight is 3 
 divided by twice the square of E. 
 
 2. Let the probability of error decrease uniformly 
 as the magnitude increases, the greatest possible errors 
 being -f- E and E ; which implies that the chance 
 of an error lying between go and -\- x is the product 
 of x and the remainder of 2 E divided by the square of 
 E. For instance, if E be 10, and this law of facility 
 prevail, the chance of an error lying between 2 and 
 -j- 2, is the product of 2 and 18 divided by 10 times 
 10, or -ff-Q. In this case a large number of observ- 
 ations must be treated as if the weight of each observ- 
 ation were 3 divided by the square of E. 
 
 3. If the weight of the observations be considered 
 
A0~ ESSAY ON PROBABILITIES. 
 
 to be 5 divided by twice E 2 , the preceding methods 
 will show the chances of a large number of observ- 
 ations, upon a supposition intermediate between the 
 two last, and coinciding nearly with the first when the 
 errors are small, and with the second when they are 
 considerable. 
 
 I now proceed to the method of combining the 
 results of observation, and deducing the mean risks of 
 error. Suppose, for example, that A and B are two 
 results of a large number of observations, of which the 
 product is required. Nothing can be more erroneous 
 than to suppose that the mean risk of this product 
 will be the product of the mean risks of its factors. 
 By the mean risk here is meant the same thing as 
 before : imagine the product of A X B formed in every 
 possible way from the single results of the several 
 observations. Each error will be as likely to be positive 
 as negative, if the errors of the original observations 
 be the same. Take the average of all the several 
 errors, neglecting their signs, and one half of this 
 average will be (in the long run) what is called 
 the mean risk, explained in the same manner as in 
 p. 135. 
 
 The risk of the result will be modified by the 
 manner in which the operation makes one quantity 
 affect the error of the other. Suppose, for example, 
 that observation gives A 100, and B = 150. If 
 these were certainly true, the required result would be 
 accurately 100 x 150, or 15,000. But if 100 be 
 wrong to any amount, the product will be wrong by 
 150 times that amount, on that account only : while, 
 if B be wrong, its error will be multiplied 100 times 
 in the result ; besides which, there will be an additional 
 error, the product of the two errors of A and B. On 
 the other hand, if observation give A too large and B 
 too small, the opposite errors may either compensate 
 each other exactly, and give the product precisely what 
 it ought to be, or may make some approach towards this 
 compensation. The product then may be rendered 
 
ON ERRORS OP OBSERVATION. 153 
 
 much more erroneous than the observations, or much 
 less so ; both of which possible cases are considered, in 
 all their extent, in the investigations which give the 
 following results. In all of them, except the first, the 
 mean risks are supposed to be small. 
 
 1. To find the mean risk of the sum or difference 
 of any number of quantities determined by observation, 
 add together the squares of all their mean risks, 
 and extract the square root of the result. Thus, if 
 the mean risks of two quantities be 3 and 4, that 
 of their sum or difference is the square ro s ot of 16 + 4, 
 that is, 5. If the mean risks be all equal, the rule 
 may be simplified into that of multiplying the mean 
 risk of one by the square root of the number of 
 quantities. Thus the mean risk of the sum of 100 
 observed quantities of equal risk is 10 times that of 
 one of them. 
 
 EXAMPLE. Given the mean risks of A, B, and C, 
 namely, 1, 2, and 3, required that of 10 A + 9 B - 4 C. 
 Here every error which can happen in A is made 
 tenfold in 10 A, and the mean risk oflOAislOxl 
 or 10. Similarly the mean risks of 9 B and 4 C, are 
 9x2 and 4 X 3 or 18 and 12. The squares of 10, 
 18, and 12, added together, give 568, the square root 
 of which is 23*8, the mean risk required. 
 
 It may seem strange at first sight that, cceteris paribus, 
 the mean risk of a sum and difference should be the 
 same. But a little consideration will show that, posi- 
 tive and negative errors being equally likely, the errors 
 of a difference may be as large as those of a sum : 
 and that no combination of errors can affect a sum, 
 without an equal probability of another equally pro- 
 bable combination affecting the difference in the same 
 way. 
 
 2. To find the mean risk of the product of any 
 number of quantities A, B, C, &c. Take the fraction 
 which each mean risk is of its quantity : add the 
 squares of these fractions^ and multiply the square 
 root of the result by the product itself. This rule is 
 
154 ESSAY ON PROBABILITIES. 
 
 only to be trusted when the mean risks are small. 
 Let A= 100, B = 150, and let the mean risks of A 
 and B be 1 and 2. Then 1 is -01 of 100, and 2 is 
 0133 of 150. The squares of -01 and -0133 are -0001 
 and -00017689, the sum of which is -00027689, the 
 square root of which is -Ol6'7. This multiplied by 
 100 x 150, or 15,000, gives 250-5, which is the 
 mean risk of the product 15,000. 
 
 4. To find the mean risk of a fraction, or of the 
 quotient of a division, multiply each term (numerator 
 and denominator, or dividend and divisor) by the mean 
 risk of the other, add the squares of these products and 
 extract the square root of the sum: divide this by the 
 square of the denominator or divisor ; the result is 
 the mean risk required. But if the fraction be very 
 small, it is sufficient to divide the mean risk of the 
 numerator by the denominator ; while if the fraction 
 be very great, it is sufficient to multiply the fraction by 
 the risk of the denominator, and to divide the result by 
 the denominator. 
 
 The preceding will serve as specimens of the manner 
 in which complicated results of operation can have 
 those probabilities investigated which depend upon the 
 probabilities of error in their constituent parts. It 
 would be impossible to lay before a reader unacquainted 
 with the differential calculus, any such digest of rules 
 as would enable him to treat all cases with facility. 
 Any one of the mean risks obtained above will serve to 
 determine, as in p. 139-, the weight of the result, from 
 which its law of error may be investigated, as in 
 p. 143. 
 
 It appears that the chances of error may be con- 
 siderably multiplied in the course of the operations to 
 which the results of observation are subjected. It 
 must, therefore, be the object of an inquirer not only 
 to make good observations, but also to select such 
 methods of observing, and such methods of treating 
 the observations (the latter generally depending upon 
 the former), as will render the final error the least 
 
ON ERRORS OF OBSERVATION. 155 
 
 possible. The considerations necessary foF this pur- 
 pose form a great part of the application of mathe- 
 matics to the sciences of observation : in which it 
 frequently happens that good methods of observing 
 are rendered useless by the multiplication of error which 
 the methods consequent upon them involve : and con- 
 versely, that formulae good in other respects, are in- 
 admissible from the tendency to error in the observations 
 which they require. And it has happened before now 
 that mistakes of serious amount have arisen from the 
 use of mathematical methods in which the errors of the 
 observations are much multiplied. 
 
 I could hardly close such a chapter as the present 
 without some mention of the celebrated method of least 
 squares, on which the astronomy of the last thirty 
 years has depended for much of the increase of ac- 
 curacy which has been its characteristic. But as 
 any development of this very interesting subject is 
 impracticable without recourse to mathematical symbols 
 and reasoning, I content myself with a description 
 of one particular case, which is of very frequent 
 occurrence. 
 
 Suppose a number of results to be obtained by observ- 
 ation, from which a consequence is to be drawn by mathe- 
 matical reasoning. If the observations were all correct, 
 the consequence deduced from any one would be the 
 same as that from any other ; but owing to the 
 errors of the observations, such agreement is of course 
 unattainable. It is, therefore, a question what method 
 of combining the several results should be adopted : 
 and mathematical analysis shows that the object is 
 attained by choosing such an intermediate result as 
 shall make the sum of the squares of the errors the 
 least possible. It might seem as if, positive and nega- 
 tive errors being equally probable, the average of results 
 is the most probable truth ; and this is the case when 
 the observation is itself made directly upon the result 
 which is required, or when there is only one datum 
 into which the uncertainty of observation is intro- 
 
156 ESSAY ON PROBABILITIES. 
 
 duced. But even in such a case we have no right to 
 say that the average is preferred to the result of the 
 method of least squares ; for the former is then a par- 
 ticular case of the latter. Let three observations give 
 9, 11, and 16, the average of which is 12. The 
 errors, taking this average as the truth, are 3, I and 4, 
 the sum of the squares of which is 26. This is the least 
 possible sum of the squares. To try this, assume 11 as 
 the most probable truth : the errors are then 2, 0., and 
 5, the sum of the squares of which is 29 : assume 13, 
 and the errors are 4t, 2, and 3, the sum of the squares 
 of which is 29- To avoid introducing fractions, I 
 have only assumed whole numbers, but if I take 12*1 
 or 11 '9, I find the sum of the squares of the errors to 
 be in the first case 26-03, and in the second 26-03. 
 So that when one result only is in question, a direction 
 to take the average is equivalent to a direction to 
 make the sum of the squares of the errors the least 
 possible. 
 
 Let us now suppose two results of observation, say 
 that we wish to know the fraction which A is of B, 
 where both A and B are subject to errors, the positive 
 and negative being equally likely. Suppose, for ex- 
 ample, that we ask for the proportion of the population 
 of a country which is buried in a year. Statistical 
 returns will furnish the population of each year, and 
 the burials, both subject to errors. There are now 
 obviously two ways of taking an average ; I may 
 either divide the average burials by the average popu- 
 lation, or find the proportion which the first is of the 
 second in each year, and take the average of the pro- 
 portions. One not versed in mathematics would sup- 
 pose that these must give the same results, but any 
 simple instance will show the contrary. The average 
 dividend and the average divisor do not give the average 
 quotient. For instance, let dividends be 12, 13, and 
 17, and divisors 20, 22, and 30. The fractions -J, 
 -LJ, and -J,are'6, -591, and '567, the average of which 
 is '586. The average dividend is 14, the average 
 
ON ERRORS OP OBSERVATION. 157 
 
 denominator 24 and | i s not -586, but '583. The 
 results nearly coincide, but so do the data, for which 
 reason the most probable result may be very nearly 
 found, or of two results which differ very little, one 
 may be much more probable than another. When 
 observations give magnitudes so nearly coinciding as '6 
 591 and '567, it is worth while to examine the relative 
 probabilities of methods which give results so nearly 
 equivalent as '586 and '583. Which of the two 
 preceding methods is most entitled to confidence ? 
 Analysis points out that this question is useless, because 
 there is a third method which is more safe than any 
 other. The method of least squares in the case before 
 us leads to the following rule ; W r hen both the 
 numerator and denominator of a fraction are to be 
 determined by observation, and various corresponding 
 observations of both are made, multiply each numerator 
 and denominator by the denominator, and divide the 
 sum of the numerators so formed by the sum of the 
 denominators. Thus in the preceding instance, it is 
 
 12x20 + 13x22 + 17x30 1036 
 
 or '581 
 
 20 x 20 + 22 x 22 + 30 x 30 1 784 
 
 which is more probable than either '586 or '583. 
 
 If the mean risks of all the observations be the same., 
 the mean risk of the preceding result is found by adding 
 1 to the square of the result obtained ('58 1) dividing 
 by the denominator which produced it (1784), extract- 
 ing the square root of the quotient, and multiplying 
 the mean risk of each observation by this square. 
 Thus -581 X -581 is -337561, which divided by 1784 
 gives -000189, the square root of which is -014. The 
 mean risk of the result, therefore, is less than one 
 seventieth part of that of each of the observations. 
 
 The method of least squares is an extension of that 
 of taking an average, or rather it indicates the most 
 probable average in cases which, by reason of more 
 results of observation than one being involved, an in- 
 finite number of different averages exists. It is not 
 
158 ESSAY ON PROBABILITIES. 
 
 yet introduced into the affairs of common life, though 
 many cases occur in which it might be made useful. 
 But many things which are only demonstrable by the 
 higher branches of mathematics are looked upon as 
 useless by those who do not understand them ; nor 
 is this result of ignorance only to be looked for 
 among the uneducated. While the Reform Bill was 
 in its progress through the House of Commons,, a 
 method was suggested by a man of science, with whom 
 the government advised upon the subject, for esti- 
 mating the relative importance of boroughs by con. 
 sidering their population and contributions to the 
 revenue combinedly. This method, to the efficiency of 
 which most of those who examined it gave strong testi- 
 mony, was ridiculed by some members of the house, 
 partly because it involved decimal fractions, and partly 
 because another and a more simple (but palpably 
 wrong) method gave, in that particular case, nearly the 
 same results. When legislators are neither able to see 
 that erroneous methods may sometimes lead to truth, 
 being not therefore one bit the less erroneous, nor that 
 the truth of a result is the same, whether decimal or 
 common fractions be employed, it is little to be won- 
 dered at if useful applications of abstract reasoning 
 are looked upon with suspicion and introduced with 
 difficulty. 
 
 CHAPTER VIII. 
 
 ON THE APPLICATION OF PROBABILITIES TO LIFE CON- 
 TINGENCIES. 
 
 WHEN questions connected with life contingencies were 
 first considered, it was regarded as most deliberate 
 gambling to be in any way concerned in buying or 
 selling such articles as annuities, or any interests 
 depending upon them. Before we can well enter upon 
 
ON LIFE CONTINGENCIES. 159 
 
 the question of the truth or falsehood of the preceding 
 notion, it will be necessary to ask what laws the 
 duration of human life follows, and whether it fol- 
 low any laws at all ? Take two separate hundreds 
 of persons, each aged twenty, is there any reason to 
 conclude that the united lives of all the first hundred 
 will make an amount of years nearly equal to that of 
 the second ? 
 
 In order to try this point, I shall take another question, 
 yet more unfavourable to the result which I wish 
 to establish. In 100 persons all aged twenty, we know 
 that there is but a very slight chance that any given 
 one of them shall reach the age of eighty; and we 
 may consider it a certainty (or of an extremely high 
 probability), that none of them will see the age of a 
 hundred and twenty. We will consider it therefore as 
 given, that no one shall live to the last-mentioned age, 
 and we will even suppose that all ages of death between 
 20 and 120 are equally probable. This of course 
 very much increases our chance of fluctuation : but 
 even with this supposition it is not very great. 
 
 Let us suppose a lottery in which there are counters 
 marked with every possible number or fraction inter- 
 mediate between and E : so that the drawing may 
 have any mark whatsoever. If then we draw out 100 
 counters, the least possible amount of drawings will be 0, 
 the greatest 100 times E : and if all drawings be 
 equally probable, we have no reason to suppose that 
 our amount will exceed 50 times E, which does not 
 equally apply in favour of its falling short of that 
 quantity. That we shall have exactly 50 times E, is 
 an event of which the chance is infinitely small : but 
 that the amount shall lie between limits which are 
 tolerably near 50 times E, is very probable. 
 
 PROBLEM. Let there be counters, in equal numbers, 
 with every possible mark between and E. What is the 
 probability that the average of n drawings shall not differ 
 from the half of E, one way or the other, by more than Jc. 
 
 RULE. Multiply k by the square root of six times 
 
160 ESSAY ON PROBABILITIES. 
 
 w, and divide the product by E. Call the quotient t ; 
 then the value of H (Table I.) is the probability re- 
 quired. 
 
 EXAMPLE. In 600 drawings, each of which may 
 be any thing between and 100., required the proba- 
 bility that the average of all the drawings shall lie 
 between 50 + 5 and 50 5. 
 
 n 600, E = 100, k = 5 ; the square root of 6 times 
 600 is 60, and 5 times 60 divided by 100 is 3. The 
 first table does not contain values of t higher than 2 : 
 an event being almost certain, or of a very high pro- 
 bability when t is equal to 2. Table II., however, 
 furnishes us with an extension of Table I. ; the K oppo- 
 site to any value of t in that table. being always nearly 
 the H which belongs to half that value of t. Conse- 
 quently, the H belonging to t = 3, is the K belonging 
 to t = 6. But the second table only goes to t = 5 ; 
 in which case K is *999- It is then more than 999 to 
 1 that the average of the 600 drawings is within the 
 limits specified. If we take k = 1, in which case 
 t = '6, we find it is 3 to 2 that the average is con- 
 tained between 49 and 51. 
 
 If then there were 600 infants born, and if it were 
 the law of human life that any individual is as likely to 
 die at one age as another, for any age not exceeding 100 
 years, even then, and with so much more scope for 
 fluctuation than is actually found, it would be more than 
 999 to 1 against the average life of the 600 infants 
 exceeding 55, or falling short of 45 years,; and more 
 than 3 to 2 that the same average should fall between 
 49 and 51 years. If such be the case, it is obvious that 
 the chances of fluctuation are much diminished by the 
 superior chances of death happening at some periods of 
 life rather than at others ; as well as by the smaller limits 
 of human life, which need not for any practical purpose 
 be supposed to extend as far as one hundred years. 
 
 To suppose that the duration of human life is regu- 
 lated by no laws, would be to make an assumption of a 
 most monstrous character, a priori, and most evidently 
 
ON LIFE CONTINGENCIES. l6l 
 
 false. For it is a law, were it the only one, that no 
 individual shall attain the age of 200 years. So much 
 is known to all ; but to those who consider the subject 
 more closely, by the aid of recorded facts, it may be 
 made as evident as the existence of a limit to human life, 
 that the casualties of mortality are distributed among 
 mankind in so uniform a manner, that the average 
 existence of a thousand infants will differ very little 
 from that of another thousand born in the same country 
 and station of life. It is true that differences of race, 
 climate, manner of living, &c., &c., produce marked 
 effects upon the duration of life; which is no more 
 than might be expected : but it is equally true that the 
 notorious individual uncertainty of life cannot be dis- 
 covered in the results of observations made upon masses 
 of individuals. 
 
 There are various results of observation, which are 
 called tables of mortality, which differ only in the 
 methods of presenting the same sets of facts. Firstly, 
 we have what may be called tables of the numbers living. 
 These show, for a given number born, how many attain 
 each year of age. Thus, in the Carlisle table, opposite 
 to and 50, we find 10,000 and 4*397, indicating that, 
 according to observations made at Carlisle, the propor- 
 tion of those born to those who saw their fiftieth birth- 
 day, was that of 10,000 to 4397. Again, opposite to 
 60, we find 3643, meaning, that of 4397 persons aged 
 50, 3643 attain the age of 60. Secondly, we have tables 
 of yearly decrements, in which the same number of per- 
 sons are supposed to be alive at every age, and the pro- 
 portion who die in the next year is set down in the table. 
 Thus in the government annuity tables, opposite to 50 
 and 60, we find l6l and 315, meaning that, according 
 to the observations from which these tables were con- 
 structed, of 10,000 persons aged 50, l6l died before 
 completing the next year of life ; and of 10,000 persons 
 aged 60, 315 died before attaining the age of 6l. 
 Thirdly, we have tables of mean duration of life (com- 
 monly called expectation of life), which show the average 
 M 
 
162 
 
 ESSAY ON PROBABILITIES. 
 
 number of years enjoyed by individuals of every age. 
 This, in another variety of the Carlisle tables, opposite 
 to 50 and 60, we find 21*11 and 14*34 ; meaning that, 
 according to these tables, persons aged fifty live, one 
 with another, 21*11 years more, and persons aged 60, 
 14*34 years more. 
 
 Until observations of human mortality become more 
 extensive and correct, I prefer the tables of mean 
 duration to all others. The events of single years are 
 subject to considerable error, and generally present 
 such varieties of fluctuation, that it has become usual to 
 take some arbitrary and purely hypothetical mode of 
 introducing regularity. This practice cannot be too 
 strongly condemned, since the tables thereby lose some 
 of their value as representations of physical facts, 
 without any advantage ultimately gained. For if by 
 using the raw result of experiments, tables of annuities 
 were rendered unequal and irregular, it would be as 
 easy, and much more safe, to apply the arbitrary method 
 of correction to the money results themselves, than to 
 introduce it at a previous stage of the process. It is 
 not, however, a matter of much consequence as to the 
 annuities, &c., deduced from the tables : and as yet, the 
 rudeness of the original observations renders the effect 
 of any such alteration not so great as the probable 
 errors of the observations themselves. 
 
 The mean duration of life is approximately calculated 
 as follows. Suppose (taking an instance from the Car- 
 lisle tables) that 75 persons are alive at the age of 92, 
 of whom are left at the successive birthdays, 54, 40, 
 30, 23, 18, 14, 11, 9, 7, 5, 3, 1, 0. Consequently, in 
 their 93d year, 54 persons enjoy a complete year of life, 
 and 21 die, whom we may suppose, one with another, 
 to live through half the year, and 54 years and 21 half 
 years make 64^ years, which is the total life of 75 per- 
 sons for that year. Proceeding in this way, we find 
 that there are, 
 
 in the 93d year 54 + i of 21 years. 
 94th 40 + ^ of 14 
 
ON LIFE CONTINGENCIES. l63 
 
 95th 
 
 30 + of 10 
 
 96th 
 
 1 2S + \ of 7 
 
 97th 
 
 18 + i of 5 
 
 98th 
 
 14 + i of 4 
 
 99th 
 
 11 + |of 3 
 
 100th 
 
 9 + of 2 
 
 101st 
 
 7 + i of 2 
 
 102d 
 
 5 + | of 2 
 
 103d 
 
 3 + i of 2 
 
 104th 
 
 1 + I of 2 
 
 105th 
 
 + ^ of 1 
 
 Total, 215 + 
 
 Hence 75 individuals, aged 92, enjoy 215 -f- -J- of 75 
 years, and each has, one with another, the 75th part of 
 this, or 3*37 years. 
 
 RULE. To find the mean duration of life from a 
 table of the living at every age out of a given number 
 born, add together the numbers in the table for all the 
 ages above the given age, divide by the number at the 
 given age, and add half a year to the result. 
 
 The preceding rule is mathematically incorrect, being 
 only an approximation to the truth, even supposing the 
 tables perfectly correct. The error of computation 
 may be found, nearly, as follows. Divide the number 
 who die in the year next following the given age by 
 twelve times the number in the table at that age, and 
 diminish the result of the preceding rule by the quotient. 
 Thus, in the instance before us, 21 divided by 12 times 
 75 is -02, so that 3'35 is nearer the truth. This error, 
 however, is immaterial for practical purposes. 
 
 A more important question is that of the degree of 
 confidence which may be placed in tables of mean 
 duration, the errors of observation, being supposed to be 
 as likely to be positive as negative. In order to 
 estimate this, we must compute the mean square of the 
 duration of life ; that is, multiplying the time which 
 each individual lives by itself, we must add the results 
 together and divide by the whole number of individuals. 
 To make a rough approximation to this in the case 
 before us, remember that 
 
 M& 
 <9 
 
164 ESSAY ON PROBABILITIES. 
 
 21 individuals live J a year | x \ = \ giving L 
 
 ] 4 f~~" 2 X 2~ = 4 T~ 
 
 i/- 5 5 v 5 25 250 
 
 10 3- 2~ x 2 ~ 4 ~r 
 
 7 ?x? = f - ^ 
 
 5 - !x|- 
 
 4 - ^xu=m 
 
 405 
 
 13 13*13 = 169 .. 507 
 
 3 2 2*24 4 
 
 O M It x 15 = 225 450 
 
 * 2 224 4 
 
 o II 17 v 11 289 . 578 
 
 58 2 22 4" ~T 
 
 o 18 _ 1.9 x 15 = 3M 722 
 
 * 2 224 4 
 r> 21 _ 21 y 21 441 882 
 
 a 22 X T r ~r 
 
 O 23 _ 23 v 23 529 1058 
 
 2 2*24 T~ 
 
 T " T X ^2 4~ T" 
 
 75 (Average square 21-5) ^1 
 
 RULE. From the mean square of the duration of 
 life at any age, subtract the square of the mean duration 
 at that age : divide the difference by the number of 
 lives of the given age from which the table was made, 
 and extract the square root of the quotient. Take 
 four tenths (more correctly -39894), of this square 
 root, which gives the mean risk of error, and '67 of 
 the square root gives the probable error. 
 
 Suppose that in the case before us, the number of 
 lives aged 92 was 40 *, from which the preceding table 
 was made. We have then, 
 
 Mean square of durations 21-5 
 
 Square of 3 *37, the mean duration 1 1 -36 
 
 40)10-14 
 
 254 ^-254 =-504 
 504 x *67 = '33 of a year, the probable error. 
 
 The same process may be applied to any other case, 
 and the result of the whole is, that observation of a 
 number of lives which is not very great, will be suffi- 
 
 * This is nearly the number of lives at that age among those from 
 which the Carlisle table was formed, but the arbitrary help introduced 
 from other tables at the older ages, on account of presumed insufficiency of 
 data, makes the result of this example of no greater value than a nume- 
 rical instance arbitrarily chosen. 
 
ON LIFE CONTINGENCIES. l65 
 
 cient to give the mean duration of life with considerable 
 approach to exactness. This is confirmed by the 
 results of various tables, from which it appears that 
 when the individuals composing an observation are of 
 the same country, and under the same general circum- 
 stances, the results of such tables come very near to each 
 other. 
 
 The reader who desires to know the history of tables 
 of mortality should consult the articles MORTALITY and 
 ANNUITIES in the new edition of the Encyclopaedia 
 Britannica, both from the accurate pen of Mr. Milne, the 
 author of the Carlisle tables. I cannot, in this work, 
 pretend to give more than a slight summary of results 
 connected with life contingencies, such as may guide the 
 reader who understands the main points of the theory 
 of probabilities to safe conclusions. 
 
 From some tables made from observations at Breslau, 
 De Moivre concluded that the following hypothesis, 
 namely, that of 86 persons born one dies every year 
 till all are extinct, would very nearly represent the 
 mortality of the greater part of life, and that its errors 
 would nearly compensate one another in the calculation 
 of annuities. The Northampton tables of Dr. Price, 
 which have been used by most of the insurance offices, 
 very nearly represent this hypothesis at all the middle 
 ages. But both give much too large a mortality for 
 the circumstances of the last half century, as is proved 
 by all the tables which have been lately constructed. 
 The greater part of the difference, I have no doubt, is 
 due to the real improvement of life which has taken 
 place, from the introduction of vaccination, more tem- 
 perate habits of life *, better medical assistance, and 
 greater cleanliness in towns. We may now state, as a 
 much nearer approximation to the mortality of the 
 
 * I must be understood, here, as speaking particularly of the middle 
 classes, in English towns and cities. Most of the tables have a majority of 
 this class, and there is not any very precise information on the mortality 
 of the labouring classes, or in the inhabitants of the country as distin- 
 guished from those in towns. With regard to the point on which this 
 note is written, all old persons remember the time when what we should 
 now call hard drinking was almost universal. 
 M 3 
 
166 ESSAY ON PROBABILITIES. 
 
 middle classes, that from the age of 15 to that of 65, 
 the average may be represented as follows : of 100 
 persons aged 15, one dies every year till the age of 65. 
 But the mean duration of life will serve to give a more 
 precise idea, and a simple rule may be given, which 
 will, for rough purposes, represent the Carlisle table 
 between the ages of 10 and 60. Of persons aged 10 
 years, the average remaining life is 49 years, with a 
 diminution of 7 years for every 10 years elapsed ; thus 
 of persons aged 20 years, the average remaining life is 
 49-7 or 42 years ; at 30 years of age, 35 years. The 
 following list of tables will be followed by some notice 
 of each. 
 
 oJ 
 to 
 
 i 
 
 1 
 
 
 o 
 
 
 i . 
 
 f 
 
 | 
 
 o 
 
 g.s 
 
 
 1 
 
 3 
 
 1 
 
 S 
 
 EJ 
 g| 
 
 o 
 
 1 
 
 II 
 
 I 
 
 fl 
 
 a 
 
 ! 
 
 O 
 
 o"~ 
 O 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 43 
 
 25-2 
 
 . _ 
 
 38*7 
 
 - 
 
 m 
 
 - 
 
 
 
 5 
 
 40-5 
 
 40-8 
 
 - _ 
 
 51-3 
 
 _ 
 
 48-9 
 
 54-2 
 
 5 
 
 10 
 
 38 
 
 39-8 
 
 - - 
 
 48-8 
 
 48-3 
 
 45 6 
 
 51-1 
 
 10 
 
 15 
 
 35-5 
 
 36-5 
 
 - . 
 
 45-0 
 
 45-0 
 
 41-8 
 
 47-2 
 
 15 
 
 20 
 
 33 
 
 33-4 
 
 36-6 
 
 41-5 
 
 41-7 
 
 38.4 
 
 44-0 
 
 20 
 
 25 
 
 30-5 
 
 30-9 
 
 34-1 
 
 37-9 
 
 38-1 
 
 35'9 
 
 40-8 
 
 25 
 
 30 
 
 28 
 
 28-3 
 
 31-1 
 
 34'3 
 
 34-5 
 
 33-2 
 
 37-6 
 
 30 
 
 35 
 
 25-5 
 
 25-7 
 
 27-7 
 
 31-0 
 
 30'9 
 
 30-2 
 
 34-3 
 
 35 
 
 40 
 
 23 
 
 23-1 
 
 24-4 
 
 27-6 
 
 27-4 
 
 27-0 
 
 31-1 
 
 40 
 
 45 
 
 20-5 
 
 20-5 
 
 21-1 
 
 24-5 
 
 23-9 
 
 23 -8 
 
 27-8 
 
 45 
 
 50 
 
 18 
 
 18-0 
 
 17-9 
 
 21-1 
 
 20-4 
 
 20-3 
 
 24-4 
 
 50 
 
 55 
 
 15-5 
 
 15-6 
 
 15-1 
 
 17-6 
 
 17-0 
 
 17-2 
 
 20-8 
 
 55 
 
 60 
 
 13 
 
 13-2 
 
 12-5 
 
 14-3 
 
 13-9 
 
 14-4 
 
 17-3 
 
 60 
 
 65 
 
 10-5 
 
 10-9 
 
 9-9 
 
 11-8 
 
 11-1 
 
 11-6 
 
 14-0 
 
 65 
 
 70 
 
 8 
 
 8-6 
 
 7-8 * 
 
 9-2 
 
 8-7 
 
 9-2 
 
 11.0 
 
 70 
 
 75 
 
 5-5 
 
 6-5 
 
 6-2 
 
 7-0 
 
 6-6 
 
 7-1 
 
 8-5 
 
 75 
 
 80 
 
 3 
 
 4-8 
 
 5'0 
 
 5-5 
 
 4-8 
 
 4-9 
 
 6-5 
 
 80 
 
 85 
 
 0-5 
 
 3-4 
 
 4-0 
 
 4-1 
 
 3-4 
 
 3-1 
 
 4-8 
 
 85 
 
 90 
 
 - - 
 
 2-4 
 
 2-9 
 
 3-3 
 
 2-6 
 
 2-0 
 
 2-8 
 
 90 
 
 95 
 
 - - 
 
 0-8 
 
 1-4 
 
 35 
 
 1-1 
 
 1-2 
 
 1-6 
 
 95 
 
 100 
 
 
 
 
 o.<? 
 
 
 
 
 TOO 
 
 
 
 
 
 
 
 
 
 
ON LIFE CONTINGENCIES. l7 
 
 1. De Moivre's hypothesis was suggested by H alley's 
 Breslau tables, made from observations of the mortality 
 of that town in the years 1687 1691- It confessedly 
 errs considerably at the beginning and end of life. 
 
 2. The Northampton tables were constructed by 
 Dr. Price from the mortality of that town, in the years 
 1741 1780, the numbers of male and female deaths 
 being very nearly equal. These tables were, and are, 
 almost universally used by the assurance offices, and 
 are those by which legacy duties are estimated in the 
 act of parliament, 36 Geo. III. cap. 52. 
 
 3. The Amicable Society's table was formed some 
 years ago by Mr. Finlaison, at the suggestion of that 
 gentleman and myself to the directors, and as a means 
 of furnishing information upon points * as to which 
 they had consulted us. The Amicable society was 
 founded in 1705, and the table is formed from the 
 experience of more than half the subsequent period 
 ending in 1831. 
 
 4. The Carlisle table, formed by Mr. Milne from 
 the observations of Dr. Heysham upon the mortality of 
 that town, in the years 17791787- They are to be 
 considered the best existing tables of healthy life which 
 have been constructed in England. The relative pro- 
 portions of the sexes are 9 females to 8 males. 
 
 5. The Equitable table (published by the Equitable 
 society in 1834) gives the results of the experience of 
 that society from 1762 to 1829. The total number 01 
 deaths recorded is upwards of 5000. 
 
 6. The Government tables (male and female life 
 separately). These tables were constructed by Mr. Fin- 
 laison, actuary of the national debt office, from various 
 tontines, &c., of which the records are in the possession 
 of the government. Each table contains about 5000 
 
 * In mentioning this suoject, I may be allowed to state my full approval 
 of the plan subsequently adopted by the society, and my conviction -that 
 the errors of their ancient system have entirely disappeared. 
 
 The mean durations above given were computed by myself, from the 
 tables of decrements circulated by the directors among the members. 
 
168 
 
 ESSAY ON PROBABILITIES. 
 
 deaths. These are the tables on which the commis- 
 sioners for the reduction of the national debt grant life 
 annuities in lieu of stock. 
 
 I will now add some deductions made by myself 
 from the tables contained in the Recherches sur la 
 Reproduction, S$c., fyc. Brussels, 1832, by M. Que- 
 telet and Smits; republished in the treatise Sur I'Homme, 
 S$c. of the former. They are founded upon the 
 statistical returns of the whole of Belgium, made in 
 three successive years, and distinguish not only the 
 sexes but the residences of the parties, whether in 
 towns or in the country. The middle table is the 
 general average of the whole country, whether male or 
 female, in town or country. 
 
 Age. 
 
 Toi 
 Males. 
 
 vns. 
 Females. 
 
 Both. 
 Both. 
 
 Cou 
 Males. 
 
 ntry. 
 Females. 
 
 Age. 
 
 
 
 29-2 
 
 33-3 
 
 32-2 
 
 32-0 
 
 32-9 
 
 
 
 5 
 
 45-0 
 
 47-1 
 
 45-7 
 
 46*1 
 
 44-8 
 
 5 
 
 10 
 
 42-9 
 
 45-0 
 
 43-9 
 
 44-4 
 
 42-9 
 
 10 
 
 15 
 
 39-0 
 
 41.3 
 
 40-5 
 
 41-2 
 
 40-0 
 
 15 
 
 20 
 
 35-4 
 
 38-0 
 
 37.3 
 
 38-1 
 
 37-0 
 
 20 
 
 25 
 
 33-1 
 
 35-0 
 
 34-7 
 
 35-7 
 
 34-2 
 
 25 
 
 30 
 
 30-4 
 
 32-1 
 
 32-0 
 
 33-0 
 
 31-5 
 
 30 
 
 35 
 
 27-5 
 
 29-2 
 
 28-9 
 
 29-7 
 
 28*7 
 
 35 
 
 40 
 
 24-4 
 
 26-5 
 
 25'8 
 
 26-0 
 
 25-9 
 
 40 
 
 45 
 
 21-5 
 
 23-3 
 
 22-7 
 
 22-5 
 
 23-2 
 
 45 
 
 50 
 
 18-3 
 
 20-1 
 
 19'5 
 
 19-1 
 
 20-0 
 
 50 
 
 55 
 
 15-5 
 
 17-1 
 
 16-4 
 
 16-2 
 
 16*9 
 
 55 
 
 60 
 
 12 8 
 
 14-0 
 
 13-4 
 
 13-3 
 
 13-7 
 
 60 
 
 65 
 
 10-4 
 
 11-2 
 
 10-8 
 
 10-6 
 
 JO-9 
 
 65 
 
 70 
 
 8-2 
 
 8-6 
 
 8-4 
 
 8-2 
 
 8-5 
 
 70 
 
 75 
 
 6-3 
 
 6-6 
 
 6'4 
 
 6-3 
 
 6-5 
 
 75 
 
 80 
 
 4-8 
 
 5-1 
 
 5-0 
 
 5-0 
 
 5-1 
 
 80 
 
 85 
 
 3-7 
 
 4-0 
 
 3.8 
 
 3-8 
 
 3-8 
 
 85 
 
 90 
 
 2-9 
 
 3-0 
 
 3-1 
 
 3-1 
 
 3-2 
 
 90 
 
 95 
 
 1-8 
 
 2-0 
 
 2-1 
 
 2'2 
 
 1-9 
 
 95 
 
 100 
 
 o-o 
 
 0-5 
 
 1-3 
 
 05 
 
 0-5 
 
 100 
 
ON LIFE CONTINGENCIES. l 
 
 Most tables in which the sexes are distinguished 
 unite in presenting this result, that female life is ma- 
 terially better than male life. But this fact is much 
 more distinctly apparent in towns than in the country, 
 and in the Belgian tables the phenomenon is reversed, 
 so that while female life is decidedly better than male 
 life in the towns, it is not so good in the country. 
 Mr. Milne has remarked that in Stockholm the dif- 
 ference between male and female mortality was three 
 times as large a per centage of the whole as it was in all 
 Sweden. The probable reason for this discordance 
 is the different employment of women in town and 
 country ; all the tables yet constructed which distin- 
 guish the sexes, and include rural life, having been 
 made from a great preponderance of the working 
 classes. The only tables which separate the sexes, and 
 which are formed from the middle classes, are those of 
 Mr. Finlaison ; and here the difference is greatest of 
 all. 
 
 This consideration is very material in comparing the 
 tables which I have given. If a table of male life 
 should fall short of one of female life, all other cir- 
 cumstances remaining the same, it is no more than we 
 might expect ; while at the same time the true pro- 
 portions of male and female life, as well as the manner 
 in which they depend on local or other circumstances, 
 are very imperfectly known. But if a table of male life 
 only should present the same results as one of mixed 
 lives, we are then sure that the former represents a 
 longer duration of existence. For instance, the table 
 of the Equitable insurance office, which is almost entirely 
 composed of males, is almost identical with the Carlisle 
 table in which there are more females than males. 
 This shows that the select male lives of the office are 
 much better than the male lives of the Carlisle table : 
 but that the male lives of the office, constantly recruited 
 as they have been with selected lives of all ages, are 
 no better than the mixed lives of the Carlisle table. 
 Similarly, the males of the Amicable table are very 
 
170 ESSAY ON PROBABILITIES. 
 
 much better than those of the Northampton table. 
 The male and female lives of the latter are nearly equal 
 in number ; the former is almost entirely founded upon 
 male lives : while the former, with its male lives only, 
 gives a longer duration of life than the latter. For 
 old lives, however, the Northampton table gives a some- 
 what longer duration than the Amicable. This is only 
 one fact out of many which show that the Northampton 
 table, while it gives much too great a mortality to the 
 younger class of lives, errs in the other extreme as to 
 the older. Of thirty-five tables, made in different 
 countries and at different times, and including all of 
 any celebrity which had appeared before 1830, I find 
 that the Northampton table is the eighth from the 
 lowest at the ages of 10 25, and the tenth from 
 the highest at the age of 65. The same may be said 
 of De Moivre's hypothesis, which the Northampton 
 table closely follows. 
 
 The Northampton and Amicable tables are decidedly 
 older as to the period at which their members lived, 
 than the Carlisle and Equitable. Life is shorter in the 
 former pair than in the latter, while both of the former 
 agree in presenting the older lives comparatively better 
 than the younger ones, as compared with the latter 
 pair. By the Northampton table, the duration at 65 
 years is about a third part of that at 25 ; while the 
 same proportion is decidedly less in the same ages of 
 the Carlisle table : a similar result appears in the 
 Amicable and Equitable tables. I remember remarking 
 the same phenomenon in the results of a comparison of 
 the lives of naval officers. There can be little doubt that 
 the reason is as follows : in circumstances which create 
 a large mortality at the younger ages, all the feeble 
 constitutions are prevented from attaining old age, so 
 that the lives which really arrive at advanced years are 
 the remains of the very best lives. I saw the ocur- 
 rence of the same disproportion in the lives of officers 
 of the Anglo-Indian army; in which, however, it was 
 probably increased by the residence of many of the 
 
ON LIFE CONTINGENCIES. 171 
 
 officers in question in England during the latter years 
 of their lives. 
 
 The Equitable society has the character of having 
 been much more careful in the selection of its lives than 
 was the Amicable society during the earlier part of its 
 existence. This, together with the gradual improve- 
 ment of human life, serves to explain the very great 
 difference between the results of their experience. The 
 latter years of the Amicable society do not exihibit any 
 very decided difference of the sort. 
 
 The state in which we stand with respect to tables 
 of human life is singular, considering the enormous 
 amounts which daily pass from hand to hand in the 
 purchase of life interests. I may have occasion to 
 speak more at length on this subject in the sequel : 
 in the mean time let the reader observe the difference 
 between the various tables, and remember that each has 
 its votaries. If the late Mr. Morgan (whose name 
 stands very high as an authority on such matters) had 
 been requested to state the value of an annuity on the life 
 of a female aged 40, and the same for a male of the 
 same age, he would have replied that there was no 
 material difference between male and female life, and 
 that both belong to a class whose average existence is 
 23 years ; and he would accordingly have used the 
 Northampton tables of annuities. At the national debt 
 office, it would have been answered by Mr. Finlaison 
 that the male and female life are two very distinct 
 cases, and that the two different classes to which they 
 belong have severally the average lives of 27 and 
 31 years. That such differences should exist, is a 
 proof of insufficiency of information upon the sub- 
 ject : a want which nothing but the government can 
 supply, but which no government ever will attempt to 
 supply until increasing knowledge among the commu- 
 nity at large creates an influential body of remonstrants. 
 
 Having given a table of mean durations, it is easy to 
 find the proportion who die in one of the intermediate 
 periods, on the supposition that the deaths are equally 
 
172 
 
 ESSAY ON PROBABILITIES. 
 
 distributed through the period. This supposition is 
 not actually true, though for a long course of ages the 
 amount of mortality does not vary much from year to 
 year. The main feature of De Moivre's hypothesis, 
 equal decrements, appears in some measure at the adult 
 and middle ages of life in all tables. I do not, however, 
 know of any observations in which the numbers dying 
 at every age are large enough to produce much confidence 
 in the details of the tables of decrements, though the 
 fluctuations may compensate each other in the deter- 
 mination of the mean durations of life. Tables which 
 agree in the latter point may differ materially as to the 
 former. As an instance, I give the following com- 
 parison of the Carlisle and Equitable tables, which 
 agree more closely than any others in their mean 
 durations. The first column shows the common age of 
 10,000 persons, the second and third the number who 
 die in the following year in the Carlisle (C.) and 
 Equitable (E.) tables. 
 
 A. 
 
 C. 
 
 E. 
 
 A. 
 
 C. 
 
 E. 
 
 A. 
 
 C. 
 
 E. 
 
 A. 
 
 C. 
 
 E. 
 
 10 
 
 45 
 
 72 
 
 35 
 
 103 
 
 92 
 
 60 
 
 335 
 
 315 
 
 85 
 
 1753 
 
 2210 
 
 15 
 
 62 
 
 75 
 
 40 
 
 130 
 
 110 
 
 65 
 
 411 
 
 428 
 
 90 
 
 2606 
 
 2686 
 
 20 
 
 71 
 
 73 
 
 45 
 
 148 
 
 127 
 
 70 
 
 516 
 
 639 
 
 95 
 
 2333 
 
 5566 
 
 25 
 
 73 
 
 76 
 
 50 
 
 134 
 
 150 
 
 75 
 
 955 
 
 931 
 
 
 
 
 30 
 
 101 
 
 81 
 
 55 
 
 179 
 
 208 
 
 80 
 
 1217 
 
 1329 
 
 
 
 
 According to the Carlisle table, of 10,000 persons 
 aged 30, 101 die before attaining the next birthday ; 
 while one fifth less die in the Equitable table. And 
 yet, one with another, the average lives of two sets of 
 10,000 do not differ by more than their 170th part. 
 I now compare the actual tables of decrements, writing 
 opposite to each age the survivors of 10,000 births who 
 attain that age. (A, age ; C, Carlisle; E, Equitable.) 
 
 A. 
 
 C. 
 
 E. 
 
 A. 
 
 C. 
 
 E. 
 
 A. 
 
 C. 
 
 E. 
 
 
 
 10,000 
 
 
 35 
 
 5362 
 
 5292 
 
 70 
 
 2401 
 
 2310 
 
 5 
 
 6797 
 
 
 40 
 
 5075 
 
 5034 
 
 75 
 
 1675 
 
 1572 
 
 10 
 
 6460 
 
 6460 
 
 45 
 
 4727 
 
 4751 
 
 80 
 
 953 
 
 898 
 
 15 
 
 6300 
 
 6192 
 
 50 
 
 4397 
 
 4441 
 
 85 
 
 445 
 
 354 
 
 20 
 
 6090 
 
 5956 
 
 55 
 
 4073 
 
 4069 
 
 90 
 
 142 
 
 86 
 
 25 
 
 5879 
 
 5733 
 
 60 
 
 3643 
 
 3588 
 
 95 
 
 30 
 
 21 
 
 30 
 
 5642 
 
 5524 
 
 65 
 
 3018 
 
 3002 
 
 100 
 
 9 
 
 
ON LIFE CONTINGENCIES. 173 
 
 In the Equitable table, 5000 persons are supposed, 
 each aged 10 years; this I have altered to 6460, to 
 make the two tables agree at their outset. 
 
 The successive quinquennial decrements of the Car. 
 lisle table from the age of 20 are 211, 237, 280, 287, 
 348, 330, 324, 430, &c. If these deaths be supposed 
 to take place at equal or nearly equal intervals during 
 the five years, if, in fact, we may suppose each of the 
 individuals who die .in a period to enjoy, one with 
 another, half that period of existence, we may ascertain 
 the law of mortality from the table of mean durations 
 in the following manner. 
 
 RULE. To the mean duration at the end of the 
 period add the term elapsed and subtract the mean 
 duration at the beginning; divide by the smaller 
 duration increased by half the term, and the quotient 
 is the fraction which expresses the proportion dying 
 during the term. For example : the mean durations of 
 life at 25 and 30 in the Carlisle table are 37'9 and 
 34-3; and 34-3 + 5 37'9 is 1-4; which, divided by 
 34-3 + 2-5 or 36'8, gives ^ or ^ T ; so that of 184 
 persons aged 25, 7 die before attaining 30 years. In 
 the table, we have -/-/^ while -^ is about -g-gfo. 
 
 If we were to take any table now existing on English 
 lives, and ask, (as in p. 92-), what is the probability 
 that a large number of lives, say 1000, should drop 
 nearly in the same manner as those from which the 
 table was formed, we should find the resulting chance 
 not strong enough to make it prudent to risk much 
 money in such contingencies. Nevertheless, the appli- 
 cation of this theory to pecuniary risks has always been 
 in a more forward state than the physical theory of 
 human life. The reasons will be explained when we 
 come to treat on the grounds of the confidence to which 
 a contingency office is entitled. In the mean while, 
 supposing a table to represent perfectly the average of 
 a large number of the lives of the class to which an in- 
 dividual belongs, I proceed to show the method of using 
 such a table 
 
174 ESSAY ON PROBABILITIES. 
 
 Persons who are desirous of using tables of life on a 
 larger scale, are referred to the standard works of Messrs. 
 Morgan, Baily, and Milne, on life insurance. In the pre- 
 sent work I assume that it will be sufficient to be within 
 two years and a half of any age which may be named, and 
 I have given the several tables for intervals of five years. 
 The extremes which are used by actuaries generally 
 being contained in the Carlisle and Northampton tables, 
 and having given the former, I now add the latter. 
 The first column contains the age, the second the table 
 of decrements, the third the number out of 10,000 who 
 die in one year after completing the age in the first 
 column. 
 
 Age. 
 
 
 10,000 
 
 
 Age. 
 35 
 
 3437 
 
 187 
 
 Age. 
 70 
 
 1056 
 
 649 
 
 5 
 
 5356 
 
 
 40 
 
 3116 
 
 209 
 
 75 
 
 713 
 
 962 
 
 10 
 
 4864 
 
 92 
 
 45 
 
 2784 
 
 240 
 
 80 
 
 402 
 
 1343 
 
 15 
 
 4648 
 
 92 
 
 50 
 
 2449 
 
 284 
 
 85 
 
 159 
 
 2204 
 
 20 
 
 4399 
 
 140 
 
 55 
 
 2098 
 
 335 
 
 90 
 
 39 
 
 2609 
 
 25 
 
 4080 
 
 158 
 
 60 
 
 1747 
 
 402 
 
 95 
 
 3 
 
 7500 
 
 30 
 
 3759 
 
 171 
 
 65 
 
 1399 
 
 490 
 
 
 
 
 Supposing the tables perfectly accurate, the following 
 simple questions will show the nature of the first steps 
 which occur in their application. The Carlisle tables 
 are used throughout. 
 
 Question 1. What is the chance that an individual 
 aged 35 will live to the age of 50 ? Of 5362 persons 
 aged 35, 4397 live to be 50; hence the chance in 
 question is -|--||-J or '82. Answer, 41 to 9 for the 
 event. 
 
 Question 2. What is the chance that A aged 45 
 and B aged 50, shall both be alive in ten years ? The 
 chance for A, by the last question, is |4f|-f> or '862 
 and that for B ||-^| or '829 ; the product of these 
 (p. 43.), or '715, is the chance required. Again, the 
 chances of A and B dying during the ten years are 
 1 -862 and 1 '829, or -138 and -171 ; whence, 
 
ON LIFE CONTINGENCIES. HJ5 
 
 The chance is 
 
 That both shall live '862 x -829 
 
 That A shall live and B die -862 x -171 
 That A shall die and B live -138 x -829 
 That both shall die '138 x -171 
 
 Question 3. What is the chance that A aged 25, 
 shall die between the ages of 60 and 65 ? Of 5879 
 persons aged 25, 36433018 or 625,, die between the 
 ages of 60 and 65 ; hence / is the chance re- 
 quired. 
 
 Questions of this kind are readily solved, the only 
 impediment being the arithmetical operation. It fre- 
 quently happens, however, that the probability of one 
 individual surviving another is required, which though 
 an even chance when the individuals are of the same 
 age, is a matter of considerable calculation when one is 
 older than the other. Suppose, for example, that the 
 chance of A (aged 25) surviving B (aged 30) is required. 
 The survivorship, as it is called, meaning the period 
 during which A lives after the death of B, may begin 
 in any one year of A's age. For each year the pro- 
 bability of the survivorship beginning in that year must 
 be calculated. To make this calculation for one in- 
 dividual year, say that in which A is between 49 and 
 50, two cases must be considered : either B may die 
 between 54 and 55, and A may attain 50 complete 
 years (of which the chance may be found as in the 
 preceding questions), or both may die in the same year 
 (that is A between 49 and 50, and B between 54 and 
 55), but B may die first. If the chance of both dying 
 in that year be, say,'012,it is sufficiently correct to con- 
 sider the half of this chance, or '006, as being that which 
 expresses the chance of A's survivorship both beginning 
 and ending in that year : # supposition which is quite 
 correct only when the deaths of the year are equally 
 distributed through it. The result of this calculation 
 is arranged in tables, of which I here give a brief 
 abstract. 
 
176 
 
 ESSAY ON PROBABILITIES. 
 
 1 
 
 Younger. 
 
 Northamp- 
 ton. 
 
 Carlisle. 
 
 1 
 
 
 Younger. 
 
 Northamp- 
 ton. 
 
 1 
 
 15 
 
 5 
 
 447 
 
 400 
 
 80 
 
 50 
 
 136 
 
 093 
 
 20 
 
 10 
 
 415 
 
 383 
 
 85 
 
 55 
 
 113 
 
 -094 
 
 25 
 
 15. 
 
 420 
 
 381 
 
 90 
 
 60 
 
 097 
 
 113 
 
 30 
 
 20 
 
 423 
 
 375 
 
 95 
 
 65 
 
 037 
 
 147 
 
 35 
 
 25 
 
 417 
 
 372 
 
 45 
 
 5 
 
 252 
 
 177 
 
 40 
 
 30 
 
 409 
 
 366 
 
 50 
 
 10 
 
 206 
 
 146 
 
 45 
 
 35 
 
 402 
 
 360 
 
 55 
 
 15 
 
 201 
 
 135 
 
 50 
 
 40 
 
 394 
 
 350 
 
 60 
 
 20 
 
 193 
 
 119 
 
 55 
 
 45 
 
 385 
 
 329 
 
 65 
 
 25 
 
 172 
 
 110 
 
 60 
 
 50 
 
 376 
 
 315 
 
 70 
 
 30 
 
 148 
 
 097 
 
 65 
 
 55 
 
 360 
 
 323 
 
 75 
 
 35 
 
 124 
 
 081 
 
 70 
 
 60 
 
 339 
 
 322 
 
 80 
 
 40 
 
 102 
 
 075 
 
 75 
 
 65 
 
 317 
 
 303 
 
 85 
 
 45 
 
 082 
 
 059 
 
 80 
 
 70 
 
 300 
 
 320 
 
 90 
 
 50 
 
 069 
 
 052 
 
 85 
 
 75 
 
 Qf\ 
 
 292 
 
 332 
 
 95 
 
 55 
 
 025 
 
 078 
 
 95 
 
 OU 
 
 85 
 
 161 
 
 462 
 
 55 
 
 60 
 
 5 
 10 
 
 195 
 144 
 
 125 
 091 
 
 25 
 
 5 
 
 377 
 
 .307 
 
 65 
 
 15 
 
 136 
 
 085 
 
 30 
 
 10 
 
 344 
 
 283 
 
 70 
 
 20 
 
 125 
 
 .071 
 
 35 
 
 15 
 
 845 
 
 279 
 
 75 
 
 25 
 
 103 
 
 061 
 
 40 
 
 20 
 
 343 
 
 270 
 
 80 
 
 30 
 
 082 
 
 056 
 
 45 
 
 25 
 
 331 
 
 263 
 
 85 
 
 35 
 
 064 
 
 046 
 
 50 
 
 30 
 
 317 
 
 251 
 
 90 
 
 40 
 
 051 
 
 044 
 
 55 
 60 
 
 35 
 
 303 
 
 nc o 
 
 231 
 
 95 
 
 45 
 
 018 
 
 049 
 
 65 
 70 
 75 
 80 
 85 
 90 
 95 
 
 45 
 50 
 55 
 60 
 65 
 70 
 75 
 
 ZOO 
 
 269 
 246 
 218 
 189 
 166 
 157 
 072 
 
 194 
 177 
 177 
 190 
 174 
 191 
 300 
 
 65 
 70 
 75 
 80 
 85 
 90 
 95 
 
 5 
 10 
 15 
 20 
 25 
 30 
 35 
 
 143 
 087 
 078 
 069 
 053 
 041 
 014 
 
 086 
 054 
 048 
 040 
 034 
 033 
 039 
 
 35 
 40 
 45 
 50 
 55 
 
 5 
 10 
 15 
 20 
 25 
 
 314; 
 
 273 
 271 
 265 
 249 
 
 235 
 207 
 202 
 189 
 174 
 
 75 
 80 
 85 
 90 
 95 
 
 5 
 10 
 15 
 20 
 
 25 
 
 098 
 044 
 037 
 035 
 012 
 
 057 
 028 
 024 
 012 
 029 
 
 60 
 
 30 
 
 230 
 
 158 
 
 85 
 
 5 
 
 063 
 
 040 
 
 65 35 
 
 209 
 
 143 
 
 90 
 
 10 
 
 022 
 
 017 
 
 70 
 
 40 
 
 186 
 
 126 
 
 95 
 
 15 
 
 007 
 
 023 
 
 75 
 
 45 
 
 160 
 
 104 
 
 95 
 
 5 
 
 021 
 
 036 
 
ON LIFE CONTINGENCIES. 177 
 
 The quantity found in this tahle is the probability of an 
 elder life surviving the younger. The difference of ages 
 differs in the various compartments of the table ; in the 
 first it is ten years, in the second twenty years, and so 
 on. The two results accompanying each pair of ages 
 are those of the Northampton and Carlisle tables. Thus, 
 according to the Northampton table, the chance of a 
 life of sixty surviving one of thirty is *23 ; that of the 
 younger surviving the elder is therefore 1 -23 or *77- 
 According to the Carlisle table, the same chances are 
 158 and -842. 
 
 Almost universally, the Northampton table gives a 
 greater chance of the elder life beating the younger than 
 the Carlisle. This is a consequence of that undue 
 degree of comparative goodness which the former table 
 gives to older lives, and to which I have already ad- 
 verted. 
 
 If De Moivre's hypothesis were correct, it would be 
 sufficient to divide the mean duration of B's life by 
 twice that of A, and the result would be the chance 
 which B has of surviving A, B being the elder of the 
 two lives. This process, applied to the Northampton 
 table, will give results very near the truth, when 
 neither of the lives is very young. The same rule 
 would give comparatively but a very rough guess at the 
 result of the Carlisle table. If, however, the chance be 
 calculated which the younger life possesses of dying in 
 the average term of the elder, the result will be an 
 approximation to the probability of the elder surviving 
 the younger, when neither of the lives is very young, 
 and when their ages are not nearly equal. Thus, the 
 mean duration of a life of 50 being 21 years, and the 
 chance of a life of 30 years surviving 21 years being 
 '769, the chance of the same life not surviving 21 years 
 is -231 ; while in the table, the chance of a life of 50 
 .surviving one of 30 is *251. 
 
 I shall, in the next chapter, consider the application 
 of the tables to pecuniary questions, and shall now 
 
 N 
 
1?S ESSAY ON PROBABILITIES. 
 
 proceed to point out the connexion between a table of 
 mortality and one of population. 
 
 The whole number of persons inhabiting any country 
 is in continual state of increase from births and immi- 
 gration, and of decrease from deaths and emigration. 
 There are few countries in which immigration and 
 emigration produce any serious effect upon the popu- 
 lation, and, in times of very moderate quiet and pros- 
 perity, the births always exceed the deaths : so that, 
 generally speaking, the number of people alive in a 
 given country is yearly augmented by the excess of the 
 births over the deaths. If accurate registers of births 
 and deaths (with the ages at death) were kept for a 
 a century and a half, accompanied, if need were, by a 
 register of incomers and outgoers, with their ages, the 
 community would be in possession of a complete history 
 of its statistical changes, from which the law of mor- 
 tality might be deduced, and its fluctuations noted, if 
 any. 
 
 Again, if in any one year a complete census were made, 
 registering the age of every individual, and if the deaths 
 which took place in the 365 days next following the 
 day of the census were noted, the law of mortality 
 could be deduced. In such a case, the numbers of the 
 living at every age would be so large that the propor- 
 tion of deaths among them in a single year could be 
 safely depended on for pointing out, with great nearness, 
 the law which regulates the mortality of large masses of 
 people. 
 
 No such statistical means exist in this country, 
 partly from the defective manner in which the censuses 
 of population are made, partly from the circumstance of 
 the registries of births and deaths having been, almost 
 up to the time of writing this work, connected with 
 the religious ceremonies of the established church, which 
 has had the effect of excluding many dissenters from 
 registration. In the absence of all specific information, 
 recourse was had to the registers of burials, which are 
 usually accompanied by a statement of the age of the 
 
ON LIFE CONTINGENCIES. 179 
 
 parties, though without any sufficient guarantee for the 
 accuracy of the information. The hypothesis upon which 
 alone registers of burials will give a correct law of mor- 
 tality, requires that one of two alternatives should exist : 
 either a permanent law of mortality, with a knowledge 
 of the population in every year, and of the number, of 
 emigrants and immigrants, with their ages ; or a 
 stationary population, with the same number of births 
 and of deaths in each year, and a permanent law of 
 mortality. This latter supposition is never exactly 
 true ; but, as many societies have made a near approach 
 to it, and as many tables have been constructed by its 
 means, it will be worth while to explain the conse- 
 quences of the supposition. 
 
 If the Carlisle law of mortality remained in unin- 
 terupted operation for a century, and if 10,000 infants 
 were born alive in every year, the time would come 
 when the number of the living at any age in that table 
 would express the number alive at that age in the 
 society in question. Thus the number of persons aged 
 25 would be the 5879 survivors of those who were 
 born 25 years ago ; and the number of the living 
 at every age and upwards would be found by multi- 
 plying the number alive at that age by the mean 
 duration of life in the table in p. 166. If, then, the 
 law of mortality of such a society were required, it 
 would be found written in the burial registers of any 
 one year. For the numbers of births and deaths being 
 equal, there would be found for each year 10,000 
 burials ; which, if the law of mortality were permanent, 
 would be found distributed among the different ages 
 according to the table. Hence the number, out of 
 10,000 born, who attain a given age, would be found by 
 adding the number buried after that age* 
 
 But let us now suppose a population uniformly 
 increasing from year to year, say at the rate of 2 per 
 cent, per annum ; such a population would double 
 itself in 35 years ; and the younger lives would always 
 exist in a greater proportion to the older ones than 
 N 2 
 
180 ESSAY ON PROBABILITIES. 
 
 would be indicated by a correct table of mortality. 
 The burials at the younger of two ages would therefore 
 occur in too large a proportion to those at the older. 
 Suppose, for instance, that 350 deaths take place at 
 the age of 4041, and 1200 at the age of 56; 
 we are not therefore to conclude, that out of 10,000 
 individuals born, the deaths at 40 and 5 would be as 
 350 to 1200 : for since the population doubles itself 
 in 35 years, those who now die aged 5, are part of 
 twice as great a number of such lives as were of the 
 same age 35 years ago : consequently, of the set from 
 whom 350 died at the age of 40, 600 died at the age 
 of 5. If, then, a table were constructed from burials 
 alone, without paying any attention to the rate of 
 increase of the population, the older lives would appear 
 too good of their kind ; that is, relatively to the younger 
 ones of the same society. This, as already observed, is 
 the case in the Northampton table; whereas, in the 
 formation of the Carlisle table, proper attention was 
 paid to the variation in question. The difference is 
 very perceptible in comparing each of these tables with 
 that of the insurance office which it most resembles. 
 At 25 years of age, the mean duration of the North- 
 ampton table is 30*9? and that of the Amicable 34*1. 
 If the proportions of the mean durations remained 
 nearly the same, (as generally happens,) then the 
 Amicable table at 6*0 giving 12*5, the Northampton 
 table should give 11-4 ; instead of which it gives 13 2. 
 The preceding supposes, that while the population 
 changes, the law of mortality remains stationary. It is 
 very unlikely that such should be the case ; and observ- 
 ation, so far as it goes, tends to confirm the a priori 
 suspicion. When provisions are cheap, or wages high, 
 when, in fact, it is easy to maintain a family, marriages 
 are more frequent, and are contracted at earlier ages. The 
 same abundance of nourishment which tends to production, 
 also tends to preservation, both of parents and children ; 
 the consequence of which is, that a rapid increase of 
 population is often accompanied by a diminution of the 
 
ON ANNUITIES. 181 
 
 proportionate mortality. On the other hand, and from 
 contrary causes, a diminution of the rate of population 
 may be attended by an increase of the mortality. 
 
 'As this work does not profess to enter further into 
 statistics than is necessary to exemplify the principles 
 of the theory of probabilities, I shall here close what I 
 have to say on the rate of mortality, considered inde- 
 pendently of the most important pecuniary applications. 
 The next chapter will point out in what way money 
 calculations are made. 
 
 CHAPTER IX. 
 
 ON ANNUITIES AND OTHER MONEY CONTINGENCIES. 
 
 IF money could make no interest, the principles of 
 this chapter would be simplified, and the details 
 of calculation connected with it would be somewhat 
 reduced in amount. It will first be requisite to point 
 out the effect of compound interest, and to show how 
 to make computations connected with it. The funda- 
 mental calculation may be saved, for all such purposes 
 as this work is intended to answer, by the following 
 table, which may be described as follows. Opposite 
 to any year in the column headed Y, and under the 
 rate of interest in question (which is in numerals 
 at the head), will be found, within ten shillings, the 
 number of pounds which wil! 3 in such number of years, 
 at such rate of interest, produce a thousand pounds. 
 v Thus, opposite to 23 years, in the column headed 3, we 
 see 507 ; that is, 507^ (or, more strictly, something 
 between 5l6l. 10s. Od. and 517/. 10s. Od.,) will, when 
 improved at 3 per cent, for 23 years, produce 1000/. 
 N 3 
 
182 
 
 ESSAY ON PROBABILITIES. 
 
 Y. 2 
 
 a 
 
 3 
 
 ^ 
 
 4 
 
 <4 
 
 Y. 
 
 1 
 
 980 
 
 976 
 
 971 
 
 966 
 
 962 
 
 957 
 
 1 
 
 2 
 
 961 
 
 952 
 
 943 
 
 934 
 
 925 
 
 916 
 
 2 
 
 3 
 
 942 
 
 929 
 
 915 
 
 902 
 
 889 
 
 876 
 
 3 
 
 4 
 
 924 
 
 906 
 
 888 
 
 871 
 
 855 
 
 839 
 
 4 
 
 5 
 
 906 
 
 884 
 
 863 
 
 842 
 
 822 
 
 802 
 
 5 
 
 6 
 
 888 
 
 862 
 
 837 
 
 814 
 
 790 
 
 768 
 
 6 
 
 7 
 
 871 
 
 841 
 
 813 
 
 786 
 
 760 
 
 735 
 
 7 
 
 ' 8 
 
 853 
 
 821 
 
 789 
 
 759 
 
 731 
 
 703 
 
 8 
 
 9 
 
 837 
 
 801 
 
 766 
 
 734 
 
 703 
 
 673 
 
 9 
 
 10 
 
 820 
 
 781 
 
 744 
 
 709 
 
 676 
 
 644 
 
 10 
 
 11 
 
 804 
 
 762 
 
 722 
 
 685 
 
 650 
 
 616 
 
 11 
 
 12 
 
 788 
 
 744 
 
 701 
 
 662 
 
 625 
 
 590 
 
 12 
 
 13 
 
 773 
 
 725 
 
 681 
 
 639 
 
 601 
 
 564 
 
 13 
 
 14 
 
 758 
 
 708 
 
 661 
 
 618 
 
 577 
 
 540 
 
 14 
 
 15 
 
 743 
 
 690 
 
 642 
 
 597 
 
 555 
 
 517 
 
 15 
 
 16 
 
 728 
 
 674 
 
 623 
 
 577 
 
 534 
 
 494 
 
 16 
 
 17 
 
 714 
 
 657 
 
 605 
 
 557 
 
 513 
 
 473 
 
 17 
 
 18 
 
 700 
 
 641 
 
 587 
 
 538 
 
 494 
 
 453 
 
 18 
 
 19 
 
 686 
 
 626 
 
 570 
 
 520 
 
 475 
 
 433 
 
 19 
 
 20 
 
 673 
 
 610 
 
 554 
 
 503 
 
 456 
 
 415 
 
 20 
 
 21 
 
 660 
 
 595 
 
 538 
 
 486 
 
 439 
 
 397 
 
 21 
 
 22 
 
 "647 
 
 581 
 
 522 
 
 469 
 
 422 
 
 380 
 
 22 
 
 23 
 
 634 
 
 567 
 
 507 
 
 453 
 
 406 
 
 363 
 
 23 
 
 24 
 
 622 
 
 553 
 
 492 
 
 438 
 
 390 
 
 348 
 
 24 
 
 25 
 
 610 
 
 539 
 
 478 
 
 423 
 
 375 
 
 333 
 
 25 
 
 26 
 
 598 
 
 526 
 
 464 
 
 409 
 
 361 
 
 318 
 
 26 
 
 27 
 
 586 
 
 513 
 
 450 
 
 395 
 
 347 
 
 305 
 
 27 
 
 28 
 
 574 
 
 501 
 
 437 
 
 382 
 
 333 
 
 292 
 
 28 
 
 29 
 
 563 
 
 489 
 
 424 
 
 369 
 
 321 
 
 279 
 
 29 
 
 30 
 
 552 
 
 477 
 
 412 
 
 356 
 
 308 
 
 267 
 
 80 
 
 35 
 
 500 
 
 421 
 
 355 
 
 300 
 
 253 
 
 214 
 
 35 
 
 40 
 
 453 
 
 372 
 
 307 
 
 253 
 
 208 
 
 172 
 
 40 
 
 45 
 
 410 
 
 329 
 
 264 
 
 213 
 
 171 
 
 138 
 
 45 
 
 50 
 
 372 
 
 291 
 
 228 
 
 179 
 
 141 
 
 111 
 
 50 
 
 55 
 
 337 
 
 257 
 
 197 
 
 151 
 
 116 
 
 089 
 
 55 
 
 60 
 
 305 
 
 227 
 
 170 
 
 127 
 
 095 
 
 071 
 
 60 
 
 65 
 
 276 
 
 201 
 
 146 
 
 107 
 
 078 
 
 057 
 
 65 
 
 70 
 
 250 
 
 178 
 
 126 
 
 090 
 
 064 
 
 046 
 
 70 
 
 75 
 
 226 
 
 157 
 
 109 
 
 076 
 
 053 
 
 037 
 
 75 
 
 80 
 
 205 
 
 139 
 
 094 
 
 064 
 
 043 
 
 030 
 
 80 
 
 85 
 
 186 
 
 123 
 
 081 
 
 054 
 
 036 
 
 024 
 
 85 
 
 90 
 
 168 
 
 108 
 
 070 
 
 045 
 
 029 
 
 019 
 
 90 
 
 95 152 
 
 096 
 
 060 
 
 038 
 
 024 
 
 015 
 
 95 
 
 100 ! 138 
 
 085 
 
 052 
 
 OS 2 
 
 020 
 
 012 100 
 
ON ANNUITIES. 
 
 183 
 
 Y. 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 Y. 
 
 1 
 
 952 
 
 943 
 
 935 
 
 926 
 
 917 
 
 909 
 
 1 
 
 2 
 
 907 
 
 890 
 
 873 
 
 857 
 
 842 
 
 826 
 
 2 
 
 3 
 
 864 
 
 840 
 
 816 
 
 794 
 
 772 
 
 751 
 
 3 
 
 4 
 
 823 
 
 792 
 
 763 
 
 735 
 
 708 
 
 683 
 
 4 
 
 5 
 
 784 
 
 747 
 
 713 
 
 681 
 
 650 
 
 621 
 
 5 
 
 6 
 
 746 
 
 705 
 
 666 
 
 630 
 
 596 
 
 564 
 
 6 
 
 7 
 
 711 
 
 665 
 
 623 
 
 583 
 
 547 
 
 513 
 
 7 
 
 8 
 
 677 
 
 627 
 
 582 
 
 540 
 
 502 
 
 467 
 
 8 
 
 9 
 
 645 
 
 592 
 
 544 
 
 500 
 
 460 
 
 424 
 
 9 
 
 ]0 
 
 614 
 
 558 
 
 508 
 
 463 
 
 422 
 
 386 
 
 10 
 
 11 
 
 585 
 
 527 
 
 475 
 
 429 
 
 388 
 
 350 
 
 11 
 
 12 
 
 557 
 
 497 
 
 444 
 
 397 
 
 356 
 
 319 
 
 12 
 
 13 
 
 530 
 
 469 
 
 415 
 
 368 
 
 326 
 
 290 
 
 13 
 
 14 
 
 505 
 
 442 
 
 388 
 
 340 
 
 299 
 
 263 
 
 14 
 
 15 
 
 481 
 
 417 
 
 362 
 
 315 
 
 275 
 
 239 
 
 15 
 
 16 
 
 458 
 
 394 
 
 339 
 
 292 
 
 252 
 
 218 
 
 16 
 
 17 
 
 436 
 
 371 
 
 317 
 
 270 
 
 231 
 
 198 
 
 17 
 
 18 
 
 416. 
 
 350 
 
 296 
 
 250 
 
 212 
 
 180 
 
 18 
 
 19 
 
 396 
 
 331 
 
 277 
 
 232 
 
 194 
 
 164 
 
 19 
 
 20 
 
 377 
 
 312 
 
 258 
 
 215 
 
 178 
 
 149 
 
 20 
 
 21 
 
 359 
 
 294 
 
 242 
 
 199 
 
 164 
 
 135 
 
 21 
 
 22 
 
 342 
 
 278 
 
 226 
 
 184 
 
 150 
 
 123 
 
 22 
 
 23 
 
 326 
 
 262 
 
 211 
 
 170 
 
 138 
 
 112 
 
 23 
 
 24 
 
 310 
 
 247 
 
 197 
 
 158 
 
 126 
 
 102 
 
 24 
 
 25 
 
 295 
 
 233 
 
 184 
 
 146 
 
 116 
 
 092 
 
 25 
 
 26 
 
 281 
 
 220 
 
 172 
 
 135 
 
 106 
 
 084 
 
 26 
 
 27 
 
 268 
 
 207 
 
 161 
 
 125 
 
 098 
 
 076 
 
 27 
 
 28 
 
 255 
 
 196 
 
 150 
 
 116 
 
 090 
 
 069 
 
 28 
 
 29 
 
 243 
 
 185 
 
 141 
 
 107 
 
 082 
 
 063 
 
 29 
 
 30 
 
 231 
 
 174 
 
 131 
 
 099 
 
 075 
 
 057 
 
 30 
 
 35 
 
 181 
 
 130 
 
 094 
 
 068 
 
 049 
 
 036 
 
 35 
 
 40 
 
 142 
 
 097 
 
 067 
 
 046 
 
 032 
 
 022 
 
 40 
 
 45 
 
 111 
 
 073 
 
 048 
 
 031 
 
 021 
 
 014 
 
 45 
 
 50 
 
 087 
 
 054 
 
 034 
 
 021 
 
 013 
 
 O09 
 
 50 
 
 55 
 
 068 
 
 041 
 
 024 
 
 015 
 
 009 
 
 005 
 
 55 
 
 60 
 
 054 
 
 030 
 
 017 
 
 010 
 
 006 
 
 003 
 
 60 
 
 65 
 
 042 
 
 023 
 
 012 
 
 007 
 
 004 
 
 002 
 
 65 
 
 70 
 
 033 
 
 017 
 
 009 
 
 005 
 
 002 
 
 001 
 
 70 
 
 75 
 
 026 
 
 013 
 
 006 
 
 003 
 
 O02 
 
 001 
 
 75 
 
 80 
 
 020 
 
 009 
 
 004 
 
 002 
 
 001 
 
 000 
 
 80 
 
 85 
 
 016 
 
 007 
 
 003 
 
 001 
 
 001 
 
 000 
 
 85 
 
 90 
 
 012 
 
 005 
 
 002 
 
 001 
 
 000 
 
 000 
 
 90 
 
 95 
 
 010 
 
 004 
 
 002 
 
 001 
 
 000 
 
 000 
 
 95 
 
 100 
 
 008 
 
 003 
 
 O01 
 
 000 
 
 000 
 
 000 
 
 100 
 
 N 4 
 
184 ESSAY ON PROBABILITIES. 
 
 When 000 appears in the table, the sum requisite is less 
 than ten shillings. Thus, less than ten shillings im- 
 proved for 85 years at 10 per cent., will produce a 
 thousand pounds. 
 
 There are six primary results which will be neces- 
 sary. 
 
 1. The present value of 1L 9 meaning that fraction of 
 a pound which will, when improved at interest, produce 
 11. in a certain number of years. Since the present 
 value of WOOL is tabulated, that of 11. is virtually so, 
 and may be found by placing the decimal point before 
 any result of the preceding table. Thus, *458/. is the 
 fraction of a pound which will, when improved at 5 per 
 cent, compound interest for sixteen years, produce 11.; or, 
 in technical language, the present value of 11. due sixteen 
 years hence, at 5 per cent. To find the present value 
 for any year not in the table, multiply the present 
 values for any years in the table the sum of which is 
 the number of years in question. Thus, to find the 
 present value of 11. due fifty-seven years hence, at 3 per 
 cent., multiply together '337 and *96l, the present 
 values for fifty-five and two years, the result of which 
 is *324, the fraction of 11. required. 
 
 2. The present value of a perpetuity of I/., or the 
 sum which will, wj^en improved at interest, pay 11. at 
 the end of every 3 65 days from the present time, for 
 ever. The number of pounds required is found by 
 dividing 1001. by the rate of interest ; being, in fact, 
 nothing but the sum which will at interest produce 11. 
 per annum. The results are in the following table : 
 
 Rate P.V. of Perp. I Rate P.V. of Perp. Rate P.V. of Perp, 
 
 3 
 
 50 
 40 
 
 25 
 J 22f 
 20 
 
 9 11 9 
 10 10 
 
 Thus 20J. improved at 5 per cent, yields 11. a year foi 
 ever, provided the first payment be due at the interval 
 of a year from the time of putting the money out at 
 
ON ANNUITIES. 185 
 
 interest. This is also said to be 20 years' purchase, 
 meaning the sum which would buy at once the annual 
 payments of twenty years. 
 
 3. The present value of an annuity certain of 11. 
 (so called to distinguish it from a life annuity.) By an 
 annuity of II. is meant, a right to receive II. at the 
 expiration of every complete year after the creation of 
 the annuity,, which is said to commence a year before 
 any payment is made; (or a term before payment is made, 
 whether the term be yearly, half-yearly, &c.) Thus 
 an annuity certain of five years, commencing January 1, 
 1838, is paid on the first days of 1839, 1840, 1841, 
 
 4 1842, and 1843. 
 
 When an annuity is to be designated which makes 
 one payment immediately, I shall call it an annuity 
 due; and a perpetuity of which one payment is to 
 be made immediately, a perpetuity due. Thus on 
 January 1, 1839, the preceding annuity becomes an 
 annuity due of four years, comprising an immediate 
 payment of II. and a commencing annuity of four 
 years. To find the present value of an annuity of I/., 
 use the following 
 
 RULE. Multiply the perpetuity (present value of 
 the perpetuity) by the excess of one pound over the 
 present value of the last payment. Thus, for an 
 annuity of five years, at 4 per cent, multiply 25 by the 
 excess of 1 over '822, or by -178; the result is 
 4-45/., nearly 
 
 4. The amount of II. at compound interest, meaning 
 the sum to which II. with its interest will amount in 
 any number of years. It is found by dividing 1 by 
 the present value of II. due that number of years hence. 
 Thus II in 25 years, at 6 per cent., will amount to 
 (11. -s- -233), or 4-29/. 
 
 5. The amount of an annuity at compound interest, 
 meaning the sum of which the annuitant would be 
 possessed immediately after receiving the last annual 
 payment, if he had made compound interest upon every 
 preceding payment. This amount is found by multi- 
 
186 ESSAY ON PROBABILITIES. 
 
 plying the number of years' purchase in the perpetuity 
 by the interest (not the amount) of one pound im- 
 proved during the existence of the annuity- Thus, at 
 6 per cent., and in 50 years, l^will become (l/.-r-'054), 
 or 18*5/., so that the interest is 17'5/. This multiplied 
 by l6f, or ^%-, gives 292/. The more correct answer 
 is 29O34/., but the error is not ten shillings in a hun- 
 dred pounds. 
 
 6. The present value of an annuity for a term of years, 
 to begin after the expiration of another term; tech- 
 nically called a deferred annuity. To find this, multiply 
 the present value of such an annuity created immediately 
 by the present value of II. due at the end of the term 
 of deferment. Thus an annuity of WL for five years, at 
 four per cent., to commence at the end of twenty years, 
 is thus found : were it to commence immediately, it 
 would (see last page) be worth 4'45 X 10/. or 44*5/., and 
 the present value of II. to be received twenty years hence 
 is '456/. Multiply 44*6 by "456, which gives 2O3/., the 
 present value required. 
 
 ' Extensive tables of the results of the preceding pro- 
 cesses are given in all works on interest or annuities ; 
 the present treatise is not meant to contain more than 
 enough to enable the student to exercise himself in first 
 principles, or the proficient to obtain approximate re- 
 sults, when no more extensive work is at hand. I now 
 pass to the consideration of annuities in general. 
 
 Though the term annuity be generally understood as 
 meaning a sum of money paid yearly or half-yearly to 
 an individual, yet it is important that the reader should 
 consider the word as implying any sum of money paid 
 at fixed intervals, for any term, definite or indefinite 
 provided only that the payment can never be suspended 
 and resumed again. Thus II. to be paid every year 
 'for ten years provided A live as long, with an ad- 
 ditional condition that payment is to cease if C in the 
 term should die, B being alive at the time of C's death, 
 is an annuity within the meaning of the word : nothing 
 can make the payment cease at all, without making it 
 cease altogether. But II. to be paid as long as A, B, 
 
ON ANNUITIES. 187 
 
 and C are alive, to cease when one dies, and to re- 
 commence when a second dies, ceasing finally with the 
 death of the third, is not one annuity, but two annuities. 
 And by whatever name periodical payments may be 
 called, whether rent, salary, rent-charge, interest, &c. 
 &c., they are considered in this subject as annuities. 
 
 An annuity may be in possession or in reversion. 
 In the first case, a payment will become due at the end 
 of a term after the creation of the annuity, unless, in 
 the mean time, one of the conditions on which payment 
 depends cease to exist. In the second, the annuity is 
 not to begin to be paid until some circumstances happen, 
 or cease to happen, which are named in the agreement. 
 Sometimes the term reversionary annuity implies that 
 there is a contingency in the time of its beginning ; 
 and an annuity not to commence till after a fixed time 
 has elapsed, would be called a deferred annuity. There 
 is, however, a little variation in the use of these words : 
 reversion sometimes implies a remainder of something 
 existing, while a deferred interest is one which does not 
 commence till a future time. Thus if B be to take an 
 annuity on his own life as soon as A is dead, A en-* 
 joying it during his own life, B's reversion is a certain 
 part of an annuity on his own life, reckoned from the 
 present time. But if no annuity at all were paid until 
 A's death, then B's interest might be called a deferred 
 annuity. No difference arises from these distinctions in 
 formulae or calculations ; and they are useful in de- 
 scribing the circumstances of complicated problems. 
 
 A reversion, and also a deferred annuity, may be 
 certain or contingent : and the same of a reversionary 
 or deferred fixed sum. Thus A may have B's estate 
 after his death, provided he survive B ; or else A and 
 his heirs or assigns may have the same : in the first 
 case the reversion is contingent, in the second certain. 
 A life insurance may be of one or the other kind : thus 
 A may covenant for his executors to receive 100/. at 
 his death, in any case, or else if B should be then sur- 
 viving. And if, in return for such insurance, A should 
 engage to pay a yearly premium, making the first pay- 
 
188 ESSAY ON PROBABILITIES. 
 
 ment (as is usually done) at the time of contracting, 
 then the premiums altogether constitute an annuity due 
 upon A's life. 
 
 All problems relating to this subject might be solved 
 from first principles, as will be shown ; but, in such case, 
 even the most simple of them would be attended with 
 laborious calculation. Tables are therefore constructed 
 of such results as will most facilitate the solutions of all 
 problems : and these tables give the values of annuities 
 on single lives, and on two joint lives. The annuity 
 presumed is always II. per annum, from which the 
 value of any other annuity is found by simple multi- 
 plication. 
 
 By the value of an annuity is meant the sum which 
 must be paid down in order to enable the grantor of 
 the annuity to pay it as it becomes due during the 
 term of its continuance. If money made no interest, 
 the average duration of such sets of circumstances as 
 are conditions for the payment of the annuity would 
 immediately point out its value. For instance, accord- 
 ing to the Carlisle tables, persons aged 40 live, one 
 with another, 27*6 years. Now, remembering that 
 half a year was added (p. 163.), as being the part of 
 their last year which, one with another, they pass 
 through, but which will not (in our meaning of the 
 term annuity) entitle them to any payment, we see that 
 27*1 is the average number of payments which such 
 annuitants, aged 40, will receive; that is, one hundred 
 with another, 100 annuitants will receive 2710 pay- 
 ments. Consequently, money making no interest, each 
 of them must pay 27*1 01. for a life annuity of I/., 
 or 27*10 years' purchase. But if each had been 
 entitled to his fraction of the sum which would have 
 become due had he lived to the end of the year, then 
 2 7' 60 years' purchase would have been necessary. 
 
 Let us now suppose money to make interest. It 
 has before now seemed, to more than one writer, that 
 the value of a life annuity must be the same as that of 
 an annuity certain during the average duration of the 
 life. Many of my readers will not see the fallacy of 
 
ON ANNUITIES. 189 
 
 supposing that, money making 4 per cent., l6'7l., the 
 value of an annuity certain for 28 years, must be but very 
 little more than the value of an annuity on a life aged 
 40. Such a rule, if it be absolutely true, must be true 
 in any extreme case, however physically impossible. 
 Suppose, then, that of 101 persons aged 40, one lives for 
 ever, and the rest die between the ages of 41 and 42 : 
 whence 100 of them will receive only one payment, 
 and the remaining one will receive a perpetuity of \L 
 The present value of an annuity of 11. for one year, at 
 4 per cent., is *96/., and $6l. increased by the value 
 of I/, for ever, is 121 1., the present value of all pay- 
 ments. If, then, it be not known which is to have the 
 perpetuity, -j-y/. is what each should pay, or 1*2 II. 
 very nearly. This result is undoubtedly correct : but 
 the average life of the 101 persons is infinitely great, 
 since there is no number of years which is not ex- 
 ceeded by them all put together. Each, then, would 
 have to pay for a perpetuity, if the preceding fallacy 
 were admitted ; or all together would pay 25 x 101/._, 
 that is 2525/., more than 200 times the truth. It is 
 true that, applied to any actually existing law of life, 
 the incorrect notion cannot produce results so grossly 
 false as the preceding: but it is quite sufficient for 
 us to know that a rule is incorrect in its principle, to 
 make it necessary to apply correct reasoning before we 
 can attempt to say how far it is wrong. 
 
 The rule for calculating the real value of an annuity 
 is made up of a collection of individual cases,, not 
 more complicated than the following. A is to receive 
 one pound ten years hence, if a halfpenny which is 
 to be thrown up give a head; what is the present 
 worth of his chance ? If he were to receive the money 
 immediately, in case of winning, its value, on the prin- 
 ciples in chapter V., would be '51. ; but the present 
 value of II. deferred for ten years at 4 per cent, is 
 ?09/. : whence that of -51. similarly deferred is '3551. 
 This simple process contains the method of valuing 
 the sum which must be now paid down to secure the 
 tenth payment of an annuity to a life belonging to a 
 
190 ESSAY ON PROBABILITIES. 
 
 class of which just one half die in ten years. A 
 repetition of a similar process for every year in which 
 the annuity may become payable gives the present 
 values of the other payments. The sum of all the 
 present values of the different payments is the present 
 value of the whole annuity. Thus if we take a very 
 old life, and suppose that of 10 alive at the present 
 time there will be left at the end of successive years, 
 7, 5, 3, 0, there are three possible payments of the 
 annuity, and the chances of having to make them are 
 j-?,-, -, and -p () . But the first, if made at all, is not 
 made for a year, and the second and third are not made 
 for two and three years. If then a, b, and c be the 
 present values of II. to be received at the end of one, 
 two, and three years, the values of the several payments 
 are 1 7 fT of a, T 5 - of &, and -^ of c, the sum of which is 
 the value of an annuity of 11. on any one life of the kind 
 in question. 
 
 By the status of an annuity, I mean the state or 
 condition of things during the continuance of which 
 the annuity is to be paid. This status may be simple 
 or complicated : in the latter case the method of find- 
 ing the chances of its continuance or termination will 
 also be complicated ; but this does not affect the simple 
 rule by which those chances, when found, are made, in 
 conjunction with the rules of compound interest, to give 
 the value of the annuity. Thus the status in question 
 may be of a compound character, allowing of several 
 distinct changes. For instance, A is to enjoy an 
 annuity to the end of his life, unless B should die 
 before C, in which case it is to cease. This annuity will 
 be enjoyed as long as either of the following status exist. 
 
 A, B, and C all living. 
 
 A and B living, and C dead. 
 
 A living, and B and C dead, C having died first. 
 
 Tables of the values of annuities are constructed in 
 the case of single lives and two joint lives; that is to 
 say, the mere inspection of the tables will enable us to 
 
ON ANNUITIES. Ipl 
 
 answer the questions what is the value of an annuity 
 to be paid so long as A shall live,, and what is the 
 value of an annuity to he paid so long as A and B 
 shall both be ?live, but to cease when either of them 
 dies. The authorities upon the subject have been 
 already mentioned ; namely, Mr. Milne * for the Car- 
 lisle tables., and Dr. Price f and Mr. Morgan J for the 
 Northampton tables. Besides these we have the work 
 of Mr. Baily ,, now completely out of print; in con- 
 nexion with which must be mentioned the treatises || 
 on interest and on leases by the same author, which are 
 also out of print and scarce. The work of Mr. Davies ^[ 
 contains, in connexion with the Northampton tables, a 
 very considerable amount of results^ not elsewhere pub- 
 lished. The work of Mr. Baily on insurances has 
 lately been translated into French.** The results of 
 the Carlisle and Northampton tables have been lately 
 published on one large sheet, by Mr. M'Kean.ft All 
 the preceding are valuable works, and very much sur- 
 pass, in useful application, all that has been done in 
 any other country, or in all countries put together. 
 
 I now give such a brief abstract of the results of the 
 Carlisle and Northampton tables as is consistent with 
 the plan of this work, in regard to annuities upon 
 single lives, and upon two joint lives. 
 
 * A Treatise on the Valuation of Annuities and Assurances, &c. &c., by 
 Joshua Milne. London. Longman and Co. 1815. 
 
 t Observations on Reversionary Payments, &c. &c., by Richard Price, 
 D.D. F.R.S.; seventh edition, edited by William Morgan, F.R.S. London. 
 Cadell and Davies, 1812. 
 
 t The Principles and Doctrines of Assurances, Annuities, &c., by Wil- 
 liam Morgan, F.R.S. London. Longman and Co. 1821. 
 
 $ The Doctrine of Annuities and Insurances, &c., by Francis Baily. 
 London. Richardson, 1816. 
 
 The Doctrine of Interest and Annuities, &c., by Francis Baily Lon- 
 don. Richardson, 1808. 
 
 Tables for the purchasing and renewing of Leases, &c.,by Francis Baily 
 London. Richardson, 1807, (second edition). 
 
 If Tables of Life Contingencies, &c., by Griffith Davies. London. Long- 
 man and Co., and Richardson, 1825, 
 
 ** Theorie des Annuite? Viageres, &c., traduit de 1' Anglais par Alfred 
 de Courcy. Paris. Bachelier, 1836. 
 
 tf Practical Life Tables, by Alex. M'Kean. London. Richardson, Butter- 
 worth, &c., 1S37. The annuities on two lives are given for every difference 
 or age ; whereas, in the works of Messrs. Morgan and Milne, they are only 
 given for multiples of five years, and the rest must be supplied by inter- 
 polation. This sheet is a beautiful specimen of typography. 
 
192 
 
 ESSAY ON PROBABILITIES. 
 
 Values of an Annuity o 
 made a decimal) o 
 
 y or (with the last figure 
 upon a single Life. 
 
 
 Northampton. 
 
 Carlisle. 
 
 Age. 
 
 Age 
 
 3 p. c. 
 
 4 p. c. 
 
 5p.c. 
 
 6 p. c. 
 
 3 p. c. 
 
 4 p. c. 
 
 5 p. c. 
 
 6 p. c. 
 
 
 
 123 
 
 103 
 
 89 
 
 76 
 
 173 
 
 143 
 
 121 
 
 104 
 
 
 
 5 
 
 205 
 
 172 
 
 148 
 
 130 
 
 237 
 
 196 
 
 166 
 
 143 
 
 5 
 
 10 
 
 207 
 
 175 
 
 151 
 
 133 
 
 235 
 
 196 
 
 167 
 
 144 
 
 10 
 
 15 
 
 197 
 
 168 
 
 146 
 
 129 
 
 226 
 
 190 
 
 162 
 
 141 
 
 15 
 
 20 
 
 186 
 
 160 
 
 140 
 
 124 
 
 217 
 
 184 
 
 158 
 
 138 
 
 20 
 
 25 
 
 178 
 
 154 
 
 136 
 
 121 
 
 207 
 
 176 
 
 153 
 
 135 
 
 25 
 
 30 
 
 169 
 
 148 
 
 131 
 
 117 
 
 196 
 
 169 
 
 147 
 
 130 
 
 30 
 
 35 
 
 159 
 
 140 
 
 125 
 
 112 
 
 184 
 
 160 
 
 141 
 
 126 
 
 35 
 
 40 
 
 148 
 
 132 
 
 118 
 
 107 
 
 171 
 
 151 
 
 134 
 
 120 
 
 4O 
 
 45 
 
 137 
 
 123 
 
 111 
 
 101 
 
 159 
 
 141 
 
 126 
 
 114 
 
 45 
 
 50 
 
 124 
 
 113 
 
 103 
 
 94 
 
 143 
 
 129 
 
 117 
 
 106 
 
 50 
 
 55 
 
 112 
 
 102 
 
 94 
 
 87 
 
 124 
 
 113 
 
 103 
 
 95 
 
 55 
 
 60 
 
 98 
 
 90 
 
 84 
 
 78 
 
 105 
 
 97 
 
 89 
 
 83 
 
 60 
 
 65 
 
 83 
 
 78 
 
 73 
 
 68 
 
 89 
 
 83 
 
 78 
 
 73 
 
 65 
 
 70 
 
 67 
 
 64 
 
 60 
 
 57 
 
 71 
 
 67 
 
 63 
 
 60 
 
 70 
 
 75 
 
 52 
 
 50 
 
 47 
 
 45 
 
 55 
 
 52 
 
 50 
 
 48 
 
 75 
 
 80 
 
 38 
 
 36 
 
 35 
 
 34 
 
 44 
 
 42 
 
 40 
 
 39 
 
 80 
 
 85 
 
 26 
 
 25 
 
 25 
 
 24 
 
 32 
 
 31 
 
 30 
 
 29 
 
 85 
 
 90 
 
 18 
 
 18 
 
 17 
 
 17 
 
 25 
 
 24 
 
 23 
 
 23 
 
 90 
 
 95 
 
 2 
 
 2 
 
 2 
 
 2 
 
 28 
 
 27 
 
 26 
 
 25 
 
 95 
 
 100 
 
 
 
 
 
 
 
 
 
 17 
 
 17 
 
 16 
 
 16 
 
 100 
 
 According to this first table it appears,, for instance, 
 that at the Northampton rate of mortality, the value 
 of an annuity upon a life aged 45, at 3 per cent., is 
 13*7 years' purchase, or 131. 14s. Od. for an annuity of I/., 
 1371. for an annuity of Wl., and 13701. for one of 100/. 
 per annum. Or rather, since the table is correct within 
 one twentieth of a year's purchase, we should say that 
 the value of. such an annuity is between 13*65 and 
 13*75 years' purchase. The same remarks apply to 
 the table of annuities on joint lives. 
 
OX ANNUITIES. 
 
 393 
 
 Values of an annuity of a?10_, or (with the last figure 
 made a decimal) of 1, upon two joint lives. 
 
 
 Northampton. 
 
 Carlisle. 
 
 
 Ages. 
 
 
 
 Ages. 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 3 
 
 4 
 
 5 
 
 6 
 
 
 5 
 
 5 
 
 156 
 
 136 
 
 120 
 
 107 
 
 198 
 
 168 
 
 145 
 
 127 
 
 ~ 
 
 5 
 
 10 
 
 10 
 
 163 
 
 143 
 
 127 
 
 113 
 
 200 
 
 170 
 
 148 
 
 130 
 
 10 
 
 10 
 
 15 
 
 15 
 
 152 
 
 134 
 
 120 
 
 108 
 
 189 
 
 163 
 
 142 
 
 126 
 
 15 
 
 15 
 
 20 
 
 20 
 
 141 
 
 125 
 
 112 
 
 102 
 
 180 
 
 156 
 
 137 
 
 122 
 
 20 
 
 20 
 
 25 
 
 25 
 
 134 
 
 119 
 
 108 
 
 98 
 
 169 
 
 148 
 
 131 
 
 117 
 
 25 
 
 25 
 
 30 
 
 30 
 
 126 
 
 113 
 
 103 
 
 94 
 
 158 
 
 139 
 
 124 
 
 112 
 
 30 
 
 30 
 
 35 
 
 35 
 
 117 
 
 106 
 
 97 
 
 89 
 
 147 
 
 131 
 
 118 
 
 107 
 
 35 
 
 35 
 
 40 
 
 40 
 
 108 
 
 98 
 
 90 
 
 83 
 
 135 
 
 121 
 
 110 
 
 100 
 
 40 
 
 40 
 
 45 
 
 45 
 
 98 
 
 90 
 
 83 
 
 77 
 
 124 
 
 112 
 
 103 
 
 94 
 
 45 
 
 45 
 
 50 
 
 50 
 
 87 
 
 81 
 
 75 
 
 70 
 
 109 
 
 101 
 
 93 
 
 86 
 
 50 
 
 50 
 
 55 
 
 55 
 
 77 
 
 72 
 
 67 
 
 63 
 
 91 
 
 85 
 
 79 
 
 74 
 
 53 
 
 55 
 
 60 
 
 60 
 
 66 
 
 62 
 
 59 
 
 56 
 
 73 
 
 69 
 
 65 
 
 61 
 
 60 
 
 60 
 
 65 
 
 65 
 
 55 
 
 52 
 
 50 
 
 47 
 
 60 
 
 57 
 
 55 
 
 52 
 
 65 
 
 65 
 
 70 
 
 70 
 
 43 
 
 41 
 
 39 
 
 38 
 
 46 
 
 44 
 
 42 
 
 40 
 
 70 
 
 70 
 
 75 
 
 75 
 
 31 
 
 30 
 
 29 
 
 28 
 
 32 
 
 31 
 
 30 
 
 9 
 
 75 
 
 75 
 
 80 
 
 80 
 
 21 
 
 21 
 
 20 
 
 20 
 
 25 
 
 24 
 
 23 
 
 23 
 
 80 
 
 80 
 
 85 
 
 85 
 
 13 
 
 13 
 
 13 
 
 12 
 
 17 
 
 16 
 
 16 
 
 15 
 
 85 
 
 85 
 
 90 
 
 90 
 
 9 
 
 9 
 
 9 
 
 9 
 
 11 
 
 11 
 
 10 
 
 10 
 
 90 
 
 90 
 
 5 
 
 10 
 
 160 
 
 139 
 
 123 
 
 110 
 
 199 
 
 169 
 
 146 
 
 129 
 
 5 
 
 10 
 
 10 
 
 15 
 
 158 
 
 138 
 
 123 
 
 110 
 
 194 
 
 166 
 
 145 
 
 128 
 
 10 
 
 15 
 
 15 
 
 20 
 
 147 
 
 130 
 
 116 
 
 105 
 
 184 
 
 159 
 
 140 
 
 124 
 
 15 
 
 20 
 
 20 
 
 25 
 
 137 
 
 122 
 
 110 
 
 100 
 
 174 
 
 152 
 
 134 
 
 120 
 
 20 
 
 25 
 
 25 
 
 30 
 
 130 
 
 116 
 
 105 
 
 96 
 
 163 
 
 143 
 
 127 
 
 114 
 
 25 
 
 30 
 
 30 
 
 35 
 
 121 
 
 109 
 
 100 
 
 91 
 
 152 
 
 135 
 
 121 
 
 109 
 
 30 
 
 35 
 
 35 
 
 40 
 
 112 
 
 102 
 
 93 
 
 86 
 
 140 
 
 126 
 
 114 
 
 103 
 
 35 
 
 40 
 
 40 
 
 45 
 
 102 
 
 94 
 
 86. 
 
 80 
 
 129 
 
 116 
 
 106 
 
 97 
 
 40 
 
 45 
 
 45 
 
 50 
 
 92 
 
 85 
 
 79 
 
 74 
 
 116 
 
 106 
 
 97 
 
 90 
 
 45 
 
 50 
 
 50 
 
 55 
 
 82 
 
 76 
 
 71 
 
 67 
 
 99 
 
 92 
 
 85 
 
 80 
 
 50 
 
 55 
 
 55 
 
 60 
 
 71 
 
 67 
 
 63 
 
 59 
 
 81 
 
 76 
 
 71 
 
 67 
 
 55 
 
 60 
 
 60 
 
 65 
 
 60 
 
 57 
 
 54 
 
 51 
 
 66 
 
 62 
 
 59 
 
 56 
 
 60 
 
 65 
 
 65 
 
 70 
 
 48 
 
 46 
 
 44 
 
 42 
 
 52 
 
 50 
 
 47 
 
 45 
 
 65 
 
 70 
 
 70 
 
 75 
 
 36 
 
 35 
 
 33 
 
 32 
 
 38 
 
 37 
 
 35 
 
 34 
 
 70 
 
 75 
 
 75 
 
 80 
 
 25 
 
 24 
 
 24 
 
 23 
 
 28 
 
 27 
 
 26 
 
 25 
 
 75 
 
 80 
 
 80 
 
 85 
 
 16 
 
 16 
 
 16 
 
 15 
 
 20 
 
 19 
 
 19 
 
 19 
 
 80 
 
 85 
 
 85 
 
 90 
 
 11 
 
 11 
 
 10 
 
 10 
 
 13 
 
 13 
 
 13 
 
 13 
 
 85 
 ~ 
 
 90 
 
 5 
 
 IT 
 
 154 
 
 135 
 
 120 
 
 107 
 
 193 
 
 165 
 
 143 
 
 126 
 
 
 "IF 
 
 10 
 
 20 
 
 152 
 
 134 
 
 119 
 
 107 
 
 189 
 
 163 
 
 142 
 
 126 
 
 10 
 
 20 
 
 15 
 
 25 
 
 142 
 
 126 
 
 113 
 
 102 
 
 178 
 
 155 
 
 136 
 
 121 
 
 15 
 
 25 
 
 20 
 
 30 
 
 133 
 
 119 
 
 107 
 
 97 
 
 167 
 
 147 
 
 130 
 
 116 
 
 20 
 
 30 
 
 25 
 
 35 
 
 125 
 
 112 
 
 102 
 
 93 
 
 157 
 
 138 
 
 124 
 
 111 
 
 25 
 
 35 
 
 30 
 
 40 
 
 116 
 
 105 
 
 96 
 
 88 
 
 144 
 
 129 
 
 116 
 
 105 
 
 30 
 
 40 
 
 35 
 
 45 
 
 106 
 
 97 
 
 89 
 
 
 133 
 
 120 
 
 109 
 
 100 
 
 35 
 
 45 
 
 40 
 
 50 
 
 96 
 
 88 
 
 82 
 
 76 
 
 120 
 
 109 
 
 100 
 
 92 
 
 40 
 
 50 
 
 45 
 
 55 
 
 86 
 
 79 
 
 74 
 
 69 
 
 104 
 
 96 
 
 89 
 
 82 
 
 45 
 
 55 
 
 50 
 
 60 
 
 75 
 
 70 
 
 66 
 
 62 
 
 87 
 
 81 
 
 76 
 
 71 
 
 50 
 
 60 
 
 55 
 
 65 
 
 63 
 
 60 
 
 57 
 
 54 
 
 72 
 
 68 
 
 64 
 
 61 
 
 55 
 
 65 
 
 60 
 
 70 
 
 51 
 
 49 
 
 47 
 
 45 
 
 56 
 
 53 
 
 50 
 
 48 
 
 60 
 
 70 
 
 65 
 
 75 
 
 40 
 
 38 
 
 37 
 
 35 
 
 43 
 
 41 
 
 39 
 
 38 
 
 65 
 
 75 
 
 70 
 
 80 
 
 28 
 
 28 
 
 27 
 
 26 
 
 32 
 
 31 
 
 30 
 
 29 
 
 70 
 
 80 
 
 75 
 
 85 
 
 19 
 
 19 
 
 18 
 
 18 
 
 22 
 
 22 
 
 21 
 
 20 
 
 75 
 
 85 
 
 80 
 
 90 
 
 13 
 
 13 
 
 13 
 
 12 
 
 16 
 
 16 
 
 15 
 
 15 
 
 80 
 
 90 
 
194 
 
 ESSAY ON PROBABILITIES. 
 
 
 Northampton. 
 
 Carlisle. 
 
 
 Ages. 
 
 
 
 Ages. 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 3 
 
 4 
 
 5 
 
 6 
 
 
 5 
 
 20 
 
 148 
 
 130 
 
 116 
 
 104 
 
 187 
 
 161 
 
 140 
 
 124 
 
 5 
 
 20 
 
 10 
 
 25 
 
 147 
 
 130 
 
 116 
 
 105 
 
 182 
 
 158 
 
 139 
 
 123 
 
 10 
 
 25 
 
 15 
 
 30 
 
 137 
 
 122 
 
 110 
 
 100 
 
 171 
 
 149 
 
 132 
 
 118 
 
 15 
 
 30 
 
 20 
 
 35 
 
 127 
 
 114 
 
 104 
 
 95 
 
 160 
 
 141 
 
 126 
 
 113 
 
 20 
 
 35 
 
 25 
 
 40 
 
 119 
 
 107 
 
 98 
 
 90 
 
 148 
 
 132 
 
 119 
 
 107 
 
 25 
 
 40 
 
 30 
 
 45 
 
 109 
 
 100 
 
 91 
 
 84 
 
 137 
 
 123 
 
 111 
 
 101 
 
 30 
 
 45 
 
 35 
 
 50 
 
 99 
 
 91 
 
 84 
 
 78 
 
 123 
 
 112 
 
 102 
 
 94 
 
 35 
 
 50 
 
 40 
 
 55 
 
 89 
 
 82 
 
 77 
 
 71 
 
 107 
 
 98 
 
 90 
 
 84 
 
 40 
 
 55 
 
 45 
 
 60 
 
 78 
 
 73 
 
 68 
 
 64 
 
 91 
 
 84 
 
 78 
 
 73 
 
 45 
 
 60 
 
 50 
 
 65 
 
 66 
 
 62 
 
 59 
 
 56 
 
 77 
 
 72 
 
 68 
 
 64 
 
 50 
 
 65 
 
 55 
 
 70 
 
 54 
 
 51 
 
 49 
 
 4? 
 
 60 
 
 57 
 
 54 
 
 52 
 
 55 
 
 70 
 
 60 
 
 75 
 
 42 
 
 40 
 
 39 
 
 37 
 
 45 
 
 43 
 
 41 
 
 40 
 
 60 
 
 75 
 
 65 
 
 80 
 
 31 
 
 30 
 
 29 
 
 28 
 
 35 
 
 34 
 
 33 
 
 32 
 
 65 
 
 80 
 
 70 
 
 85 
 
 21 
 
 20 
 
 20 
 
 19 
 
 25 
 
 24 
 
 24 
 
 23 
 
 70 
 
 85 
 
 75 
 
 90 
 
 14 
 
 14 
 
 14 
 
 14 
 
 18 
 
 17 
 
 17 
 
 16 
 
 75 
 
 90 
 
 5 
 
 25 
 
 143 
 
 126 
 
 113 
 
 102 
 
 180 
 
 156 
 
 137 
 
 121 
 
 5 
 
 25 
 
 10 
 
 30 
 
 142 
 
 126 
 
 113 
 
 102 
 
 174 
 
 152 
 
 134 
 
 120 
 
 10 
 
 30 
 
 15 
 
 35 
 
 132 
 
 118 
 
 107 
 
 97 
 
 163 
 
 143 
 
 128 
 
 115 
 
 15 
 
 35 
 
 20 
 
 4<) 
 
 121 
 
 109 
 
 99 
 
 91 
 
 151 
 
 134 
 
 121 
 
 109 
 
 20 
 
 40 
 
 25 
 
 45 
 
 112 
 
 102 
 
 93 
 
 86 
 
 140 
 
 125 
 
 113 
 
 103 
 
 25 
 
 45 
 
 30 
 
 50 
 
 102 
 
 93 
 
 86 
 
 80 
 
 126 
 
 114 
 
 104 
 
 96 
 
 30 
 
 50 
 
 35 
 
 55 
 
 91 
 
 84 
 
 78 
 
 73 
 
 109 
 
 100 
 
 92 
 
 86 
 
 35 
 
 55 
 
 40 
 
 60 
 
 80 
 
 75 
 
 70 
 
 66 
 
 92 
 
 86 
 
 80 
 
 74 
 
 40 
 
 60 
 
 45 
 
 65 
 
 69 
 
 65 
 
 61 
 
 58 
 
 79 
 
 74 
 
 70 
 
 66 
 
 45 
 
 65 
 
 50 
 
 70 
 
 56 
 
 53 
 
 51 
 
 48 
 
 63 
 
 60 
 
 57 
 
 54 
 
 50 
 
 70 
 
 55 
 
 75 
 
 44 
 
 42 
 
 40 
 
 39 
 
 48 
 
 46 
 
 44 
 
 42 
 
 55 
 
 75 
 
 60 
 
 80 
 
 32 
 
 31 
 
 30 
 
 29 
 
 37 
 
 36 
 
 34 
 
 33 
 
 60 
 
 80 
 
 65 
 
 85 
 
 22 
 
 22 
 
 21 
 
 21 
 
 27 
 
 26 
 
 26 
 
 25 
 
 65 
 
 85 
 
 70 
 
 90 
 
 15 
 
 15 
 
 15 
 
 15 
 
 20 
 
 19 
 
 19 
 
 18 
 
 70 
 
 90 
 
 5 
 
 30 
 
 138 
 
 122 
 
 110 
 
 99 
 
 172 
 
 150 
 
 132 
 
 118 
 
 5 
 
 30 
 
 10 
 
 35 
 
 135 
 
 121 
 
 109 
 
 99 
 
 166 
 
 146 
 
 130 
 
 116 
 
 10 
 
 35 
 
 15 
 
 40 
 
 125 
 
 112 
 
 102 
 
 93 
 
 153 
 
 136 
 
 122 
 
 110 
 
 15 
 
 40 
 
 20 
 
 45 
 
 114 
 
 103 
 
 94 
 
 87 
 
 142 
 
 127 
 
 115 
 
 105 
 
 20 
 
 45 
 
 25 
 
 50 
 
 104 
 
 95 
 
 87 
 
 81 
 
 128 
 
 116 
 
 106 
 
 97 
 
 25 
 
 50 
 
 30 
 
 55 
 
 93 
 
 86 
 
 80 
 
 75 
 
 111 
 
 102 
 
 94 
 
 87 
 
 30 
 
 55 
 
 35 
 
 60 
 
 82 
 
 77 
 
 72 
 
 67 
 
 94 
 
 87 
 
 81 
 
 76 
 
 35 
 
 60 
 
 40 
 
 65 
 
 70 
 
 66 
 
 62 
 
 59 
 
 80 
 
 75 
 
 70 
 
 66 
 
 40 
 
 65 
 
 45 
 
 70 
 
 57 
 
 55 
 
 52 
 
 50 
 
 65 
 
 61 
 
 58 
 
 55 
 
 45 
 
 70 
 
 50 
 
 75 
 
 45 
 
 43 
 
 41 
 
 40 
 
 50 
 
 48 
 
 46 
 
 44 
 
 50 
 
 75 
 
 55 
 
 80 
 
 33 
 
 32 
 
 31 
 
 30 
 
 39 
 
 38 
 
 36 
 
 35 
 
 55 
 
 80 
 
 60 
 
 85 
 
 23 
 
 22 
 
 22 
 
 21 
 
 28 
 
 27 
 
 26 
 
 26 
 
 60 
 
 85 
 
 65 
 
 90 
 
 ~ 
 
 16 
 
 16 
 
 15 
 
 15 
 
 21 
 
 21 
 
 20 
 
 20 
 
 65 
 
 90 
 
 5 
 
 
 131 
 
 117 
 
 106 
 
 96 
 
 164 
 
 144 
 
 128 
 
 114 
 
 5 
 
 35 
 
 10 
 
 40 
 
 128 
 
 115 
 
 104 
 
 95 
 
 156 
 
 138 
 
 124 
 
 112 
 
 10 
 
 40 
 
 15 
 
 45 
 
 117 
 
 106 
 
 97 
 
 89 
 
 144 
 
 129 
 
 116 
 
 106 
 
 15 
 
 45 
 
 20 
 
 50 
 
 105 
 
 96 
 
 89 
 
 82 
 
 130 
 
 118 
 
 107 
 
 98 
 
 20 
 
 50 
 
 25 
 
 55 
 
 95 
 
 88 
 
 81 
 
 76 
 
 113 
 
 103 
 
 95 
 
 88 
 
 25 
 
 55 
 
 30 
 
 60 
 
 84 
 
 78 
 
 73 
 
 68 
 
 95 
 
 88 
 
 82 
 
 76 
 
 30 
 
 60 
 
 35 
 
 65 
 
 72 
 
 67 
 
 64 
 
 60 
 
 81 
 
 76 
 
 71 
 
 67 
 
 35 
 
 65 
 
 40 
 
 70 
 
 59 
 
 56 
 
 53 
 
 50 
 
 65 
 
 62 
 
 58 
 
 55 
 
 40 
 
 70 
 
 45 
 
 75 
 
 46 
 
 44 
 
 42 
 
 40 
 
 51 
 
 49 
 
 46 
 
 44 
 
 45 
 
 75 
 
 50 
 
 80 
 
 34 
 
 32 
 
 31 
 
 30 
 
 41 
 
 39 
 
 37 
 
 36 
 
 50 
 
 80 
 
 55 
 
 85 
 
 23 
 
 23 
 
 22 
 
 22 
 
 30 
 
 29 
 
 28 
 
 27 
 
 55 
 
 85 
 
 60 
 
 PO 
 
 16 
 
 16 
 
 16 
 
 15 
 
 22 
 
 21 
 
 21 
 
 20 
 
 60 
 
 90 
 
ON ANNUITIES. 
 
 195 
 
 
 Northampton. 
 
 Carlisle. 
 
 
 Ages. 
 
 
 
 Ages. 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 3 
 
 4 
 
 5 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 40 
 
 124 
 
 112 
 
 101 
 
 92 
 
 154 
 
 136 
 
 122 
 
 110 
 
 ~T 
 
 40 
 
 10 
 
 45 
 
 120 
 
 109 
 
 99 
 
 91 
 
 146 
 
 131 
 
 118 
 
 107 
 
 10 
 
 45 
 
 15 
 
 50 
 
 108 
 
 99 
 
 91 
 
 84 
 
 131 
 
 119 
 
 108 
 
 99 
 
 15 
 
 50 
 
 20 
 
 55 
 
 96 
 
 89 
 
 82 
 
 76 
 
 114 
 
 105 
 
 96 
 
 89 
 
 20 
 
 55 
 
 25 
 
 60 
 
 85 
 
 79 
 
 74 
 
 69 
 
 97 
 
 89 
 
 83 
 
 77 
 
 25 
 
 60 
 
 30 
 
 65 
 
 73 
 
 68 
 
 64 
 
 61 
 
 82 
 
 77 
 
 72 
 
 68 
 
 30 
 
 65 
 
 55 
 
 70 
 
 60 
 
 57 
 
 54 
 
 51 
 
 66 
 
 62 
 
 59 
 
 56 
 
 35 
 
 70 
 
 40 
 
 75 
 
 47 
 
 45 
 
 43 
 
 41 
 
 51 
 
 49 
 
 47 
 
 44 
 
 40 
 
 75 
 
 45 
 
 80 
 
 34 
 
 33 
 
 32 
 
 31 
 
 41 
 
 39 
 
 38 
 
 36 
 
 45 
 
 80 
 
 50 
 
 85 
 
 24 
 
 23 
 
 23 
 
 22 
 
 30 
 
 29 
 
 28 
 
 28 
 
 50 
 
 85 
 
 55 
 
 90 
 
 17 
 
 16 
 
 16 
 
 16 
 
 23 
 
 22 
 
 22 
 
 21 
 
 55 
 
 90 
 
 5 
 
 45 
 
 116 
 
 105 
 
 96 
 
 88 
 
 144 
 
 129 
 
 116 
 
 105 
 
 "~T 
 
 ~45~ 
 
 10 
 
 50 
 
 110 
 
 101 
 
 93 
 
 85 
 
 133 
 
 120 
 
 110 
 
 100 
 
 10 
 
 50 
 
 15 
 
 55 
 
 99 
 
 91 
 
 84 
 
 78 
 
 115 
 
 105 
 
 97 
 
 90 
 
 15 
 
 55 
 
 20 
 
 60 
 
 86 
 
 80 
 
 75 
 
 70 
 
 98 
 
 90 
 
 84 
 
 78 
 
 20 
 
 60 
 
 25 
 
 65 
 
 74 
 
 69 
 
 65 
 
 62 
 
 83 
 
 78 
 
 73 
 
 69 
 
 25 
 
 65 
 
 SO 
 
 70 
 
 60 
 
 57 
 
 54 
 
 52 
 
 67 
 
 63 
 
 60 
 
 56 
 
 30 
 
 70 
 
 35 
 
 75 
 
 47 
 
 45 
 
 43 
 
 42 
 
 52 
 
 49 
 
 47 
 
 45 
 
 35 
 
 75 
 
 40 
 
 80 
 
 35 
 
 33 
 
 32 
 
 31 
 
 41 
 
 39 
 
 38 
 
 36 
 
 40 
 
 80 
 
 45 
 
 85 
 
 24 
 
 24 
 
 23 
 
 22 
 
 31 
 
 30 
 
 29 
 
 28 
 
 45 
 
 85 
 
 50 
 
 90 
 
 17 
 
 17 
 
 16 
 
 16 
 
 24 
 
 23 
 
 22 
 
 22 
 
 50 
 
 90 
 
 5 
 
 50 
 
 107 
 
 97 
 
 89 
 
 82 
 
 131 
 
 118 
 
 108 
 
 99 
 
 5 
 
 ~50~ 
 
 10 
 
 55 
 
 101 
 
 93 
 
 86 
 
 80 
 
 117 
 
 107 
 
 98 
 
 90 
 
 10 
 
 55 
 
 15 
 
 60 
 
 88 
 
 82 
 
 76 
 
 71 
 
 99 
 
 91 
 
 84 
 
 79 
 
 15 
 
 60 
 
 20 
 
 65 
 
 74 
 
 70 
 
 66 
 
 62 
 
 84 
 
 79 
 
 74 
 
 69 
 
 20 
 
 65 
 
 25 
 
 70 
 
 61 
 
 58 
 
 55 
 
 52 
 
 67 
 
 61 
 
 60 
 
 57 
 
 25 
 
 70 
 
 30 
 
 75 
 
 48 
 
 46 
 
 44 
 
 42 
 
 52 
 
 50 
 
 47 
 
 45 
 
 30 
 
 75 
 
 35 
 
 80 
 
 35 
 
 34 
 
 33 
 
 32 
 
 41 
 
 40 
 
 38 
 
 37 
 
 35 
 
 80 
 
 40 
 
 85 
 
 24 
 
 24 
 
 23 
 
 23 
 
 31 
 
 30 
 
 29 
 
 28 
 
 40 
 
 85 
 
 45 
 
 90 
 
 17 
 
 17 
 
 16 
 
 16 
 
 24 
 
 23 
 
 22 
 
 22 
 
 45 
 
 90 
 
 ~ 
 
 ~5*T 
 
 97 
 
 89 
 
 83 
 
 , 77 
 
 115 
 
 105 
 
 96 
 
 89 
 
 5 
 
 55 
 
 10 
 
 60 
 
 90 
 
 83 
 
 78 
 
 73 
 
 100 
 
 92 
 
 85 
 
 79 
 
 10 
 
 60 
 
 15 
 
 65 
 
 76 
 
 71 
 
 67 
 
 63 
 
 85 
 
 79 
 
 74 
 
 70 
 
 15 
 
 65 
 
 20 
 
 70 
 
 61 
 
 58 
 
 55 
 
 53 
 
 68 
 
 64 
 
 61 
 
 57 
 
 20 
 
 70 
 
 25 
 
 75 
 
 48 
 
 46 
 
 44 
 
 42 
 
 53 
 
 50 
 
 48 
 
 46 
 
 25 
 
 75 
 
 30 
 
 80 
 
 35 
 
 34 
 
 33 
 
 32 
 
 42 
 
 40 
 
 38 
 
 37 
 
 30 
 
 80 
 
 35 
 
 85 
 
 25 
 
 24 
 
 23 
 
 23 
 
 31 
 
 30 
 
 29 
 
 28 
 
 35 
 
 85 
 
 40 
 
 90 
 
 17 
 
 17 
 
 16 
 
 16 
 
 24 
 
 23 
 
 22 
 
 22 
 
 40 
 
 90 
 
 5 
 
 60 
 
 86 
 
 80 
 
 75 
 
 70 
 
 98 
 
 90 
 
 84 
 
 79 
 
 5 
 
 60 
 
 10 
 
 65 
 
 77 
 
 72 
 
 68 
 
 64 
 
 85 
 
 80 
 
 75 
 
 70 
 
 10 
 
 65 
 
 15 
 
 70 
 
 63 
 
 59 
 
 56 
 
 54 
 
 68 
 
 64 
 
 61 
 
 58 
 
 15 
 
 70 
 
 20 
 
 75 
 
 48 
 
 46 
 
 44 
 
 42 
 
 53 
 
 50 
 
 4 
 
 46 
 
 20 
 
 75 
 
 25 
 
 80 
 
 36 
 
 34 
 
 33 
 
 32 
 
 42 
 
 40 
 
 39 
 
 37 
 
 25 
 
 80 
 
 30 
 
 85 
 
 25 
 
 24 
 
 23 
 
 23 
 
 31 
 
 30 
 
 29 
 
 28 
 
 30 
 
 85 
 
 35 
 
 90 
 
 17 
 
 17 
 
 17 
 
 16 
 
 24 
 
 23 
 
 23 
 
 22 
 
 35 
 
 90 
 
 5 
 
 65 
 
 74 
 
 70 
 
 65 
 
 62 
 
 84 
 
 78 
 
 73 
 
 69 
 
 5 
 
 65 
 
 10 
 
 70 
 
 63 
 
 60 
 
 57 
 
 54 
 
 69 
 
 65 
 
 61 
 
 58 
 
 10 
 
 70 
 
 15 
 
 75 
 
 49 
 
 47 
 
 45 
 
 43 
 
 53 
 
 51 
 
 48 
 
 46 
 
 15 
 
 75 
 
 20 
 
 80 
 
 36 
 
 34 
 
 33 
 
 32 
 
 42 
 
 41 
 
 39 
 
 37 
 
 20 
 
 80 
 
 25 
 
 85 
 
 25 
 
 24 
 
 24 
 
 23 
 
 31 
 
 30 
 
 29 
 
 28 
 
 25 
 
 85 
 
 30 
 
 90 
 
 17 
 
 17 17 
 
 16 
 
 24 
 
 23 
 
 23 
 
 22 
 
 30 
 
 90 
 
 o 2 
 
196 
 
 ESSAY ON PROBABILITIES. 
 
 Ag 
 
 5 
 
 10 
 15 
 20 
 25 
 
 es. 
 
 "TO" 
 
 75 
 80 
 85 
 90 
 
 Northampton. 
 
 Carlisle. 
 
 Ag 
 
 5 
 10 
 15 
 
 20 
 25 
 
 es. 
 
 "TO" 
 
 75 
 80 
 85 
 90 
 
 3 
 
 4 
 
 5 
 
 6 
 
 3 
 
 4 
 
 5 
 
 6 
 
 61 
 50 
 36 
 25 
 17 
 
 58 
 47 
 35 
 24 
 17 
 
 55 
 45 
 34 
 
 24 
 17 
 
 52 
 44 
 33 
 23 
 16 
 
 67 
 54 
 
 42 
 31 
 
 24 
 
 64 
 51 
 41 
 
 30 
 23 
 
 60 
 49 
 39 
 29 
 23 
 
 57 
 46 
 38 
 28 
 22 
 
 5 
 
 10 
 15 
 20 
 
 75 
 
 80 
 85 
 90 
 
 48 
 36 
 25 
 17 
 
 46 
 35 
 25 
 17 
 
 44 
 34 
 24 
 17 
 
 42 
 93 
 23 
 16 
 
 52 
 43 
 31 
 24 
 
 50 
 
 41 
 30 
 24 
 
 48 
 39 
 29 
 
 23 
 
 45 
 
 38 
 28 
 
 22 
 
 5 
 10 
 15 
 
 20 
 
 75 
 80 
 85 
 90 
 
 5 
 10 
 15 
 
 80 
 85 
 90 
 
 IF 
 90 
 
 
 
 
 
 
 
 42 
 32 
 24 
 
 40 
 31 
 
 24 
 
 38 
 30 
 23 
 
 37 
 29 
 22 
 
 5 
 
 10 
 15 
 
 80 
 85 
 90 
 
 IF 
 90 
 
 5 
 
 10 
 
 
 
 31 
 
 25 
 
 30 
 24 
 
 29 
 23 
 
 28 
 
 22 
 
 5 
 10 
 
 The preceding table contains the values of annuities 
 upon two lives, for all ages which are multiples of 5. 
 Thus, for the ages 25 and 40, look to that part of the 
 table in which the ages differ by 15 years, and there, 
 opposite to 25 40, will be found, under 4 per cent., 
 132 in the Carlisle table and 107 in the Northampton. 
 That is, money making 4 per cent., an annuity of 10/., 
 which is to continue as long as lives of 25 and 40 are 
 both in being, is worth something between 1 311. 10s. Od. 
 and 132/. 10s. Od. according to the Carlisle tables, and 
 something between 1067. 10s, Od. and 107/. 10s. Od. 
 according to the Northampton tables. 
 
 When the two required lives have ages which do 
 not end with the figures or 5, proceed as follows: Let 
 the value of annuity be required on joint lives of 38 
 and 47, (Carlisle tables at 3 per cent.). First take 35 and 
 45, and 40 and 45, and between the corresponding annuities 
 insert such a mean as would represent 38 and 45 upon the 
 supposition uniformly diminishing values. Then be- 
 tween 35 and 50 and 40 and 50 insert such a mean 
 as answers to 38 and 50. Having then 38 and 45 
 and 38 and 50, find such a mean as answers to 38 and 
 47. This process, which will be intelligible to a 
 reader who has practised similar ones before, will only 
 
ON ANNUITIES. 197 
 
 "be comprehended (if at all) by others from the 
 example. 
 
 ' 35 and 45 133 35 and 50 123 
 
 40 and 45 129 40 and 50 120 
 
 4 3 
 
 3 3 
 
 5) 12 5) 9 
 
 2 2 
 
 38 and 45 131 38 and 50 121 
 38 and 50 121 
 
 10 
 
 2 38 and 47 127 Ans. 
 
 5) 20 
 
 Before proceeding further, I shall describe the 
 notation of which I intend to make use. It was not 
 the practice of the earlier writers to invent any dis- 
 tinctive notation of different contingencies, the first 
 attempt at which is found in the work of Mr. Baily. 
 Here, however, it was not carried to the full extent, and 
 Mr. Milne endeavoured to organise a system which 
 should take in every case, in which he succeeded perfectly 
 as far as distinct representation of all the cases which 
 occur. His symbols, however, are complicated and 
 strange, though I am clearly of opinion that they are 
 much preferable to the attempt to dispense with 
 notation altogether. The new principle which the 
 notation I now propose involves, lies in the treatment 
 of terms of years certain as lives not subject to con- 
 tingencies. Thus, if AB represent an annuity on the 
 joint lives of A and B, meaning that it is to cease 
 when either A or B dies, then B may represent an 
 annuity on the joint term of t years and B's life, to 
 cease with the first which expires ; or what would be 
 o 3 
 
198 ESSAY ON PROBABILITIES. 
 
 called a temporary annuity on the life of B to last t 
 years, provided B should live t years. 
 
 1. Any simple status on the existence or termination 
 of which a benefit depends, is denoted by juxtaposition 
 of large and small letters, the large letters denoting 
 separately the values of annuities on given lives or status, 
 the small letters (or rather certain small letters, m, n, t, 
 for the most part) denoting given terms of years. 
 Thus A B C t is in existence as long as t years last, 
 provided A, B, and C (or the persons on whom 
 annuities now granted have these values), remain alive 
 all the time. 
 
 2. A compound status, or one which exists as long 
 as either of two or more status remain, is denoted by 
 colons placed between the symbols of the simple status ; 
 thus A : B : t is in existence as long as A, or B, or t 
 years, any or all, are in existence. The symbol is un- 
 meaning between two certainties : thus, n: n + t is 
 
 3. A bar placed over two status indicates that the 
 one is to succeed the other and that the compound 
 symbol denotes a status in being as long as the one, or the 
 
 other after it shall exist : thus A P denotes a status 
 which remains in being during the life of A, and also 
 during a life to be named at the end of the year in 
 which A dies, having then the value P. If there be 
 occasion, a thicker bar, or one with an accent, or a 
 double bar, may be used where there are two successions 
 involved, between which it is necessary to distinguish. 
 
 4. The symbol | in all cases gives the whole symbol 
 the meaning of the present value of a benefit to be re- 
 ceived, and a figure attached, as in | 4 , denotes'the rate 
 per cent, at which the value is to be calculated. This 
 benefit is always 11. at each one or more payments. The 
 description of the status which must end before the benefit 
 begins will be found on the left of | ; and the description 
 of the status during which the annual payments of the 
 said benefit are to last will be found on the right. It 
 is further to be understood that the first payment made 
 
ON ANNUITIES. 1^9 
 
 under A|B will take place at the end of the year in 
 which A drops, provided B be then in existence : thus 
 
 A|l denotes * the present value t of an annuity of I/., 
 the first (and only) payment of which is to be made at 
 the end of the year in which A drops ; while Ajl denotes 
 the present value of II. , to be paid in a year from this 
 time, if A be then dead. 
 
 5. The last moment of a term certain is a part of 
 that term, unless the contrary be expressed by symbols: 
 
 thus 6|1 refers to a pound payable at the end of seven 
 years, and is 6|7 But when the last moment of a 
 term is considered as having followed the end of the 
 term, a small hyphen (considered as an abridged nega- 
 tive sign) is placed after the term in question : thus, 
 
 t-\n denotes an annuity of n payments, the first of 
 which is to be made at the end of t years ; and is the 
 
 same with t l|n. 
 
 6. The absence of symbols on the left of | indicates 
 that the first payment takes place in a year from the 
 present time; but -| indicates an annuity now due. 
 The absence of symbols on the right of | indicates 
 a perpetuity in reversion ; and | itself indicates a perpe- 
 tuity, the first payment of which is to be made in a 
 year ; while -| indicates a perpetuity now due. 
 
 7. Dots between two symbols of status indicate 
 that the joint status shall be held to exist throughout 
 the year in which the first is determined, provided 
 the second remain at the end of the year; and 
 dots placed under several status denote that the 
 succeeding benefit is not due unless all those status 
 shall drop in the same year : thus |A...B denotes an 
 annuity on the joint lives of A and B, payable also 
 at the end of the year in which A dies, if B be then 
 
 alive ; and A : B|l is the value of I/, at the end of the 
 life of the longest survivor of A and B, provided they 
 
 * One bar may be omitted in very simple cases. 
 
 f Remember particularly that in A 1 1, 1 means one year, not one pound. 
 
 o 4 
 
200 ESSAY ON PROBABILITIES. 
 
 both die in the same year; also |AB... denotes the 
 value of an annuity on the joint lives to be paid in 
 addition at the end of the year in which the joint 
 existence fails. 
 
 8. When the condition is that a given status shall be 
 in existence at the moment in which another status 
 drops (whether the first last to the end of the year or 
 
 not), single dots are placed over the two : thus A|l B 
 means the present value of II., to be received at the end 
 of the year in which A dies, provided B be alive at that 
 
 moment; while A|lB means the same, provided B be 
 alive when the payment is to be made. 
 
 9. When it is a condition that deaths are to happen 
 in a specified order, it must be represented by writing 
 small figures under the status. Thus, A : B|C means 
 
 i 2 
 
 an annuity on the life of C, to begin at the end 
 of the year in which B dies, provided A have died 
 
 before B ; and A : (BC)|1 denotes the present value of 
 
 I/, payable at the end of the year in which the longest 
 of the two status A and BC drops, provided that the 
 status BC is determined by the death of B. 
 
 10. When the joint existence of one number of lives, 
 out of a larger number, is a condition, a figure may be 
 annexed as follows : ( A B C D) indicates a status which 
 exists as long as any two out of the four are alive. 
 
 1 ] . The double sign || indicates the premium which 
 is to be paid during the continuance of the status on 
 the left, in consideration of the deferred benefit described 
 on the right ; premium being always interpreted as an 
 annuity due. And where a specific event, as distin- 
 guished from the duration of a status, is a condition, the 
 premium is to be held payable as long as any status exists 
 out of which that event may happen. Thus A : B||C 
 
 1 2 
 
 denotes the premium which should be paid as long as 
 A lives with B, or B after A, to secure an annuity to 
 C when both are dead, provided A die first. 
 
ON ANNUITIES. 
 
 201 
 
 I now proceed to some further instances : 
 
 | n means the present value of an annuity of IL to last 
 n years. 
 
 m-\n the present value of an annuity of I/., which is to 
 commence payment at the end of m years, and then 
 to last n 1 years, or n I more payments. 
 
 n\ the present value of a perpetuity of II. , com- 
 mencing at the end of n years, or first paid at the 
 end of n -f 1 years. 
 
 n-\ the present value of a perpetuity, first paid at 
 the end of n years. 
 
 | A the present value of an annuity on the life of A. 
 
 JAB the present value of an annuity on the joint 
 lives of A and B, to cease with either. 
 
 t \ A the present value of an annuity on the life of A, 
 to begin in t years ; that is, the first payment to 
 be made at the end of t -j- 1 years, if A should then 
 be alive. 
 
 \At the present value of an annuity on A's life, or 
 t years, whichever drops first. 
 
 A | the present value of U. for ever, to be first received 
 at the end of the year in which A dies. 
 
 AB| the present value of IL for ever, to be first re- 
 ceived at the end of the first year in which A or B 
 dies. 
 
 A 1 1 the value of an annuity for t years, payment to 
 begin at the end of the year in which A dies. 
 
 AB|C the present value of an annuity which begins 
 payment at the end of the year in which either 
 A or B dies (the first), provided C be then alive, 
 and which continues during the life of C. 
 
 | A : B signifies the present value of an annuity which 
 is to be paid as long as either A or B is alive. 
 
 AB : C|D : E the present value of an annuity which is 
 not to be paid as long as A and B are both alive, 
 nor as long as C is alive, but which begins when 
 the joint existence of A and B, and that of C, are 
 both terminated; and continues as long as either 
 D or E are alive. 
 
202 ESSAY ON PROBABILITIES. 
 
 Brackets, as distinguished from colons,, will serve the 
 same purpose as in algebra, namely,, to give compound 
 terms the meaning of single ones. Thus, 
 
 (AB) C|D : E denotes the last-mentioned annuity, 
 on the supposition that payment is to begin when either 
 of two events happens, the failure of the joint existence 
 of A and B, or that of C. 
 
 AB|A : B the present value of an annuity to begin 
 when one of the two, A and B, dies, and to continue 
 during the life of die survivor. There is in this par- 
 ticular case the expression of an event which cannot 
 happen ; for if B die first, it is only A who can receive 
 the annuity. Thus B|B is an expression for nothing ; 
 for the present value of an annuity on the life of B, to 
 begin at the death of B, is nothing. In the expression 
 A B | A : B, part is nothing, and the rest has a value. 
 
 A 1 1 is the present value of 1 1. to be received at the 
 end of the year in which A dies; -|l is the present 
 
 value of I/, due t years hence; AB|lC is the pre- 
 sent value of an annuity which, commencing with the 
 first death out of the two, A and B, lasts till either one 
 payment, or the life of C drops: that is to say, the 
 value of I/, to be received at the end of the year in 
 which the joint existence of A and B fails, provided C 
 be then alive. 
 
 Let a colon placed after the final letter denote that 
 a perpetuity is one of the status during which the 
 annuity is to last. Thus, 
 
 A|(C:) denotes the present value of an annuity to 
 last for ever, after the death of A. The symbol denotes 
 that C being alive at the time of the first payment is a 
 necessary condition. This being satisfied, the longest 
 of the two, C, or a perpetuity, is of course a perpetuity. 
 
 The presence of the colon always indicates the 
 longest of the two status, and when the colon is a final 
 symbol, one of the status is an infinite number of years, 
 or a perpetuity. 
 
 A! : B 2 |C denotes the present value of an annuity 
 
ON ANNUITIES. 
 
 203 
 
 which is to be paid during the life of C, after the 
 deaths of A and B, if A die before B. 
 
 A : B : Cjl E is the present value of 11. to be paid 
 
 at the end of the year in which the last of the lives, 
 A, B, C, drops,, on condition that B shall have died second 
 or jhircL, and that E shall be alive. 
 
 A|P denotes the present value of an annuity which 
 is to begin payment at the end of the year in which A 
 dies, and to last during the life which shall then have 
 the value P. If there be several conditions, put a symbol 
 over the status which ends and before the one which 
 begins. Thus, 
 
 | ABC : P denotes the value of an annuity which is 
 to be paid during the joint existence of A, B, and C, 
 and further during that of a life which is to be 
 nominated (then having the value P) at the end of 
 the year in which either of the three drops : while 
 ABC|P denotes that part of the annuity which is paid 
 during the life of P. An author's interest in his 
 works, which is now denoted by | A : 28 was proposed, 
 in a bill lately before the House of Commons, to be 
 changed into 'A : 60. 
 
 The preceding notation will admit of almost any 
 degree of extension, and will be found perfectly capable 
 of expressing any case which now occurs in practice. 
 It must receive some generalisation before it can be 
 applied to an indefinite number of lives, in the manner 
 of Mr. Milne. But since it very rarely happens that 
 more than three lives occur in a practical question, I 
 shall leave farther extension to those who may find the 
 want of it. I shall merely now add, that any case 
 must admit of expression by means of a notation which 
 provides for the conditions under which the benefit 
 begins, the number of years which it is to last, and the 
 conditions of discontinuance : and also that in problems 
 of any degree of complexity, the invention of the 
 notation will be a useful preliminary to the actual solu- 
 
204 ESSAY ON PROBABILITIES. 
 
 tion of the problem. Thus, suppose it required to 
 represent the value of an annuity which is to continue 
 as long as any two of the three, A, B, and C, are alive. 
 This must be done thus : 
 
 | AB: AC: BC 
 
 This might be abbreviated into | (ABC) , or any such 
 symbol; which, however, I should recommend to no 
 one who is not very familiar with the developed form. 
 
 A method of making the notation of chances analogous 
 to that of annuities was devised by Mr. Milne, which is 
 much too ingenious and efficient to allow of its being 
 dispensed with in any future system. To adapt the 
 principle of this connexion to the system which I have 
 proposed, let n^a express the chance that A shall be 
 alive in n years, in which the smah 1 letters answering to 
 the capitals which denote lives refer to the chances 
 of those lives, and certain letters m, n, t (or more 
 if necessary), are reserved to signify terms of years. 
 Then all which precedes the sign t (or any other which 
 may be preferred), refers to lives or terms expired, and 
 those which follow the sign, to lives or terms which 
 are to be then in existence. Where no given term of 
 years is included, the chance must be understood to 
 refer to the whole continuance of all the status mentioned. 
 Thus, 
 
 t& is the chance that A shall die before B. 
 
 a wt6 is the chance that one of the two, A or n years, 
 shall fail before B. 
 
 a : n^b is the chance that both A and n years shall 
 fail before B ; that is, that B shall outlive A, and also 
 live more than n years. 
 
 a^n the chance that A shall not survive n years. 
 
 a : n^b the chance that A shall die in n years, and 
 
 that B shall outlive that term. 
 
 e chance that the joint existence of A and 
 B (or both A and B) shall outlast n years. 
 
 n: a 4- (71 -fl) the chance that A shall die in the 
 
 1 2 
 
 (n -}- 1 )th year from this time. 
 
ON ANNUITIES. 205 
 
 The tables, of which an abstract has been given, 
 will enable us to find |A and |AB, the values of an 
 annuity on one or two lives, by simple inspection or 
 easy interpolation, but not |ABC, the value of an 
 annuity on three joint lives. A sufficient approximation 
 to | ABC is found by finding the single life Z, whose 
 annuity is equal, or as nearly equal as the tables will 
 give, to the annuity on the joint lives of the two elder 
 of A, B, and C. This new life must be placed instead 
 of those of B and C, which reduces the three lives to 
 two. That is |Z being equal to |BC (A being the 
 youngest of the three) | A B C is | AZ very nearly. Thus, 
 the ages of A, B, and C being 45, 50, and 65, we 
 have |BC (Carlisle table, 5 per cent.) = 6*8, and in 
 the table of single lives 7*8 and 6*3 are the values of a 
 single life at 65 and 70 years, giving a diminution of 
 1-5 in five years or '3 per year of age. Hence 7*8 
 will become 6*8 in about three years, or 68 is the age 
 of Z. Again, writing the ages at the feet of the letters 
 signifying lives, !A 45 Z 65 is 7'0 and |A 45 Z 70 is 5'8, 
 giving a diminution of '24 for every year of Z's age, so 
 that |A 45 Z 68 will be 7*0 '7 or 6'3, which is near 
 the value of |A 45 B 50 C f)5 . 
 
 The values of annuities upon single or joint lives 
 being thus found, it is easy to solve many other pro- 
 blems of annuities. Such annuities may either be re- 
 quired for a temporary purpose, or as a provision for 
 future years, or for some parties after the death of others. 
 I write the result required in symbols at the beginning 
 of each question, and leave the reader to refresh his 
 memory of them by observing the demonstration. 
 
 PROBLEM. An annuity of II. on the life of A, aged n, 
 is to be bought, to begin payment at the end of t years, 
 if A should live so long : required its value. 
 
 The question is proposed in the most usual form, 
 but a little change will facilitate its solution. The 
 right of A is evidently an annuity to begin in t 1 
 years, should he live so long. And the rule is expressed 
 thus, 
 
206 ESSAY ON PROBABILITIES. 
 
 At the end of tl years, if A be then alive (of 
 which find the chance), he enters upon an annuity of 
 11. y being then n -f t 1 years of age. The value of an 
 annuity at that age, multiplied by the chance of attain- 
 ing that age, and by the present value of I/., to be 
 received tl years hence, is, on the principles ex- 
 plained in p. 189-, tne present value of the annuity. 
 
 Example w = 45, =11 (Carlisle tables 5 per cent.) 
 10fa 45 = $|f=-862 9(1 =-614 |A 55 =10-3 
 862 x *614 x 10-3 = 5-45 5 '45 Answer. 
 
 PROBLEM. An annuity of II. is granted to A, aged n, 
 for t years, provided he live so long : what is its value ? 
 \A n t=A n -t\A n 
 
 The last equation is obvious ; for the whole annuity 
 on A's life is made up of a contingent annuity for 
 t years, and a contingent annuity to commence payment 
 in t -f 1 years. Consequently, from the whole value of 
 an annuity for A's life subtract that of an annuity to 
 commence payment in t -f 1 years, if he should be then 
 alive, and the remainder is the value of an annuity for 
 t years, if he should live so long. 
 
 Example w = 45, t\Q 
 
 By last problem 10|A 45 = 5'5, |A 45 =12-6. 
 Therefore |( A 10)= | A 45 - 10|A 45 = 7'1 7 '1 Answer. 
 
 PROBLEM. To find A m I B w? the value of an annuity 
 on the life of B, aged n y the first payment of which is 
 to be made at the end of the year in which the life of 
 A, aged m, fails. This is called a survivorship annuity, 
 since it can never be paid unless B survive A. To 
 give this annuity is evidently to give a complete annuity 
 to B, on condition that he shall restore it as long as A 
 is alive ; that is, 
 
 A|B=|B-|AB; 
 
 or, from the value of an annuity on B's life subtract 
 that of an annuity on the joint lives. Thus (Carlisle 
 tables 4> per cent.), the value of an annuity on the life 
 
ON ANNUITIES. 207 
 
 of B, aged 30, to commence after the death of A aged 
 35, is l6'9 13-5,, or 3'4 years' purchase. 
 
 PROBLEM. To find AB|A:B, the value of an 
 annuity on the life of the survivor, whichever it may 
 be, after the death of the other. 
 
 AB|A:B=|A + |B-2(|AB) 
 
 This is the same thing as giving an annuity to both, 
 on condition that both restore it as long as both are 
 alive. From the sum of the annuities, therefore, on 
 the lives of A and of B, subtract twice the value of an 
 annuity on the joint lives. 
 
 PROBLEM. To find ABC|AB : BC : CA, or 
 ABC|(ABC) , the present value of an annuity to 
 begin payment at the end of the year in which one of 
 the three dies, and to continue as long as both of 
 the other two are alive. Give each pair an annuity on 
 their joint existence, and withdraw all three annuities 
 as long as all three are alive. 
 
 ABC|(ABC) 2 =|AB + |BC + |CA-3(|ABC) 
 
 If the annuity be to commence immediately, without 
 waiting for the death of one, this is an additional grant 
 of |ABC or 
 
 |AB:BC: CA=|AB + |BC + |CA-2(|ABC) 
 
 PROBLEM. To determine |A : B, the present value 
 of an annuity to be continued as long as eitlier A or B 
 shall be alive. Give both an annuity, but withdraw it 
 from one as long as both shall be alive. 
 
 |A:B=|A + |B-|AB 
 
 PROBLEM. To determine |A : B : C, the present 
 value of an annuity to be continued so long as any one 
 of the three shall be alive. Give each an annuity, and 
 withdraw one of the annuities from any pair as long as 
 that pair shall be both alive : but as this would take 
 away the annuity during the joint continuance of the 
 
208 
 
 ESSAY ON PROBABILITIES. 
 
 three lives, grant an additional annuity of II. on that 
 joint continuance. Thus, 
 
 JA:B: C=|A + |B + |C-|AB-|BC-|CA + |ABC; 
 
 in the process of finding which it becomes evident that 
 
 ABC|A: B: C= | A + |B + |C- | AB- |BC- |CA 
 
 It may he worth while to point out how, in every pos- 
 sible case, the preceding grants and withdrawals produce 
 the required effect ; namely, one annuity to be paid as 
 long as any one life lasts. The following are the pay- 
 ments and withdrawals : 
 
 Living. 
 
 Dead. 
 
 Pay. 
 
 Withdraw. 
 
 Balance 
 
 ABC 
 
 
 1 +1+1+1 
 
 1+1+1 
 
 
 AB 
 
 C 
 
 1+1 
 
 1 
 
 
 BC 
 
 A 
 
 1 +1 
 
 1 
 
 
 CA 
 
 B 
 
 1 + 1 
 
 1 
 
 
 A 
 
 BC 
 
 1 
 
 
 
 
 B 
 
 AC 
 
 1 
 
 
 
 1 
 
 C 
 
 BA 
 
 1 
 
 
 
 1 
 
 PROBLEM. Required C| A B, the value of an annuity 
 on the joint lives of A and B as long as they shall sur- 
 vive C. Grant a complete annuity on the joint lives, and 
 withdraw it while all three are alive. 
 
 C|AB = |AB-|ABC 
 
 PROBLEM. Required AB|C, the value of an annuity 
 on the life of C, to commence after the failure of the 
 joint existence of A and B. Grant an annuity on the 
 life of C, and withdraw it as long as all three are alive. 
 Thus, 
 
 AB|C=|C-|ABC 
 
 PROBLEM. Required the value of an annuity B : C|A, 
 to be paid to A after B and C are both dead. Grant 
 
ON ANNUITIES. 209 
 
 to A an annuity on his own life, withdrawing it as long 
 as either B or C survives with A ; that is, withdrawing 
 
 |AB: AC or | AB + | AC- | ABC. Hence, 
 
 B:C|A=|A-|AB-|AC 
 
 PROBLEM. Required C| A : B, the value of an annuity 
 to be paid as long as either A or B shall survive C. 
 Grant one annuity to be paid as long as A or B is alive 
 (| A : B), and withdraw it as long as A or B lives with 
 C; that is, withdraw |AC : BC. Hence, 
 
 C|A: B=|A:B-|AC: BC 
 
 = |A + |B-|AB-(|AC 
 = |A+|B-(|A-B + |AC 
 
 The same case also amounts to granting |A : B : C and 
 withdrawing |C. 
 
 PROBLEM. An annuity C|A : B, payable as long as 
 either A or B shall survive C, is to be divided equally 
 between them, while they both live, and is then to go to 
 the survivor. What is the value of the interest of 
 each ? 
 
 The interest of A is an annuity of half a pound 
 while both A and B survive C, and of a whole pound 
 as long as he shall survive both C and B : or 
 But, 
 
 i(C|AB)HK|AB)-J(|ABC) 
 
 C: B|A=|A-|AB-|AC+|ABC 
 the sum of which is 
 
 |A-i(|AB)-|AC+J(|ABC) 
 and similarly, the value of B's interest is . 
 |B-i(|AB)-|BC+4(|ABC) 
 
 the sum of which makes up, as it should do, the value 
 of C|A : B, as given above. 
 p 
 
210 ESSAY ON PROBABILITIES. 
 
 PROBLEM. ,An annuity on the longest survivor of 
 A and B, or | A : B, is to be equally divided between them 
 during their joint lives, and afterwards to go to the 
 survivor. What is the value of the interest of each ? 
 That of A is evidently \ (|AB) + B|A, which is 
 
 J(|AB) + |A-|ABor|A-J(|AB) 
 Similarly that ofBis |B-J(|AB) 
 
 which results are obtainable by a yet more evident 
 process, since the interest of each is evidently an annuity 
 on his own life, with half of it withdrawn as long as 
 both are alive. 
 
 PROBLEM. To determine (BC)|A, the value of an 
 
 annuity on the life of A, to commence with the failure 
 of the joint existence of B and C, provided it be B who 
 dies first. 
 
 There are no tables for the accurate solution of this 
 problem ; but the following reasoning leads to a result 
 which cannot be far wrong, unless some of the lives be 
 very old, and which will always be near enough for the 
 species of application contemplated in this work. The 
 interest of A may be divided into two annuities, one 
 of which is B|AC, and the other a portion of B : C|A. 
 For A is certain of an annuity during C's life after the 
 death of B, and of another after both are dead, provided 
 B die first. Suppose A's interest in the latter annuity 
 to be worth one half of it, which is strictly true if B 
 and C be of the same age, and not much beside the 
 truth for considerable differences of age, particularly 
 when A is the oldest of the three. On this supposition 
 A's interest is 
 
 B|AC+J(B: C|A) 
 
 or |AC-|ACB+i(|A- |AB- |AC + |ABC) 
 
 =(BC)|A 
 
 which is obtained by supposing 
 
ON ANNUITIES. 
 
 211 
 
 The facility with which the preceding rules may he 
 applied, in every part of the process except that of 
 finding the annuities on three lives, makes it unnecessary 
 to present examples. In order that examples may be 
 readily obtained in cases involving three lives, some 
 partial tables are given both by Mr. Morgan and Mr. 
 Milne, from which the following selection is made : 
 
 Values of an Annuity of \Q (or, with the last figure 
 made a decimal, of &!,) on the joint continuance 
 of three lives of equal ages, from the Northampton 
 and Carlisle tables ; in the former at 4 per cent., and 
 in the latter at 5 per cent. 
 
 
 d 
 
 
 
 C 
 
 
 1 
 
 II 
 
 l! 
 
 
 
 
 |1 
 
 r 
 
 1* 
 
 1 S3 
 
 6 
 
 11 
 
 ! 
 
 5 
 10 
 
 112 
 122 
 
 129 
 134 
 
 45 
 50 
 
 71 
 63 
 
 88 
 79 
 
 15 
 
 113 
 
 127 
 
 55 
 
 56 
 
 65 
 
 20 
 
 103 
 
 122 
 
 60 
 
 48 
 
 51 
 
 25 
 
 98 
 
 115 
 
 65 
 
 39 
 
 42 
 
 30 
 
 92 
 
 108 
 
 70 
 
 30 
 
 32 
 
 35 
 
 86 
 
 102 
 
 75 
 
 21 
 
 21 
 
 40 
 
 79 
 
 94 
 
 80 
 
 14 
 
 16 
 
 The following is from the Northampton table, at 
 4 per cent., the annuity being ^lO, and the ages as 
 specified. 
 
 Ages. 
 
 Annuity. 
 
 Ages. 
 
 Annuity. 
 
 5, 15, 25 
 
 107 
 
 45, 55, 65 
 
 51 
 
 10, 20, 30 
 
 104 
 
 50, 60, 70 
 
 42 
 
 15, 25, 35 
 
 97 
 
 55, 65, 75 
 
 33 
 
 20, 30, 40 
 
 90 
 
 60, 70, 80 
 
 24 
 
 25, 35, 45 
 
 83 
 
 65, 75, 85 
 
 16 
 
 30, 40, 50 
 
 76 
 
 70, 80, 90 
 
 11 
 
 35, 45, 55 
 
 68 
 
 75, 85, 95 
 
 2 
 
 40, 50, 60 
 
 60 
 
 
 
212 
 
 ESSAY ON PROBABILITIES. 
 
 The following is from the Carlisle table, at 5 per 
 cent : 
 
 Ages. 
 
 Annuity. 
 
 Ages. 
 
 Annuity. 
 
 5, 30, 35 
 
 Ill 
 
 45, 70, 75 
 
 33 
 
 10, 35, 40 
 
 106 
 
 50, 75, 80 
 
 25 
 
 15, 40, 45 
 
 99 
 
 55, 80, 85 
 
 18 
 
 20, 45, 50 
 
 91 
 
 60, 85, 90 
 
 12 
 
 25, 50, 55 
 
 80 
 
 65, 90, 95 
 
 11 
 
 30, 55, 60 
 
 66 
 
 70, 95, 100 
 
 9 
 
 35, 60, 65 
 
 55 
 
 
 
 40, 65, 70 
 
 44 
 
 
 
 I shall proceed in the next chapter to consider the 
 methods of finding the value of reversionary interests, 
 including life insurances. 
 
 CHAPTER X. 
 
 ON THE VALUE OF REVERSIONS AND INSURANCES. 
 
 THE distinction between the problems of this chapter 
 and the last, lies more in names and in the circumstances 
 under which they occur for solution, than in difference 
 of methods, principles, or (according to the scheme 
 which I have suggested), even of notation. Every 
 interest, the symbol of which has any thing preceding 
 the |, is properly a reversion, being something of which 
 the benefit is not to begin until the happening of some 
 event, or the determination of some existing status. 
 
 It may be proper here again to remark, that all the 
 rules in the preceding chapter, though the status men- 
 tioned are technically called lives, are equally true for 
 any species of circumstances, temporary or permanent, 
 certain or contingent. Thus an annuity for t years, to 
 
ON THE VALUE OF REVERSIONS. 
 
 begin after n years, signified by n\n + t, or by n\t, is de- 
 termined by the same formula as an annuity on the life 
 of B, to begin after the death of A, signified by A|B. 
 n|n + *=|n + *-|(n + A|B = |B-|AB 
 
 but in the first I write \n instead of \(n n-\-i) since 
 the joint continuance of n and n+t years must be n 
 years. 
 
 The only difficulty of this notation is the necessity of 
 remembering the distinction by which it appears whether 
 n\n-Jrt means that a payment is to be made at the end 
 of the nth year or of the (n-hl)th. In the case of 
 A|B the first payment is to be made at the end of the 
 year in which A dies, which is always intelligible, 
 since it is considered as an infinitely small probability 
 that A should die at the moment which divides two 
 years. But since the term of n years does expire at 
 such a moment, the analogy which connects the symbols 
 of terms certain and contingencies points out no rule. 
 Let n | stand for a perpetuity of I/., the first payment to 
 be due at the end of n+ 1 years, then n 1 1 stands for a 
 similar perpetuity due at the end of n years. But since 
 the introduction of a new figure may turn the attention 
 off n, the datum of the problem, and since analogy 
 does not require us to strike off a whole year from the 
 term, let -| signify a perpetuity deferred for a term 
 (no matter how little) short of n years, that is, payable 
 in n years. And by analogy -| A will signify an annuity 
 due on the life of A. The symbol n-\n + t must be 
 treated as if it were 
 
 n l\n + t or n 1\ t+I 
 
 The symbol of a perpetuity of II to commence 
 from the present time (that is, payment at the end of 
 a year), is simply | . This is found as in p. 1 84., 
 and by subtracting | A, the value of an annuity on the 
 life of A, we obtain | | A, the value of the reversion 
 of a perpetuity, of which payment is made at the end 
 of the year in which A dies. At the end of that year, 
 the holder of the reversion will have in possession and 
 p 3 
 
214 ESSAY ON PROBABILITIES. i 
 
 expectation, an equivalent to -| or 1 -f |, or the value 
 of a perpetuity due. Consequently, since the present 
 value of 1 + 1 to be received at the death of A is | | A, 
 that of II. will be found by dividing the latter by the 
 former : or, 
 
 The present value of 11. to be received at the end of 
 the year in which A dies, is found by subtracting the 
 present value of an annuity on the life of A from that 
 of a perpetuity, and dividing the remainder by the 
 present value of a perpetuity due, or one year's pur- 
 chase more than the present value of a perpetuity. 
 
 EXAMPLE. (Northampton tables, 3 per cent.) What 
 is the value of 11. , to be received at the end of the 
 year in which a life of 30 shall fail ? 
 
 Perpetuity of l at 3 per cent. 33 -3 
 Value of annuity on life of 30 l6'9 
 
 34-3) 16-4(-478 
 
 In the preceding rule any status may be substituted 
 for a single life, and the value of the annuity which is 
 to be paid as long as the status lasts is connected with 
 the present value of I/, to be received at the end of 
 the year in which the status fails, by the preceding 
 simple rule. 
 
 The premium which should be paid (first down, and 
 afterwards at the end of each year), is an annuity due 
 upon the life or status, and is therefore worth -| A or 
 1 + | A year's purchase. Consequently the premium which 
 should be paid for the II. above described is the pre- 
 ceding present value divided by one year's purchase more 
 than the annuity is worth. In the example, divide '478 
 by 1 + 16*9 or 17'9, which gives -026?, so that 21. 13*. 
 6d. is the premium for insuring 100/. at the end of the 
 year in which a life of 30 fails. 
 
 The following rule is somewhat shorter, in the case 
 in which the premium only is required, and not the 
 present value. 
 
 QUESTION. To find the premium which should be 
 paid (first down, &c.), during the continuance of a 
 
ON THE VALUE OF REVERSIONS. 
 
 215 
 
 status, to insure II at the end of the year in which that 
 status drops. 
 
 RULE I. From the quotient of a perpetuity divided 
 by a perpetuity due, subtract that of an annuity on the 
 status divided by an annuity due. 
 
 34-3) 33-3 ('9708 
 9441 
 
 17-9) 16-9 (-9441 
 
 0267 Answer, as before. 
 
 RULE II. Divide 1 by the value of an annuity due, 
 and by that of a perpetuity due, the difference of the 
 quotients is the premium required. 
 
 34 -3)1 (-0292 
 
 17 '9)1 ('0559 
 0292 
 
 0267 Answer, as before. 
 
 A perpetuity divided by a perpetuity due, is the 
 present value of II. to be received a year hence, and 
 may be taken from the following table : 
 
 p. c. 
 
 
 p. c. 
 
 
 p. c. 
 
 
 2 
 
 9804 
 
 4 
 
 9569 
 
 9 
 
 9174 
 
 24 
 
 9756 
 
 5 
 
 9524 
 
 10 
 
 9091 
 
 3 
 
 9709 
 
 6 
 
 9434 
 
 
 
 3J 
 
 9662 
 
 7 
 
 9436 
 
 
 
 4 
 
 9615 
 
 8 
 
 9259, 
 
 
 
 From the preceding rule an illustration of the reason 
 of it may be derived, which I give professedly as an 
 exercise of ingenuity to those who may be beginners in 
 the subject. Let there be two persons, one of whom 
 holds a perpetuity and the other a life annuity, each of 
 I/. Both the perpetuitant * and the annuitant desire 
 
 * If the holder of an annuity be an annuitant, the extension of lan- 
 guage is justifiable, by which the holder of a perpetuity may be called a 
 perpetuitant. 
 
 p 4 
 
216 
 
 ESSAY ON PROBABILITIES. 
 
 to commute their interests for interests due : that is, the 
 perpetuitant, instead of 11. a year hence and so on, 
 desires to receive a fraction of a pound now, and the 
 same fraction at the end of every year ; and the same 
 for the annuitant. Say the value of money is four per 
 cent.., then the perpetuitant desires to change an interest 
 which is worth twenty-five years' purchase into an 
 equivalent interest worth twenty-six years' purchase 
 (or income) ; consequently his year's income (now due, 
 &c.) must be only -||/. Say that the annuity is worth 
 ten years' purchase; then by the same reasoning the 
 yearly income of the annuitant (now due, c.) must 
 be only ^l. The second is less than the first ; whereas 
 the original incomes were the same, both I/. But 
 there must be some consideration which the com- 
 mutation gives to the annuitant, and for which this 
 greater diminution of his income is the payment ; and 
 it is as follows : Since the commutation forestalls each 
 successive payment, giving it (or the substitute for it) 
 a year before it becomes due, the annuitant would 
 receive, if his income were made equal to that of the 
 perpetuitant, the II. which, had he lived, would have 
 become due at the end of his last year, but which his death 
 hinders from becoming due. This difference of income 
 (If "~~TrX * s therefore equivalent to preventing his 
 receiving M. at the end of the year in which he dies, 
 and it is taken from him now and in every succeeding 
 year of his life. Consequently it is the premium which 
 such an annuitant should pay to receive 11. at the end 
 of the year in which he dies ; and it is also the result 
 of the first preceding rule. 
 
 The second rule may receive an explanation of a 
 similar kind. I now reverse the problem, and ask the 
 following 
 
 QUESTION. If an office charge the premium p for 
 insuring 11. at the end of the year in which a life (or 
 other terminable status) drops, what should we infer 
 that they suppose to be the greatest possible value of 
 an annuity to continue during the remainder of that life 
 
ON THE VALUE OF REVERSIONS. 21? 
 
 or status ; that is, what is the value of an annuity on 
 that status, which is such that the office must be ruined 
 if the truth falls below it ? 
 
 RULE. Take the rate of interest which money 
 really makes *, and subtract the premium for II. from 
 the present value of I/, to be received at the end of a 
 year (see last table) : divide the remainder by the 
 excess of unity over the remainder, and the quotient is 
 the number of years' purchase in the present value of 
 the annuity. 
 
 EXAMPLE. A society professes to insure lives of 
 35 at a premium of 3 per cent, on the nominal sum 
 insured ; what is the lowest value of the annuity on 
 such lives at which this can be done ? 
 
 At 4 per cent. |l = -9615 
 
 (page 200.), A||l ='0300 
 
 0685)-9315(13-6 -0685 
 
 ANSWER. Such an office cannot permanently stand 
 (as far as this one species of bargain is concerned), 
 unless the value of an annuity on lives of 35 (at 4 per 
 cent.) be more than 13'6 years' purchase. 
 
 Generally speaking, contracts of insurance are not made 
 for the end of the year in which the party dies, but for 
 payment at a given number of months after the parties' 
 death is proved and the claim made. If this agreement 
 were always made for six months after the real death, 
 the office would, one party with another, neither gain 
 nor lose, while for every month less than six, the office 
 gives that month's interest to the parties' executor, while 
 for every month more than six by which payment is 
 deferred, the office takes a month's interest. I believe 
 no office defers its payments more than six months after 
 the claim is made ; and the difference is rendered im- 
 material by the probable errors of the tables, which 
 require too large a covering profit to make it worth 
 while to take such a circumstance into account. 
 
 * This is an essential element, but cannot be very accurately determined : 
 something above the truth should be assumed. 
 
218 ESSAY ON PROBABILITIES. 1 
 
 The premium demanded by an office is that charged 
 by their tables at the age which the party will attain at 
 his next birthday ; thus if a person desire to insure his 
 life the day after he attains 31 years complete, he will 
 be required to pay the same as if he had deferred 
 completing the insurance till the day before his thirty- 
 second birthday. This is, one party with another, a 
 gain of half a year to the office. Thus, the North- 
 ampton table at 3 per cent, giving 16*7 and l6'5 as 
 the values of annuities at the above-mentioned ages, all 
 parties who have passed 31 years at their last birth day 
 are considered as having lives worth 16*5, whereas they 
 are worth, one with another, l6'6. The tables are not 
 sufficiently accurate to make the effect worth caring for. 
 
 A party having made an insurance, and paid one or 
 more premiums, the instrument by which the right to 
 receive the stipulated sum at death on payment of a 
 stipulated premium is conveyed, is called a policy of 
 insurance. The value of this policy is then easily 
 determined ; at least what we may call its office value, 
 supposing the tables of the office to be perfectly correct. 
 A person aged thirty insures for 100/., for which he pays, 
 say 31. ; he continues to pay this premium until the 
 age of fifty, at which time, if he had began to insure, 
 the annual premium would have been, say 51. Suppose 
 that the holder of the policy wishes to sell his interest 
 just before he would otherwise have had to pay another 
 premium, it is plain that he then offers for an in- 
 surance on the life of 50, a better bargain than the 
 office would offer, since the buyer of the policy (who 
 pays all future premiums) will acquire, in consider- 
 ation of an annuity due of 31. upon the life of A, that 
 which the office would not sell for less than an annuity 
 due of 51. upon the same life. The difference, or an 
 annuity due of 2/. upon the same life, is the value of 
 the policy. 
 
 RULE. To find the present value of a policy of insur- 
 ance, at the moment before a premium becomes due, sub- 
 tract the premium which is to be paid from the premium 
 
ON THE VALUE OF REVERSIONS. 219 
 
 which would be paid if the same party made the same 
 insurance at the present time. Find the present value 
 of an annuity on the life of the party insured, of the 
 same yearly amount as the preceding difference,, and this 
 value, increased by one year's purchase, is the present 
 value of the policy. 
 
 To find the value of the policy immediately after 
 a premium is paid, add the premium just paid to the 
 result of the preceding rule. It would not be worth 
 while, in the present work, to give a rule for any in- 
 termediate value. (See Milne, p. 283.) 
 
 Bat in finding the real value of a policy, there are 
 one or two circumstances to be considered, of which no 
 mention is made in the preceding rule. The buyer of 
 the policy, being uncommitted by any previous act of 
 his own, is not bound to consider the premium of any 
 one office as a standard. Suppose that in the preceding 
 example, another office of equal solvency can be found, 
 which will insure a life of 50 at 4 per cent, instead 
 of 5 : the buyer, therefore, may consider that the 
 seller offers him for 31. a year during his life a benefit 
 which he might buy elsewhere for 4/., and that he 
 should therefore pay only the value of an annuity due 
 of IL instead of 2. But since the two offices cannot 
 be together parties to any transfer of policy, the pre- 
 ceding case will only serve to show that it may be 
 more prudent for a person who has money to invest, to 
 lay it out at once in insuring lives in a cheap office, 
 than in buying existing policies in a dear one. It is to 
 be remembered that the lower premium in the preceding 
 rule is to be paid, by bargain already existing, while 
 the higher one is hypothetical, depending on the 
 buyer's opinion of tables of mortality. That the office 
 which demanded and obtained the SL would demand 
 the 51. for an insurance now to commence, must be no 
 consideration for a person who is merely thinking how 
 to lay out his money to the best advantage ; it may be 
 by buying the policy which is offered to him, or by 
 insuring his own life, or that of some one else, in the 
 
220 ESSAY ON PROBABILITIES. 
 
 same or another office. It is his business to consider 
 what he is likely to have to pay, in the shape of future 
 premiums, and not what an office, which must he on 
 the safe side, has thought fit to suppose it will have to 
 receive. Putting out of view the state of health * of 
 the party insured, I should think it most advisable to 
 calculate the value of policies by finding the present 
 value of the sum insured, and also that of the premiums 
 to be paid, from the tables which best represent healthy 
 life, and using the rate of interest which money will 
 really obtain, rather above than below ; that is, I should 
 use the Carlisle tables at 4 per cent. The profits 
 guaranteed by the office, if any, should be duly con- 
 sidered. Thus suppose a person at the age of 30 had 
 insured for 10001. in an office which demands 25 /. 
 premium for that insurance, and returns no profits, 
 and suppose that twenty years have elapsed, so that the 
 life insured is now at the age of 50, what is the real 
 value of his policy? The value of II. to be received at 
 the death of a person aged fifty, by the Carlisle tables 
 at 4 per cent., is (p. 214.), 25 12'9 divided by 25 -4- 1 
 or '465 : that of WOOL is therefore 465/. If a premium 
 be just becoming due, the present value of all the 
 premiums is therefore 1 -f 12*9 or 13'9 years' purchase ; 
 and 13'9 X 25/. is 347'5/. Consequently 465- 347|, or 
 117^., is what I should consider to be the value of that 
 policy. But if I took the tables of the same office, 
 which require a premium of 47 /. at the age of fifty, 
 and which, with some variation, are derived from the 
 Northampton tables at 3 per cent., I should find by 
 the rule in p. 218., (1 + 12-4) x (47-25), or 295/. 
 nearly. So great is the difference between policies 
 valued by the nearest approximation which exists to the 
 actual truth, and then valued by the tables which 
 offices adopt for their own security. 
 
 The office itself, which takes an advantage of the 
 buyer when the policy is first created, may reasonably 
 
 * Of course the policy of a person whose health has very much declined 
 since he effected the insurance, is of higher value on that account : but 
 this cannot be made the subject of calculation. 
 
ON THE VALUE OF REVERSIONS. 221 
 
 allow that advantage to the insured, if he afterwards 
 desire to sell his policy to the office itself. I am not 
 aware of the exact rule which is followed by the offices 
 in this respect, except in one or two cases, in which 
 the plan is, or was, to follow their own tables, with a 
 certain deduction from the result, and to give the dif- 
 ference to an insured party who desired to sell his 
 policy. This is well enough in the case of offices which 
 return profits ; but if such a rule be followed by those 
 which do not, it may amount to a contradiction of their 
 profession in the case of the sale of policies; and may 
 become in effect an allowance of that share in the profits 
 to those who desire to leave the office, which they re- 
 fuse to grant to those who continue. To prevent such 
 a result, I believe the offices who would be liable to it, 
 make a large deduction from the value of policies, as in- 
 dicated by their tables. 
 
 All the preceding rules apply to any given status as 
 well as to a given life. Thus, to effect an insurance on 
 the survivor of two lives, the present value and the 
 premium (payable as long as either, is alive) are to be 
 found by using |A + |B - |AB for the value of the an- 
 nuity, instead of |A. I now proceed to some simple 
 cases of insurance, where the payment on one party's 
 death is made conditional upon another party being alive 
 to receive it. 
 
 The symbol A| (IB) denotes the value of an annuity 
 upon the joint continuance of one year and the life of B, 
 payment being made at the end of the year in which A 
 dies. It is therefore necessary that B should be alive 
 at the end of the year in which A dies. But in the 
 usual conditions of contingent insurances, it is sufficient 
 that B should be alive at the moment in which A dies. 
 
 Let this be expressed by A] IB ; it is then evident that 
 
 A | IB is greater than A| IB. The following preliminary 
 considerations will be necessary. 
 
 PROBLEM. Required the value of an annuity on the 
 joint lives of A and B, to be paid at every end of a 
 year at which B shall be alive, provided A were alive 
 
222 ESSAY ON PROBABILITIES. 
 
 at the beginning of the year. This may he denoted 
 by | (A...B). 
 
 The condition that a life shall be alive at the begin- 
 ning of the year must be, in tables of averages,, the 
 same as that of a life a year younger being alive at the 
 end of the year. For example, suppose that of 500 
 persons of the age of n I, 493 attain to that of n, 
 then 500 annuities granted on the lives of A n -\ will 
 be equivalent to 493 granted to A w , if the latter be 
 payable at the end of any year in which A shall have 
 been alive at the beginning ; and the same for any 
 joint lives combined with both. Thus 500 annuities 
 granted on the joint lives of A n _i and B, payable as 
 usual, are equivalent to 493 on the joint lives of A n 
 and B, payable as long as B is alive at the end, and A 
 was alive at the beginning, of a year. Hence | (A...B) is 
 .5.00. o f |A,B, where A / stands for a life a year younger 
 than A. Hence the following RULE. To solve the 
 preceding question, multiply the value of a joint an- 
 nuity on B and one year younger than A by the number 
 alive at that younger age in the table, and divide by 
 the number alive at A's age : the result is the present 
 value required. Or more concisely, divide | (A X B) by 
 the chance which A, has of living a year. 
 
 Now let us ask, by how much does |(A...B) as above 
 described exceed | AB. The only possible case in which 
 a payment will ever be made upon the first annuity, and 
 not upon the second, is when A dies before B, for both 
 are determined by the death of B. When A dies be- 
 fore B, the annuity will be paid at the end of the year 
 upon |(A....B), if B be alive, but not upon | AB. Con- 
 sequently the excess of the first over the second is the 
 value of l to be received at the end of the year in 
 which A dies, provided B be then alive. 
 
 r '_ 
 
 A|TB=|(A...B)-|AB, B|lA=|(B...A)-|AB 
 
 Also the present value of an annuity to be paid at the 
 end of any year in which both A and B were alive at the 
 
ON THE VALUE OP REVERSIONS. 
 
 223 
 
 beginning, or |AB..., is evidently |AB increased by AB|l, 
 the present value of an insurance of one pound on the 
 joint lives. 
 
 PROBLEM. Required the present value of ^l to be 
 paid at the end of any year, provided that both A and 
 B die in that year ; which has been signified by A : B|l. 
 
 Grant the following annuities ; 
 
 |AB, on the joint lives of A and B. 
 
 |(AB...) the same as |AB, but to be also paid at 
 the end of the year in which the joint existence fails. 
 
 Take in exchange the following annuities : 
 
 |(A...B) and |(B... .A) two annuities to be paid 
 during the joint lives, and also at the end of the year 
 in which the joint existence fails, provided B in the first, 
 and A in the second, be alive at the end of the year. 
 
 The balance of this transaction will be^l to be paid 
 at the end of any year, provided B and A both die in 
 the year. For as long as both are alive, two annuities 
 are payable each way ; if A die and B remain alive 
 till the end of the year, |AB has ceased, |(AB...) is 
 payable, but |(B...A)has ceased, and (A...B) is payable ; 
 similarly in the case of B dying and A remaining alive. 
 If both die in one year |AB has ceased, but |(AB...) 
 is payable, while JB...A and |A...B have both ceased. 
 Consequently, the only possible payment which the 
 grantor has to make, over and above those which he 
 receives, is the ^1 in the question proposed ; or 
 
 We are now in a condition to solve the final PROBLEM. 
 Required the value of \ to be paid at the end of the 
 year in which A dies, if B should have been alive at the 
 moment of A's death. This is denoted by A|1B. 
 
 When A dies before B, either B survives till the 
 end of the year, or dies in the intermediate time. The 
 insurance on the first risk is worth A| IB determined 
 in p. 222.; on tbe second it is worth half the result of 
 the last problem, if it be considered that the chances of 
 
224? ESSAY ON PROBABILITIES. 
 
 A dying before and after B, in any one given year, are 
 equal. We have therefore 
 
 Or, 
 
 But, (p. 223.) 
 
 IAB...-JAB is AB|I 
 
 Whence the final result is, 
 
 RULE. For determining the value of 11. payable at 
 the end of the year of the death of A, provided B be 
 alive at the moment of A's death. 
 
 (1.) From the value of a perpetuity subtract the 
 value of the joint annuity, and divide by that of a per- 
 petuity due. 
 
 (2.) Multiply the value of the joint lives of B and a 
 year younger than A by the number alive in the table 
 at the younger age, and divide by the number alive at 
 the age of A. 
 
 (3.) Repeat the preceding process, substituting B 
 for A, and A for B. 
 
 (4.) From the sum of the results of (1) and (2) 
 subtract that of (3), and half the difference is the pre- 
 sent value required. The premium payable during the 
 joint lives is found by dividing the result by the value 
 of an annuity due on the joint lives. 
 
 Though I have deduced the preceding rule at length 
 as an instance of the very improving exercise of deducing 
 the more complicated results of this subject by what 
 we may call the balance of annuities, yet for the rough 
 purposes of this work, if not for others still more exact, 
 the more simple process implied in finding A...B |AB 
 is sufficient. It must be obvious that the fraction of 
 the whole value of a survivorship insurance, which 
 depends on the risk of both parties dying in the same 
 year, is a small one ; so that it is nearly sufficient (and 
 certainly within the probable errors of tables, &c.) to 
 
ON THE VALUE OF REVERSIONS. 
 
 225 
 
 consider this question independently of the double risk, 
 and as if it were certain that both parties would not die 
 
 in the same year. Instead of A) IB, we then employ 
 A|fB (page- 222). Mr. Milne (in his page 352. &c.) 
 has given three examples, which I here repeat by the 
 latter formula, taking the data from the work cited : 
 1. A is aged 1 7, B is aged 57 : Carlisle Tab., 4 per cent. 
 
 A...B = 10-009 
 |AB = 9-923 
 
 1 B= -086, which in work cited 
 ["is -08656. 
 
 ( 0-086 
 
 2. A is aged 45, B is aged 35 ; do. do. 5 per cent. 
 
 = 10-912 
 
 0-260 
 
 Ajl B = -260, which in work cited 
 [is -266331. 
 
 3. A is aged 64, B is aged 19; do. do. 5 per cent. 
 
 7-593 
 0-522 
 
 1 B = -522, which in work cited 
 [is -52556. 
 
 The present value of a pound, to be received at the 
 end of the year in which A dies, provided he survives 
 
 B, or B:A 
 
 1, is readily found from the preceding: 
 
 for since A must either die before or after B, the sum 
 of the two must be the present value of II. to be received 
 at the death of A, independently of B. That is, 
 
 7 A 
 
 FB rz A 
 
 T Al B very nearly. 
 
 The present value of an insurance is also that of the 
 
 reversion of a fixed sum ; since it is the same thing whe- 
 
 ther I/, is to be received from an office, or conditionally 
 
 under a will, or in any other way. The reversion of a 
 
 Q 
 
22 ESSAY ON PROBABILITIES. 
 
 perpetuity should be treated as that of the value of a 
 perpetuity due at the end of the year in which the life 
 drops. 
 
 PROBLEM. Required the present value of II. to be re- 
 ceived at the end of the year in which A dies, provided 
 that event take place before the expiration of t years 
 
 from the present time : signified by A| 1 t. 
 
 Suppose a person to have the certain reversion of a 
 perpetuity due at the end of t years, or sooner, if A die 
 before t years are expired : the reversion of the perpetuity 
 after the failure of the joint existence of A and t years, 
 is | | A ty which can be found from page 206, But 
 this is more than that fraction of a perpetuity due at the 
 end of the year in which A dies, which will pay for the 
 chance of entering on it before t years are expired : for 
 part of it expresses the value of a perpetuity due, which, 
 though A should be alive, is to be entered on by the 
 failure of t years. If t'\~ a be the chance that A is 
 alive at the end of t years, then t] a X t\ must be 
 deducted, as being expressed twice in the preceding : 
 consequently 
 
 |-|A t t ta X t\ 
 
 is the present value of a perpetuity due, or -|, to be 
 entered upon at the end of the year in which A dies^ if 
 before t years. The preceding then divided by -j gives 
 the present value of 11. to be received under the condi- 
 tions of the question. But a perpetuity created at the 
 end of t years, or t\, divided by a perpetuity now due^ 
 gives the present value of I/, to be received at the end of 
 t+1 years; which gives the following 
 
 RULE. From the value of a perpetuity subtract that of 
 an annuity on the given life for t years, and divide by 
 the value of a perpetuity due. From the quotient sub- 
 tract the present value of one pound to be received at 
 the end of t -j- 1 years, if the life be in being at the end 
 of t years: the difference is the present value of 11. to 
 be received at the end of the year in which the life drops, 
 if before t years have expired. The present value just 
 
ON THE VALUE OF REVERSIONS. 227 
 
 found, divided by that of an annuity due on the given 
 life for t years, gives the requisite premium. 
 
 And since the present value of such an insurance as 
 the preceding, together with the present value of II. to 
 be received at the end of the year in which A dies, if 
 after t years, make up the present value of IL to be re- 
 ceived in any case at the death of A, the third diminished 
 by the first will give the second. But it will be prefer- 
 able to make an independent investigation of this case. 
 
 PBOBLEM. Required t : A 
 
 1 the present value of I/. 
 
 to be received at the end of the year in which A dies, 
 provided t years shall have previously expired. 
 
 If from the present value of a perpetuity deferred for 
 t years, we deduct that of an annuity on the life of A de- 
 ferred for t years, we have the value of a deferred perpe- 
 tuity, further suspended during the term by which A out- 
 lasts t years, and to commence at the end of the year in 
 which A dies : or not to be suspended at all if A should 
 die in less than t years. Take away the value of a per- 
 petuity beginning from the end of t years, if A should 
 have died in the interval, and we have remaining the 
 present value of a perpetuity due at the qnd of the year 
 in which A dies, if that be deferred beyond t years. 
 This last is therefore 
 
 t\ t\A. -(l-t t a) X t\ 
 
 where t t a is the chance of A living t years. This can 
 be reduced to 
 
 t t a x t\ *|A 
 
 Divide this by the value of a perpetuity due, and we 
 have the present value of 1 /. receivable on the same con- 
 ditions. But t\ divided by -| gives t \ I, as before; 
 whence the following 
 
 RULE. Multiply the present value of II. receivable at 
 the end of t -f- 1 years by the chance which A has of 
 living t years; and from the product subtract the quotient 
 
228 ESSAY ON PROBABILITIES. 
 
 of a deferred annuity on A's life, divided by a perpetuity 
 due : the remainder is the present value of*U. to be re- 
 ceived^ if A outlive t years, at the end of the year in 
 which he dies. 
 
 The preceding pages contain all those cases which 
 most usually occur in practice : but it is to be noticed 
 that various modifying circumstances will present them- 
 selves in different problems, for which no general rule 
 can be given. I now proceed to problems in which 
 successions occur ; that is, in which a benefit depends 
 upon the continuance of a status which does not begin 
 until another status is finished. 
 
 In the case of an annuity for a certain term of years 
 t t to begin payment at the end of the year in which A 
 dies, we have obviously to consider a benefit which, at 
 the end of the said year, will amount to an annuity due, 
 of t payments; or II. augmented by the present (as it 
 will be then) value of t 1 future payments. This 
 then value can be found as in page 185., and, being mul- 
 tiplied by the present value of II. receivable at the end of 
 the year in which A dies (found in page 214.), gives the 
 present value of this contingently deferred annuity. 
 
 In the case of A|B, an annuity on the life of B after 
 the death of A, we certainly have a succession, but it is 
 one which may never exist. To make a problem which 
 may come under the present division of our subject, we 
 must imagine that, at the end of the year in which 
 A dies, a new life may be nominated at pleasure, which 
 is then to be of a given age. If P be the value of an 
 annuity upon such a life, then, according as the benefit 
 is an annuity, or an annuity due at the end of A's year 
 of death, we find the present value of P or 1 -f P to be 
 received at the end of that year. The result is the pre- 
 sent value of the succession. This problem includes 
 that of finding the value of the next presentation to a 
 living. The patron of a living of 500/. a year may con- 
 sider that he gives the clergyman whom he presents 
 100/. a year (or whatever may be called liberal remuner- 
 ation for a curate) for work and labour, and the remain- 
 
ON THE VALUE OF REVERSIONS. 
 
 ing 400/. as a free gift. If he sell the next presentation, 
 he must therefore consider that he sells 400/. a year (not 
 500/., since that would he to allow the clergyman no 
 salary* for his labour), to he paid yearly during the con. 
 tinuance of a life to be named by the buyer,, at the de- 
 cease of the present incumbent. And, since the right to 
 name new incumbents of 24 years of age is part of the 
 bargain, the patron will require a sum corresponding to 
 the value of an annuity upon a life of that age. De- 
 ducting a sum for first fruits, probability of expenses 
 from dilapidations, &c., which must be determined by 
 the circumstances of each case, the remainder is the net 
 present value of the living. It would probably be most, 
 fair to value the interest of the purchaser as if the new 
 incumbent would come into half a year's revenue at the 
 end of the year in which the present incumbent dies. 
 
 QUESTION. What is the present value of the next 
 presentation to a living of which the average annual 
 income is s, the salary of a curate t being v, and/ the 
 estimated expenses at entry. Let A be the value of the 
 incumbent's life, and P that of a life of 24 years of age. 
 Find the present value of P + J, to be received at the end 
 of the year in which A dies (p\ 214.), and multiply the 
 result by the excess of s over v ; from this deduct the 
 present value of/, to be received at the end of the year 
 in which A dies, and the remainder is the net present 
 value of the next presentation. 
 
 The perpetual value of an advowson (that is, of the 
 right to nominate the incumbent in all time to come, 
 after the decease of the present one) is generally valued 
 as the reversion of the net income after the death of the 
 present incumbent. But the expenses of entry, first 
 fruits, &c., should be considered as a fine levied on the 
 property at the death of every tenant, in diminution of 
 
 * The right of selling livings is therefore u bond, fide right to alienate 
 all the church property which is in private hands, with the exception only 
 of that minimum which will obtain a curate. 
 
 f If the living be one on which a curate must be kept by the incumbent, 
 the salary of two curates should be deducted from ths yearly revenue in the 
 valuation. 
 
 Q 3 
 
230 ESSAY ON PROBABILITIESo 
 
 the total value. To the problems connected with this 
 subject I now pass. 
 
 A great many interests are held in this country on 
 the consideration of rents, fines, or whatever they may 
 be called, which are not paid at any fixed time, but at 
 the deaths of successive lives which are named, each life 
 being nominated, and the rent or fine paid, at the death 
 of the preceding nominee. Leases held under ecclesi- 
 astical and other corporations, copyholds, &c., are in- 
 stances. By a statute of Henry VIII., corporations are 
 permitted to lease lands for three lives, or twenty-one 
 years; so that it may be suspected the legislature ima- 
 gined the average term of the duration of three lives to 
 be 21 years; or that, any three mature lives being 
 named in one set, and a large number of such sets being 
 taken, and each set being considered as a status to last 
 as long as any one of its lives was in being, the average 
 duration of such a status was 21 years. If this were 
 the opinion, and grounded upon any thing like experi- 
 ence, the value of life in that day must have been incre- 
 dibly below what it is at present: but it must be 
 remembered that in that day of insecurity few people 
 would venture on the life of a child or a woman ; and 
 that in all probability the lives contemplated were those 
 of men of middle age. However this may be, since that 
 time the tenure of lease upon lives has become ex- 
 tremely common, it being understood that the lives 
 which drop are renewable upon the payment of a fine, 
 either fixed or at the discretion of the lessor. 
 
 It is, of course, the interest of the lessor that the lives 
 should be as bad, and of the lessee that they should be 
 as good, as it is possible : but the lessee, having the no- 
 mination of the lives, will choose the best the tables 
 afford. The rate of interest being settled, the highest 
 life annuity in the tables gives the age which the life 
 nominated ought to have. The best age in the North- 
 ampton tables is 8 years, and in the Carlisle 7 years : 
 for which ages I subjoin the values of annuities at va- 
 rious rates of interest, adding also the age of 24 3 which 
 
ON THE VALUE OF REVERSIONS. 
 
 231 
 
 will be useful in the calculation of the values of advow 
 sons, with the correction above proposed. 
 
 Northampton. 
 
 Carlisle. 
 
 Ages. 
 
 3 p. c. 
 
 4 p. c. 
 
 5 p. c 
 
 Sp.c.'sp.c. 
 
 4 p. c. 
 
 5 p. c. 
 
 6 p. c. 
 
 Ages. 
 
 8 
 
 20-9 
 
 17-7 
 
 15-2 
 
 13-3 
 
 23-9 
 
 19-8 
 
 16-8 
 
 14-5 
 
 7 
 
 24 
 
 18-0 
 
 15-6 
 
 137 
 
 12'1 
 
 20-9 
 
 17'8 
 
 15-4 
 
 13-5 
 
 24 
 
 QUESTION. At the end of the year in which A dies, 
 a fine of 11. is to be paid, and a new life nominated, of 
 which the value will then be P : at the end of the year 
 in which P dies, another fine of II. is to be paid, and a 
 new life P nominated, and so on for ever : what is the 
 present value of all the fines, or what present money 
 must a person be considered as paying who receives an 
 estate charged with the preceding liabilities ? 
 
 This problem, as will be more fully explained in the 
 second appendix, was incorrectly solved by every writer 
 on the subject, down to the time of Mr. Milne, whose 
 solution, though perfectly correct, is in a difficult form. 
 The coincidence of the rule I now give with that of 
 Mr. Milne will be shown in the appendix cited. 
 
 Let us suppose a fine of IL per annum, first payable 
 at the end of the year in which A dies. If, then, a 
 receiver P were appointed for his life, his interest in the 
 fines, at the end of the year in which A dies, would be 
 1 + | P ; and if at his death a second receiver were ap- 
 pointed, of the same age at which the first was when his 
 term began, the interest of this second receiver at his 
 entrance would also be 1 -f |P, and so on. But if the 
 tenant compounded with each receiver on his entrance, 
 for the; rents payable during the life of that receiver, it 
 would evidently be equivalent to paying a fine of 1 -j- |P 
 at the end of the year in which each dies, and also at 
 the end of the year in which A dies. But the present 
 value of all the rents is a perpetuity diminished by the 
 value of an annuity on A's life, or | |A, And if this 
 be the value of a fine of 1 + |P, then | | A, divided by 
 Q 4 
 
232 ESSAY ON PROBABILITIES. 
 
 1 4- |P, gives the value of a fine of 11. in the same cir- 
 cumstances. Hence the following 
 
 RULE. From the value of a perpetuity subtract that 
 of an annuity on A's life, and divide the remainder hy 
 the value of an annuity due on the renewal life at the 
 time of renewal. 
 
 If there be several lives in the lease, apply the pre- 
 ceding rule to each life, and add the results: for the 
 several contingencies do not interfere with or depend 
 upon each other, nor will the case of more lives than 
 one in one lease differ from that of several leases each on 
 one life. The most convenient method is as follows : 
 
 RULE (for several lives). Multiply the value of a 
 perpetuity by the number of the lives, and subtract the 
 sum of the values of the annuities on the different lives: 
 divide the result by the value of an annuity due on the 
 renewal life at the time of renewal. 
 
 QUESTION. An estate of the clear annual value of 
 a per annum is to be leased on n lives, A, B, C, &c., 
 with liberty to renew at the end of each year in which 
 a life drops, the best life in the tables being P : what 
 fine should be paid, on the supposition that the pur- 
 chaser is to have a given rate of interest for his money? 
 
 RULE. Find the value of the perpetuity of a per 
 annum; multiply it by the value of an annuity due on 
 the renewal life at the time of renewal, and divide by 
 the excess of n times the value of a perpetuity of I/, 
 over the sum of the values of annuities on the lives of 
 A, B, C, &c. : the quotient is the value of each fine 
 required. But if a sum s be paid down, and the rest 
 of the value of the estate is to be paid in fines, then 
 subtract s from the perpetuity of a per annum, before 
 using it in the preceding rule. $% 
 
 EXAMPLE 1. The lives in possession, A, B, and C, 
 are 35, 48, and 60 years of age, and the fine paid on 
 renewal is 3007. What is the present value of all the 
 fines, using the Carlisle tables, and interest at 4 per 
 cent ? * 
 * I have taken Mr, Milne's example,.in order to show the accordance of 
 
ON THE VALUE OP REVERSIONS. 233 
 
 = 25 |A = 16-041 |P=19-792 
 
 3 |B = 13-419 1 
 
 |C= 9-663 
 
 75 20-792)35 '877(1 -7255 
 
 39-123 39-123 300 
 
 35-877 517-65 
 
 In Milne 517-6296 
 
 The present value of every single pound of the fine is 
 1'7255L, which,, multiplied by 300, gives 517'65L 
 
 EXAMPLE 2. All things remaining as above, the pre- 
 ceding lease, worth 120/. per annum, is purchased for 
 2500/. and a contract for fines on renewal. What 
 should the fine be ? 
 
 |= 25 l + |P = 20-792 
 
 120 500 
 
 3000 35-877)10396(289-77 
 
 2500 289-782 in Milne. 
 
 500 
 
 The answer is 289'77/. 
 
 QUESTION. What yearly rental should the fines be 
 considered as amounting to ; and what should be paid 
 by the lessee annually to an insurance office which would 
 undertake to pay all the fines as they become due ? 
 
 These two questions are the same, and the answer to 
 both is, the yearly interest upon the present value of 
 the fines. Thus, in the first preceding example, the 
 lessor's interest, at 4 per cent., is worth 20'7/. per 
 annum for ever ; which the lessee might either pay to 
 his landlord, as a commutation of fines, or to an insur- 
 ance office, which should take them upon itself. 
 
 QUESTION. What is the present value of the next 
 fine upon the renewal of the first life which drops of 
 the three, A, B, C ? 
 
 the rules. The slight difference arises from Mr. Milne's rule requiring an 
 interpolation, which he has very properly thought it not worth while to 
 make. I have taken more decimal places than those previously given, iw 
 order to show the accordance more clearly. (Milne, page 365.) 
 
234 
 
 ESSAY ON PROBABILITIES. 
 
 This is evidently the present value of 11. to be re- 
 ceived upon the failure of the joint existence of A, B, 
 and C, and is to be found (page 214.) by subtracting 
 (ABC, the value of an annuity on the joint lives, from 
 that of a perpetuity, and dividing by the present value 
 of a perpetuity due. 
 
 QUESTION. If the tenant wish to exchange one life 
 for another and a better, how much should he pay to be 
 allowed to do so ? 
 
 RULE. If the fine be I/., subtract the value of the 
 inferior life from that of the better one, and divide the 
 difference by the value of a perpetuity due on the re- 
 newal life at the time of renewal. To exchange two 
 lives, or three lives, use the sums of the values of the 
 better lives, diminished by that of the inferior ones. 
 
 QUESTION-* If the lessor have only a life interest in 
 the estate on which he grants leases for lives, what is the 
 value of his interest ? 
 
 In strict justice to future holders, it ought not to be 
 worth more than the rental calculated in the last ques- 
 tion but three, continued for his life. But it is the 
 nature of this species of property, that the life interest 
 of the holder is subject to considerable fluctuations of 
 value, the preceding annuity being at one time less and 
 at another greater. A lessor, for instance, who enters 
 when the lives in possession are very old, himself being 
 very young, has nearly a certainty of one fine on account 
 of each, and riot much less than an even chance of a 
 second, while his prospect of a third may be worth 
 calculating. But a lessor who enters at an advanced 
 age, against lives which are very young, has a present 
 interest in each coming fine, which may be determined 
 by finding the present value on/., to be paid when the life 
 drops,on condition the lessor survives(page 223). Indeed, 
 whatever may be the lives in the lease, provided the 
 lessor enters at an advanced age, his interest is deter- 
 
 * This problem, properly treated, would be of extreme complication, and 
 I do not remember having seen it proposed. The method in the text is an 
 approximation. 
 
ON THE VALUE OF REVERSIONS. 235 
 
 mined with sufficient accuracy by finding the values of 
 insurances on the lives in possession, on condition of the 
 lessor surviving. But when the lessor is young, I am 
 not aware of any rule to which I would trust, as making 
 as good an approximation to the value of his life interest 
 as can be made in other cases. Each case must be deter- 
 mined by its own details ; and it will always be safe to 
 begin by calculating what we may call the mean value,, 
 namely, the annuity first mentioned; which may, for 
 any thing I see to the contrary, be a perfectly correct 
 mode of proceeding in all cases. 
 
 QUESTION. If the lessor should refuse to renew, and 
 if it be pretty certain that his successor will adopt the 
 same course, what is the present value of the tenant's 
 interest ? 
 
 Evidently the clear annual value of the estate, con- 
 sidered as an annuity upon the longest of the three lives, 
 the value of which is determined in page 208. 
 
 This awkward contrivance for limiting the rights of 
 corporators over property is prejudicial in its effects, both 
 upon the tenants and the lessors. The former, holding 
 an interest of a comparatively precarious character, have 
 not the same inducement to improve their property which 
 is felt by leaseholders for fixed terms of years. On 
 the other hand, the lessor, in all cases in which he has a 
 personal interest in the proceeds of the estate, has two 
 distinct periods of temptation to an act of equivocal 
 morality. If he be young, he may, as it is called, run 
 his life against those in possession ; that is, refuse all 
 renewals, upon the prospect of a large ultimate gain 
 from the falling in of the old leases : if he be old, he 
 may induce the tenants, by offering easy terms, to change 
 their old lives for young ones, thus impoverishing the 
 successor, by leaving him nothing but long leases, or 
 leases on young lives. 
 
 It must very often be a question for the lessee, 
 whether it would not be his wisest plan to refuse all 
 renewal, and to insure a certain sum upon the last sur- 
 vivor of the three lives by which he now holds. The 
 
236 ESSAY ON PROBABILITIES. 
 
 prudence of such a step must depend upon the fine 
 demanded for a renewal. In the case>f church leases, 
 I believe it could not often be desirable : and certainly 
 not if they are let so much below their value as has 
 been asserted. But if the fine demanded should be 
 exorbitant, it would then become cheaper to insure the 
 longest of the lives in possession than to pay the demand. 
 The premium for such insurance would be found as in 
 page 214, the value of an annuity on the longest of the 
 lives having been previously found in page 208. 
 
 QUESTION. The average value* of a living is s per 
 annum, and the proper allowance to the incumbent for 
 the performance of the duties is v } the unavoidable 
 expenses at entrance are /for each new incumbent; and 
 the value of the life of the present incumbent is |A, 
 while that of the new incumbent will be |P. What is 
 the value of the perpetual advowson of such living? 
 
 RULE. The value of a reversion of 3 /. per annum 
 after the death of A being found (by subtracting !A 
 from the value of a perpetuity), and multiplied by the 
 excess of s over v, will give the value of the perpetual 
 advowson, as it would be but for the expenses at entry. 
 For these, deduct the present value of a fine/, payable 
 at the end of the year in which each incumbent dies, 
 the value of each pound of which determined by di- 
 viding the reversion aforesaid by one year's purchase 
 more than |P. The difference is the net present value 
 of the advowson required. According to a frequent 
 practice of valuing advowsons, in which the expenses 
 at entrance are neglected, the buyer pays them twice 
 over, 
 
 * This should include all real profits : for instance, the value of the par- 
 sonage house as a residence, considered as taken on a strict repairing lease. 
 
237 
 
 CHAPTER XL 
 
 ON THE NATURE OF THE CONTRACT OF INSURANCE, AND 
 ON THE RISKS OF INSURANCE OFFICES IN GENERAL. 
 
 IN laying down the following considerations, I think it 
 right to state most explicitly, that I intend no direct 
 reflection upon any office now in existence, or whose 
 establishment is contemplated. In a set of societies so 
 numerous and varied as those in question, there must be 
 details in one and another, of which any individual, who 
 turns his attention to the subject, must disapprove ; but 
 the studied exclusion of the name of every office what- 
 soever will, I hope, be taken as earnest of my desire to 
 confine myself to the enabling other persons to discover 
 the grounds of censure, without directing their attention 
 to the quarter in which they are to be found. I have 
 not much fear that any part of this chapter could have 
 been misconstrued into allusion ; and perhaps even the 
 present disclaimer may have no other effect than to make 
 some imagine that there must be more meaning some- 
 where than is openly expressed. Leaving such lovers 
 of mystery to their search, I proceed to the subject be- 
 fore me. 
 
 The avowed levellers in politics, a rare and scanty 
 sect among educated persons, would have an argument 
 of some force, from considerations of general expe- 
 diency, if it could not be shown that any attempt 
 to equalise property would be attended with a vast di- 
 minution of the fund itself, so that the great majority* 
 
 * The only great alteration of property which is likely to be agitated, is 
 the question of the national debt, the entire abolition of which is not with- 
 out its advocates. This enormous sum, as it appears, is really little more 
 than one year's income of the country, and perhaps not so much, if all 
 the colonies be considered. The honesty of a sponge not being considered, 
 there would still remain'.this question : Would the ultimate loss occasioned 
 by the subversion of such a debt, amount to a year's income of the country ? 
 If so, there would be no gain arising from the abolition of the debt. 
 
238 ESSAY ON PROBABILITIES. 
 
 would really have even less than they now possess, and 
 also less facilities of increasing their stock. The differ- 
 ences of talent and of life would still remain, constantly 
 working towards a restoration of the ancient inequalities, 
 in which they would almost instantaneously show their 
 power. A division of property, to be permanent, must 
 be accompanied by a division of intellect, a division of 
 manual skill, and a division of life; nor would the sum of 
 the parts make up the whole in anyone of the four, except 
 the last. A law which should tax the property of all who 
 live beyond a certain time of life, to provide an addi- 
 tion to the maintenance of the widows and children of 
 those who die before it, would not be so utterly im- 
 practicable, nor so pernicious, as an attempt at equaliza- 
 tion of fortune, intellect, or skill. Such a law would, 
 however, fail in its operation, by the mere difficulty of 
 arranging its enormous details, the frauds to which it 
 would give rise, and the temptation to idleness which it 
 would hold out to the young. A small community, 
 consisting of members of known honesty, living under 
 a government in which they reposed entire confidence, 
 and possessing sufficient inducements not to relax in 
 their exertions, by the certainty of a provision for their 
 families, might live under such a law : and such com- 
 munities actually do exist, under the name of insurance 
 companies. 
 
 If a large number of persons, all of the same income 
 and prospects, and all certain of the same duration of 
 life, were to choose a common bank in which to de- 
 posit their savings, each laying by a given proportion 
 of his income, it is obvious that each would receive the 
 same sum as the rest at his decease ; but, if the lives 
 were of unequal and uncertain duration, this result 
 would no longer be produced. It might, however, be 
 attained by a covenant, that all sums paid in should 
 remain till all were dead, and then be equally divided 
 among the executors of the parties. Such a bank might 
 be called an equalization office, and it would present 
 
ON THE NATURE OF INSURANCE. 239 
 
 the first approximation towards an insurance office such 
 as those which at present exist. 
 
 As yet we have not mentioned the interest of money. 
 Suppose the equalization office to pay no interest ; and 
 suppose all the lives to be 20 years of age,, such as are 
 described in the Carlisle tables, the average duration of 
 which is 41^ years. If, then, every person pay 11. per 
 annum, each will ultimately receive 41-J^? which is the 
 mere compensation of the inequality of life. Such per- 
 sons would enter into a mutual covenant, by which 
 those who live beyond the average term would divide 
 the surplus of their savings among those who fall short 
 of it. 
 
 Probably, if the following question were put to all 
 those whose lives are now insured, What is the advantage 
 which you derive from investing your surplus income 
 in an insurance office ? more than half would reply, 
 The certainty of my executors receiving a sum at my 
 death, were that to take place to-morrow. This is but 
 half an answer ; for not only does the office undertake 
 the equalization of life,, as above described, but also the 
 return of the sums invested, with compound interest. 
 
 No one can form an accurate idea of such an esta- 
 blishment, who does not consider it as a savings bank^ 
 yielding interest, and interest upon interest. This is 
 the reason why an office which charges for its insurance 
 more than it is worth, as an insurance, may nevertheless 
 put its contributors in a better position than they could 
 have held if there had been no such institution. To 
 make this clear, let us consider the working of a simple 
 investment office. A large number of individuals sub- 
 scribe a sum, which they intrust to an individual or a 
 company to employ, yielding them the return at some 
 fixed, but distant, period. Let each share be 1001. 
 The best thing which an individual could do with such 
 a small sum, so as to have perfect security for its return, 
 would be to invest it in the funds, at 3 per cent. He 
 might also invest the interest, and thus obtain com. 
 pound interest : but it is not easy for an individual to 
 
240 ESSAY ON PROBABILITIES. 
 
 do this. Unless he provide an agent to draw the divi- 
 dends immediately on their becoming due, various cir- 
 cumstances will happen to prevent the immediate in- 
 vestment of the interest. It is not at all an unfair 
 calculation to suppose that, upon each half yearly divi- 
 dend a month will he lost, so that nominal compound 
 interest for 42 years will only be really for 35 years. 
 A single pound, therefore, laid up by a man of 20, 
 and improved for the average term of his life, at 3 
 per cent., would only become 3^1. ; while, in the hands 
 of a person who lost no time, it would become 4^/., or 
 nearly a pound more. On the other hand, a company, 
 or a skilful individual who can command large sums of 
 money, can always make the best interest which the 
 market will afford. The funds, from the security of 
 their tenure, and the conveniences which they offer, 
 will always, in ordinary times, represent the lowest rate 
 of interest which money will yield. Other investments, 
 which offer better interest, are generally only accessible 
 to those who can command considerable sums, and are 
 frequently attended with risk ; so that it requires know- 
 ledge to distinguish between the sound and the unsound. 
 A company, employing the whole time of a person or 
 persons skilled in money matters, and having continual 
 large investments to make, can realise not only more 
 interest, but so much more, that there shall remain a 
 surplus worth considering, after the skill employed has 
 been paid for. It is not assuming too much to say 
 that, all expenses paid, they can command 3-^ per cent, 
 compound interest. More than this, they can obtain 
 such interest without any delay in investing the interest. 
 The process is extremely simple : it' is not difficult to 
 ascertain what sum should lie permanently at the 
 banker's, in order to meet current expenses, so that the 
 banker has general directions to buy stock as soon as the 
 balance in his hands exceeds that sum ; and all cash 
 received is paid into the bank at the close of each day. 
 Suppose it should happen that ten individuals paid 1001. 
 into the office on account of life insurance premiums, in 
 
ON THE NATURE OF INSURANCE. 241 
 
 the same hour in which the executors of a deceased 
 contributor received a claim of 100/. The hundred 
 pounds,, which, in the theory of the process, should be 
 sold out, or otherwise set free, to meet the claim, is in 
 its practice supplied by the new premiums, so that the 
 premiums of those contributors are making interest from 
 the hour in which they are paid. But there is always an 
 unemployed sum lying at the banker's. This is true; but 
 the interest of that sum is the salary of an officer of the 
 institution, namely, the banker himself. All such ex- 
 penses paid, I believe it may be stated, with correctness, 
 that an investment office can net 3^ per cent, compound 
 interest. Hence I/., improved during the average life 
 of an individual aged 20 years, would become 4*/. 
 The institution we have hitherto described is simply 
 an office for the investment of premiums and the equal- 
 ization of results : it becomes an insurance office when 
 it undertakes to pay a fixed sum for a fixed premium, 
 at the end of a given time after the decease of the party. 
 It then begins to incur a risk of a twofold character : 
 in the first place, the lives which it undertakes to insure 
 may not die, one with another, in or near the same 
 manner as those from which the tables were constructed ; 
 in the second place, the rate of interest, upon which it 
 calculates the premiums, may be higher than it is after- 
 wards able to obtain. According to the Carlisle table, 
 the premium which should now be paid to insure 100/. 
 upon the life of an individual aged 20, is one pound 
 seven shillings, or 1-32/., at four per cent. According 
 to the Northampton table, and at three per cent., the 
 same premium should be 2'2/. Taking the first pre- 
 mium, and assuming its table, the office will not be 
 sure of avoiding loss, until the party has lived 35 years; 
 by which time the premiums, with their accumulated 
 interest, will have passed 100/. It is a little more 
 than 2 to 1 that a life of such an age shall live beyond 
 32 years after the contract. Taking the premium of 
 the Northampton table, the party must live 28 years 
 before the office can gain by him ; and it is about 10 to 
 
 R 
 
242 ESSAY ON PROBABILITIES, 
 
 7 that he will outlive this term. We have now to 
 ask, What are the principles which should guide the 
 office in the determination of its premiums, it being 
 remembered that there is an absolute security required, 
 and that the remote chance of bankruptcy, which is 
 almost essential to the ordinary run of commercial affairs, 
 is not to be encountered ? 
 
 The basis of the tables is the observation of the lives 
 of a comparatively small number of individuals; it being 
 well known that the value of life varies considerably in 
 passing from one class of society to another. Now, we 
 have seen (page 91-). ^ at we cannot depend upon a 
 law of probability, derived from a limited number of 
 instances, with the same degree of confidence, as upon 
 one which we know to exist a priori, if we were sure 
 beforehand that the great average of life in England 
 was according to the Northampton, or any other table, 
 we might rely upon such a document as being extremely 
 likely to exhibit, with small fluctuations, the future 
 course of the lives of the two thousand or ten thousand 
 persons insured in any given office. Let such a table 
 be assumed, and let the premiums be so calculated, that 
 it shall be a thousand to one against any ruinous amount 
 of fluctuation, taking the law of the tables as that which 
 will certainly prevail in the long run. Then return 
 from the hypothesis to the truth, and, taking the num- 
 ber of lives from which the table was actually formed, 
 say 5000, suppose another 5000 persons to have com- 
 menced an insurance office. The degree of fluctuation 
 within which it was 1000 to 1 that the future re- 
 sults should be contained, is now larger than before, 
 in the proportion of the square root of 2 to 1, or 
 in that of 14 to 10, nearly. Larger premiums would 
 then be required to make ruinous fluctuation as unlikely 
 as upon the preceding supposition. These considerations, 
 which may easily be reduced to calculation by the rules 
 in chapters IV. and V., will serve to show that there 
 may be danger in the assumption of any table formed 
 from experience ; and they ought to operate powerfully 
 
ON THE NATURE OF INSURANCE. 23 
 
 as a caution against lightly admitting a change of pre- 
 miums, on the authority of any small number of facts. 
 But more particularly should they be attended to in the 
 formation of new varieties of contingency offices, the 
 chances of which have not yet stood the test of ex- 
 perience. 
 
 But there are reasons why the premiums of an in- 
 surance office need not be so high as the very limited 
 number of data in their tables might seem to require. 
 If the fluctuations from the average, which are within 
 the most cautious definition of reasonable probability _, 
 were all to be encountered at once, or might be en- 
 countered at once, it is difficult to say what premiums 
 should be considered as too high. But this cannot be 
 the case, unless, indeed, a pestilence should single out 
 the members of an insurance office, or an earthquake 
 should, by one extraordinary event, swallow them all up 
 in the place where, by a most remarkable coincidence, 
 they were all assembled together. Such extreme cases 
 are not worth consideration ; and we may take the 
 chances of life and death as distributed over a large 
 number of years. In the meanwhile the surplus fund 
 increases at compound interest; and the problem is, not 
 whether a given number of lives will, on the whole,, 
 drop so much before the predicted time that a given 
 fund will be destroyed, but whether this can happen so 
 fast, that it will outrun the increase of the fund at 
 compound interest. If, indeed, there were compound 
 mortality to set against compound interest ; that is, if 
 the number of deaths must become larger from year to 
 year, or if the rate of mortality were increasing, the 
 fear of such a result might be entertained ; but all 
 experience is on the other side, and tends to show that 
 the value of life is increasing, instead of decreasing. 
 
 The tables of an insurance office must be considered 
 as collections of limited data, the premiums deduced 
 from which are increased by a percentage, to meet the 
 possible fluctuations of mortality. As soon as these 
 tables are formed, and the directors have published 
 
244 ESSAY ON PROBABILITIES. 
 
 their proposals, an insurance office is created, with all 
 these fundamental characters already described, and 
 which are but ill represented by the term. The word 
 insurance or assurance has given rise to some wrong 
 notions, and it will be worth while to examine the 
 nature of the contract. 
 
 A and Co. engage with B that, in consideration of 
 11. a year, paid by him during his life, they will pay 
 20/. to his representatives as soon as he shall be dead. 
 Both parties run a risk ; A and Co. that of having to 
 pay B more than they receive ; B, that of paying more 
 than will at his death produce 20/. But the risk of 
 the office is of immediate loss, and that of B, of deferred 
 loss : that of the former is also continually lessening, and 
 that of the latter increasing ; until, should B live long 
 enough, both risks become certainties. If the in- 
 surance be only for a term of years, B runs the risk of 
 losing his premiums altogether. 
 
 The office does not inquire what reason B may have 
 for insuring his own life or that of another person, nor 
 do any possible contingencies, except those of life, affect 
 the office calculations. We cannot, therefore, be too 
 much surprised at the ignorance shown by that judge* 
 f who declared that life insurance t was of its own nature 
 a contract of indemnity ; that is to say, if, by any 
 lucky chance, B can be proved to have accomplished 
 the object for which he insured by other means, he has 
 no claim upon the office. The circumstances are as 
 follows ; and the absurd conclusion is law, and 
 would be practice, if the insurance offices had not re- 
 fused to acknowledge the decision^or protect themselves 
 by the precedent. A and Co. covenanted with B to 
 pay 500/., if C should die within the term of seven 
 years next ensuing, in consideration of the usual pre- 
 mium. C did die within the term ; and A and Co., in 
 
ON THE NATURE OF INSURANCE* 245 
 
 answer to a claim of 500/., replied, that the intention 
 of B in insuring the life of C, was to obtain security 
 for the payment of a debt of 5001., due by C to B, 
 which debt had been already paid by C's executors: 
 consequently they owed nothing to B. An action was 
 brought by B, and defended by A and Co. on the above 
 plea ; and a special case being made, the point was de- 
 cided by the court of King's Bench against the plain- 
 tiffs ; thereby establishing the principle, that life in- 
 surance is a thing similar to fire or ship insurance ; 
 namely, a contract of indemnity, to be fulfilled with 
 allowance for salvage. 
 
 The defendants' case rested upon the asserted nature of 
 the contract, and the statute 14Geo. III. c. 48., which 
 enacts, that " no greater sum shall be recovered from 
 the insurers than the amount or value of the interest 
 of the insured in such life." The act does not state 
 at what time this interest is to be reckoned, but the 
 plaintiffs contended that the time of death was the 
 meaning of the statute ; the defendants averred, and 
 the court decided, that the time of bringing the action 
 was to be understood. The plaintiffs contended that 
 the debt was not the object of insurance, but the life 
 of the insured ; the court decided, that fc This action is, 
 in point of law, founded upon a supposed damnification 
 of the plaintiffs, occasioned by the death, existing and 
 continuing to exist at the time of the action brought ; 
 and, being so founded, it follows, of course, that if, 
 before the action was brought, the damage which was 
 at first supposed likely to result to the creditor were 
 wholly obviated and prevented by the payment of his 
 debt, the foundation of any action on his part, on the 
 ground of such insurance, fails." This sentence con- 
 tains nothing but very good sense, and, no doubt, very 
 good law : but the application of it was accompanied 
 by a mistake as to the nature of the damnification 
 which the plaintiffs had sustained. The counsel on 
 both sides, the court, the insurance office, and the 
 plaintiffs themselves, showed a very partial knowledge 
 u 3 
 
24(5 
 
 ESSAY ON PROBABILITIES. 
 
 of the nature of the contract ; and I make no doubt, 
 that almost every person who heard it agreed with the 
 court, however much they might impugn the decision 
 on other grounds, that the damage* to the creditor 
 ff was wholly obviated and prevented by the payment 
 of his debt." 
 
 In order to show that such was not the case, we 
 must suppose that an exactly similar transaction had 
 taken place before any insurance office existed. How 
 this could have been may not be apparent, if we 
 take the notion which the law formerly entertained of 
 such an office ; namely, that it is a species of gambling 
 house: but if we prefer to consider it as a savings 
 bank, with an equalization system (page 238.), which is 
 unquestionably the correct notion, we may return to the 
 circumstances which the case would have presented had 
 there been no insurance. C, a person whose credit has 
 become doubtful, is indebted to B to an amount which B 
 could not afford to lose ; consequently, B, knowing that 
 his chance of payment is precarious, resolves to diminish 
 his expenses, hoping by economy to restore to his family 
 the sum which he may have lost by his engagements with 
 C. He collects, accordingly, a small fund, which he 
 places with his banker, avowing the purpose of its 
 collection. In the mean time C dies, and some friends 
 pay off his debts, and that due to B among the rest. 
 The latter having now no further occasion for such 
 economy, draws upon his banker for the amount, and 
 is answered, that, since the purpose of the saving was 
 fulfilled by the payment of C's debt, he, B, has no 
 further claim upon his own money. An action is 
 brought, and the courts decide that the banker is right, 
 and that B, having really attained his object in one 
 way, has no right of property in the proceeds of another 
 attempt to serve the same purpose. 
 
 * The defendants paid into court a sum somewhat less than the amoun 
 of the premiums they had received from the plaintiffs, doubtless as a pre- 
 caution, incaselhe court or jury should think the premiums ought to 
 have been returned. 
 
ON THE NATURE OF INSURANCE. 247 
 
 The only distinction between the case just put and 
 that which actually occurred is, that the banker was a 
 person who gained his profits by receiving such savings 
 during a contingent term, and guaranteeing a fixed 
 sum ; standing the loss, if there were any, and paying 
 himself for it out of the gain which would accrue in 
 another instance: the premium having been calculated 
 so as to insure a moral certainty of profit upon the 
 average of similar cases. It is not pretended, on either 
 side, that the chance of indemnification at the hands of 
 C's executors was made to lessen the consideration paid 
 by B for the guarantee ; and the legal iniquity of the 
 decision may, I think, be made clear, as follows : - 
 
 It will hardly be disputed, firstly, that the legislature 
 is the judge of what shall constitute valuable consider- 
 ation ; and, secondly, that a consideration which is ex- 
 pressly allowed to be good in a statute, should be ad- 
 mitted as such in the decisions of the courts. Now, 
 the contract of insurance, be it gambling, or be it not, 
 rests entirely upon the permission given by the law to 
 consider a high chance of a small sum as good con- 
 sideration for a low chance of a large sum. If I now 
 pay 2/. of premium for 100/., in case I should die in 
 a year, and if my executors can maintain an action for 
 100/., it must be because the law sanctions the notion 
 that 2/., nearly certain, may, with consent of parties, be 
 considered as an actual equivalent for a distant chance 
 of 100/., as much so as one weight of silver for another 
 of bread, or food, clothing, and wages for personal 
 service. It is true that the same law, fearing certain 
 reputed immoral practices, to which the power of 
 making a particular bargain offers temptations, may 
 limit the circumstances under which it will permit 
 such bargains to be made ; but this is equally true in 
 regard to the other sort of contracts mentioned : 
 indeed, there is no sort of bargain which is not 
 under regulation. The law, then, allows risks, and 
 permits unequal chances to be compensated by giving 
 odds ; the courts declare that, after the cast shall have 
 R 4 
 
248 ESSAY ON PROBABILITIES. 
 
 been made, and one of the parties shall have stood his 
 risk, which turns out in his favour, the other party 
 shall receive an ex post facto release from the condi- 
 tions of his bargain, because circumstances afterwards 
 arise, which, had they existed* at the time of making 
 the bargain, would have made it illegal. The several 
 principles on which the decision was founded, well 
 carried out, as they say in parliament, would require 
 that the previous contracts of a man who becomes 
 insane should be null and void ; that the meat which 
 a man buys for his dinner should be returnable to the 
 butcher under the costt, if a friend should invite him 
 in the mean time ; and, in the case before us, supposing 
 that C should have outlived the term, and his debt 
 were paid, as before, then B might have brought his 
 action against the office, for the return of the premiums ; 
 alleging that, as it turned out, the office would have been 
 indemnified, and, therefore, should be considered as 
 having run no risk. 
 
 But, said the judge, the damage was " wholly " 
 obviated and prevented by the payment of the debt. 
 To try this point, let us make a debtor and creditor 
 account of the whole transaction. The following is 
 the way in which it will stand. 
 
 Cr. Dr. 
 
 500 worth of goods fur- ^500 paid by C's executors, 
 nished to C. 
 
 Certain small premiums Those same premiums re- 
 paid to an insurance office, turned by the office, instead 
 with imminent risk of their of ^500. 
 entire loss; such premiums, 
 multiplied by the risk of loss, 
 as in chapter V. , being good 
 legal consideration for a re- 
 mote chance of gaining 500, 
 and so considered by both 
 parties. 
 
 * This is admitting more than is absolutely necessary ; for, unless there 
 were mathematical certainty that a third party would step in and pay C's 
 debts, it is difficult to see how B's insurable interest would cease. 
 
 f The sum paid into court by the insurance office, was less than the 
 amount of the premiums,: but the plaintiffs waived that point. 
 
ON THE NATURE OF INSURANCE. 24$ 
 
 The advantage of the moral security which a contract 
 of insurance gives is obvious in the transaction which 
 led to this decision ; namely, the insurance of the life of 
 a creditor by a debtor at his own expense. Commercially 
 speaking, such a transaction is literally this: C owes 
 500/. to B, who, doubting his chance of payment if the 
 debtor should die, buys 500/. from a third person, and 
 makes believe that it is the 500/. which C owes him. 
 Morally speaking, it is the determination of B to retrench 
 his own expenditure, as soon as he finds that a part of 
 his property consists in bad debts. This the office 
 enables him to do in a manner which will make the 
 retrenchment proportional to the necessity for it. In 
 the mean time, it is much to be wished that the law of 
 life insurance were settled upon a fixed basis, which 
 should proceed upon such a definition of the contract as 
 has been here explained, and not on the notions which 
 have been drawn from a supposed analogy between it 
 and the insurance of a ship or a house. The effect of 
 the present state of the law is, that the offices have no 
 law except that of honour, which, though it more than 
 suffices for the protection of the insured, yet may at 
 any time involve the offices in the necessity of paying 
 really questionable policies, without having the means of 
 submitting to open examination the point on which 
 they wish to resist. Policies of insurance are sold daily 
 to persons who have no interest in the lives of the 
 insured parties, on the faith of the good conduct of the 
 offices. If an office were to resist the payment of a 
 policy so transferred, say on the ground of fraudulent 
 representation, the parties so resisted might give out 
 that the opposition of the office arose out of an intention 
 to cover themselves by the present letter of the law. 
 Neither could such a case be carried into court without 
 proof that the plaintiffs possessed that insurable interest 
 in the life of the deceased which the law requires. 
 
 The nature of the contract, both in law and usage, 
 having been laid down, we must next ask what are the 
 means which the offices employ to reduce the risk so as 
 
250 ESSAY ON PROBABILITIES. 
 
 to render themselves safe against fluctuation. The state 
 of opinion upon this matter is somewhat unsettled ; one 
 party advocating the practice of approaching near to the 
 line which separates security from insecurity ; another 
 insisting upon what appears to the first a most super- 
 fluous degree of caution. Without expressing an opinion,, 
 I will describe the various risks, and the method of 
 avoiding them which has usually prevailed. 
 
 1. The insecurity of data, that is,, of existing tables 
 of mortality. This divides itself into two parts ; that 
 relating to the young and middle aged,, and that relating 
 to old lives. With regard to the first, the data might 
 probably be obtained in sufficient numbers to justify a 
 considerable degree of confidence in them as to the 
 chances of a single life, or even of a considerable number ; 
 but when the number of lives is to be as great as the 
 number of persons who may choose to offer themselves, the 
 considerations in Chapter IV., again adverted to in page 
 242, present themselves in force. I am not aware that 
 any writer on the subject in this country has formally 
 taken into consideration the uncertainty of tables, arising 
 from their limited numbers, except Mr. Lubbock, who 
 has made use of (Cambridge Phil. Trans., and Treatise 
 on Probability Lib. Usef. Know.) the correction which 
 the probability of living a given number of years should 
 receive on that account. But, considering the probable 
 errors of the data, this correction is small, and the 
 question how far an office proceeding upon such data 
 can deal with the public to any amount is yet in its 
 infancy, though the necessity for its consideration is 
 approaching, and it is one of vital importance to the 
 interests of the middle and lower classes. 
 
 The constructor of tables of mortality draws a number 
 of balls from an urn which contains an infinite number, 
 and, having sorted them into red, blue, black, &c., 
 presents them to the world as a necessary representation 
 (or very nearly so) of the proportions in which those 
 colours are scattered throughout the whole urn. He 
 commits an error which is in all probability very small, 
 
ON THE NATURE OF INSURANCE. 251 
 
 and which has hitherto been carefully guarded in the 
 deduction of office results. But there is a much more 
 important question behind. Suppose the calculator had 
 undoubtedly succeeded in exhibiting the real law of 
 mortality, and that it were quite certain the next 
 hundred million inhabitants of Great Britain would die 
 in the manner pointed out in a table. In such a 
 case, many will say, the office may charge the real 
 premiums deduced from the table, with a very slight 
 addition for expenses of management. They may leave 
 the fluctuations to take care of themselves, and trust in 
 the long run. This assertion I now proceed to discuss. 
 
 If the banker of a gaming-table were to follow the 
 same plan, that is, if he were to stake against ah 1 comers 
 with only just enough of advantage to cover expenses, 
 he would infallibly be ruined at last. It might not be 
 in this year nor the next, nor in this century nor the 
 next; but ruined at some time or other he must be 
 (see page 110, and also Appendix I.). If the case 
 of the office managers were precisely analogous to that 
 of the bankers of the gaming-table, I would repeat 
 with as much confidence of the former what I have said 
 of the latter. But, in the first place, the fluctuations of 
 mortality are not, by very much, so great as those which 
 take place in the assortment of cards, nor even so great 
 as those which take place in harvests, in the price of 
 provisions, &c. This is much in favour of the insurance 
 office; but who can say how much? 
 
 In the second place, the fluctuations of mortality 
 have of themselves a tendency to create opposite fluc- 
 tuations. Thus, a very sickly season carries off the weak, 
 and deprives the succeeding years of those who were 
 most likely to have died; causing, therefore, a season of 
 remarkable health. This is a very important item in 
 the theory of the fluctuations of mortality, and there is 
 nothing similar to it in the case of the gaming house. 
 It reduces annual fluctuation itself to a species of regu- 
 larity, and is, perhaps, the sufficient reason for the slight- 
 ness of the total fluctuations. 
 
252 ESSAY ON PROBABILITIES. 
 
 In the third place, with a merchant "or a banker, the 
 liability to a demand and the demand itself come so 
 nearly upon one another, that real insolvency and bank- 
 ruptcy are never far asunder. When credit cannot be 
 sustained by monthly, and even daily, proofs of substance, 
 it takes its departure altogether : but it is not necessarily 
 so with an insurance office, of whose existence it is the 
 essence to be always receiving consideration for bills 
 which, one with another, have a long time to run. 
 Such an establishment, as will presently more distinctly 
 appear, may be in reality insolvent many years before 
 the symptoms of bankruptcy come on. As no large 
 concern of the kind has hitherto failed, it is difficult to 
 say how they would finally come on : but this much is 
 certain, that an insurance office which could really pay 
 only ten shillings in the pound might, by introducing a 
 better system, or by mere force of circumstances, not 
 only recover its ground, but ultimately become exceed- 
 ingly profitable. But I throw this part of the argument 
 (though it shows a strong principle of vitality inherent 
 in the constitution of such offices) out of the question ; 
 for, surely, no sane and honest person would trifle with 
 important matters so far as to assert that the possibility 
 of temporary insolvency, to be redeemed by the chapter 
 of accidents, or prudence, when it was wanted, should 
 enter into the deliberate calculations on which men 
 should be invited to stake the subsistence of their 
 children. . 
 
 If the last contingency be rejected, that is, if it be 
 held absolutely necessary to calculate on permanent 
 solvency, both real and apparent, then I assert that 
 there is not sufficient ground to gainsay the conclusion, 
 that any insurance office charging only real* premiums 
 (increased for expenses of management) must inevitably 
 have its phases of solvency and insolvency, at the very 
 best. Begin by considering the office as identical in 
 principles with the gaming house, and beat down the 
 
 * By real premiums I mean those which only cover the risks of life. 
 
ON THE NATURE OF INSURANCE. 253 
 
 certainty of ruin which is thus known to exist, if they 
 play upon equality of chance, by allowing for the first 
 two of the three preceding considerations. There must 
 still remain more risk than it is safe to face of insolvency, 
 either temporary or permanent. And though, in con- 
 sequence of the smallness of the portion which the office 
 risks upon one hazard, a very small mathematical ad- 
 vantage might he sufficient, yet, so long as the necessity 
 for such an advantage exists, and its absolute amount is 
 unknown, so long must an office guard itself by requiring, 
 in the first instance, a sensible addition to the real 
 premiums. 
 
 With regard to the old lives, there is an additional 
 ground of insecurity. Not only are the probable errors 
 of the tables exceedingly large with respect to them, 
 but, from the smallness of the number which will 
 enter an office, there will be a liability to great fluc- 
 tuations in the results of transactions with them. The 
 first circumstance would prevent the second from be- 
 coming ruinous, but at a risk of loss to the capital 
 invested by younger lives : it is usual, therefore, to 
 exclude all lives above a certain age from entering the 
 office, upon the principle that no risks are to be taken 
 of which the numerical amount is not well understood, 
 and of which the number is not large enough to secure 
 an average. But, since the tables of old lives are only 
 a very unsatisfactory approximation, and since the pre- 
 miums payable by young lives depend in part on the 
 chances of those lives becoming very old, how does it 
 happen that the insecurity of the latter part of the 
 tables does not affect the premiums throughout? It 
 does affect them, but not sensibly, for the following 
 reason. If, assuming the Northampton table, we sup- 
 pose a person aged 40 to insure his life, we see that 
 the portion of the present value of his insurance which 
 depends upon his dying in his 85th year is very small, 
 on two accounts : firstly, because the chance of his 
 living to the age of 84 is very small; and, secondly, 
 because the present value of a sum to be received 
 
254 ESSAY ON PROBABILITIES. 
 
 45 years hence is small, compared with that of a sum 
 now due, or receivable soon. This last consideration 
 works as follows: When^a percentage comes to be added 
 to the whole present value or to the premium deducible 
 from it, for the security' of the office, that percentage 
 being made upon a much larger sum than the present 
 value just mentioned, a very trifling deduction from 
 the whole additional sum will cover a very serious 
 mistake in the mortality of the older years. For ex- 
 ample, in the Northampton tables, the chance of 
 40 living to 85 is about T / (T , and the present value of 
 11. due in 45 years is about 5*. at 3 per cent. From 
 this it follows that 100/., to be paid if a person aged 40 
 dies after 85, cannot be worth so much as 1'25/. But 
 the present value of the whole insurance is 53*8/. ; and 
 if this be the real value, and 10 per cent, be added for 
 security, then 5'38/. is added; so that if T25/. were 
 considered as added solely for the chances after 85 years, 
 it follows that we might consider ourselves as having 
 allowed for not being able to calculate the chances on 
 old lives within one half, and as having added 8 per 
 cent, to the whole present value besides. Thus, it ap- 
 pears that our comparatively little knowledge of old 
 life, though not unimportant, yet can be made to be of 
 less importance than might have been expected by 
 one who has not considered the matter. Of course, 
 the preceding reasoning must be considered only as 
 addressed to a person to whom, for any thing he sees 
 to the contrary, it is of as much consequence to know 
 the entire law of mortality in the insurance of young 
 lives as of old ones. 
 
 There is one use of the table of old lives, by which 
 an insurance office might make its existence very pro- 
 blematical, to use a gentle term ; namely, by inverting 
 the order of security, and selling guaranteed* benefits, 
 which are to increase with the age of the party, and to 
 be accumulated solely out of his premiums. To take 
 
 * This, of course, does not apply to divisions of profit gained, but to 
 contracts for sums to be accumulated after the date of the engagement. 
 
ON THE NATURE OF INSURANCE. 255 
 
 an extreme case, suppose an office should name a pre- 
 mium for which it would undertake to pay 100/., if the 
 party dies in the subsequent year ; 200/, if he dies in 
 the second subsequent year ; 300/. if he dies in the 
 third ; and so on. In this case, every fluctuation which 
 bears the appearance of lengthened life, were it only to 
 amount to deferring one death for a single year, would 
 be a new claim of WQL upon the office. , The fluc- 
 tuations which are observable in the very old lives, would 
 become matters of extreme importance; and though, 
 assuming a given table fairly to represent the average, 
 premiums might be calculated which should be sufficient 
 in the long run, yet there is no possibility of saying 
 what capital might become necessary to meet the fluc- 
 tuations of half a century. Such an attempt as tjie 
 preceding can be compared to nothing but gambling, 
 and its stability to nothing but that of a ship running 
 before the wind, with all the heavy cargo lashed to the 
 topgallant mast. Other cases might be mentioned, 
 which should partake of the same species of danger in 
 a less degree ; but every attempt to guarantee increased 
 benefits with increasing life should be looked at with 
 caution, as being of its own nature the addition of 
 risks in which the errors of the unsafe part of the 
 tables are, or may be, multiplied into importance. 
 There is an opposite plan, which I am not aware has 
 been tried, but which I should strongly recommend to 
 any new insurance office, as being of a safe character, 
 and also meeting the views under which many insure 
 their lives. It is that of insuring decreasing sums, 
 ; upon either fixed or decreasing premiums. Many 
 i persons are so situated, that they will be able to provide 
 for their families if they live a few years. To provide 
 for the hazardous period, they are under the necessity 
 either of insuring for their whole lives, that is, of buying 
 more insurance than they want; or of insuring for a 
 fixed term of years, which does not meet several con- 
 tingencies ; or of making complicated survivorship in- 
 surances. But, if a person so circumstanced found, by 
 
256 
 
 ESSAY ON PROBABILITIES. 
 
 estimation of his income, that he should want 5000/. if 
 he died in one year, 4800/. if he died in the second, 
 and so on, it would be desirable that he should be able 
 to insure for these several sums, contingently upon his 
 dying in any of the several years to which they are 
 made to belong. Various modifications of this scheme 
 might be proposed, all having this difference from the 
 usual plans, in that the latter enable a person to make 
 a provision for his family, while the former would only 
 supply the deficiencies which his death would leave in 
 the proceeds derived from other sources. In an ap- 
 pendix (on the value of increasing annuities) will be 
 found the method of calculating the present values of 
 such insurances. 
 
 2. The possible fluctuations of the rate of interest. 
 These may be either general and national fluctuations, 
 or alterations in the value of the property held by the 
 office. The former cannot be guarded against or pre- 
 dicted ; and, as the rate of interest has been slowly 
 falling for centuries, there is some reason to suppose 
 that this depreciation of money may continue. But 
 this gradual sinking of the rate of interest may be only 
 partly dependent upon the fall of profits, and part may 
 be due to the increase of security. I question whether 
 the political economist has found the historical materials 
 for determining this most important element ; namely, 
 the extremes of interest at which loans were contracted 
 in the different periods of our history. The legal 
 maximum of interest, at the beginning of the reign of 
 James I., was 10 per cent., and at the end of the cen- 
 tury, 6 per cent. But, at the beginning of the century, 
 land was commonly bought at 20 years' purchase, and 
 never at less than 1 6 years' purchase ; while at the end 
 of the century it was still at 20 years' purchase. No 
 method of proving such a point is better than the 
 examination of the works on interest which appeared 
 during the century. If, then, we suppose, with Adam 
 Smith, and I believe with most others, that the changes 
 in the legal maximum of interest followed, and did not 
 
ON THE NATURE OF INSURANCE. 257 
 
 precede, those of the market, there is good ground for 
 imagining that the diminution of the rate of interest 
 between borrower and lender (from 10 to 4 per cent.) 
 has arisen more from the increase of security than 
 from any other cause. If such be the case, there is 
 strong presumption that the fall is near its end. But, 
 if the preceding surmise should not be well founded, 
 and if (as was the case in Holland during a part of the 
 last century) the rate of interest should fall until go- 
 vernment can borrow at 2 per cent., and others at 3 per 
 cent., the change may happen in a manner which will 
 seriously affect the insurance offices, unless it should 
 come about so gradually that they will have time to 
 introduce new premiums for incomers, and surplus to 
 meet the claims of those to whom they are already 
 engaged. It is, in the meanwhile, a question well 
 worth the attention of those connected with them, what 
 the causes have been which have determined the rate of 
 interest, and the rapidity and amount of its variations. 
 
 The offices depend for the existence of their present 
 system upon the national debt ; and they are differently 
 situated from the government which owes the debt, in 
 that the engagements of the latter are all maxima, 
 while theirs are minima. If the rate of interest should 
 really fall, the government will have the means of 
 reducing the interest of the debt, never to rise again ; 
 while the offices have, in fact, guaranteed to their ex- 
 isting customers a rate per cent., which is never to fall 
 during their lives. The rate assumed by the offices 
 should, therefore, never be above that at which the 
 government can borrow. 
 
 With respect to the second reason for a variation in 
 the rate of interest, as experienced by the office, namely, 
 a depreciation in its own property, such an establish- 
 ment, not being allowed to run the usual risks of 
 mercantile life, should not deal in any but the most 
 secure investments, and those which depend on the 
 personal security of others should be altogether avoided. 
 The only point which it is incumbent to mention, in 
 s 
 
258 ESSAY ON PROBABILITIES. 
 
 addition to general cautions, is a mistake to which such 
 offices are subject in the valuation of their property ; 
 namely, the estimation of different items by their re- 
 puted worth, or by the price which was given for them, 
 instead of the actual income which they produce. We 
 shall see the effect of such a mistake in considering 
 the proper method of inquiring into the state of their 
 affairs. 
 
 The precedent are the contingent risks to which an 
 office is subject : its certain expenses are the ordinary 
 charges of management, including rent, salaries, interest 
 of sums lying at the banker's, &c., advertisements, and 
 the commission, as it is called, which most of the offices 
 pay to those who bring them business. 
 
 Commission, in general, means either a per centage 
 paid to a factor for the transaction of business, or a volun- 
 tary relinquishment in favour of the person who brings 
 business of a part of the profit which the said person, 
 being honourably free to choose between one competitor 
 and another, has brought to the trader who, there- 
 fore, allows the commission. It answers to the profit 
 which the retail dealer is allowed by the wholesale mer- 
 chant from whom he buys. But, when an insurance 
 office announces to the solicitor, attorney, or agent of a 
 party desiring to insure, that they will allow him a 
 liberal commission, the term has a different meaning. 
 As between one office and another, the attorney is in a 
 judicial capacity ; and, as regards his client, he is 
 already the paid protector of the interests of another per- 
 son. He has, therefore, no liberty of choice between one 
 office and another, but is already bound to choose that 
 which he judges best for his client. All who have 
 written on the subject of late years have attacked this 
 'bribe, for such it is ; but they have directed all their cen- 
 sures against the offices, as if they were the only parties 
 to blame. If, indeed, the bribe had been offered to the 
 needy and ignorant only, this partial distribution of 
 blame might have been allowed; but when the parties 
 who receive the bribe are men of education, and moving 
 
ON THE NATURE OF INSURANCE. 259 
 
 in those professions which bring the successful to afflu- 
 ence, I do not see the justice of allowing them to escape. 
 I have little doubt that an increasing sense of right and 
 wrong will banish this unworthy practice, either by 
 failure of givers or receivers. A barrister cannot offer 
 an attorney commission on the briefs which he brings, 
 nor can a physician pay an apothecary for his recom- 
 mendation ; a jury never receives a hint that the plaintiff 
 will give commission on the damages which they award ; 
 and the time will come when the offer of money to a 
 person whose unbiassed opinion is already the property 
 of another, will be deemed to be what it really is, 
 namely, bribery and corruption. It is one among many 
 proofs how low is the standard of collective morality ; 
 and how easy it is for honourable individuals to do in 
 concert that from which they would separately shrink. 
 
 It appears, then, from all which precedes, that the 
 ordinary risks of an insurance office are alterations of, 
 and mistakes in determining, the rate of mortality, and 
 reduction of the rate of interest : which are guarded 
 against by assuming a rate of mortality beyond all 
 question greater than exists, and a rate of interest below 
 that which the funds will yield. At the peace of 1815, 
 every insurance office used the Northampton table at 
 3 per cent. This was at a time when the real rate of 
 interest was higher than at present, and the offices must 
 have made considerable profit. It was well known that 
 they did so ; and, accordingly, new offices were formed, 
 and have continued to be formed up to the present 
 time, some upon lower premiums than others, and most 
 of them returning all or part of the profits to the in- 
 sured. At the same time, an opinion has become very 
 prevalent, that it is possible for such offices to maintain 
 their ground at much lower rates of premium than 
 those in use ; a notion which I proceed to examine. 
 
 Mr. Finlaisgn, whose experience in such matters is 
 well known to the public, and for whose opinion I en- 
 tertain a high respect, stands foremost among those 
 who contend for low rates of premium, having pub- 
 s 2 
 
260 ESSAY ON PROBABILITIES. 
 
 lished a table, which he certifies to be " abundantly safe 
 and practicable," and " so high as to insure, beyond all 
 doubt, a surplus of profit ;" which table charges pre- 
 miums at the ages in which most insurances are made, 
 falling short of those actually in use about 15 per cent. 
 These premiums are supposed not chargeable with the 
 management of the office, and at a rate of interest of 
 3-J- per cent. I take Mr. Finlaison's proposition as a 
 modified one, for there are some which go beyond it. 
 
 On the other hand, the late Mr. Morgan could never 
 be persuaded that it was safe to abandon the North- 
 ampton table ; and considered that the superior vitality 
 of the members of the Equitable was altogether a con- 
 sequence of their being select lives. He seems to have 
 thought that, whatever run of success an office might 
 have, it should always be on the look out for reverses ; 
 and that even the enormous accumulations of his office 
 were no more than the seven good harvests, a provision 
 for other seven of a different character. 
 
 In holding an opinion which comes between that of 
 these two authorities, I form it on a ground on which 
 neither would have rested the truth or falsehood of his 
 own. I consider the fluctuations of mortality as very 
 little to be feared, compared with those of the rate of 
 interest. It has long been matter of observation, that 
 the phenomena of the natural state of man vary but 
 little compared with those of his social condition. The 
 price of provisions swings to and fro like a pendulum ; 
 the variations of mortality which follow its changes^, 
 though sensible, bear no proportion to the magnitude of 
 their cause. The rate of interest has been halved 
 within the memory of man, and a heavy war might 
 double it again. That same war, with all its casualties, 
 direct and indirect, included, would not alter the mor- 
 tality of the country by any serious amount. I con- 
 sider it, then, as next to certain, that the insurance 
 offices have more to look for, whether as matter of hope 
 or fear, from the fluctuations of the rate of interest, 
 than from those of mortality. If the interest of money 
 
ON THE NATURE OF INSURANCE. 2l 
 
 could be made as stable as the duration of human life, 
 I could then see no objection to an immediate and con- 
 siderable reduction of the premiums charged, to an 
 amount at least equal to that proposed by Mr. Finlai- 
 son. But here lies the difficulty; that these tables, at 
 3-J- per cent., already involve a rate of interest which 
 the office cannot much exceed, if at all ; so that the secu- 
 rity which the precautions, nominally made against mor- 
 tality, really afforded against fluctuations of interest, is 
 partially or wholly destroyed, while no safeguard is in- 
 troduced to supply its place. 
 
 An office raises its premiums either because its pre- 
 vious notions of existing mortality were wrong, or be- 
 cause it finds that it had calculated upon too high a 
 rate of interest. A mistake on either of these points 
 might be compensated by a contrary mistake as to the 
 other. Now, though the offices which existed during 
 the war have demonstrated that the mortality and rate 
 of interest together yielded a large profit, it by no 
 means follows that one of those causes of profit may be 
 fully corrected, while the other has been correcting 
 itself. To make both perfectly accurate, would bring 
 the office to the very line which divides security from 
 insecurity ; a position which it would not be safe to en- 
 deavour to maintain. We are already in a very dif- 
 ferent position as to the rate of interest, which has been 
 gradually falling since the war. The opinion as to the 
 extent to which tables of mortality may be safely cor- 
 rected, is formed upon arguments which dwell on the 
 favourable rate of mortality, without sufficiently consi- 
 dering the counterpoise (for, as far as it goes, it is a 
 counterpoise) existing in the alteration of the value of 
 money. 
 
 Assuming the necessity of calculating upon a rate of 
 interest something less than that which can actually be 
 attained, I should think that no office would be justi- 
 fied in supposing more than 3 per cent., with tables 
 which are sufficiently high to come any ways near to the 
 actual experience of mortality. With regard to one 
 s 3 
 
252 ESSAY ON PROBABILITIES. 
 
 point, and that of fundamental importance,, namely, the 
 possibility of a still further fall in the rate of interest, 
 it may even be doubted whether, with such tables, a still 
 lower rate of interest should not be allowed. But I 
 am not here advocating one result or another, but only 
 the necessity of taking into consideration all the pos- 
 sible sources of danger. To those who would use tables 
 of greater vitality, I concede that, so far as mortality 
 alone is concerned, the alteration is admissible ; and for 
 this reason, that experience shows human life to be 
 of a higher value than formerly ; but the concession 
 is accompanied by the requisition of a lower rate of 
 interest, and for the selfsame reason, that experience 
 shows the value of money to be less than it was. 
 
 The preceding conclusion is reinforced by the con- 
 sideration, that the worst is to be made of every cir- 
 cumstance in our previous calculations. When mor- 
 tality is diminishing, the whole diminution is not to be 
 allowed ; but when it is increasing, a larger increase is 
 to be contemplated. A person who would walk dry- 
 shod on the sea shore, must not advance so fast as the 
 ebb, and must retreat faster than the flow. Upon this 
 consideration, the necessity of providing for a further 
 fall in the interest of money is increased ; or, which 
 amounts to the same thing, the amount by which the 
 favourable alteration in the rate of mortality may be 
 allowed to affect the premiums is less than it would be 
 if it were certain that the value of money would remain 
 unaltered. 
 
 A very common security or guarantee to the public 
 is the announcement of a large subscribed capital, either 
 paid up in whole or part, or liable to be called for. 
 This is equivalent to the personal security of a number of 
 shareholders, collectively making themselves answerable 
 for the engagements of the office up to a certain amount. 
 Such a provision in itself is an obvious good; but, it being 
 remembered that this security must be paid for, it be- 
 comes a question how much it is worth, and whether it 
 may not be bought at too high a price. It is easily 
 
ON THE NATURE OF INSURANCE. 263 
 
 understood that the consideration which tempts men to 
 lend names or money to an insurance office, is the offer 
 of payment for the risk, or of higher than market interest 
 for the money. If the capital be paid up, the office 
 makes common interest upon it, which is returned, with 
 an augmentation, to the proprietors : if the capital be 
 only paid in part, or merely nominal, still the office has 
 to pay something more than it receives. 
 
 Now, I take it for granted that an office charging 
 premiums* such as are commonly demanded, managed 
 with prudence and economy, and successful in obtaining 
 business, will not ultimately need any capital at all: 
 firstly, because the premiums are such as must, in the 
 long run, realise a profit after paying the expenses of 
 management ; so that the only use of the capital would 
 be as a provision against extraordinary temporary fluc- 
 tuation: secondly, because a sufficient supply of business 
 renders the probability of ruinous fluctuation extremely 
 small, and altogether beneath consideration, j- Now, 
 since it is well known that the premiums are sufficient, 
 it follows that the only need which a commencing 
 insurance has of capital is for safeguard against the early 
 expenses of management, and against failure of business; 
 as follows. 
 
 The expenses of carrying on an insurance office, 
 though they vary somewhat with the amount of business, 
 yet do not by any means increase as fast. In the first 
 year of its existence, it would not be surprising if all 
 the premiums paid were swallowed up by house-rent, 
 salaries, &c. ; while, in process of time, increase of 
 business might reduce such expenditure to 2 per cent. 
 upon the yearly premiums. Some capital, therefore, is 
 necessary at the commencement ; for, if there be none, 
 
 , 
 
 least improbable, is a national bankruptcy. Any contingency, then, which 
 is much less likely than a national bankruptcy, need not be considered. 
 
 B4 
 
264 ESSAY ON PROBABILITIES. 
 
 those who first insure their lives are entirely dependent 
 upon the future success of the office. But this capital 
 need not be large : in the present state of things, an 
 engaged capital of one hundred thousand pounds is 
 certainly above the mark, even for an office which is 
 entirely without connection, and starts without one 
 single life insured. If, as very often happens, a tolerably 
 large number of customers has been obtained before the 
 prospectus of the office is announced, then a capital, the 
 interest of which will cover the expenses of management, 
 is sufficient. But here it must be observed that the 
 proprietors of this capital run some risk of losing a 
 portion of their principal, and a still greater one of 
 losing the interest for a limited time. This risk is the 
 greater the smaller the original subscription, and it 
 must be paid for accordingly. At the same time, it must 
 be remembered that the mere existence of the capital 
 diminishes the risk, by making it the interest of every 
 proprietor to procure business for the office. The con- 
 nection thus created is the secret of the successful start 
 which has frequently been made ; and it may be con- 
 sidered as very unlikely that an office will fail, from 
 want of business, which is so well supported in the first 
 instance as is implied when a capital of the preceding 
 amount is announced. 
 
 There is, however, one case in which a larger capital 
 is desirable, and even requisite ; that is, where an office 
 is established which is to insure some new and yet 
 untried risk. Whatever pains may be taken in such a 
 case to procure facts and deduce proper tables, there is 
 always a risk that the experience of the office may be at 
 variance with the facts of the tables. When, for instance, 
 the general conclusions drawn from the mortality of 
 towns were first applied to the insurance of life, it was 
 a risk of unknown amount as to whether the lives of 
 those who would come to insure would be of the same 
 class as those from which the tables were made. They 
 might turn out better, or worse. This risk has been 
 tried, and found to be in favour of the offices; but in 
 
ON THE NATURE OF IJS 7 SRANCE. 265 
 
 another speculation, of another kind, the same species 
 of risk might give a contrary result. 
 
 Among the sources from which the insurance offices 
 have drawn profit, we must reckon lapsed policies. It 
 has frequently happened that an individual insuring his 
 life has continued to pay the premiums for a few years, 
 and then, either through incapacity to continue the 
 payment, or because the object of his insurance was 
 otherwise attained, has allowed his policy to lapse to 
 the office by non-payment. The office, of course, is 
 benefited, but not, as might be supposed, by the total 
 amount of his premiums. What they have received 
 does not all become profit by the lapse of the policy, 
 but only that portion by which the premium for the 
 whole life exceeds the premium for a temporary in- 
 surance. Every premium which is paid by an insurer 
 contains the consideration given for the chance of his 
 dying in each and every subsequent year. If, then, he 
 remain a member of the office, and stand the risk of 
 death during a certain number of years, all such part of 
 his premiums as was consideration for the risks of those 
 years became due to the office, and was taken by the 
 office, as compensation for those risks, and cannot 
 therefore be said to fall to them as profit upon the lapse 
 of the policy. Two individuals, A and B, go to the 
 office on the same day, and insure their lives for the 
 same sum, A upon his whole life, and B for seven years. 
 A pays, say 101. of premium, and B 11. At the end of 
 seven years, A allows his policy to lapse, just at the 
 time when B's policy expires by its own construction. 
 What does the office gain by the lapse ? Evidently the 
 temporary annuity of 31. , by which the two premiums 
 differ. The 11. paid by A out of 101. is not more than 
 sufficient to pay his share of the claims which arose 
 during the years which he continued in the office : the 
 remaining 31. was a reserve for future years, which 
 becomes profit to the office on his declining to stand 
 the risks of those years. 
 
 Perhaps no part of the subject is less understood than 
 
266 
 
 ESSAY ON PROBABILITIES. 
 
 this. Persons having insured for their whole lives, and 
 being afterwards desirous to discontinue, are surprised 
 to find that they cannot get for their policies even as 
 much as the amount of their premiums, to say nothing 
 of interest. Each of them reasons thus : Since I did 
 not die, the office lost nothing by me, and, as it has 
 turned out, ran no risk : why, then, should they not 
 restore me the premiums which I have paid ? To which 
 it should be answered: Because the risk, which turned 
 out favourably in -your case, did not produce the same 
 result in another case ; and it is the very essence of an 
 insurance office, that those who live pay for those who 
 die. If you can induce the executors of those who have 
 died during your tenure of your policy to refund what 
 they have received from the office, with compound 
 interest, then the office will repay you your premiums, 
 also, with compound interest. The above-mentioned 
 reasoning of the insured party is much on a par with 
 that of the judge in Godsal's case. 
 
 A respectable weekly newspaper has lately allowed 
 the following doctrine to be promulgated in its columns ; 
 namely, that it is an undeniable fact, demonstrable by 
 the books of any insurance office, that very much the 
 larger portion of their profits has always arisen from 
 lapsed policies ! Till I saw that article, I could hardly 
 have believed that even a newspaper would have admitted 
 so palpable a mistake. On the supposition (no matter 
 how false) that all the back premiums of a lapsed policy 
 are, as they say in book-keeping, to be carried to profit 
 and loss, how could such an assertion be made, in the 
 face of the well-known fact, that premiums are deduced 
 from a table of much higher mortality than that actually 
 experienced ? Those persons who, one with another, 
 were expected to live twenty years, have lived twenty- 
 four years. A small proportion of them have allowed 
 their policies to lapse, enough to give, perhaps, a per- 
 ceptible profit to the office, but not enough materially 
 to increase its funds ; for it must be remembered that, 
 though the number of policies allowed to lapse bears a 
 
MANAGEMENT OF AN INSURANCE OFFICE. 267 
 
 proportion to the whole which might give some colour 
 to the preceding assertion, yet the value of these policies 
 is generally small. It is seldom that a policy is aban- 
 doned which involves a large sum, or on which many 
 premiums have been paid. If, instead of comparing 
 the number abandoned with the whole number of poli- 
 cies, we were to calculate the value of those policies, 
 and compare them with the value of all the liabilities 
 of the office, the former would be found a very small 
 portion of the latter. It is well that it has been so, for 
 this source of profit is diminishing as the subject 
 becomes better understood. It is known that a policy 
 of a very few years' standing is worth something, and had 
 better be sold at any price than abandoned. 
 
 All that precedes has reference to the relation in which 
 the office stands to the public, and to the collective body 
 of the insured. All dangers, and all remedies, have been 
 considered merely with reference to the general security 
 of the establishment, and without inquiring into the 
 effect produced on the relative interests of the insured. 
 Since it is the first principle that no interest of one or 
 the other class of -insurers must be consulted to the de- 
 triment of the whole, the order of discussion which I 
 have followed is necessary to the subject. It now re- 
 mains to treat of the internal management of an office, 
 and to this subject I proceed in the next chapter. 
 
 CHAPTER XII. 
 
 ON THE ADJUSTMENT OF THE INTERESTS OF THE 
 DIFFERENT MEMBERS IN AN INSURANCE OFFICE. 
 
 THERE is~not a circumstance against which it is necessary 
 to guard in the general management of an office, but what 
 
268 
 
 ESSAY ON PROBABILITIES. 
 
 is accompanied by this inconvenience, that the measures 
 adopted, whether of precaution or remedy, may be made 
 to press unequally upon the different classes of insurers. 
 If we take, for instance, a fixed rate of interest, suffi- 
 ciently below that which can really be obtained, we find 
 that many of those insured must pay their premiums at 
 a time when interest is comparatively higher, and vice 
 versa. With regard to the tables of mortality, most 
 probably (it has always so happened) a table which is 
 generally too high will be unequally too high ; so that 
 some classes of insurers will contribute more largely to 
 the safety fund than others. And even in the distri- 
 bution of the profits, however good the will may be to 
 apportion them duly, there are yet practical difficulties in 
 selecting an equitable method out of those which do 
 not require calculations of insupportable minuteness. 
 
 It will only here be necessary to dwell upon two 
 points, the distribution of the premiums, and the method 
 of appropriating the profits. 
 
 In the last chapter, in speaking of the use of too 
 high a table of mortality, as a safeguard, I was merely 
 considering the collective security of the office. There 
 are two different ways of answering the same end : 
 either by using a table of mortality confessedly too high, 
 or constructing premiums from a true table of mortality, 
 and increasing these by such a percentage as will pro- 
 duce the same receipts to the office. For general security, 
 these two plans are equally good ; but they may produce 
 very different consequences upon the relative state of the 
 members. For instance, the Northampton table, which 
 is the basis of most of those now in use, is certainly, as 
 already noticed, too favourable to the older lives. Mr. 
 Morgan gives the following table *, exhibiting the num- 
 ber who did die, and those who should have died, if the 
 Northampton table had been correct, all in the twelve 
 years preceding 1828. 
 
 * View of the Rise and Progress of the Equitable Society, London, 182S, 
 age 42. 
 
MANAGEMENT OF AN INSURANCE OFFICE. 269 
 
 Age. ( 
 
 Number. 
 
 Of whom 
 did die. 
 
 Of whom should 
 have died. 
 
 20 30 
 
 4,720 
 
 29 
 
 68 
 
 3040 
 
 15,951 
 
 106 
 
 243 
 
 40 50 
 
 27,072 
 
 201 
 
 506 
 
 50 60 
 
 23,307 
 
 339 
 
 545 
 
 60 70 
 
 14,705 
 
 426 
 
 502 
 
 7080 
 
 5,056 
 
 289 
 
 290 
 
 80 95 
 
 701 
 
 99 
 
 94 
 
 From this comparison, Mr. Morgan concluded that 
 the superior vitality of the young and middle ages was 
 the effect of selection, which wore out, so to speak, after 
 the age at which no new members were admitted; thereby 
 proving, in his opinion, at once the effect of selection, 
 and the excellence of the Northampton table. Now, it 
 obviously cannot prove both of these things : granting 
 the latter, it would certainly go a great way to prove the 
 former ; and granting the former, it does not impugn 
 the latter : which is all that can be said. But, if it 
 should happen that the mortality of the Northampton 
 table is near the truth at the older ages, and very much 
 above it at the younger, the sort of result shown in the 
 preceding comparison would follow of course ; and this 
 circumstance, demonstrated as it is by other and inde- 
 pendent tables, is, no doubt, the true explanation. 
 
 If such be the case, where is the fairness of using a 
 table which demands premiums very much larger than 
 the real risks from the young, while it admits older lives 
 on more easy terms ? Ought the older lives to enjoy 
 any privilege in this respect ? Quite the reverse ; for, 
 (page 253.) belonging to a class which is less known, 
 and entering also in smaller numbers, with results there- 
 fore more subject to fluctuation, the percentage, added to 
 the premiums deduced from a true table, ought rather to 
 be larger in the case of old lives than in that of young 
 ones. The best customers, both in number and quality, 
 ought not to come worst off. 
 
270 ESSAY ON PROBABILITIES. 
 
 The proposed table of Mr. Finlaison (page 259) affords 
 a striking illustration of this point. It is accompanied 
 by a table representing the average premiums of all the 
 offices. At the age of thirty, Mr. Finlaison proposes 
 to demand 17 per cent, less than the average of what is 
 now asked by the offices ; at the age of 60, this same 
 able and strenuous advocate of reduction would only 
 reduce the average premium of the offices by 3J per 
 cent. I now put down the present value of 100/., pay- 
 able at the end of the year in which a life drops, from 
 the Northampton and Carlisle tables, at 3 per cent., 
 and for different ages, together with the percentage 
 which must be taken from the former to reduce it to the 
 latter. 
 
 Age. 
 
 Northampton. 
 
 Carlisle. 
 
 Percentage of 
 difference. 
 
 20 
 
 42-8 
 
 33-9 
 
 20-8 
 
 30 
 
 47*8 
 
 40-0 
 
 16-1 
 
 40 
 
 53-8 
 
 47-1 
 
 12-5 
 
 45 
 
 57-2 
 
 50-8 
 
 11'2 
 
 50 
 
 60-9 
 
 55-4 
 
 9-0 
 
 55 
 
 64-6 
 
 60-9 
 
 5-7 
 
 60 
 
 68-6 
 
 66-5 
 
 3-1 
 
 65 
 
 729 
 
 71-1 
 
 2-5 
 
 In offices, then, which continue to use the Northamp- 
 ton table throughout, the safety rate is levied upon those 
 who enter at the age of 20, to the amount of 2 1 per cent, 
 out of the total sum they pay; while on those aged 65 
 it only amounts to 2^ per cent. The Carlisle table 
 represents the experience of the Equitable Society very 
 nearly. 
 
 Again, the Amicable Society now charges premiums 
 deduced from its own experience, and in which the 
 fundamental inequality of the Northampton table is 
 corrected. It will be worth while to compare the aver- 
 age of all the offices given by Mr. Finlaison, with the 
 
MANAGEMENT OF AN INSURANCE OFFICE. 
 
 271 
 
 present premiums charged by the Amicable. The sup- 
 position is for 100/. insured. 
 
 Age. 
 
 Average. 
 
 Amicable. 
 
 Mr. F.'s pro- 
 posed Premiums. 
 
 20 
 
 2 -02 
 
 ^2-03 
 
 #1*76 
 
 30 
 
 2-50 
 
 2-53 
 
 2-07 
 
 40 
 
 3-26 
 
 3'25 
 
 2-78 
 
 50 
 
 4.47 
 
 4-83 
 
 4-06 
 
 55 
 
 5-38 
 
 5-90 
 
 5-00 
 
 60 
 
 6-58 
 
 7'33 
 
 6-25 
 
 From such comparisons as the preceding, I have long 
 been of opinion that, safe as the offices are, each con- 
 sidered as a whole, the proportions of the premiums 
 demanded at different ages are, in the first instance, 
 inequitable. To a certain extent, the young are made 
 to work for the old ; that is to say, the person who 
 insures early in life, the more prudent of the two, is 
 made to pay a part of the premium of the one who 
 does not begin till he is old. 
 
 The evil is not so great as it might at first sight 
 appear, for two reasons : firstly, because those who enter 
 at the older ages are few in number compared with 
 those who begin between 30 and 50 years of age; 
 secondly, because many offices make compensation to the 
 younger members in the division of the profits. Still, 
 however, the inequality is of a sufficient magnitude to 
 demand alteration, which will be brought about in an 
 obvious way ; namely, by the younger insurers giving 
 the preference to those offices in which, premiums and 
 returns considered together, the inequality is the least. 
 
 There is another point, though not of so much con- 
 sequence, in which an inequality falls more heavily 
 upon the young than upon the old ; namely, the method 
 of paying the expenses of management. The yearly 
 contribution of every member to this fund ought to be 
 the same. Suppose, then, that from every premium a 
 given sum is subtracted, to answer this end, the in- 
 
272 ESSAY ON PROBABILITIES. 
 
 equality of the remainders is increased ; it being obvious 
 that any disproportion which exists between two numbers 
 is made larger by taking away the same from both. 
 
 The way to correct the inequality, without altering 
 the actual receipts of the office, is as follows. The pro- 
 portions in which the different ages exist in the office at 
 any one time can be pretty nearly found. Let the office 
 table of premiums be taken, and from it let an average 
 premium be formed, by taking into account as well the 
 several premiums, as the numbers who pay them, Suppose, 
 for instance, that A persons pay the premium a, B pay 
 b, &c. &c. ; then the average premium is found by 
 dividing the sum of the products of A and a, B and b, 
 &c., by the sum of A, B, &c. Let the actual average 
 premium be called P ; and let the average premium, 
 formed in the same manner from a true table of mor- 
 tality (in which a, b, c. are different, but A, B, &c. 
 the same as before), be Q. Let P exceed Q by k per 
 cent, of Q ; then the premiums given by the true table, 
 increased by k per cent, are those which should be sub- 
 stituted for the existing premiums, in order that all 
 inequalities may be corrected, without diminishing the 
 receipts of the office. It matters nothing, in the pre- 
 ceding rule, whether the premiums of what has been 
 called the true table are correct or not, so long as their 
 proportions are correct ; and one office might, by this 
 rule, adopt the proportions of another, without altering 
 its own receipts. 
 
 If such a process as the preceding were performed, 
 deducting from the receipts required by the office the 
 whole expense of management, and afterwards adding 
 the last-mentioned item in equal shares to all the poli- 
 cies, the distribution of the premiums would be theo- 
 retically perfect. It remains to consider the more difficult 
 part of the question, the method of dividing the profits. 
 
 Hitherto, I have had no occasion to speak of a most 
 important difference of system which distinguishes one 
 office from another ; the distinction of mutual and pro- 
 prietary. The former have no capital, except what arises 
 
MANAGEMENT OF AN INSURANCE OFFICE. 273 
 
 from their own accumulations, and each member is a 
 guarantee to the rest for the fulfilment of all engagements. 
 If the office possess a charter, this guarantee operates no 
 further than to pledge the premiums already paid by any 
 member for the discharge of all claims which arise be- 
 fore his own, since a corporation is considered in law 
 as an individual. If, on the other hand, there be no 
 charter, the whole fortune of every member is pledged for 
 the discharge of all claims. The risk, however, at the com- 
 mencement is not great in character, and small in amount; 
 and the quantity of risk diminishes so much faster than 
 the amount increases, that it may safely be said there is 
 nothing in the commercial world which approaches, even 
 remotely, to the security of a well established and pru- 
 dently managed insurance office. 
 
 A proprietary insurance office has a capital, the 
 proprietors of which may or may not be insured in 
 the office, and for which such a bonus is paid, in ad- 
 dition to the market rate of interest, as is mentioned 
 in p. 263. It would perhaps be difficult, at the pre- 
 sent time, to establish a new proprietary office with a 
 very large capital. The public now begins to see 
 that much capital is not necessary, and that nearly 
 all the bonus which is paid for its use is so much taken 
 away from the savings of the insured, without any ade- 
 quate benefit received in return. One by one, the pro- 
 prietary offices must (as some have done) admit the 
 insured to a share in the profits : the necessity for which 
 will be taught by the decline of business, if not previ- 
 ously learnt. 
 
 The question as to how profits should be divided^ is 
 of the same nature in both species of offices; the differ- 
 ence being, that the offices which are partly proprietary 
 have less to distribute among the insured than those 
 which are mutual. The first inquiry must be, What is 
 the profit of an insurance office; and how is the amount 
 to be ascertained ? Firstly, as to the profit which an 
 insurance office may be expected to realise, judging by 
 the premiums they receive, and the mortality they have 
 T 
 
274 ESSAY ON PROBABILITIES. 
 
 hitherto experienced. Certain limits may be obtained,, 
 which may sometimes serve as a useful check. 
 
 Perhaps the average age of admission to an insurance 
 office is about 40, as many entering younger as older. 
 The average premium charged by the offices at that age 
 is 3'26/. per cent. Now the most extreme supposition 
 which can be imagined in favour of the insured is, that 
 the Carlisle table should be taken as the law of mortality, 
 and 4 per cent, as the interest of money. Upon these 
 suppositions, the accumulations of the office would 
 amount, upon a premium of 3*26/., at the death of 
 parties aged 40, one with another, to 13?^. But this 
 pushes every favourable supposition to its extreme, and 
 moreover allows nothing for expenses of management. 
 I am inclined to think, however, that the usual pre- 
 miums will, as long as the rate of mortality continues at 
 its present amount, yield about 1251. for 1001. nominally 
 insured, and perhaps something more. 
 
 It must not be left out of sight, that the offices con- 
 sider every person as having the age which he will attain 
 at his next birth-day. If, for instance, a person who 
 attains 40 years of age on the 12th of March were to 
 insure his life on the 13th, he would be said to be 41 
 years of age, and would have to pay accordingly. The 
 effect of this very proper * regulation is, that, one party 
 with another, all are half a year younger than their office 
 age. Again, all the tables are computed on the supposition 
 that interest is made yearly, whereas in fact it is made 
 quarterly. Circumstances of this sort, trivial as they 
 appear, do nevertheless produce a sensible effect in a 
 large number of years. To the above we must add the 
 profits arising from the purchase of policies, which is 
 always done by the offices on terms very favourable to 
 themselves ; fines for non-payment of premiums ; the 
 profits of lapsed policies ; and so on. 
 
 Leaving all speculation as to the probable profits, 
 I now proceed to show how to ascertain, from the 
 
 * Proper as long as there is no subdivision of a year. I think the offices 
 might very rationally divide the year into quarters. 
 
MANAGEMENT OF AN INSURANCE OFFICE. 275 
 
 actual statistics of an office, what its real condition 
 is. And here I must observe, that though in the con- 
 struction of premiums, a table of more than the real 
 mortality must be used, yet no such thing is absolutely 
 necessary in the valuation of its liabilities and assets. 
 Here truth, and not security, is the object ; and if by 
 any means a true table can be obtained, its results 
 should be calculated ; though I do not say that in the 
 declaration of profit, such results should be admitted 
 to their full extent. The most simple theoretical way 
 of conducting the process, is to ascertain the value of 
 every policy, as in page 218.; that is, to ascertain how 
 much should be given to the holder of each policy to 
 renounce his claim, the office also abandoning the future 
 premiums. When this- is done, it is obvious that the 
 office is not solvent, unless the assets arising from the 
 accumulations of former years be sufficient to pay the 
 values of all the policies, and thus to buy them all up. 
 Supposing the office able to do this, with a capital 
 remaining larger than would be necessary to create a 
 permanent fund for the expenses of management, the 
 surplus of that capital is profit. Otherwise, calculate 
 the present value of all premiums due to the office, and 
 also the present value of all claims to which it is liable. 
 To the former add the sum total of the assets of the 
 office, and to the latter add the present value of a per- 
 petuity equal to the expenses of management. Thus, let 
 
 P = present value of all premiums. 
 
 C present value of all claims. 
 
 A = total assets of the office. 
 
 M = present value of all expenses of management. 
 If then P and A together exceed and M together, 
 the office is solvent, and the excess is profit. 
 
 On each of these items a few remarks may be made. 
 
 (P.) All the parties who are of the same office age, 
 may have their several policies considered as one col- 
 lective policy, in respect of which the sum of the 
 premiums is paid as one premium, and the sum of the 
 T 2 
 
276 ESSAY ON PROBABILITIES. 
 
 possible claims is one claim. But as these premiums 
 are payable at all periods of the year, they may be con- 
 sidered as, one with another, due at six months after 
 the valuation, at which time the present office age of 
 the parties may be considered to be their real age. 
 
 (C.) All bonuses whicli have actually been added to 
 policies (if any) must be included in the claims ; and 
 the value of each claim must be carefully found, with 
 reference to the time after death at which it is paid. 
 (See Appendix the Second.) 
 
 (A.) The principal of the assets must be deduced 
 entirely by means of the income it yields, and must be 
 ascertained from the income by means of the rate of 
 interest assumed. On this subject, which contains a dif- 
 ficulty of a peculiar character,' see the Sixth Appendix. 
 
 (M.) Against the expenses of management may be 
 set, as far as they go, the incidental profits, when they 
 can be tolerably well ascertained. 
 
 The profit being thus found, and that share of it 
 which belongs to the insured (if the office be not 
 mutual), it remains to inquire, What principle of division 
 should be adopted ? And, firstly, it may be doubted 
 whether the whole of the profit is immediately divisible, 
 consistently with prudence. To use an astronomical 
 phrase, the increase of the surplus is partly secular, and 
 partly periodic ; that is to say, instead of a steady and 
 uniform increase, there is a fluctuating rate of aug- 
 mentation, compounded of that permanent rate which 
 the largeness of the premiums necessarily gives, and 
 the alternate accelerations and retardations occasioned 
 by the departures of the incidents of the several years 
 from the average. The only way of obtaining the per- 
 manent part of the surplus is by estimating it on the 
 average of a considerable number of past years, regard 
 being had to the relative, not the absolute, surplus. 
 Let us suppose, for instance, that the present value of 
 all claims is ascertained to be one million, and the present 
 value of all premiums 700,000/,, the office possessing 
 besides (clear of charges of management) 500, OOO/. : 
 
MANAGEMENT OF AN INSURANCE OFFICE. 277 
 
 there is then a surplus of 200,000/. ; which having been 
 accumulated out of premiums, and profits having been 
 regularly paid up to the present time, it may be pre- 
 sumed that the premiums themselves are capable of 
 maintaining this rate of surplus. The office must then be 
 presumed able to pay 120/> for every 100/. insured. 
 
 But it is important to note, that the present rate of 
 profit must not always be assumed as that which can be 
 permanently maintained. Suppose, for instance, an office 
 which begins for the first time to divide profits : its 
 accumulations are therefore, in part, the reserves of profit 
 which should have been added to former claims, had any 
 division of surplus previously existed. The same remark 
 may be necessary when any change is made in the way 
 of dividing profits, since the surplus existing at the 
 moment of the change is the result of a former state of 
 things. Thus, an office which has proceeded injudici- 
 ously, in making too large divisions, may possibly, when 
 it adopts a more prudent system, be justified in forming 
 a system which would require a larger surplus than the 
 one which it actually possesses at the time of discover- 
 ing the error ; for the then existing surplus has been 
 unduly weakened, and is not to be considered as repre- 
 senting the permanent effect of the improved mode of 
 proceeding. 
 
 I have stated, that the percentage which can be added 
 to each 1001. insured should be determined by the 
 average of a number of years. If this number be too 
 great, the incidental fluctuations of mortality may be 
 compensated ; but at the same time the real and secular 
 changes of mortality may be prevented from producing 
 their proper effect. As long as the value of life is in- 
 creasing, too long an average is a defect on the safe side : 
 but if it were diminishing, it might happen that the 
 mean of a number of preceding years would present a 
 higher result than would be consistent with security. 
 As yet the offices have had nothing to encounter except 
 the diminution of mortality, and its consequences ; but 
 T 3 
 
278 ESSAY ON PROBABILITIES. 
 
 in constructing rules for their own guidance, they should 
 be careful not to fall into such errors on the safe side 
 as become errors on the wrong side when circum- 
 stances change. I hold an opinion which I think, from 
 his writings, was also that of the late Mr. Morgan ; 
 namely, that an insurance office must consider the last 
 half century as having been a period of circumstances 
 singularly favourable for the formation and growth of 
 such institutions, more so than it would be wise to expect 
 for the future. Perhaps from five to ten years is the 
 length of time for which the preceding average should 
 be computed. 
 
 The valuations should, if possible, be made yearly. 
 No check which can be devised is so likely to be useful 
 as yearly valuation ; and it is absolutely necessary to any 
 system which gives the real amount of their premiums 
 to the insured. In a mutual insurance office, starting 
 without much capital, it would be madness to rest upon 
 any tables and to neglect valuations ; unless, as before 
 remarked, the returns made to the insured are meant to 
 be very much below their payments. And in conjunction 
 with yearly valuations should come yearly divisions of 
 profits, or something equivalent. There is, I believe, a 
 prejudice against frequent divisions in the minds of 
 many who have derived their ideas on the subject from 
 the former practice of offices. But surely, provided that 
 the proper amount of profit be divided yearly, and no 
 more, it matters nothing whether the apportionment be 
 made seven times in seven years, or once only, as far as 
 security is concerned. For it is to be remembered, that 
 yearly division of profits does not imply an annual expen- 
 diture, but only an annual distribution of future expen- 
 diture. In septennial divisions, one of two things always 
 takes place : either the profits are made contingent upon 
 a party surviving one or more periods of division, which 
 creates great inequalities between the lot of different 
 persons (the very thing an insurance office was intended 
 to avoid); or it declares beforehand, what the profits 
 shall be during periods of seven years. In the latter 
 
MANAGEMENT OP AN INSURANCE OFFICE. 279 
 
 case the annual division is unquestionably the more safe ; 
 since it is easier to predict the capabilities of one year 
 than of seven. 
 
 In writing upon any point connected with insurance, 
 the practice of the Equitable Society naturally suggests 
 itself. Nevertheless, I always consider that society as a 
 distinct and anomalous establishment, existing at this 
 moment under circumstances of an unique character. 
 It is the result of an experiment which it was most 
 important to try ; but which having been tried, need not 
 be repeated. Its history is briefly this : The Ami- 
 cable Society, which, in the year 1760, was the only 
 one existing, was originally founded rather on princi- 
 ples of mutual benevolence, than of mutual insurance, 
 as now understood. A certain number of persons (the 
 only restriction being that their ages should be between 
 twelve and forty-five), each paying the same sum 
 yearly, the whole fund of each year (or the greater 
 part) was divided among the representatives of those 
 who died within the year.* The Equitable Society 
 was founded upon the principle of apportioning the 
 payments to the risk of life. The tables were con- 
 structed by Dodson, who, as Mr. Morgan remarks, 
 fe for greater security assumed the probabilities of life 
 in London, during a period of twenty years ; which, 
 including the year 1 740, when the mortality was almost 
 equal to that of a plague, rendered such premiums 
 much higher than they ought to have been, even ac- 
 cording to the ordinary probabilities of life in London 
 itself." The truth of this remark will sufficiently ap- 
 pear, from comparing the average of the present office 
 premiums with the original Equitable premiums, as 
 given in the following table. And even these pre- 
 miums were increased on the most frivolous pretexts. 
 Thus female life and young life were considered as 
 more than usually hazardous, and paid for accordingly. 
 
 *~The Amicable Society now retains only one of its original characters ; 
 namely, that all members, whatever may be their age at death, or the term 
 of their continuance in the society, participate equally in the profits. 
 T 4 
 
280 
 
 ESSAY ON PROBABILITIES. 
 
 Age. 
 
 Equitable 
 Premium, 1771. 
 
 Equitable 
 Premium, 1779. 
 
 Average present 
 Premium. 
 
 
 s. d. 
 
 s. d. 
 
 s. d. 
 
 14 
 
 2 17 
 
 255 
 
 
 20 
 
 394 
 
 2 12 10 
 
 200 
 
 25 
 
 3 14 
 
 306 
 
 240 
 
 30 
 
 3 18 7 
 
 3 8 11 
 
 2 10 
 
 40 
 
 4 17 9 
 
 4 7 11 
 
 350 
 
 49 
 
 625 
 
 5 10 2 
 
 460 
 
 Mr. Morgan says, "that for the first twenty years, 
 the society possessed such an excess of income, that 
 being suffered to accumulate without interruption, it 
 contributed, in a great measure, to form the basis of its 
 future opulence." This circumstance, with the great 
 number of policies which were abandoned* in the 
 early stages of its career, and the increase of interest 
 during the war, are quite sufficient to explain the 
 wealth which the Equitable Society has accumulated : 
 to these must be added the parsimony with which, at 
 first, additions were made to the policies. The whole 
 was an experiment, on a graduated scale of premiums, 
 made with a caution, which, though it turned out to be 
 superfluous, could not be known to be such, except by 
 the result. It was at the same time a venture, and by 
 many considered as a hazardous one ; for instance, the 
 law officers of the Crown refused a charter, on account of 
 the lowness of the premiums. The hazard having been 
 run, and having turned out profitably, the proceeds be- 
 long to those who ran it, and to those who, by their 
 own free consent, became their lineal successors. Nor 
 is it the least remarkable circumstance connected with 
 this society, that the immense funds at its disposal have 
 been always opened, though under restrictions, to the 
 public. Though this has been done in a way which 
 renders the participation of the new insurer in the 
 
 * Perhaps Mr. Morgan's statement on this point may have led to the 
 statement alluded to in page 266. 
 
MANAGEMENT OF AN INSURANCE OFFICE. 281 
 
 previous accumulations a remote contingency, still it 
 is done, and by a body who might without any bar, 
 legal or moral, immediately close their doors, and divide 
 the whole among themselves. 
 
 I have made the preceding remarks, in order that it 
 may be clear how little the history or practice of the 
 Equitable Society should have any direct authoritative 
 bearing on the spirit in which the management of a more 
 modern office should be carried on. The general lesson 
 taught by it is, be cautious ; but, among other things, 
 be cautious of carrying caution so far as to leave a part 
 of your own property for the benefit of those who are 
 in no way related to you. If there be a Charybdis in 
 an insurance office, there is also a Scylla : the mutual 
 insurer, who is too much afraid of dispensing the profits 
 to those who die before him, will have to leave his own 
 share for those who die after him. Reversing the 
 fable of Spenser, we should write upon the door of 
 every mutual office but one, be wary ; but upon that one 
 should be written, be not too wary, and over it, f ' Equit- 
 able Society." 
 
 An insurance office has no existence separate from 
 that of its insurers ; and no public duty to fulfil, ex- 
 cept to collect, improve, and equalize their premiums 
 (p. 238.) : therefore, their most important object, 
 next to the fulfilment of their guaranteed engagements, 
 is the distribution of their profits in such manner that 
 every one may obtain his due share. The question now 
 becomes, What is the due share of each party ? This is, 
 in some measure, a question of previous contract, though 
 there are those who consider that there must be a right 
 and a wrong way. For instance, Mr. M'Kean, the 
 compiler of the tables alluded to in page 191.5 and of a 
 useful work * which accompanies them, says, " Our 
 conclusion, and a most important one, lies conspicuous 
 on the very surface. It is impossible that ALL the 
 
 * " Exposition of the practical Life Tables, &c. London : Butter worth, 
 Richardson, &c. 1837." This work is, I believe, sold separately. 
 
282 ESSAY ON PROBABILITIES. 
 
 offices above mentioned can be correct or just in their 
 laws for dividing tbe surplus. If the plan of the 
 Equitable is right, then most unquestionably the plan of 
 the Atlas is wrong, and great injustice is done to the 
 younger members, and so vice versa. But, is this a state 
 of things in which so important a system as that of life 
 insurance, based, as that system is, on mathematical 
 science, ought or can continue to exist ? Certainly not." 
 On this I observe, that though life insurance be an 
 application of (not based upon) mathematical science, 
 yet that the entrance of exact numerical reasoning is 
 subsequent to the admission of certain principles, and the 
 experimental acquisition of certain facts. It is not by 
 mathematics we learn that life is uncertain in individual 
 cases, but nearly certain in the mass that it is the 
 duty of every one to provide for his family and that 
 this can be done without contingency, if those who 
 survive the average term agree to surrender a part of 
 their substance to those who do not. Calculation will 
 point out the amount which, upon any given principle 
 of division, belongs to one or another of the insured ; 
 but before we can come to this point, it must be settled 
 with what intention the surplus was paid ; which may 
 be different in different offices. The following con- 
 siderations might be addressed to any person who in- 
 tends to insure his life : You are aware that the pre- 
 mium demanded of you is, avowedly, more than has 
 hitherto been found sufficient for the purpose, the reason 
 being, that it is impossible to settle the exact amount, on 
 account of our not knowing whether the future and the 
 past will coincide in giving the same law of mortality, 
 and the same interest of money. The surplus arising 
 from this overcharge, for the future existence of which 
 it is hundreds to one, is now at your own disposal, and 
 you must choose between one office and another, accord- 
 ing to your intentions with regard to its ultimate des- 
 tination. Firstly, if you doubt the general security of 
 the plan of insurance, and are desirous of an absolute 
 guarantee, independently of accumulations from pre- 
 
MANAGEMENT OF AN INSURANCE OFFICE. 283 
 
 miums, there are offices which will, in consideration of 
 the surplus aforesaid, pledge their proprietary capitals 
 for the satisfaction of your ultimate demand upon them. 
 Secondly, if, being of the opinion aforesaid, you think 
 the whole surplus too much to pay for the guarantee, 
 there are proprietary offices which retain a part of the 
 profit in consideration of the risk of their capital, and 
 return the remainder. Thirdly, if you wish the surplus 
 premium, as fast as it is proved to be such, to be ap- 
 plied in obviating the necessity of any further over- 
 charges, there are offices which divide the profits during 
 the life of the insured, by means of a reduction of pre- 
 mium. Fourthly, if you wish the surplus to accumulate, 
 and, feeling confidence in your own life, are willing to 
 risk losing it (the surplus, remember) entirely if you 
 die young, on condition of having it proportionally in- 
 creased if you live to be old, there are offices which 
 divide all or most of the profits among old members. 
 Fifthly, if you would prefer a certainty of profit, die 
 when you may, there are offices which at once admit 
 new members who die early to a full participation in all 
 advantages. The choice between these several modes 
 must be made by yourself, according to your own 
 inclinations, views of fairness, or particular circum- 
 stances. 
 
 There are three modes of division which deserve par- 
 ticular notice; namely, periodical additions to the policies, 
 periodical diminutions of premium, and addition to the 
 policy at death to an amount depending upon the assets 
 of the office, without reference to the time during which 
 the insured has paid premiums. I may, perhaps, be 
 thought to treat this subject with prolixity ; notwith- 
 standing, knowing that this part of the subject has 
 created more discussion of late years than any other, I 
 think an attempt to compare the principles of different 
 plans not out of place. 
 
 The considerations which follow will apply to all 
 offices which divide any profits whatever : the inquiry 
 being, not how much surplus should be divided, but in 
 
284 ESSAY ON PROBABILITIES. 
 
 what proportions a given sum should be divided among 
 the insured. 
 
 Let us return to the original constitution of an in- 
 surance office (page 238.). derived from the statement 
 of its main object ; namely, that it is a savings' bank 
 with a power of equalizing those results in which the 
 different durations of life would cause differences, Sup- 
 pose that such an office sets out with premiums imagined 
 to be no more than sufficient, but which are afterwards 
 found to be more than sufficient, leaving an admitted 
 amount of surplus in hand. The first thought would 
 be of restitution; namely, rendering back to each indi- 
 vidual the amount which he had bond fide contributed 
 towards the surplus. To do this properly, it must first 
 be settled whether the insurance office is one or many. 
 Does each age insure itself, or do the separate ages in- 
 sure both themselves and each other ? If the premiums 
 were properly proportioned, there would be no occasion 
 to ask this question : but if the incomers of one age pay 
 unduly as compared with those of another, then it is but 
 fair that they should receive in proportion. In the dis- 
 tribution of premiums, which I have described in p. 270., 
 it is equitable that a remedy should be provided, by 
 virtue of which those who enter the office young should 
 receive more than the rest. And it is, for this reason, 
 desirable that the proportions of the division should be 
 regulated by a true table of mortality. 
 
 Let P be the real premium, and P -j- p the office pre- 
 mium ; and let the death of an individual take place 
 after he has been n years insured, and just before the 
 (w+l)th premium is paid. If the office had been a 
 compound interest savings' bank, the deceased would, 
 at his death, have been entitled to the following amount. 
 
 P + p improved at compound interest for n years 
 P+p n-l 
 
 P -f p 1 year. 
 
 But under the conditions of insurance, the part P, with 
 
MANAGEMENT OF AN INSURANCE OFFICE. 285 
 
 its accumulations,, is the consideration for the sum in- 
 sured ; the remaining part p, with its accumulations, is 
 due under the name of profit or restitution, in a strictly 
 mutual office. 
 
 The application of the preceding method would require 
 that a calculation should be made once in every year of 
 the quantity p and its accumulations, for every individual 
 insured. This having been done, and the surplus 
 A -f P C having been calculated from a true table of 
 mortality, it is then known in what proportion any two 
 individuals insured are claimants upon this fund. Sup- 
 pose that p and its accumulations amount, in the case of 
 the persons X and Y, to 1001. and 150/. Suppose that 
 A + P C is 100,000/., and that the sum of all the 
 excesses of premium with their accumulations, of which 
 the 1 001. and 150/. just mentioned are items, is 120,000/. 
 It matters nothing that the last sum is greater than 
 100,000/., since we are not speaking of a fund on which 
 there are definite claims, but of one the nature of 
 which it is to be of uncertain amount. The use of the 
 items IOOL and 1501.., and of the sum total of 120,000/., 
 is to enable us to divide the real fund of 100,000/. among 
 those who raised it, in the proportions in which they 
 contributed towards it. Thus if X and Y were to die 
 in the year of the valuation, it would be fair that they 
 should receive such proportions of the 100,000/. as IOOL 
 and 150J. are of 120,000/.; that is, five-sixths of 1001. 
 and 150/. This method proceeds upon the principle that 
 all the excess of premium is taken in trust as a guarantee 
 for the main fund, and is to be returned if not wanted, or 
 such proportion of it as is not wanted. It confines the 
 insurance, or provision against the uncertainty of life, 
 entirely to a stipulated sum, and regards all that part of 
 the premium which is not really wanted to provide this 
 sum, for one man with another, as paid into a com- 
 mon savings' bank, in which no equalization is supposed. 
 The labour of making the calculations would, I ima- 
 gine, prevent any office from adopting the preceding 
 plan, so as to carry it into execution yearly. With a 
 
286 
 
 ESSAY ON PROBABILITIES. 
 
 good system, however, the difficulty of managing the 
 details of such a scheme would not be so great as at 
 first sight might be supposed. Upon its principle hang 
 the two first plans of division mentioned ; namely, 
 periodical additions to the policies, and periodical di- 
 minutions of premuim. In both of these, the advantage 
 of the insured is increased by the length of his life ; 
 that is to say, the excesses of his premiums are placed 
 to his credit in the first, and considered as having been 
 prospective payments of his future premiums in the 
 second. But nevertheless there runs through the offices 
 which adopt these plans more or less of a practice which 
 prevents the surplus from being divided among the in- 
 . sured in equitable proportions. Suppose that there is a 
 septennial bonus, as it is called, which was declared in 
 the year 1830. Immediately after the award, two 
 persons, A and B, aged 30 and 60, enter the office each 
 upon a policy of 1001. , and were both alive when the 
 bonus of 1837 was declared. This bonus is generally a 
 percentage, not upon the amount of premiums paid, but 
 upon the sum insured, and both would have the same 
 addition made to the 1 001. for which they have insured. 
 But have both contributed to the accumulations of the 
 office in the proportion which would render this mode of 
 division equitable ? To consider this point, remember 
 that a promise to pay, say 5/., at the death of a person 
 aged 67? is of much more value than the same at the 
 death of a person aged 37. The older life therefore 
 receives much more than the younger life. But he has 
 paid much more. That is true ; but at the same time 
 he has occasioned a greater risk to the office, and it is 
 the excess of his premium above the risk (and not the 
 whole premium) which the office acknowledges in de- 
 claring the bonus. From page 270. it sufficiently ap- 
 pears that the premiums of the older ages are already 
 too small in comparison with those of the younger : 
 this mode of dividing the surplus, therefore, only tends 
 to increase the existing injustice. The only remedy is, 
 to make use of the process laid down in the preceding 
 
MANAGEMENT OF AN INSURANCE OFFICE. 287 
 
 page ; and having ascertained the amount of what each 
 person has paid over and above what was necessary, to 
 consider each person as entitled to the sum which his 
 overplus would purchase at his death, if the bonus be 
 made by addition to his policy ; or to a diminution of 
 premium answering to the annuity on his life, which the 
 overplus would buy, if the bonus be made by diminu- 
 tion of premium. 
 
 The knowledge, therefore, of the real premium is 
 necessary for an equitable distribution of the surplus, 
 upon the supposition that the said distribution is made 
 on the principle of dividing the surplus fund among the 
 contributors in proportion to their contributions. Every 
 plan which, ceteris paribus, makes equal additions to the 
 policies of different ages, is inequitable. I repeat again, 
 that in the preceding cases, the principle of division 
 ought to be considered as arising from the combination 
 of an insurance office and a savings' bank ; the por- 
 tion of premium which covers the risk of life being 
 paid to the former, and the remainder to the latter. 
 
 The third method of division supposes the establish- 
 ment to be entirely an insurance office, and not at all 
 a savings' bank. Its object is to make the returns to 
 the different members both equal and equitable. Con- 
 sidering that the real risk of life is not perfectly ascer- 
 tained, and that if it were it would not be safe to re- 
 duce the premiums to the lowest theoretical safety-point, 
 such an office, instead of demanding a premium avowedly 
 too high for the sum insured, and engaging to return 
 all or part of the surplus, considers the sum insured as 
 indefinite, except only in so far as a minimum is named., 
 below which it is not to fall. Thus such an office, 
 receiving, say 3L of premium, from a person aged 34, 
 for what is called, in compliance with custom, a policy 
 of 100/., does in fact make the following bargain: The 
 office engages to return, at the death of the party, let 
 that take place when it may, such a sum as will represent 
 the average accumulation of an annuity of 31. continued 
 during the life of a person aged 34, be that sum more 
 
288 ESSAY ON PROBABILITIES. 
 
 or less ; with this additional limitation, that the office 
 undertakes that the said accumulation shall not be less 
 than 100/. This last guarantee,, though necessary for 
 the satisfaction of the public,, is in truth so certain,, from 
 the amount of the premium demanded, that a person 
 acquainted with the subject looks upon the possibility 
 of the funds of the society suffering from it as an ex- 
 tremely remote chance. 
 
 In order, however, to make the proceedings of such an 
 office equitable, the proportions of the premiums paid by 
 parties of different ages must be fairly regulated. On 
 the supposition that the inequality pointed out in page 
 2?0. is allowed to exist, the preceding methods of divi- 
 sion may be (I do not say are) adjusted so that every 
 interest shall be consulted. But in the present plan, it 
 is impracticable to remedy any such defect of proportion, 
 at least without dividing the establishment into as many 
 different offices as there are ages, which would not be 
 easy, and perhaps not very safe. The simple rule for 
 determining the relative premiums is to make them 
 proportional to the real premiums, with the exception 
 of a given addition to each (not premium, but) policy, 
 for expenses of management. In a large office, how- 
 ever, the expenses of management may be made a part 
 of the percentage addition to the premiums. 
 
 The method of division in such an office is extremely 
 simple, and has been already described in page 276. 
 Subtracting the present value of all the claims, that is, 
 of all the minimum claims, reckoned as 1001. for each 
 tabular premium paid, from the sum of the present 
 values of all premiums, and of the assets of the office, 
 the proportion which this remainder is of the present 
 value of all the claims expresses the fraction of 1001. 
 which may be added to each 1001. insured. 
 
 Let A, the assets* of the office, be 500,000/.; P, the 
 present value of all premiums, 600,000/.; and C, the 
 prseent value of all claims, 850.000/. : then A + P- C, 
 
 * Diminished for the expenses of management, as in page 275. 
 
MANAGEMENT OF AN INSURANCE OFFICE. 289 
 
 the surplus, is 250,000/., which being 25 parts out of 
 85 of the whole claims, or 29^7- per cent., will afford 
 129 1 3 T /. for every 100/., which is guaranteed. Those 
 who die in the year of this valuation, may therefore 
 receive that sum. 
 
 The principle on which the preceding division is 
 made, is, that if the same state of things continue, 
 every one will in turn receive the same dividend. But, 
 can such a prediction be made ? Undoubtedly not, 
 for the fluctuations, both of those who come into 
 the office, and those who go out, will tend to produce 
 variations. It is very unlikely that any office should 
 maintain itself for a long series of years nearly in the 
 same position ; and, since the idea of allowing any per- 
 manent diminution of the surplus must not be ad- 
 mitted, there is no alternative except an arrangement 
 for a gradual increase, which it is the object of this mode 
 of division to make as slow as is consistent with the cer. 
 tainty of having it. But in this case, it may seem as 
 if the old system were revived, and a fund instituted 
 by the present insurers, for the sole benefit of those 
 who come after them. There is, however, an im- 
 portant difference between never paying more than the 
 guaranteed minimum, so that all the surplus goes to- 
 wards that fund, and drawing upon the surplus nearly 
 to the full amount which safety would allow, leaving 
 only such a trifle to augment the fund as is requisite to 
 avoid too large an out-going. The old principle, then, 
 which formerly prevented any bonus whatsoever, is here 
 merely applied to such an extent as to keep the bonus 
 within proper limits. 
 
 If the tables of mortality by which the profits are 
 divided, be actual representatives of existing mortality, 
 and if the number of members remain nearly the same, 
 the indications of these tables, implicitly followed, 
 would soon reduce the surplus of the office to that 
 which is barely necessary for the extreme payment 
 which the premiums will admit. To take a case : sup- 
 pose that the premiums will in the long run pay 125/. 
 u 
 
290 ESSAY ON PROBABILITIES. 
 
 for every 100?. guaranteed; the present value of all 
 the claims is 1,000,000?., that of all the premiums 
 700,000?., and the value of the assets of the office 
 600,000?. The surplus is therefore 300,000?.,, and, 
 going upon real tables, the office begins to pay 130?. for 
 every 100?. guaranteed; and this it would be able to 
 do in favour of all who are insured at the time of the 
 preceding valuation. But part of this dividend does 
 not, and, by hypothesis, cannot, arise from the pre- 
 miums: it is therefore paid entirely out of surplus, 
 and will gradually disappear. The dividend will be 
 reduced to 125?., about which it will fluctuate, being 
 sometimes a little less and sometimes a little more. An 
 increase of business in such an office would make the 
 surplus disappear more rapidly, since each new comer 
 brings in an equivalent to 125?. and those of the new 
 comers who die receive 130/. A diminution of busi- 
 ness would produce a contrary effect ; and a total ces- 
 sation of new comers would allow the dividend to re- 
 main at 130?. As far as any danger from fluctuations 
 of mortality is concerned, I do not see any objection to 
 such a division as the preceding : but when it is re- 
 membered that the possible diminution of the rate of 
 interest must also be provided for, I think it would be 
 prudent to reserve a small proportion of the surplus 
 for accumulation. 
 
 There are two ways in which this reserve may be 
 made ; firstly, by employing a table of less than the 
 real mortality in the valuation of the claims and pre- 
 miums ; secondly, by calculating the surplus from a 
 real table, and dividing as upon the supposition that a 
 given fraction of this surplus, say one eighth or one 
 tenth, should be expunged in the calculation. The 
 latter plan is the best of the two, in every respect but 
 one, as follows. The mutual insurance office must be a 
 republic, and many of its members have very little in- 
 formation upon the questions which are, from time to 
 time, submitted to them. They are easily dazzled by 
 the appearance of surplus, and are quick to believe that 
 
MANAGEMENT OF AN INSURANCE OFFICE. 29 1 
 
 a larger division might be made in their favour. Add 
 to this that the older members carry with them in the 
 discussion of questions, that influence which age na^ 
 turally and properly gives in the management of impor- 
 tant affairs; and as to which the conduct of an in- 
 surance office only forms an exception, because questions 
 arise in which the interests of the old and young 
 clash * with each other, which is nowhere else the case. 
 Under such circumstances the disposition to break in 
 upon the surplus is the fault to which the body has a 
 tendency, and it is not a bad thing to place some small 
 difficulties in the way of doing this. Now if a fraction 
 of the surplus be withdrawn from the calculation of 
 the dividend, it is very easy to change one fraction 
 into another. A vote of the general meeting, and a few 
 strokes of the actuary's pen, and the thing is done. 
 But when the requisite fraction of the surplus is de- 
 ducted by the supposition of a lower rate of vitality 
 (or of interest) than actually prevails, no change can 
 be made without the entrance of a large number of 
 important considerations, the discussion of which occu- 
 pies some time, and places a useful check in the way of 
 the restless. 
 
 But is it then proposed that every office shall be pro- 
 vided with a fund, which, though slowly, is yet inde- 
 finitely, to increase ? Not necessarily; for the reserved 
 portion of one year is not put aside, and considered as 
 inalienable, but enters into the surplus of the next year. 
 There may be, then, a limit to the increase of the sur- 
 plus, as follows. Suppose the office to be in a stationary 
 state, having arrived at the point where the influx of 
 the new members compensates the efflux occasioned by 
 death or surrender. The receipts of the office consist 
 entirely in premiums and produce of capital, the ex- 
 penditure in management and payment of claims. As 
 long as the surplus increases, the sum of the first pair 
 
 * The members of a mutual insurance office are not properly repre- 
 sented in their list of directors, unless the individuals composing it are of 
 very different ages. 
 
 u 2 
 
292 ESSAY ON PROBABILITIES. 
 
 of items will exceed that of the second; and, whatever 
 may be laid by in each year, it produces a larger sur- 
 plus, and larger payments on account of claims, in the 
 next year. If, then, the surplus could increase without 
 limit, so would the dividends ; but if the surplus have 
 a limit, the dividends also have a limit : and it is plain 
 that the limit arrives, when the yearly outgoings from 
 claims and management are equal to the receipts from 
 premiums and interest of capital. A mathematical in- 
 vestigation of the conditions necessary in order that the 
 fund may increase, but not without limit, gives the 
 following result : 
 
 Suppose an insurance office, constructed upon the 
 preceding principles, to have arrived at its stationary 
 state, with respect to influx and efflux of members, 
 and make the following suppositions : 
 
 A The assets of the office, for precision, say Janu- 
 ary 1, 1838. 
 
 P The real present value of all premiums from 
 members then in existence. 
 
 C The real present value of all claims (not includ- 
 ing additions) to which the office is then liable. 
 
 m The expenses of management till January 1, 1839- 
 
 p The amount which will accrue from premiums and 
 interest of premiums by January 1, 1839- 
 
 c The amount of claims (not including additions 
 from the surplus fund), which will be paid before 
 January 1, 1839. 
 
 r The interest of one pound for one year. 
 
 t The fraction which is taken of the tabular surplus 
 fund in the computation of the dividend. 
 
 We suppose (as must be the case in an old office), 
 C greater than P, and (as must be the case in a solvent 
 office) A and P together greater than C. 
 
 1. In order that there may be a surplus fund in- 
 creasing, but not without limit, find the fraction which 
 a year's interest on C is of c. Then t, or the fraction 
 of the surplus fund (or of A f P C), which enters 
 into the formation of the dividend, must exceed that 
 
MANAGEMENT OP AN INSURANCE OFFICE. 29# 
 
 fraction which a year's interest on C is of c, otherwise 
 the fund would increase without limit. 
 
 2. Neither can there he such a fund unless the sum 
 of m and c should fall short of the sum of p, and of a 
 year's interest on the excess of C over P. But, when 
 this is the case, the limiting surplus capital is found 
 by dividing the excess of the second total just men- 
 tioned over the first,, by a divisor obtained as follows : 
 multiply together t and c, divide the product by C, 
 and subtract r from the quotient. To this surplus 
 capital, add the excess of C over P, and the limiting 
 capital is obtained. 
 
 3. If it should happen that the limiting surplus 
 capital is less than the actually existing surplus, it is a 
 sign that the action of the preceding plan would 
 diminish the surplus towards that limit instead of 
 increasing it. In such a case, the surplus is already 
 too large for the value of t to increase it; and if t be 
 not diminished, that is, if less of the tabular surplus 
 be not taken into the computation of the dividend, the 
 fund will diminish. 
 
 It is not to be supposed that any office will ever 
 reach a stationary state ; but the approach may be near 
 enough to make the preceding process of some use, in 
 the determination of the dividends due to the insured. 
 If, following the plan which the preceding problem 
 supposes, we were to inquire what value should be 
 given to the fraction t, the answer to the question must 
 depend on the reduction of interest which is supposed 
 within the bounds of probability. Suppose the present 
 rate of interest to be 3-J per cent, and that the extreme 
 limit is supposed to be 2j per cent, in such a case 
 the value of P and C must be calculated at 2J- per 
 cent., and such a limiting surplus must be fixed upon 
 as will, at that rate of interest, enable the office to pay 
 at least its guaranteed claims But it is impossible to 
 lay down an entire system of rules for the regulation of 
 a species of undertaking which depends on the fluc- 
 tuations of the state of society. Whatever maxims 
 u 3 
 
294? ESSAY ON PROBABILITIES. 
 
 may be collected, and however sound they may be, 
 skill and judgment will always be requisite to apply 
 them to the cases which arise. In this respect the 
 offices resemble the individual problems which arise in 
 life contingencies. Many as are the cases which have 
 been described in books upon the subject, almost every 
 application of them requires attention to some cir- 
 cumstance peculiar to the instance in question. 
 
 CHAPTER XIII. 
 
 MISCELLANEOUS SUBJECTS CONNECTED WITH IN- 
 SURANCE, ETC. 
 
 THE limits of this treatise will only afford a few words 
 on several points of interest, which I will therefore 
 condense into one chapter, taking the subjects as they 
 arise. 
 
 The management of annuity offices is somewhat 
 more easy than that of insurance establishments ; and 
 the maxims of security in the former are, of course, 
 the direct reverse of those in the latter, so far as any 
 considerations of mortality are concerned. Tables 
 must be assumed of higher than the real vitality, and 
 a rate of interest somewhat below, or at least not above, 
 that which can actually be obtained. 
 
 Those who wish to buy annuities on the firmest 
 possible basis, may deal with the government. The 
 commissioners for the reduction of the national debt 
 are empowered to grant . annuities in lieu of stock, on 
 terms calculated from the government tables (page 168). 
 The rates are high ; and though a private office may 
 really be as solvent as the nation, yet confidence springs 
 
MISCELLANEOUS REMARKS. 2Q 
 
 from opinion, and the security of the national debt 
 must always be thought the very best. The patriotic 
 annuitant, too, may reflect that the profit derived from 
 him goes to the reduction of the national debt. 
 
 The distinction of male and female life becomes of 
 importance in the granting of annuities. The insurance 
 offices have not as yet, except, I believe, in one or two 
 instances, begun to recognise the distinction, which is 
 of the less consequence, since, with respect to the 
 office, it is keeping on the safe side, and, with respect 
 to the public, very few female lives are insured. But 
 the exact reverse takes place with regard to annuities ; 
 it would be insecure to grant them to females on the 
 same terms as to males ; and a very large proportion of 
 the whole number of annuitants is of the former sex. 
 
 Annuities might be granted by an office which 
 should undertake a return of profits, in the form of a 
 payment to the executors at the death of the party ; 
 and an association of mutual annuitants would not be 
 of difficult formation. The principal objection would 
 be, the smallness of the number of persons who buy 
 annuities, compared with those who insure their lives. 
 If, however, such an office were to grant reversionary 
 annuities, their field would be very much widened. 
 Several of the insurance offices grant annuities, but 
 none, I believe, in which the annuitants are sharers in 
 the profits. 
 
 The details of a Friendly Society comprise every pos- 
 sible species of life contingency. They grant weekly 
 payments during sickness, annuities in old age, and sums 
 payable at death, in consideration of weekly premiums. 
 These institutions, combined with Savings' Banks, and 
 aided by the removal of the abuses of the Poor Law, 
 will, in time, raise the labouring classes of this country 
 to a degree of independence which they have never 
 known. But, as might have been expected, the manage- 
 ment of these important institutions has, in many in- 
 stances, been wanting in prudence ; and I am afraid it 
 is hopeless to expect tbat the unity of system, which 
 u 4 
 
296 
 
 ESSAY ON PROBABILITIES, 
 
 must prevail before a thorough knowledge of the ad- 
 vantages they offer can get abroad, can be attained while 
 their several administrations are unconnected, and at 
 liberty to pursue all possible variety of plans, subject 
 only to the certificate of an actuary that each propo- 
 sition is not unsafe. But something more than safety is 
 required : an equitable distribution of benefits, and a 
 certainty of the most careful management, are as neces- 
 sary to the universal formation of these societies as an 
 opinion of their safety. The government, which 
 has within these few years been compelled, by the 
 most decided necessity, to apply a very severe and 
 searching remedy to an abuse of long standing, owes 
 the labouring classes a strong expression of sympathy 
 with the numerous cases of hardship which such a mea- 
 sure must create, and with the excellent conduct and 
 temper under its operation which has pervaded the 
 classes most immediately affected by it. It is to be re- 
 gretted that the change itself was not accompanied by 
 acts of parliament for the encouragement and aid of 
 societies such as those of which I am now speaking, in 
 addition to those which already existed for their regu- 
 lation. The most determined opponent of the protect- 
 ive principle would hardly dispute the policy of giving 
 effective help to the efforts of self-support, at the mo- 
 ment when the aid of the parish, which had been the 
 resource against poverty, became only the last security 
 against starvation. If the nation had been obliged to 
 abandon a distant colony, in circumstances of danger 
 and distress, there is no doubt that the settlers would 
 have been furnished with arms, arsenals, ships, money, 
 and all that could enable them to do whatever might be 
 done for their own defence and support. Has similar 
 help in similar circumstances been given at home ? Is 
 the labouring man, thus suddenly thrown into a position 
 where the power and the habit of depending on himself 
 are necessary to a degree of which his training never 
 implied the existence, one bit nearer to the acquisition 
 of the power or the formation of the habit, by any aid 
 
MISCELLANEOUS REMARKS. 297 
 
 of the legislature? Have even the opponents of the 
 measure, with their professions of benevolence, ever 
 pressed, or even suggested, the duty of showing the 
 labouring man, not only that by combination his class 
 can provide for itself, but that the community which 
 found it necessary to make a change involving him in 
 years of uncertainty and possible hardship, was desirous 
 that he should have that knowledge, and willing to aid 
 him in attaining its full benefits ? 
 
 It is not too late to take the necessary steps; and any 
 one who imagines a legislature able to feel, or to think, 
 will see the means of addressing himself to the first 
 faculty by such considerations as the preceding, and to 
 the second by urging the policy of giving every class a 
 share in the artificial system of property on which the 
 country now depends. At present, the property of a 
 labouring man is all tangible, and immediately at hand; 
 it would not be a great wonder if he were found to 
 have no clear opinion of the rights of a landlord, a fund- 
 holder, a mortgagee, or an annuitant. But if he him- 
 self were in possession of any of those claims which, by 
 means of law, can be created, enforced, or transferred 
 by virtue of the possession of a bit of paper still more, 
 if the support of his old age and of his sick bed were con- 
 nected with this purely legal tenure of his past savings, 
 he would then be interested in the preservation of the ex- 
 isting system by the share of it which belongs to himself. 
 
 The friendly societies, numerous as they are, are by 
 no means universally distributed ; and if they were, the 
 smallness of their several amounts of investment must 
 occasion the expenses of management to bear a larger 
 proportion to the whole than would be the case if all 
 were united. Besides which, it happens every now and 
 then that the affairs of such a society fall into disorder 
 from want of skill or care. The government has lent 
 considerable assistance by allowing their investments a 
 larger rate of interest than could elsewhere be obtained ; 
 but this aid, independently of its being but little known 
 by the class whom it most concerns, does not guarantee 
 
298 ESSAY ON PROBABILITIES. 
 
 the proper use of the funds so invested. If one large 
 office were to he established in London, having the 
 general management of the money raised, and the regu- 
 lation of its distribution, it would not be difficult to find 
 persons of station * and character throughout the country 
 who would consent to act as agents, receiving the con- 
 tributions and certifying the claims. The expense of 
 management might be borne for a few years by the 
 public purse, and this burden might be gradually thrown 
 on the establishment itself. No very great difficulties 
 could arise in the formation of such an institution, and 
 certainly none the expense of conquering which would 
 not be trifling in comparison of the greatness of the object 
 gained. The act which should establish this universal 
 Friendly Society would, in two generations, become the 
 real poor law. 
 
 The subjects of fire and of marine insurance are 
 founded on principles of great simplicity, though it is 
 not easy to procure exact data for the computation of 
 risks. As there exist no offices which are managed on 
 the republican method of a mutual Life Insurance Com- 
 pany, no publication of the results of experience has 
 been made. If every loss by fire or sea were a total 
 loss, it would only be necessary to ask what proportion 
 of all the houses or ships now existing is burnt or 
 wrecked in a year or on a voyage, and the premium for 
 insuring a house for one year, or a ship for one voyage, 
 would immediately follow. Thus, if of all the ships 
 which sail to the West Indies, one in a hundred is lost, 
 the lowest premium at which an insurance could take 
 place is one per cent., and all demanded above that pro- 
 portion would be profit. It would not, perhaps, be 
 very easy to ascertain this proportion with exactness, 
 and the difficulty is increased if ships or houses be 
 divided into different classes as to security, since the 
 risks of each class must be ascertained separately. But 
 
 * Many of the Friendly Societies now established depend almost entirely 
 upon the" superintendence of the clergy or local gentry. 
 
MISCELLANEOUS REMARKS, 299 
 
 the greatest obstacle to a satisfactory adjustment of 
 risks, lies in the necessity of taking into account the 
 chances of only partial loss, which would make the 
 tables (if they could be procured) nearly as complicated 
 as life tables. 
 
 On the subject of marine insurance, nothing is 
 known to the public, as to the experience of the 
 underwriters; and, as it is not directly interested 
 in the subject, it would be difficult to create any 
 disposition to inquiry. The mercantile world, how- 
 ever, and the underwriters themselves, have a direct 
 interest in the dissemination of such information, 
 for reasons which it is no pleasant task to state, 
 both on account of their invidious character, and their 
 obvious want of connexion with the general objects of 
 this treatise. But the latter circumstance may, perhaps, 
 not be disadvantageous, since the statement of the exist- 
 ence of an imputation, coming from a quarter in which 
 there is no interest whatever, either in the continuance 
 or discontinuance of any present condition of things, 
 need not excite any disposition, except that of calmly 
 weighing whether it is necessary or not to produce a 
 refutation. 
 
 Some years ago, I heard the following opinion stated 
 in a mixed company, in reference to a then proposed 
 attempt to render ships incapable of actually sinking, 
 however much they and their cargo might be damaged ; 
 namely, that the mercantile world would not be inclined 
 to patronise an invention which would make the seaman 
 safer than the ship. Some time afterwards, I saw an 
 article in a periodical journal, distinctly written for the 
 purpose of making its readers believe that, in conse- 
 quence of insurance, unsafe ships are allowed to be used, 
 to an extent which has caused much more loss of life 
 and property than could have been experienced if no 
 such institution had existed. Other allusions, more or 
 less direct, in various publications, have convinced me 
 that one of two things must be true, either such an 
 impression has a party who acknowledge it, or authentic 
 
300 ESSAY ON PROBABILITIES. 
 
 information upon the subject is so difficult to be ob- 
 tained, that the one, two, or ten,, who believe it, or 
 profess to believe it, feel that no answer can be made to 
 the assertion. 
 
 That there are men in the carrying world (if mer- 
 cantile world be too wide a phrase for the subject) who 
 would, from a pitiful economy, expose the seaman to 
 risks which a little outlay might prevent, is very pos- 
 sible ; there are men of such a spirit in every world : 
 that there are others who would consider such conduct 
 as little short of murder, a like analogy would equally 
 justify us in asserting. Which class has predominated 
 can only be absolutely known to the public by results, 
 without which there is but general opinion upon cha- 
 racter to aid any individual in forming his conclusion. 
 It is in human nature that the insured should not be so 
 careful as one who stands risk ; and it is, unfortunately, 
 the general experience of men acting in bodies, that they 
 are not found to be swayed by the principles which 
 would be acknowledged and acted upon by them severally. 
 Putting these things together, it is not wonderful that, 
 in any case where suspicion might attach to a body of 
 men, there should be quarters in which it does attach. 
 It would not be wonderful, either, if the suspicion were 
 found to be perfectly groundless ; but correct feeling 
 would point out the desirableness of forestalling such 
 suspicion, if possible, by the publication of all necessary 
 information. In the present instance, it would be well 
 that the proportion of loss, among insured vessels, should 
 be known ; it would not be necessary to state the values 
 of the several vessels, since the simple account of the 
 number insured, and the number on which a claim has 
 been paid, in various years, would be sufficient. The onus 
 of proving that the loss on uninsured vessels, or on 
 vessels which sailed before insurance was known, is or 
 was greater than that on insured vessels, would lie upon 
 those who make the charge. All persons, in the case of 
 any body of men, must hold every thing short of ab- 
 solute proof against them to count for nothing, when 
 
MISCELLANEOUS REMARKS. 301 
 
 they show themselves ready to communicate those ma- 
 terials out of which a misdemeanor, if there he one, 
 might be substantiated. 
 
 The offices for the insurance of fire have not given 
 any account of the proportion of insured houses upon 
 which claims have arisen. Their usual annual charge 
 is, I believe, about one part in a thousand of the sum 
 insured, upon premises of ordinary risk, such as a 
 dwelling-house in London. There are higher rates 
 for more hazardous insurances, constructed, I should 
 imagine, very much from mere estimation of the risk. 
 But the government steps in between the insurer and the 
 insured, and imposes a duty on each policy which 
 nearly trebles the annual payment upon it. This has 
 been called a tax upon prudence, and in like manner 
 the stamp duty might be called a tax upon justice. I 
 am afraid that if nothing commendable suffered under 
 an impost, reformation would thrive more than re- 
 venue ; and a deficiency of means to pay the interest 
 of the debt would be a heavier tax on prudence, justice, 
 and every thing else, than any minister has yet con- 
 templated. But it may not therefore follow that the 
 particular tax in question is politic, still less that its 
 amount is justifiable. The reason of the tax is plainly 
 this : the moral security offered by the fire office is 
 worth so much more than competition will allow them 
 to ask, that the impost is one which does not fall so 
 heavily as it would do if levied in many other quarters. 
 Nobody can question the truth of this ; but, never- 
 theless, the amount of the tax imposed by the legis- 
 lature must be owned to be excessive, and likely to act 
 as a prohibition in the case of poor persons occupying 
 small premises. 
 
 But there is a mode of overthrowing this tax, or, at 
 least, of bringing the government to terms, to which I 
 can see no impediment, practical or moral. It is the 
 application of the principle of mutual insurance by a 
 number of individuals acting in a private capacity, and 
 not opening a public office. Suppose a thousand indi- 
 
S02 ESSAY ON PROBABILITIES. 
 
 viduals, registering their names, to appoint three men of 
 undoubted character to receive contributions of one 
 guinea a year each. If the subscribers be occupiers of 
 dwelling-houses in London, tbereis no doubt that this sum 
 would be amply sufficient to insure a thousand guineas 
 to each. If three years were to elapse without a fire 
 taking place, the subscription might be suspended, until 
 circumstances should diminish the fund ; which, im- 
 proving in the mean time at interest, would become 
 every year more capable of meeting demands upon it. 
 There would be no need of any legal security, if the 
 trustees were well chosen ; and a short agreement would 
 explain the understanding on which the parties contri- 
 bute. As soon as a few such clubs were formed, the 
 inutility of imposing a tax on one particular way of 
 effecting an object would become apparent. 
 
 It would be lucky for the preceding plan, if it were 
 the decided opinion of lawyers that the courts of equity 
 would not entertain any application for inquiry into the 
 state or management of such funds ; since, in that 
 case, the law of honour would be sufficient. It has 
 always been found, that whenever the law of the land 
 refuses to protect a proceeding which is fair and equal 
 in itself, a stronger law claims jurisdiction. The parties 
 benefited in the end would be the fire offices, since such 
 a method of resisting this excessive tax would inevitably 
 procure its abolition. 
 
 There is another tax which, though not so dispro- 
 portionate in its amount, is much worse in its principle 
 than that on policies of fire insurance ; namely, the tax 
 on policies of life insurance. It must be remembered, 
 that the income of which the savings are invested in 
 this manner, has already undergone a considerable 
 amount of taxation. If any investment of such savings 
 be taxed, all should be treated alike. 
 
 The abolition of lotteries happily leaves nothing to 
 be said upon the subject of gambling, encouraged and 
 promoted by the government ; and the recent decision 
 of the French legislature, by which the public gaming- 
 
MISCELLANEOUS REMARKS. 303 
 
 houses have been suppressed, must he a source of con- 
 gratulation, both from the excellence of the measure 
 itself, and the prospect of imitation which it opens, on 
 the part of other continental powers. But, at the same 
 time, it cannot be denied that, however desirable it may 
 be that no community should give to gambling that 
 appearance of sanction which is implied in regulation, 
 the refusal of the latter is accompanied by evils, of 
 which it is never possible to say positively that they fall 
 short of those which would be produced by sufferance 
 accompanied by restriction. Jn this country, there are 
 the means of gambling open to every class of the com- 
 munity, and there can be no doubt that those who avail 
 themselves of them are subject to imposition in a degree 
 which could not be the case if the 'play were accompa- 
 nied by publicity. The classes of rank and wealth 
 have the power of forming themselves into clubs, 
 in which illegal games are played without the possibility 
 of detection, and in such institutions there can be no 
 doubt, with rare and occasional instances of exception, 
 the play is conducted at least with fairness. But no 
 such thing can be supposed with regard to the numerous 
 receptacles in London and other large towns, and which 
 are believed to exist in different forms, suited to all 
 classes of society. The difficulty of obtaining legal 
 proof renders conviction next to impossible ; and the 
 occurrences which sometimes take place at the sessions, 
 prove that, even when enough of evidence is obtained to 
 hold parties to bail, the accused can generally find the 
 means of preventing the evidence from being forthcom- 
 ing to sustain the indictment. Under such circum- 
 stances, gambling in its worst form thrives in defiance 
 of law. Nevertheless, the good consequences of dis- 
 couragement are visible throughout the country. There 
 is no people in the world among whom so little of direct 
 gambling is found. 
 
 The infatuation which leads persons to suppose that 
 they can ultimately win from a hank, which has chosen 
 a game in which the chances are against the player, is 
 
304 ESSAY ON PROBABILITIES. 
 
 one which can only be cured, if at all, by a quiet study 
 of the theory of probabilities. Perhaps some of our 
 readers may suppose, that the persons who thus court 
 ruin, do it under the notion that the results given by 
 that theory are dubious, or derived from unpractical 
 speculation, or perhaps absolutely false, So far is this 
 from being the case, that though they undoubtedly fall 
 into error by forming their notions from observation 
 unaided by theory, yet their error frequently consists 
 in representing games of chance as being more unfa- 
 vourable to themselves than they really are. Though 
 the true premises should lead them to the conclusion 
 that success is next to impossible, they cannot learn the 
 truth even from a mistake which should teach it a 
 fortiori. The author of the article " GAMING/' in the 
 Penny Cyclopcedia, states, apparently from his own 
 knowledge, that it is customary to consider the chances 
 of the bank at the game of rouge-et-noir, as 7 per 
 cent, above those of the player. Now, it can be imme- 
 diately shown, from the first appendix, that when a 
 player puts down a stake, his chances of doubling his 
 stake, of losing it, and of simply recovering it, are as 
 8903, 9122, and 1975. Now 9122 does not exceed 
 8903 by 7j per cent, of 8903. but only by about 2J per 
 cent. If, however, the preceding assertion meant that 
 the game was considered as a simple one, in which the 
 chances were as 46J to 5 3 1, the error was very large 
 indeed. So far as this one instance goes, it should 
 seem that the warning against this game, as derived 
 from observation of its results, was yet stronger 
 than that which would have been given by the 
 theory of the game. The same author adds, that he 
 heard it frequently asserted by constant frequenters 
 of the Parisian gaming-houses, that it was absolutely 
 impossible for any one to win in the long run. 
 
 Still, however, to the hopeless attempt of squaring 
 the circle, or of finding perpetual motion, we have to 
 add that of discovering a method of certainly winning 
 at play : the attempt at which has been the ruin 
 
MISCELLANEOUS REMARKS. 305 
 
 of many a speculator. The gaming banks have dis- 
 covered the secret, which is simply to embark consider- 
 able capital, and to play with chances unequally in 
 their favour. To produce in the young mind a con- 
 viction that events will happen,, in the long run, in a 
 fixed, and not in what is called a fortuitous manner, 
 should be an object of education, in order to produce 
 that soundness of views on the results of gambling 
 which is a sure protection against the temptation. By 
 trying experiments upon what are called chance events, 
 such as might easily be done with a pack of cards, or 
 a few dice, it might easily be made to appear that no 
 large number of events will present any marked devi- 
 ations from the general average which the knowledge 
 of this theory points out before-hand. Persons aware 
 of the truth of the law just stated, may often be able 
 to apply it advantageously. I received the following 
 anecdote from a distinguished naval officer, who was 
 once employed to bring home a cargo of dollars. At 
 the end of the voyage it was discovered that one of the 
 boxes which contained them had been forced ; and on 
 making further search, a large bag of dollars was dis- 
 covered in the possession of some one on board. The 
 coins in the different boxes were a mixture of all man- 
 ner of dates and sovereigns ; and it occurred to the 
 commander, that if the contents of the boxes were 
 sorted, a comparison of the proportions of the different 
 sorts in the bag with those in the box which had been 
 opened, would be strong presumptive evidence one way 
 or the other. This comparison was accordingly made, 
 and the agreement between the distribution of the se- 
 veral coins in the bag and those in the box, was such 
 as to leave no doubt as to the former having formed a 
 part of the latter. 
 
306 ESSAY ON PROBABILITIES. 
 
 ADDITION TO CHAPTER X. 
 
 The following formulae will sometimes be found 
 useful. 
 
 The present value of an insurance, which is to be 
 \ if the party die in the first year, 2 if in the second, 
 and so on, is 
 
 1 + A-rI 
 1+r 
 
 where A is the value of a common annuity ; I that of 
 an annuity which is l at the first payment, % at 
 the second, and so on; and r the interest of g@l for 
 one year. 
 
 If an office engage to pay \ at the death of an 
 individual, and also to return all the premiums at the 
 same time, that is, if they guarantee that the interest of 
 his investments shall amount to <!, the premium which 
 should be demanded is 
 
 E-A 
 
 1+A + I 
 
 where A and I are as before, and E is the value of a 
 perpetuity of 
 
APPENDIX. 
 
 APPENDIX THE FIRST. 
 
 ON THE ULTIMATE CHANCES OF GAIN OB LOSS AT PLAY, 
 WITH A PARTICULAR APPLICATION TO THE GAME 
 OF ROUGE ET NOIR. 
 
 THOUGH the first part of the following reasoning is 
 of a mathematical character, I have been induced to 
 insert it by the consideration that the results of page 109. 
 have never yet been introduced into an elementary 
 work, nor even proved to the mathematician except either 
 by incomplete or complicated trains of reasoning. Such 
 being the case, perhaps even a well-informed mathe- 
 matician might be excused for doubting some of the 
 results of chapter V., and I have therefore digested the 
 following demonstration, that no one who bears such a 
 character may be able to weaken the evidence for the 
 necessity of the pernicious results of gambling which 
 that chapter is intended to afford. 
 
 De Moivre was the first who gave a solution of the 
 following problem, and by a method of the most striking 
 ingenuity. But his demonstration has the defect of 
 assuming that one or other of the players must be 
 ruined in the long run. Laplace* and Ampere, the 
 
 * The solution of Laplace gives results for the most part in precisely 
 the same form as those of De Moivre, but, according to Laplace's usual 
 custom, no predecessor is mentioned. Though generally aware that La- 
 place (and too many others, particularly among French writers) was much 
 given to this unworthy species of suppression, I had not any idea of the 
 extent to which it was carried until I compared his solution of the problem 
 of the duration of play, with that of De Moivre. Having been instru- 
 
 x 2 
 
ii APPENDIX THE FIRST. 
 
 former in his Theorie, &c., the latter in a tract entitled, 
 Considerations sur la Theorie Mathematique du Jeu, 
 Lyons, 1802. have also solved the problem: both so- 
 lutions are of the highest order of difficulty, and cannot 
 be rendered elementary. If my memory be correct, I 
 have seen references to other solutions. 
 
 The problem is as follows : Two players, A and B, 
 the first possessed of m times and the second of n times 
 his stake,, play at a game so constituted that it is a to b 
 that A shall win any one game ; required the proba- 
 bility which each has of ruining the other, if the game 
 be indefinitely continued. 
 
 I shall first take the case where one of the players, A, 
 is possessed of unlimited means, that is, in which m is 
 infinite. Let B w> m represent the probability that 
 B having n counters, shall ruin A who has m counters. 
 Then, if m be infinite, B Wj oo will after the first game, 
 become either B w + i, oo, or B w _ lf ^ according as' 
 that game is B's or A's ; of which the chances are 
 
 b and a 
 
 a + b a+b 
 
 Consequently, 
 
 B n oo = 
 which gives 
 
 which, when C' and C" are determined, will represent 
 the probability that B will never be ruined, but will 
 continually gain more and more from A. But the 
 
 mental (in my mathematical treatise on Probabilities, in the Encyclopaedia 
 Metropolitana) in attributing to Laplace more than his due, having been 
 misled by the suppressions aforesaid, I feel bound to take this opportunity 
 of requesting any reader of that article to consider every thing there given 
 to Laplace as meaning simply that it is to be found in his work, in which, 
 as in the Mecanique Celeste, there is enough originating from himself to 
 make any reader wonder that one who could so well afford to state what 
 he had taken from others, should have set an example so dangerous to his 
 own claims. 
 
ULTIMATE RESULTS OF PLAY. Ill 
 
 same equation (1) is equally true if forB W}00) we substitute 
 either 1 B W)00j or AOO, n> which two last may be 
 different, for any thing yet proved to the contrary. In 
 fact, the equation (1) is merely the general expression 
 of the condition that n is changed into n -j- 1 or n 1 
 according as one or the other of two events happens, 
 
 b a 
 
 whose chances are and It may be seen 
 
 a -f b a + b. 
 
 however, immediately, that in the case of n = 0, in 
 which case the proposed contingency becomes an initial 
 impossibility, we must have B 0> , = 0, or C' + C" =0. 
 We have then 
 
 = "{-(9"} 
 
 This result is rational only when a is not greater 
 than by unless we suppose C" = 0. But the necessity 
 for investigating what takes place in this case is saved 
 by observing a very simple relation which exists between 
 B w> w> and B w>00 . Supposing A to have infinite means, 
 it makes no difference in the state of the question if we 
 take any number of stakes m from the stock of A, and 
 suppose that they shall be lost before the rest are touched. 
 Consequently B cannot win indefinitely from A unless 
 he first ruin A's stock of m stakes, and afterwards, be- 
 ginning from n + m stakes, win indefinitely from A's 
 remainder. That is 
 
 m,oo 
 
 -, n n \ 
 
 b a ) 
 
 Whence, applying the same reasoning to A m> Wf we 
 find that if two players A and B, possessing m and n 
 x 3 
 
APPENDIX THE FIRST. 
 
 stakes, play at a game which gives a to & in favour of A 
 at each trial, then,, in an indefinite number of trials 
 
 the chance that B sha11 ruin 
 
 A the chance that A shall ruin B 
 
 nf 7 m m *\ 
 a \^ b a ) 
 
 vt MHSUiiailvc uiai, -n. simii i uiii j 
 
 , n, ,n + m n + m 
 
 b a 
 
 The sum of these two chances is unity, from which it 
 appears that one or other must be ruined in the long run. 
 These results agree entirely with those of De Moivre, 
 and all the rules in page 109. may he easily deduced 
 from them. 
 
 It appears also, that if the conditions of any game, 
 however complicated, can be reduced, in the case where 
 one of the players (A) has unlimited means, to the 
 equation 
 
 then the ultimate results of that game exhibit probabi- 
 lities of precisely the same value as those of a simple 
 game, in which it is a to /3 for A against B. 
 
 If, besides cases in which A or B wins, there be 
 cases in which the game is drawn, no alteration is made 
 in the result (though the number of games in which 
 there is a given probability of either party winning 
 must of course be increased.) Let a, /3, and , be the 
 chances that A wins, that B wins, and that the game is 
 drawn: then (a + j3 + 8=1) 
 
 B,OQ = B w + i }00 -j-a B w _ i )00 _j_5 B w>00 or 
 
 a / j8 a \ 
 
 B == iTT5 Bw + 1 co +rT5 Bw - 1 ' co Vn5 + TH8 = V 
 
 the same as in a game which must be won or lost, and 
 in which it is a to /3 for A against B. 
 
 The following is the problem of the game of rouge et 
 noir, which I shall afterwards proceed to explain. 
 
ULTIMATE RESULTS OF PLAY. V 
 
 A and B play at a game which presents four cases, 
 
 A, B, D, and T, of which the chances are a, (3, $, and d, 
 so that a-J-/3-l--f-0=l. When A happens., the player 
 A wins ; when B happens, the player B wins ; when D 
 happens, the game is unconditionally drawn ; but when 
 T happens, the game is drawn, and in the next game 
 only the player A puts down a stake, and not the player 
 
 B. If D should follow T, or if T should happen any 
 numher of times running, or D, or successions of T and 
 D, still A's stake remains risked, without any from B : 
 "nor does B stake again, until the happening of A or B 
 recovers A's stake, or assigns it to B : after which, both 
 parties stake again. 
 
 Supposing the means of B to be unlimited, let A m 
 represent A's chance of winning indefinitely, immediately 
 before a game in which both are to stake, A having m 
 stakes in his possession: and let K' m represent A's chance 
 immediately before a game in which B does not stake. 
 Then, by the preceding method 
 
 A m = aA m + i ,_j_ /3 A m __ i -|_ SA m _j- 0A' m 
 A' OT = A m + /3 A m _ i 
 Eliminate h' m and we have 
 
 ~ (a + p)*+e& (a+j3 
 
 in which the sum of the two co-efficients is unity. 
 Hence this game is equivalent in its ultimate chances to 
 a simple game in which it is a (a -f /3) to /3 (a + /3) -f 6 ft 
 for A against B. If a=/3, the last odds are those of 
 2a to 2a + 6. 
 
 This game of rouge et noir is described in an unin- 
 telligible manner, and with material omissions, in the 
 later editions of Hoyle, from which work, and from the 
 testimony of persons who have seen it played, I give the 
 best description I can make of it, observing that the 
 most modern method of playing differs in several parti- 
 culars from that given in the book referred to. 
 
 A number of packs of cards is taken (six, it is said in 
 
VI APPENDIX THE FIRST. 
 
 Hoyle) and all the cards are well mixed. Each common 
 card counts for the number of spots on it, and the court 
 cards are each reckoned as ten. A table is divided into 
 two compartments, one called rouge, the other noir, and 
 a player stakes his money in which he pleases. The 
 proprietor of the bank, who risks against all comers, then 
 lays down cards in one compartment until the number of 
 spots exceed 30 ; as soon as this has happened, he 
 proceeds in the same way with the other compartment. 
 The number of spots in each compartment is then be- 
 tween 31 and 40, both inclusive, and that compartment 
 wins which has the lower number of spots ; so that if, 
 for instance, there should be 37 spots in the rouge, and 
 32 in the noir, those players who staked upon noir 
 would win from the bank sums equal to their stakes. If 
 the number of spots be the same in both (which is called 
 in Hoyle a refait) the game is drawn, and the parties 
 withdraw, diminish, or augment their stakes at pleasure, 
 for a new game : except only when the number of spots 
 in both compartments is 31 (called in Hoyle a refait 
 trente et un), in which case the bank is allowed to with- 
 draw its stakes, and those of the players, whatever their 
 compartment may be, are impounded (placed en prison). 
 In the next game (now called an apres), the impounded 
 stakes are played for, the players choosing their com- 
 partments as before : should the bank win it takes the 
 stake, should the bank lose the player recovers his stake. 
 Should a second refait trente et un occur, or a drawn 
 game, the stakes still remain impounded, and are not 
 released until a gain or loss arrives. In the meanwhile 
 new stakes may be put down, before the fate of the old 
 ones is decided. 
 
 The chances of this game depend, in a slight degree, 
 upon the number of packs of cards which are mixed 
 together. When, however, there are as many packs as 
 six, it is very nearly * indeed the same thing as if the 
 
 * The only ways in which 31, for example, could be obtained from an un- 
 limited number of packs, and which could not equally well be obtained from 
 six packs, are those in which more than 24 aces occur. Now the proba- 
 
ULTIMATE RESULTS OF PLAY. 
 
 Vll 
 
 number of packs were unlimited. The following tables, 
 computed upon the latter supposition, will represent, with 
 more than sufficient accuracy, the chances of the several 
 arrivals. The first three columns of the first table ex- 
 hibit the chance of arrival of each number among the 
 several sum-totals which precede the arrival of thirty-one 
 or more. Thus opposite to 4 we see *096l, which is 
 the chance that the beginning of the sequence drawn 
 shall show one of the following sets of cards (1 standing 
 for ace, &c.) : 
 
 1, 1, 1, 1 
 1, 1,2 
 1,2, 1 
 
 2,1, 
 1,3, 
 
 3, 1 
 
 2,2 
 4 
 
 1 
 
 0769 
 
 11 
 
 1200 
 
 21 
 
 1398 
 
 31 
 
 1481 
 
 2 
 
 0828 
 
 12 
 
 ;1247 
 
 22 
 
 1424 
 
 32 
 
 1379 
 
 3 
 
 0892 
 
 13 
 
 1293 
 
 23 
 
 1449 
 
 33 
 
 1275 
 
 4 
 
 0961 
 
 14 
 
 1340 
 
 24 
 
 1472 
 
 34 
 
 1169 
 
 5 
 
 1035 
 
 15 
 
 1386 
 
 25 
 
 1493 
 
 35 
 
 1060 
 
 6 
 
 1114 
 
 16 
 
 1432 
 
 26 
 
 1511 
 
 36 
 
 0950 
 
 7 
 
 1200 
 
 17 
 
 1476 
 
 27 
 
 1527 
 
 37 
 
 0838 
 
 8 
 
 1292 
 
 18 
 
 1519 
 
 28 
 
 1541 
 
 38 
 
 0723 
 
 9 
 
 1392 
 
 19 
 
 1559 
 
 29 
 
 1552 
 
 39 
 
 0607 
 
 10 
 
 3806 
 
 20 
 
 2129 
 
 30 
 
 1683 
 
 40 
 
 0518 
 
 31 
 32 
 33 
 
 0219 
 0190 
 0163 
 
 34 
 35 
 36 
 
 0137 
 0112 
 0090 
 
 37 
 38 
 39 
 
 0070 
 0052 
 0037 
 
 40 
 
 0027 
 
 D. G. 
 
 0878 
 
 The fourth column contains all the chances of those 
 points which lie between 31 and 40, that is to say, the 
 chance of each being the first which arrives. Thus 
 1060 is the chance that 35 will appear, and that it will 
 be the first which appears above 30. The second table 
 (containing the squares of the numbers in the last column 
 of the first) shows the chance of each refait, that of the 
 
 bility that out of 31 cards drawn at hazard, 24 or more shall be aces, is 
 altogether beneath consideration. It is less than one out of a million of 
 million of millions. 
 
Vlll APPENDIX THE FIRST. 
 
 refait* trente et un being '0219* Opposite to D. G. is 
 the sum of the chances of all the refaits except the first, 
 which sum is the chance of a drawn game. 
 
 If then we say that *021 is the chance of a refait 
 trente et un and *087that of a drawn game, there remains 
 892 for the chance that either the bank or the player 
 must win ; which chances being equal, give '446 for 
 the player, and the same for the bank (exclusive of the 
 benefit of the apres). Returning then to the result in 
 page v., we find a = /3='446, 0r='021, and 2a to 2a 
 + is -892 to '913, or 892 to 913. Consequently; 
 
 At the game of rouge et noir, as now played, the 
 chances of ultimate ruin to the bank or the player are 
 the same as they would be at a simple game, which 
 must be either won or lost at each throw, and in which 
 the bank has 913 chances of winning, where the player 
 has 892. 
 
 The bank, as noticed in page 1 1 0., is playing against 
 the whole public, or against a player with unlimited 
 means. Taking 892 to 913, or more correctly 8903 
 to 9122, and applying the rule in page 110., the follow- 
 ing table results, which must be thus used. Opposite 
 to 30 we find *4824, which is the chance of the ultimate 
 ruin of a bank which risks one-thirtieth of its means at 
 every game : 
 
 10 
 
 7843 
 
 110 
 
 0690 
 
 210 
 
 0061 
 
 20 
 
 6151 
 
 120 
 
 0541 
 
 220 
 
 0048 
 
 30 
 
 4824 
 
 130 
 
 0425 
 
 230 
 
 0037 
 
 40 
 
 3783 
 
 140 
 
 0333 
 
 240 
 
 0029 
 
 50 
 
 2967 
 
 150 
 
 0261 
 
 250 
 
 0023 
 
 60 
 
 2327 
 
 160 
 
 0205 
 
 260 
 
 0018 
 
 70 
 
 1825 
 
 170 
 
 0161 
 
 270 
 
 0014 
 
 80 
 
 1431 
 
 180 
 
 0126 
 
 280 
 
 0011 
 
 90 
 
 1122 
 
 190 
 
 0099 
 
 300 
 
 0007 
 
 100 
 
 0380 
 
 200 
 
 0077 
 
 400 
 
 0001 
 
ULTIMATE RESULTS OF PLAY. IX 
 
 Hitherto all our results seem in favour of the hank ; 
 that is, tending to show that its advantage is not so 
 great as is commonly supposed. No person, granting 
 a bank permission to exist, would grudge it such an 
 advantage as would make it 49 to 1 against its being 
 ruined by the possible fluctuation attendant upon an 
 unlimited duration of play. This chance of being 
 ruined, namely *02, appears from the table to be ex- 
 ceeded, unless the bank possess 160 times the sum 
 risked at each game : if this were 100/., the bank would 
 need a capital of l6,000/. But I must now request 
 attention to the other side of the question ; first, con- 
 sidering the bank against the public; and, secondly, the 
 bank against an individual player. One of the most 
 important features in this game (which springs from 
 the old game of Faro, as did the last from the still older 
 game of Basset*) is, that the bank does not risk the 
 whole sum it lays down, but only the difference between 
 those sums which the caprice of the players obliges it 
 to stake on rouge and on noir. If 20 players have each 
 staked a guinea, 12 on rouge and 8 on noir, and if rouge 
 win, the bank loses 12 guineas and gains 8, and conse- 
 quently did not risk more than four guineas. It is im- 
 possible to say what chance there is of the bank having 
 to risk a given sum in such a case, as this depends on 
 the will of the players. When the cards have several 
 times decided for rouge, those players who think the 
 run is not finished will stake on that colour, while others 
 who think differently will stake on noir. I am wholly 
 without the means of saying what average exists, but I 
 should incline to think it very unlikely that the bank 
 really risks more than one fourth of its deposits, t But 
 the advantage which it derives from the refalt trente et 
 
 * An assertion of the editor of Hoyle, which is true as to the principle of 
 the game namely, that besides equal chances for the bank and the 
 player, there are chances for a drawn game, and a case in which the bank 
 has a direct advantage amounting to half the stake but the details are 
 very different. Both games are described in De Moivre. 
 
 f The bank is evidently (its chance of the apres excepted) merely the 
 means of equalising the sums staked on the two colours. 
 
X APPENDIX THE FIRST. 
 
 un, and its consequences, is gained on the whole of the 
 opponent stakes. The following is the method of esti- 
 mating the mathematical advantage of the bank : Before 
 a common game, the prospects of the bank lie entirely 
 in the chance of that game being followed by an apres, 
 since, in all other respects, the chances of gain and loss 
 are the same. After a refait trente et un, on whichever 
 colour any player may choose to risk his impounded 
 stake, the bank has the chance a of winning that stake, 
 and none of losing. But besides this, the bank has all 
 the chances of a second refait (or + c) or 1 2a), for 
 having another trial of the same kind : if then x express 
 the fraction of a pound, which such a chance of 11. is 
 worth, we have 
 
 1 2a) r, or 
 
 The mathematical advantage of the bank is therefore, 
 the chance 0, or '0219, of being put in possession of the 
 worth of half the stakes; or 'Oil/, of all the sums 
 it deposits. This is \l. 2s. per cent.* per deposit; 
 which, to those who know the rapidity with which the 
 risks succeed one another, will appear to yield, in the 
 course of the year, an ample return, not merely to the 
 deposits, but to the sums which are reserved for security 
 against fluctuation. It is probably 4J per cent, upon every 
 real risk ; and the return in the course of a year may be 
 easily guessed at. Imagine 100 different games, played 
 on each of 100 different evenings, the sum risked by the 
 collective players on each game being 50/. The total de- 
 posits of the bank would be 500,000/., on which II. 2*. 
 per cent, is 55001. The capital required to make this spe- 
 culation much more safe than any mercantile adventure, 
 would not be larger than its probable return in one year. 
 
 * Some time after this was written I chanced to find the following 
 sentence in the lately published Theory of Probabilities of M. Poisson. 
 " Dans les jeux publics de Paris 1'avantage a chaque coupest peu consider- 
 able : au jeu de trente-et-quarante par exemple, il est un peu au-dessous 
 de "Oil de chaque mise. Voyez sur les chances de ce jeu, le memoire que 
 j'ai inseredans le journal de M. Gergonne, tome xvi. numero 6, Decembre, 
 1825." I have not seen this memoir ; the accordance of the result with my 
 own, shows that I hav described the game correctly, as it was played in 
 Paris j of which, from paucity of information, I was by no means sure. 
 
ULTIMATE RESULTS OF PLAY. XI 
 
 If any persons, aware that the preceding calculations 
 are new, should imagine that there must be some mis- 
 calculation in a result which shows that cent, per cent. 
 on the necessary capital might be gained three times in 
 a year, I reply, that the chance of a refalt trente et 
 unj as given by the editor of Hoyle, produces a result 
 of nearly as surprising a character. For "0219 read 
 01 56, or g^ and the 55001. above-mentioned becomes 
 a trifle less than 4000/. The preceding results, 
 or either of them, being admitted, it might be sup- 
 posed hardly necessary to dwell upon the ruin which 
 must necessarily result to individual players against 
 a bank which has so strong a chance of success 
 against its united antagonists. But so strangely are 
 opinions formed upon this subject, that it is not 
 uncommon to find persons who think they are in 
 possession of a specific by which they must infal- 
 libly win. The last table given will show the 
 chances which any single player has of ruining the 
 bank, and of being ruined himself, as follows : If 
 the player stake one wth part of his means * at each 
 throw, and the bank one nth part, from unity subtract 
 the number in the table opposite to n and to n-\- m y 
 and the first result divided by the second shows the 
 chance that the bank will ruin the player. Suppose, 
 for example, that the player risks 1-1 Oth and the 
 bank l-l60th of their respective resources. Then 
 opposite to 160 and 16'0-f 10, we find -0205 and *0l6l ; 
 which, being subtracted from 1, give '9795 and '9839, 
 whence tri^rf is the chance for the bank ; or it is 9795 
 to 44, or 223 to 1, that such a player will be ruined. 
 Even if both the player and the bank stake l-l60th 
 parts of their several funds, the bank will still have '98 
 or 49 to 1 in its favour against that one player. 
 
 The last -column of the table in page vii, shows that 
 the bankers at rouge et noir, by making the apres de- 
 
 * We must not here consider what the'bank stakes against the individual 
 player, but the whole sum which it risks. 
 
Xll 
 
 APPENDIX THE F1HST. 
 
 pend on the refait trente et un have chosen the most 
 favourable out of the ten cases. The following table will 
 show the effect of substituting any other refait ; the 
 first column pointing out the refait in question ; the 
 second, the simple game to which it is equivalent in the 
 chances of ultimate ruin ; the third, the benefice of the 
 bank upon every 100/. deposited : 
 
 
 
 s. d. 
 
 31 
 
 10,000 to 10,246 - 
 
 1 2 
 
 32 
 
 - 10,213 - 
 
 19 
 
 33 
 
 - 10,183 - 
 
 16 6 
 
 34 
 
 - - 10,154 - 
 
 13 6 
 
 35 
 
 - 10,126 - 
 
 0110 
 
 36 
 
 - 10,101 - 
 
 090 
 
 37 
 
 - 10,079 - 
 
 070 
 
 38 
 
 - - - 10,058 - 
 
 050 
 
 39 
 
 - 10,042 - 
 
 040 
 
 40 
 
 - 10,030 - 
 
 030 
 
 The above is a graduated scale of poisons, each one 
 being slower in its operation than the preceding ; the 
 first, or quickest of all, being that which is used at 
 present. Of all the illegal games, none that I know of 
 is less likely to lead to ruin than rouge et noir ; and the 
 results of this investigation give a sufficient notion of the 
 state of the case between the banker and his dupes. 
 
 The first table, in page vii., is calculated in the fol- 
 lowing manner: The chance of any given card at 
 a given point is T 1 j for every number which a card 
 can give, excepting 10, the chance of which is T 4 T , on 
 account of the value given to the court cards. Let x be 
 a number greater than 10, and let V x be the chance of 
 arriving at that number in the laying down of the cards. 
 Then, if (V^) signify the event of which the chance 
 is V Xi and if by (a) (6), we mean the consecutive 
 happening of the two events whose chances are a and b, 
 it follows that (V x ) when it happens, must happen in 
 one of the following ways : 
 
ULTIMATE RESULTS OF PLAY. Xlll 
 
 or V - (A) OT V - ( or 
 
 from which it is readily found that 
 
 . AV, - * ( V, *-4AV x _ 10 _V ;c . 9 . ) 
 
 The first eleven values of V x are thus determined : V t is 
 evidently ^; V 2 is ^ V t + ^ or & . fa V 8 is T ' 3 Vj + -^ V 2 
 + T3 or A (la) 2 ; anc ^ so on U P to ^10 wa ich is 
 
 V n is T 3 ( Vio + V 9 . . . + V 2 ) 
 
 The rest were then calculated by the preceding for- 
 mula for AV.J. to six places of decimals (by which the 
 accuracy of four was insured) as far as Vsi inclusive. 
 The remainder were calculated by those which preceded, 
 leaving out the terms which the necessary distinction 
 already mentioned requires to be omitted. 
 
 It may perhaps appear to some that a part of the 
 preceding reasoning is inapplicable, since it only calcu- 
 lates the chances of ruin in an indefinite succession of 
 gam es_, whereas any practicable number of games, though 
 great, may not involve the same chances as an infinite 
 number. The objection is valid in principle, but the 
 correction which is rendered necessary by it is not worth 
 consideration, if any large number of games be in ques- 
 tion. 
 
 The following are the only cases in which a simple 
 approximate rule can be given, connected with finite 
 numbers of games : 
 
 PROBLEM. Both parties have the same number of 
 stakes (which should not be less than 20), say a, and 
 the play is equal, or either has an even chance of win- 
 ning any one game. What is the chance that one or 
 other shall have been ruined before a? games have been 
 played ? (oc being a large number). 
 
XIV APPENDIX THE FIRST. 
 
 RULE. Divide 30 times x by 56 times the square of 
 a, and from the quotient subtract -1049. If the result 
 be the common logarithm of s, then ss 1 to 1 are the 
 odds in favour of the event. 
 
 EXAMPLE. The number of stakes is 45, and the 
 play equal : what is the chance that one or other is 
 ruined before 1520 games have been played ? 
 
 a = 45, oc = 1520, 56 x a x a = 1 13400 
 
 45600 
 x x 30 = 45600, = -4021 
 
 4021 -1049 = -2972 = logarithm of 1 -982. 
 Answer, -982 to 1, or 982 to 1000, 
 
 PROBLEM. The stakes being equal, and also the play, 
 as before, what is the number of games in which it is n 
 to 1 that one party or the other will have been ruined ? 
 
 RULE. To the common logarithm of one more than 
 n add *1049 : multiply the result by 56 times the square 
 of the number of stakes, and divide by 30, which gives 
 the number required, very nearly. 
 
 EXAMPLE. Both parties have 50 stakes: in what num- 
 ber of games is it 10 to 1 that one or other will be ruined? 
 
 a = 50, n= 10, n + 1 =11, log. 11 = 1-0414 
 1-0414 + -1049=1-1463, 56 x a x a = 140000 
 
 160482 
 1-1463 x 140000= 160432: ^ = 5349 
 
 Answer: in about 5349 games. 
 
 To find the number of games in which it is an even 
 chance that one or other will be ruined, from three- 
 fourths of the square of the number of stakes, subtract 
 its hundredth part. Thus, if both parties have 40 
 stakes, then 40 x 40 being 1600, three- fourths of which 
 is 1200, from 1200 subtract 12, which gives 1188 for 
 the number of games (very nearly) in which it is an 
 even chance that one or other will be ruined. 
 
 If a player with a stakes play with one of unlimited 
 means, the chances being the same for both, it is an 
 even chance that he is ruined in a number of games 
 
VALUE OF COPYHOLDS. 
 
 which is thus found: take 2J of the square of a. If 
 greater accuracy be required, add to the result one less 
 than its 7@0th part, which is sure to make it correct 
 within a single game. Thus if the number of his stakes 
 be 100, 100xlOOx2f is 23750, the ?60th part of 
 which is 31, whence 23750 + 30, or 23780, is within 
 one of the number of games required. 
 
 APPENDIX THE SECOND. 
 
 ON THE RULE FOR DETERMINING THE VALUE OF 
 SUCCESSIVE LIVES, AND OF COPYHOLD ESTATES. 
 
 THE rule given in the work is in a different form from 
 that of any writer with whom I am acquainted, though 
 it agrees with that given by Mr. Milne, as will be shown. 
 This Appendix has been rendered necessary by the fact 
 that no writer has solved the question of the value of 
 copyhold estates with absolute correctness except Mr. 
 Milne, whose solution is in a form of unnecessary diffi- 
 culty. The writers with whom I am acquainted, who 
 give the old rule, or one involving an omission of the 
 same kind, are De Moivre, Dodson, Thomas Simpson, 
 Stonehouse, Morgan, Baily, and the French translator 
 of the latter. Mr. Milne stands alone in proposing a 
 somewhat different rule, which like many results of 
 independent investigation, differs more than need have 
 been the case from the form of preceding results. 
 
 Let there be an estate held on a single life, and 
 renewable for ever upon payment of a fine of II. Let 
 it be a condition, that each renewal is to be made on the 
 1st of January next following the extinction of the 
 previous life ; it is required to find the present value of 
 all the fines. 
 
 Firstly, To find the value on the 1st of January, the 
 moment after a fine has been received, and the best life 
 which can be found has been put in. Let P be the 
 y 
 
XVI APPENDIX THE SECOND. 
 
 value of the life, or the value of an annuity upon it ; r 
 the rate of interest per pound, and F the value of all the 
 fines. Then upon the new year's day next following 
 the extinction of that life, the whole value of the fines 
 will be 1 -f F, because the person who claims the fines 
 will have one pound to receive, and will then have 
 remaining the interest which we have called F. It fol- 
 lows therefore that F is the present value of I -f F to be 
 received at the end of the year in which the life drops : 
 or, by the well known formula 
 
 where E is , the value of a perpetuity of 11. 
 
 Secondly, Let the life already in possession be of the 
 value A, the lives at each renewal having the value P, 
 as before. Consequently, the present value of the fines 
 is that of 1 + F to be received at the end of the year in 
 which the present life drops : and this is 
 
 .. 
 1+r ^ E + l 1+P 1 + P 
 
 If an estate be held on any number of lives, with a 
 fine of I/, on renewing each, it is precisely the same 
 thing as if a similar number of estates were held on single 
 lives, and the present value of all the fines, the present 
 lives being worth A, B, C, &c., n in number, is 
 
 l+P 
 
 the common rule divides by P instead of 1 + P- 
 
 A mathematical rule, when erroneous, is best exposed 
 in its extreme cases ; let us then suppose the life P 
 certain to drop in the year, that is, worth nothing. 
 There will consequently be a fine to pay every 1st of 
 January, and the present value of what I called F in 
 
 the preceding investigation, is or E. The rule I have 
 given, shows that F=E when P=0 ; that of De Moivre, 
 
VALUE OF COPYHOLDS. XV11 
 
 Simpson, &c., makes F infinite. Again, suppose the 
 life P certain to last one year and to drop in the second ; 
 1 "F 
 
 in which case its value is , or ^ -, and F is a 
 
 1 -\-r J&+ 1 
 
 perpetuity of IL receivable at the expiration of every two 
 
 E 2 E 
 
 years, or - -. If A = P ^T" p tlle new formula 
 
 becomes ; -, the old one becomes E, the value of 
 
 a perpetuity of 17. receivable at the end of every year. 
 
 I shall now show that the preceding rule agrees with 
 that of Mr. Milne; which is as follows: Let P be worth 
 an annuity certain of t years, and let v be the present 
 value of II. to be received a year hence. Then the pre- 
 sent value of all the fines, according to Mr. Milne, is 
 
 A' + B'+C' + 
 
 where A', B', C', &c. mean the present value of I/., to 
 be received at the end of the years in which the lives 
 severally drop. Since P is the value of an annuity certain 
 for t years, we have 
 
 P = E-t><E, n-P=LT- ' = ?-<, 
 
 r v 
 
 E-A A' E-A 
 
 and A = E77' whence i-r^l= TTP' 
 
 from which the coincidence of the two rules is manifest. 
 The error of the old rule, by the Northampton Tables, 
 and at 4 per cent, (the best life being worth 17*25 years 
 purchase) is, that the result is 5-J- per cent, too great. 
 
 The old rule, as Mr. Milne justly observes, is derived 
 from the supposition that the new life is put in at 
 the beginning of the year in which the old one drops, 
 instead of at the end ; which last was in the intention of 
 those who formed the rule. It may be said however, that 
 Y 2 
 
XV111 APPENDIX THE SECOND. 
 
 the old rule is nearer the truth than the new; since one 
 time with another, the renewal is made before the end 
 of the year in which the old life drops. This objection 
 must be valid to some extent ; and I proceed to inquire 
 how much weight must be allowed to it. 
 
 Let a be the fraction of a year allowed for renewal : 
 
 it is clear then that renewals (each accompanied by 
 
 a fine) may take place in the year. Let the life A drop 
 at the expiration of the fraction 9 of a year, or between 
 and 6 + dd, the chance of which is dd itself, if A be 
 supposed equally likely to drop at any period of the year. 
 At 0-fci, then, the new life is put in; and if this new 
 life drop before 1 a of the year is gone, another fine 
 must be paid, and another renewal is made, which again 
 may drop before 1 a, and so on. But since the chance 
 of each additional renewal is very much smaller than 
 that of the preceding, it will be sufficient to take the 
 first only into consideration. Let it be supposed, then, 
 that not more than one renewal shall take place within 
 the year in which A drops. 
 
 Let a be the chance that the life P drops in a year 
 after nomination, in which case we may call x a the 
 chance that it drops in any fraction x of a year. Then 
 d (i_0_ a ) a is the chance that the life A drops 
 between 9 and -f d 0, and that the next life drops 
 within the year, in which case another fine is to be paid 
 at the beginning of the next year. Consequently, neg- 
 lecting the interest of the fine in a fraction of a year, 
 the lessee has the chance d 9 (1 6 a) a of having a 
 second fine to pay, upon the contingency of A dropping 
 between 9 and 9 -f d 9. Integrate this expression from 
 6=0 to 9= l-o, and we have \a (l-) 2 for the 
 chance of a second fine : which, with the fine certain 
 upon the death of A, shows that 1 + \a (1 a) 2 is the 
 mathematical expectation of the fines to be paid, when 
 the probability of one renewal within the year is con- 
 templated, and another at the end, if necessary. 
 
 A more complicated process, proceeding on the sup- 
 
VALUE OF COPYHOLDS. XIX 
 
 position that any given number of renewals may take 
 place within the year in which A dies, gives the follow- 
 ing terms for the total probability of one or more re- 
 newals taking place before the end of the year : 
 
 a, the chance that there shall be no renewal ; 
 
 a(l-2a) 2 
 1 a - - - the chance of one only ; 
 
 0-g) f _ *(l-3) the ehance of two onl 
 
 and so on, the series being continued as long as the requisite 
 multiples of a are less than unity. Hence the chance that the 
 number of renewals shall not exceed n, is 
 
 1- 
 
 2.3.. n . n + 1 
 
 which, in the case most against us, that is, supposing instanta- 
 neous renewals, or a = 0, is 
 
 1 
 
 Let the life P be one of seven years old, in which 
 case the Northampton table gives a =^$5= '0186. 
 Neglecting interest for the fractions of the year, and 
 remembering that the number of renewals is the num- 
 ber of fines, the mathematical expectation of ah 1 the 
 fines is the sum of 
 
 (l-2a) 2 
 
 a)3\ 
 /' 
 
 2 v 2 2.3 
 
 To this add a, the chance that the renewal fine certain 
 on the death of A, shall outrun the year, and we find 
 
 a (12 cu 2 a 2 (l-3a) 3 
 
 ~T~ 2.3 
 
 for the mathematical expectation of all the fines which 
 shall be paid in the year in which the present life drops, 
 including the chance of that life dropping too late to 
 renew it within the year, and of its being therefore re- 
 newed within a period not exceeding a of the year fol- 
 lowing. If a = 0, this becomes 
 Y 3 
 
XX APPENDIX THE SECOND. 
 
 * a - 1 , c being 2 -7182818 
 a 
 
 and when a = "0186 , = 1 *019 
 Hence '01 9 -r--0186 being 1-02, it appears that 
 1 02(1 -f- r) is an enormously exaggerated representation 
 of the fine which must be substituted for I/ in the am ended 
 rule to make the correction which might be suggested 
 by the advocates of the old rule. Allowing r= *035 , 
 it may be granted that the old rule is correct, in the 
 particular case before cited, if we suppose : 1. That no 
 time whatever is allowed for renewal ; 2. That the best 
 life which can be found is such, that 1 out of 59 of such 
 lives drop in a year, and ; 3. That the lessor is to receive 
 his fines at the beginning of the year succeeding that in 
 which A drops, with 3-Jper cent, interest, reckoned from 
 the beginning of the preceding year. To take a more 
 rational supposition, let us allow six months for re- 
 newal. Here a= \ and only the first term of the series 
 can be taken, which, with the interest, gives 1 -f %. 
 
 It appears from the preceding, that the advantage of 
 the lessor over that which the rule gives him, is trivial, 
 except in this, that at the time when the rule supposes 
 him to receive one pound, he may have received and 
 improved one pound during a fraction of a year. There 
 is also another advantage which he has, and which neither 
 rule allows him. The renewal may have been made 
 before the time supposed in the rule, in which case the 
 existing life will be somewhat worse than that supposed. 
 To take all these circumstances into account, suppose 
 the life A to drop at the end of & of the year, in which 
 case the lessor is in possession of 1 -fr (0 4- (*)r at the 
 end of the year, and the lessee has a life * worth 
 
 P 1 (6 -f- a) j A P, where A P is the decrement 
 
 * Let it be remembered that the renewal may take place after the 
 beginning of the year, which is equivalent to having a better life than 
 that supposed. Algebra, as in other cases, strikes the balance of 
 positive and negative quantities^ without the necessity of introducing 
 several formula?.. 
 
VALUE OF COPYHOLDS. XXI 
 
 of the value of the life in one year. Multiplying by 
 dQ, the chance of this occurrence,, and integrating from 
 6=0 to 6 = 1, we find 1 +-| rar for the mean sum, 
 
 and P i^P-f^AP for the mean life. If the time 
 allowed for renewal be more than six months, the rule 
 (without this correction) gives an advantage to the lessor; 
 if six months, to neither ; if less than six months, to the 
 lessee. Substitute 1 + -J- r a,r and P A P -j- a A P 
 for II fine and a life P, and we have 
 
 E-A 
 
 {l+ir-ar ]; 
 
 which I believe to be the most correct rule that can be 
 given for the present value of all the fines upon a single 
 life renewable for ever, the value of the present life 
 being A, the tabular value of the best life P, that of 
 a life one year older P AP, the fraction of a year 
 allowed for renewal a, r the interest of one pound for 
 one year, and E the perpetuity. By substituting nE 
 (A + JB + C-h ---- ) for E-A, the value of all the fines 
 from an estate held on any number of lives is found. 
 Similar considerations apply to the present value of IL 
 to be received in a of a year after the death of a life A. 
 The offices frequently pay in three months after the 
 death is proved, whereas the tables are calculated for 
 the end of the year of death. Again, they rate all lives 
 as they will be at the next birthday, the parties being 
 one with another half a year younger. To a party who 
 dies at the end of of a year, the office has paid by the 
 end of the year 1+ ( 1 a 0) r which, treated as be- 
 fore, gives 1 -{--J- r ar. And the present value of I/., 
 which is computed by the office from 
 
 (where Jc is the proportion of profit demanded) may be 
 more strictly computed from 
 
 E-A IAA ,_ , . 
 
 - Ir _ f JL x (] +lr_,) 
 Y 4 
 
XX11 APPENDIX THE THIRD. 
 
 where A + AA is the value of a life one year younger 
 than the office age of the party at entry. 
 
 When an annuity is granted upon condition that the 
 executors of the party are to receive such a proportion 
 of payment for the year in which the annuitant dies, as 
 corresponds to the portion of the year during which he 
 is alive, the addition to A the value of the annuity is 
 the present value of \ + ^ r of a year's purchase, pay- 
 able at the end of the year of death. The formula 
 which then very nearly represents the result of the pre- 
 ceding correction is 
 
 A (1-5) +4-5 
 
 or to the tabular value of the annuity add the excess of 
 half a year's purchase over half a year's interest of the 
 tabular value, together with one- third of the interest of If. 
 
 APPENDIX THE THIRD. 
 
 ON THE BULB FOB DETERMINING THE PROBABILITIES 
 OF SURVIVORSHIP. 
 
 IT does not seem to have been noticed, that this rule 
 is considerably more correct than its framers could have 
 anticipated, supposing them to have contemplated no 
 higher degree of exactness than their demonstration en- 
 titled them to assert. Let< and -fyt represent the pro- 
 babilities that A and B, now alive, shall be alive at the 
 end of t years, t being whole or fractional. Then 
 \l/'t.dt is the probability that B shall die between 
 t and t + dt and ftyt. tyt dt from t=n to t=n+l 
 is the chance of a survivorship of A beginning to take 
 place somewhere in the (n-fl) th year after the present 
 time. Let a and a + Act and /3 and /3 + A/3 be the 
 chances that A and B shall be alive at the end of n and 
 years from this time : then, in the demonstration 
 
RULE FOB SURVIVORSHIP. XX111 
 
 of the rule,, <j>ti& assumed = a + A a . and i//=/3 + A/3.Z, 
 t being measured from the beginning of the (n + l^th 
 year. Hence 
 
 f$t. ^'t.dt (from #=0 to t=l) = a A/3 | AaA# 
 
 which is the common rule in a different form. 
 
 Let us now suppose (t being measured from the be- 
 ginning of the (n + 1 ) th year) 
 
 * 1 
 
 <$>t = a, +Aa.t + A 2 a. - 
 
 where the differences constitute series of rapidly di- 
 minishing terms. The only term of the second order 
 which this addition to the hypothesis introduces into 
 -/< ^'t.dtis aA 2 /3/(^-^) dt which is =0, when 
 taken from t=0 to =1. Consequently the errors of 
 the rule are all of the third order. 
 
 To give a notion of the amount of error, extend the 
 preceding formulae to terms of the third order, and form 
 the integral, reserving only the terms of the third order. 
 The final result is as follows : If x and y be the num- 
 ber of the living at the age of A and B, and if , b, c, . . . 
 be the numbers alive, at n } n + 1? years older than A, 
 and p, q, r, . . . at n, n -f 1, ... years older than B, then 
 the probability that A shall begin to survive B in the 
 course of the (n + 1) th year of the calculation, is 
 
 
 2 xy 
 
 the first term being that generally used, and the second 
 a correction which ought always to be applied in those 
 parts of the table in which the yearly decrements are 
 not equal. 
 
 The demonstration of the preceding will be easily 
 arrived at by the indication which I have given, by any 
 one acquainted with the integral calculus. To those 
 who have not that advantage, reason may be shown in 
 the result, though not for the result. The preceding 
 
XXIV APPENDIX THE FOURTH. 
 
 correction will have the positive sign when & c bears a 
 greater proportion to a b than does q r to p q: 
 that is, when the mortality in A's part of the table is in- 
 creasing faster than in B's. Now, ceteris paribus, the 
 larger the comparative mortality of the year succeeding 
 a given year, the more likely are the deaths of the latter 
 part of that given year to predominate over those of the 
 former ; consequently, the more likely is the death of 
 A, if it happen in that year, to be towards the end of it. 
 But any thing which shows that the death of A is more 
 likely than before to take place later in its year, increases 
 the probability that a survivorship commencing in that 
 year shall be in favour of A, and not of B. 
 
 APPENDIX THE FOURTH. 
 
 ON THE AVERAGE RESULT OF A NUMBER OF OB- 
 SERVATIONS. 
 
 THAT I might not further embarrass the most ab- 
 struse chapter of this work, by the introduction of an 
 isolated point of difficulty, I have chosen here to men- 
 tion some considerations connected with the value of 
 the average of observations. There is a remarkable 
 difference of principle between two problems which at 
 first sight appear identical; namely, where it is re- 
 quired to invent a method of treating observations be- 
 fore they are made, and after they are made. Positive 
 and negative errors being equally likely, and no observ- 
 ations having been made, it is easily proved that there 
 is a high probability in favour of a large number of 
 observations giving exactly or nearly the same total 
 amount of one as of the other. The case is analogous 
 to that of an urn filled with black and white balls in 
 
ON THE AVERAGE RESULT OP OBSERVATIONS. XXV 
 
 equal numbers, out of which, in a large number of 
 drawings, both sorts will come in nearly equal propor- 
 tions, 
 
 But in this, as in every other question of proba- 
 bilities, any additional knowledge of the circumstances 
 which may happen, or have happened, changes the 
 problem, and is equivalent to an extension or limitation 
 of its conditions. When the observations have been 
 made, the position of the observer is altered, since though 
 the law of facility of error be not determined, yet more 
 probability is given to some laws than to others, by 
 inspection of the observations themselves. For instance; 
 if the observations give results of very little discordance, 
 it is immediately obvious that a law of facility which 
 makes the probability of large errors very small, is 
 more likely to have been that which actually existed 
 than one of a different character. The problem now 
 presents an analogy with that of an urn, from which 
 drawings have been made and registered, so that the 
 contents of the urn are to be guessed at from the 
 drawings. 
 
 In the first problem, and supposing that a method of 
 combining the observations is to be chosen before obser- 
 vations made, it is demonstrable that the average of the 
 results is more likely to be true than any other magni- 
 tude. And the same conclusion seems probable in the 
 second case, since unassisted common sense would never 
 draw any distinction between the two problems. But the 
 results of calculation applied to the development of the 
 distinction just drawn, show that the average of obser- 
 vations made is not necessarily the most probable re- 
 sult, nor can be such for more than two observations, 
 unless one particular law of facility of error be sup- 
 posed, which law is the standard law described in 
 Chapter VII. But it is also shown, as mentioned in 
 page 142, that the results of any law of facility, when 
 applied to tolerably large numbers of observations, are 
 nearly identical with those of some variety or other of 
 the standard law; so that, practically. f ^ ~*w>v<* nf 
 
XXVI APPENDIX THE FIFTH* 
 
 observations is either the result which the strictest ap- 
 plication of sound principles would declare to be the 
 most probable truth, or else very near to it. 
 
 It is, in the meanwhile, a most remarkable circumstance 
 that a method so simple, and so conformable to common 
 sense, as that of averaging, should first turn out to be 
 incorrect, except upon a supposition never contemplated 
 in thinking of the evidence of this rule, and should after- 
 wards prove to be always nearly correct, for large num- 
 bers of observations, on account of the tendency of all 
 admissible suppositions to confound themselves, as the 
 number of observations increases, with that one parti- 
 cular supposition, which makes the common notion 
 absolutely correct. My own impression, derived from 
 this and many other circumstances connected with the 
 analysis of probabilities, is, that mathematical results 
 have outrun their interpretation : and that some simple 
 explanation of the force and meaning of the celebrated 
 integral, whose values are tabulated at the end of this 
 work, will one day be found to connect the higher and 
 lower parts of the subject with a degree of simplicity 
 which will at once render useless (except to the his- 
 torian) all the works hitherto written. 
 
 APPENDIX THE FIFTH. 
 
 ON THE METHOD OF CALCULATING UNIFORMLY DE- 
 CREASING OR- INCREASING ANNUITIES. 
 
 AN authority from which I rarely differ has spoken 
 thus, " A few writers on these subjects, of late years, 
 have employed the differential and integral calculus in 
 their investigations. We have not yet seen any fruits 
 
ON UNIFORMLY CHANGING ANNUITIES. XXV11 
 
 of this application of the calculus, which appear to us 
 of much value, nor are we at all sanguine in expecting 
 any." The tendency of such an assertion is to en- 
 courage those who study the subject, to stop short of 
 the differential calculus in their mathematical studies. 
 Now I assert, 1. That the calculus aforesaid may,, as 
 evidenced in the results of chapter IV., lead to most 
 valuable rules in the estimation of complicated proba- 
 bilities. 2. That if the calculus be not serviceable in 
 the deduction of the law of mortality, it is from defect 
 of observed data. As soon as larger and more correct 
 tables of the numbers living are obtained, the differen- 
 tial calculus is ready to furnish methods for correcting 
 those now in use. 3. That the differential calculus may 
 be made to give important simplifications of processes, 
 and to render the tables already constructed immediately 
 available for purposes to which no one now dreams of 
 applying them. 
 
 If v be the present value of I/., to be received at the 
 end of a year, and tyv be the present value of a con- 
 tingent annuity of I/., then that of an annuity which is 
 to be 11. at the end of the first year, 2/., 31. , &c., at 
 the end of the second, third, &c. years, is vfy'v, where y'v 
 is the differential coefficient of $v. Now 1 -f r being 
 the amount of II. in one year, we have 
 
 dv 1 d$v _ ^ d<f>v 
 
 ~& = (l+r)2 ' 1o = ~ 'dr ( + r > 
 
 dtyv d<j>v 
 
 and the annuity v--~ = (1+r) 
 
 dv dr 
 
 Now tables of annuities of I/, being calculated for a 
 succession of values of r differing by '01, we have 
 
 01 x = &<pv - 1 A2<Jw + I A3</w - I A4<?w + &c. 
 
 Substitute the value of thence obtained, and we 
 dr 
 
APPENDIX THE FIFTH. 
 
 have a method of finding the value of the required 
 annuity, which may be described in the following 
 
 Take out the value of an annuity of 1 /. at the given 
 rate of interest, and at several successive higher rates : 
 take the successive differences, the difference of the dif- 
 ferences, and so on. To the first difference add half of 
 the second difference, one-third of the third, and so on : 
 the sum of these, multiplied by the amount of WOL in 
 one year at the first named rate, is the value of the 
 annuity required. 
 
 I take examples from the Northampton tables, at 4 
 per cent., because Mr. Morgan has given a table of the 
 annuities required, which will serve to find verifications. 
 First suppose the age to be 5 years. 
 
 060 
 
 2-421 + ^ of -556 + of -164 + \ of -060 = 2'769 
 2-769 x 104 = 288-0 answer: in Morgan 288-4 
 
 Next suppose the age to be 80 years. 
 
 Annuity 4 p. c. 3*643 , 9R 
 
 5 3-515 if: -007 
 
 6 3-394 *. '008 
 
 7 3-281 .jlj* -006 
 
 8 3-174 
 
 128 -- \ of -007 = -132 ; -132 x 104 = 13 -7 answer 
 
 13*8 in Morgan. 
 
 PROBLEM. A life annuity is m at the end of the 
 first year, and diminishes n every year, until nothing 
 is due, after which it ceases entirely. Required its 
 present value. 
 
 RULE. When n is so small, that the annuity cannot 
 
 Annuity 4 p. c. 
 
 17-248 
 
 2*421 
 
 
 
 
 
 6 
 
 7 
 
 14-827 
 12-962 
 11-489 
 
 1-865 
 1-473 
 
 1 .1 QC 
 
 556 
 392 
 
 288 
 
 164 
 104 
 
 , 
 
 8 - 
 
 10-304 
 
 i i no 
 
 
 
ON UNIFORMLY CHANGING ANNUITIES. XXIX 
 
 be extinguished during the tabular life of the party, 
 from the value of an annuity of (m -f n) subtract n 
 times that of an increasing annuity of II. found as 
 already described. But \vhen the annuity can be ex- 
 tinguished during the life of the party (say in t years 
 exactly, so that m=nt), then to the preceding result 
 add n times the value of an increasing annuity of IL 
 on a life -f 1 years older than the party, multiplied by 
 the chance of his living t -f 1 years, and by the present 
 value of 5^ 1 due t-\-l years hence. 
 
 PROBLEM. Required the present value of m to be 
 received at the end of the year in which A dies, if in 
 a year, or (m n) if in the second year, and so on. 
 
 RULE. When n is so small that the sum insured 
 cannot be extinguished during the tabular life of the 
 party, to the value of a perpetuity of j^Pra, add that of 
 an increasing annuity of n, ^g%n, 3n, &c., and 
 subtract the value of a simple life annuity, of which 
 the yearly payment is m, increased by the product of 
 a perpetuity due, and the value of a simple annuity of 
 n : divide the difference by the value of a perpetuity 
 due, and the quotient is the present value required. But 
 if the insurance be extinguished in t years, or if m=nt : 
 find the product of an annuity due of ^1 on a life + 1 
 years older than the given life, and of a perpetuity ; 
 subtract the value of an increasing annuity of 11. on that 
 life, and having multiplied the difference by the chance 
 of the first life surviving t + 1 years, and by the present 
 value of n due t + 1 years hence, add the result to the 
 dividend in the first part of the rule, before dividing by 
 the value of a perpetuity due of l. 
 
 Various other questions will present them selves, which 
 can be easily reduced to practice by aid of the expedi- 
 tious rule for finding the values of increasing annuities. 
 This rule may be applied to the Carlisle tables (for 
 which Mr. Milne has deduced the values of annuities 
 on single lives, at rates of interest from 3 to 8 per cent., 
 
XXX APPENDIX THE FIFTH. 
 
 both included), and also to joint lives, though not 
 with so much correctness, on account of the tables not 
 containing so many rates of interest. 
 
 The following is an instance in which a deduction from 
 the calculus of differences will supply in a rough man- 
 ner the deficiencies of tables. There are none of these 
 for determining the mean duration of the joint existence 
 of two lives, but the defect may be supplied with suf- 
 ficient accuracy for many purposes, and particularly at 
 the middle and older ages, by the following RULE. 
 Let (3), (4), &c.' stand for the values of an annuity 
 on a single life, or on two joint lives, at 3, 4, &c. 
 per cent. : from twice (3) subtract (6) and reserve the 
 remainder : from (4) subtract (5), and having halved 
 the remainder, to it add the tenth part of (5), and 
 multiply the result by 9- Subtract the last product 
 from the reserved remainder, and multiply the differ- 
 ence by 10. The result increased by '5 in the case of 
 a single life, or by -25 in that of two joint lives, will 
 be something under the mean duration required. For 
 example, and to take a very unfavourable case, let the 
 Carlisle table be used, the life being 10 years old. 
 
 (3) = 23-512 (4) = 19-585 
 
 (5) = 16-669 
 
 47-024 2) 2-916 
 
 (6) = 14-448 
 
 1 -458 
 
 32-576 ^(5) 1-667 
 
 28-125 
 
 28-125 
 
 45-01 the truth being 48-82 
 
ON OFFICE VALUATIONS. 
 
 The error diminishes as the age increases, as the 
 following table will show : 
 
 Age. 
 
 Approx. 
 
 Truth. 
 
 Age. 
 
 Approx. 
 
 Truth. 
 
 
 
 34-78 
 
 38-72 
 
 60 
 
 14-13 
 
 14-34 
 
 10 
 
 45-01 
 
 48-82 
 
 70 
 
 9-33 
 
 9-18 
 
 20 
 
 39-07 
 
 41-46 
 
 80 
 
 5-57 
 
 5-51 
 
 30 
 
 33-09 
 
 34-34 
 
 90 
 
 3-43 
 
 3-28 
 
 40 
 
 27-05 
 
 27-61 
 
 100 
 
 2-36 
 
 2-28 
 
 50 
 
 20-96 
 
 21-11 
 
 
 
 
 APPENDIX THE SIXTH. 
 
 ON A QUESTION CONNECTED WITH THE VALUATION OF 
 THE ASSETS OF AN INSURANCE OFFICE. 
 
 IF an insurance office were ahout to close its doors, 
 and to buy up all the policies of its members, the pro- 
 cess of valuation would only require the assets to be ex- 
 pressed by the amount of money which they would 
 actually produce at the time of valuation. In this 
 case, the profit or surplus is properly expressed by 
 A + P C, or the amount of assets increased by the 
 present value of all the premiums, and diminished by 
 that of all the claims. 
 
 But valuation is not usually made with reference to 
 an immediate settlement ; but for the purpose of ascer- 
 taining what sum can be set apart as profit, and de- 
 clared to belong to existing policies, without anticipatory 
 injustice to future members. The preceding formula, 
 with allowance for expences of management, still repre- 
 sents the sum which may be called profit, provided that 
 the stock belonging to the office can really be improved at 
 the rate of interest assumed in the valuation. For the 
 sufficiency of this stock to answer all demands depends 
 upon its increasing at that rate of interest upon which 
 the values of P and C were found. 
 
XXX11 APPENDIX THE SIXTH. 
 
 Now, it generally happens, that the property of an 
 insurance office consists of funds invested at different 
 rates of interest, the consequence of which is, that there 
 is no absolutely rigorous method of determining the 
 profit, except by prospective calculation of the state of 
 the office for every year of the tabular duration of the 
 life of its youngest member. Supposing the insured to 
 die precisely in the manner indicated in the table, and 
 assigning the order in which the different principals are 
 to be touched, when necessary, it is then possible to 
 calculate the amount which will remain when all claims 
 are paid. The present value of this amount (the spe- 
 cies of stock in which it is to be left being known) is 
 all that can be called profit at the time of the valuation. 
 This process, however, is exceedingly laborious ; and, 
 n all probability, where yearly valuations are made, the 
 expence of making the calculation would be greater than 
 the loss prevented by taking the more simple, but less 
 accurate, method. 
 
 If money made only simple interest, and computations 
 were performed accordingly, no difficulty would arise : 
 for =^S improving at r per pound, and =S' at r' 
 per pound, is at all times equivalent to <^P(S + S'), im- 
 proving at (Sr-t-S' r x ) -r- (S-fS') per pound : so that 
 all the different stocks might be considered as lying at 
 one average rate of interest. Such, however, is not the 
 case with compound interest. 
 
 To introduce the question in a simple form, let us sup- 
 pose that all the stock of the office makes r per pound, 
 the rate assumed in the valuation, except only one sum, 
 H, which makes r' (less than r) per pound. If, then, 
 this sum were set down as H in the item A, the profit 
 would be overrated ; nor can we answer the question, 
 how much should it be estimated at, without some re- 
 ference to the time at which H, with its accumulations, 
 is to become necessary. If this will not be wanted for 
 n years, then H x (1 -f r') ra -s-(l+r) n is the value at 
 which it must be estimated. 
 
ON OFFICE VALUATIONS. XXX111 
 
 The best method of treating this case is to suppose 
 H.to stand for such a sum, that there will be no loss 
 arising from a lower rate of interest before the next 
 valuation. Accordingly, in the preceding formula, n 
 must be the number of years intervening between two 
 valuations. If such a process should give too little 
 profit at one valuation, the same item will be larger in 
 the next, and vice versa : so that there will be a con- 
 tinual tendency to correctness. If, for instance, the 
 valuations be made yearly (for which this very cir- 
 cumstance is one reason among many), then H (1 -fr x ) 
 -r-(l -j-r) should be taken for H, and the existing poli- 
 cies may have the benefit when r' is greater than r. 
 
TABLE I. 
 
 t. 
 
 H. 
 
 A 
 
 A* 
 
 t. 
 
 H 
 
 A 
 
 A2 
 
 o-oo 
 
 o-ooooo oo 
 
 1128 33 
 
 22 
 
 0-40 
 
 0-42839 22 
 
 957 68 
 
 7 82 
 
 o-oi 
 
 0-01128 33 
 
 1128 11 
 
 45 
 
 041 
 
 0-43796 90 
 
 949 86 
 
 7 95 
 
 0-02 
 
 0-02256 44 
 
 1127 66 
 
 67 
 
 0-42 
 
 0-44746 76 
 
 941 91 
 
 8 07 
 
 0-03 
 
 0-03384 10 
 
 1126 99 
 
 90 
 
 0-43 
 
 0-45688 67 
 
 933 84 
 
 8 17 
 
 0-04 
 
 0-04511 09 
 
 1126 09 
 
 1 12 
 
 0-44 
 
 0-46622 51 
 
 925 67 
 
 8 30 
 
 O'Oo 
 
 0-05637 18 
 
 1124 97 
 
 1 35 
 
 0-45 
 
 0-47548 18 
 
 917 37 
 
 8 40 
 
 0-06 
 
 0-06762 15 
 
 1123 62 
 
 1 58 
 
 0-46 
 
 0-48465 55 
 
 908 97 
 
 8 51 
 
 0-07 
 
 0-07885 77 
 
 1122 04 
 
 1 79 
 
 0-47 
 
 0-49374 52 
 
 900 46 
 
 8 61 
 
 0-08 
 
 0-09007 81 
 
 1120 25 
 
 2 01 
 
 0-48 
 
 0-50274 98 
 
 891 85 
 
 8 69 
 
 0'09 
 
 0-10128 06 
 
 1118 24 
 
 2 24 
 
 0-49 
 
 0-51166 83 
 
 883 16 
 
 8 78 
 
 0-10 
 
 0-11246 30 
 
 1116 00 
 
 2 46 
 
 0-50 
 
 0-52049 99 
 
 874 38 
 
 8 88 
 
 0-11 
 
 0-12362 30 
 
 1113 54 
 
 2 67 
 
 0-51 
 
 0-52924 37 
 
 865 50 
 
 8 96 
 
 0-12 
 
 0-13475 84 
 
 1110 87 
 
 2 88 
 
 0-52 
 
 0-53789 87 
 
 856 54 
 
 9 03 
 
 0-13 
 
 0-14586 71 
 
 1107 99 
 
 3 10 
 
 0-53 
 
 0-54646 41 
 
 847 51 
 
 9 10 
 
 0-14 
 
 0-15694 70 
 
 1104 89 
 
 3 31 
 
 0-54 
 
 0-55493 92 
 
 838 41 
 
 9 17 
 
 0-15 
 
 0-16799 59 
 
 1101 58 
 
 3 52 
 
 0-55 
 
 56332 33 
 
 829 24 
 
 9 23 
 
 0-16 
 
 0-17901 17 
 
 1098 06 
 
 3 72 
 
 0-56 
 
 0-57161 57 
 
 820 01 
 
 9 30 
 
 0-17 
 
 0-18999 23 
 
 1094 34 
 
 3 93 
 
 0-57 
 
 0-57981 58 
 
 810 71 
 
 9 35 
 
 0-18 
 
 0-20093 57 
 
 1090 41 
 
 4 14 
 
 0-58 
 
 0-58792 29 
 
 801 36 
 
 9 40 
 
 0-19 
 
 0-21183 98 
 
 1086 27 
 
 4 34 
 
 0-59 
 
 0-59593 65 
 
 791 96 
 
 9 45 
 
 0-20 
 
 0-22270 25 
 
 1081 93 
 
 4 53 
 
 060 
 
 0-60385 61 
 
 782 51 
 
 9 49 
 
 0-21 
 
 0-23352 18 
 
 1077 40 
 
 4 73 
 
 061 
 
 0-61168 12 
 
 773 02 
 
 9 53 
 
 0-22 
 
 0-24429 58 
 
 1072 67 
 
 4 92 
 
 062 
 
 0-61941 14 
 
 763 49 
 
 9 55 
 
 0-23 
 
 0-25502 25 
 
 1067 75 
 
 5 12 
 
 0-63 
 
 0-62704 63 
 
 753 94 
 
 9 59 
 
 024 
 
 0-26570 00 
 
 1062 63 
 
 5 29 
 
 0-64 
 
 0-63458 57 
 
 744 35 
 
 9 62 
 
 0-25 
 
 0-27632 63 
 
 1057 34 
 
 5 49 
 
 0-65 
 
 0-64202 92 
 
 734 73 
 
 9 63 
 
 0-26 
 
 0-28689 97 
 
 1051 85 
 
 5 67 
 
 0-66 
 
 0-64937 65 
 
 725 10 
 
 9 65 
 
 0-27 
 
 0-29741 82 
 
 1046 18 
 
 5 84 
 
 0-67 
 
 0-65662 75 
 
 715 45 
 
 9 66 
 
 0-28 
 
 0-30788 00 
 
 1040 34 
 
 6 01 
 
 068 
 
 0-66378 20 
 
 705 79 
 
 9 68 
 
 0-29 
 
 0-31828 34 
 
 1034 33 
 
 6 19 
 
 0-69 
 
 0-67083 99 
 
 696 11 
 
 9 67 
 
 0-30 
 
 0-328^2 67 
 
 1028 14 
 
 6 36 
 
 0-70 
 
 0-67780 10 
 
 686 44 
 
 9 68 
 
 0-31 
 
 0-33890 81 
 
 1021 78 
 
 6 52 
 
 0-71 
 
 0-68466 54 
 
 676 76 
 
 9 68 
 
 0-32 
 
 0-34912 59 
 
 1015 26 
 
 6 67 
 
 0-72 
 
 0-69143 30 
 
 667 08 
 
 9 66 
 
 0-33 
 
 0-35927 85 
 
 1008 59 
 
 6 84 
 
 0-73 
 
 0-69810 38 
 
 657 42 
 
 9 66 
 
 0-34 
 
 0-36936 44 
 
 1001 75 
 
 6 98 
 
 0-74 
 
 0-70467 80 
 
 647 76 
 
 9 65 
 
 035 
 
 0-37938 19 
 
 994 77 
 
 7 14 
 
 0-75 
 
 0-71115 56 
 
 638 11 
 
 9 62 
 
 0-36 
 
 0-38932 96 
 
 987 63 
 
 7 29 
 
 0-76 
 
 0-71753 67 
 
 628 49 
 
 9 61 
 
 0-37 
 
 0-39920 59 
 
 980 34 
 
 7 42 
 
 0-77 
 
 0-72382 16 
 
 618 88 
 
 9 57 
 
 0-38 
 
 0-40900 93 
 
 972 92 
 
 7 55 
 
 0-78 
 
 0-73001 04 
 
 609 31 
 
 9 56 
 
 0-39 
 
 0-41873 85 
 
 965 37 
 
 7 69 
 
 0-79 
 
 0-73610 35 
 
 599 75 
 
 9 52 
 
t. 
 
 H. 
 
 A 
 
 A2 
 
 t. 
 
 H. 
 
 A 
 
 A2 
 
 0-80 
 
 0-74210 10 
 
 590 23 
 
 9 48 
 
 1-30 
 
 0-93400 80 
 
 205 52 
 
 5 32 
 
 0-81 
 
 0-74800 33 
 
 580 75 
 
 9 45 
 
 1-31 
 
 0-93606 32 
 
 2GO 20 
 
 5 22 
 
 0-82 
 
 0-75381 08 
 
 571 30 
 
 9 41 
 
 1-32 
 
 0-93806 52 
 
 194 98 
 
 5 11 
 
 0-83 
 
 0-75952 38 
 
 561 89 
 
 9 36 
 
 1-33 
 
 0-94001 50 
 
 189 87 
 
 5 02 
 
 0-84 
 
 0-76514 27 
 
 552 53 
 
 9 31 
 
 1-34 
 
 0-94191 37 
 
 184 85 
 
 4 93 
 
 0-85 
 
 0-77066 80 
 
 543 22 
 
 9 26 
 
 1-35 
 
 0-94376 22 
 
 179 92 
 
 4 82 
 
 0-86 
 
 0-77610 02 
 
 533 96 
 
 9 21 
 
 1-36 
 
 0-94556 14 
 
 175 10 
 
 4 74 
 
 0-87 
 
 0-78143 98 
 
 524 75 
 
 9 16 
 
 1-37 
 
 0-94731 24 
 
 170 36 
 
 4 63 
 
 0-88 
 
 0-78668 73 
 
 515 59 
 
 9 09 
 
 1-38 
 
 0-94901 60 
 
 165 73 
 
 4 55 
 
 0-89 
 
 0-79184 32 
 
 506 50 
 
 9 04 
 
 1-39 
 
 0-95067 33 
 
 161 18 
 
 4 45 
 
 0-90 
 
 0-79690 82 
 
 497 46 
 
 8 97 
 
 40 
 
 0-95228 51 
 
 156 73 
 
 4 35 
 
 0-91 
 
 0-80188 28 
 
 488 49 
 
 8 91 
 
 41 
 
 0-95385 24 
 
 152 38 
 
 4 27 
 
 0-92 
 
 0-80676 77 
 
 479 58 
 
 8 83 
 
 1-42 
 
 0-95537 62 
 
 148 11 
 
 4 18 
 
 0-93 
 
 0-81156 35 
 
 470 75 
 
 8 77 
 
 43 
 
 0-95685 73 
 
 143 93 
 
 4 09 
 
 0-94 
 
 0-81627 10 
 
 461 98 
 
 8 70 
 
 44 
 
 0-95829 66 
 
 139 84 
 
 3 99 
 
 0-95 
 
 0-82089 08 
 
 453 28 
 
 8 61 
 
 45 
 
 0-95969 50 
 
 135 85 
 
 3 91 
 
 0-96 
 
 0-82542 36 
 
 444 67 
 
 8 55 
 
 4'J 
 
 0-96105 35 
 
 331 94 
 
 3 82 
 
 0-97 
 
 0-82987 03 
 
 436 12 
 
 8 46 
 
 47 
 
 96237 29 
 
 128 12 
 
 3 74 
 
 0-98 
 
 0-83423 15 
 
 427 66 
 
 8 39 
 
 48 
 
 0-96365 41 
 
 124 38 
 
 3 65 
 
 0-99 
 
 0-83850 81 
 
 419 27 
 
 8 30 
 
 49 
 
 0-96489 79 
 
 120 73 
 
 3 57 
 
 1-00 
 
 0-84270 08 
 
 410 97 
 
 8 22 
 
 50 
 
 0-96610 52 
 
 117 16 
 
 3 49 
 
 1-01 
 
 0-84681 05 
 
 402 75 
 
 8 13 
 
 51 
 
 0-96727 68 
 
 113 67 
 
 3 40 
 
 1-02 
 
 0-85083 80 
 
 394 62 
 
 8 05 
 
 52 
 
 0-96841 35 
 
 110 27 
 
 3 32 
 
 1-03 
 
 0-85478 42 
 
 386 57 
 
 7 96 
 
 53 
 
 0-96951 62 
 
 106 95 
 
 3 25 
 
 ro4 
 
 0-85864 99 
 
 378 61 
 
 7 86 
 
 54 
 
 0-97058 57 
 
 103 70 
 
 3 16 
 
 1-05 
 
 0-86243 60 
 
 370 75 
 
 7 77 
 
 55 
 
 0-97162 27 
 
 100 54 
 
 3 C9 
 
 1-06 
 
 0-86614 35 
 
 362 97 
 
 7 68 
 
 56 
 
 0-97262 81 
 
 97 45 
 
 3 01 
 
 1-07 
 
 0-86977 32 
 
 355 29 
 
 7 60 
 
 57 
 
 0-97360 26 
 
 94 44 
 
 2 94 
 
 1-08 
 
 0-87332 61 
 
 347 69 
 
 7 49 
 
 58 
 
 0-97454 70 
 
 91 50 
 
 2 86 
 
 1-09 
 
 0-87680 30 
 
 340 20 
 
 7 40 
 
 1-59 
 
 0-97546 20 
 
 88 64 
 
 2 79 
 
 1-10 
 
 0-88020 50 
 
 332 80 
 
 7 31 
 
 1-60 
 
 0-97634 84 
 
 85 85 
 
 2 73 
 
 1-11 
 
 0-88353 30 
 
 325 49 
 
 7 21 
 
 1-61 
 
 0-97720 69 
 
 83 12 
 
 2 64 
 
 1-12 
 
 0-88678 79 
 
 318 28 
 
 7 12 
 
 1-62 
 
 0-97803 81 
 
 80 48 
 
 2 59 
 
 1-13 
 
 0-88997 07 
 
 311 16 
 
 7 01 
 
 1-63 
 
 0-97884 29 
 
 77 89 
 
 2 51 
 
 1-14 
 
 0-89308 23 
 
 304 15 
 
 691 
 
 1-64 
 
 0-97962 18 
 
 75 38 
 
 2 45 
 
 1-15 
 
 0-89612 38 
 
 297 24 
 
 6 82 
 
 1-65 
 
 0-98037 56 
 
 7293 
 
 2 38 
 
 1-16 
 
 0-89909 62 
 
 290 42 
 
 6 72 
 
 1-66 
 
 0-98110 49 
 
 70 55 
 
 2 31 
 
 1-17 
 
 0-90200 04 
 
 283 70 
 
 6 61 
 
 1-67 
 
 0-98181 04 
 
 68 24 
 
 2 26 
 
 1-18 
 
 0-90483 74 
 
 277 09 
 
 6 52 
 
 1-68 
 
 0-98249 28 
 
 65 98 
 
 2 20 
 
 1-19 
 
 0-90760 83 
 
 270 57 
 
 6 42 
 
 1-69 
 
 0-98315 26 
 
 63 78 
 
 2 12 
 
 1-20 
 
 0-91031 40 
 
 264 15 
 
 6 31 
 
 1-70 
 
 0-98379 04 
 
 61 66 
 
 2 08 
 
 1-21 
 
 0-91295 55 
 
 257 84 
 
 6 22 
 
 1-71 
 
 0-98440 70 
 
 59 58 
 
 2 01 
 
 1-22 
 
 091553 39 
 
 251 62 
 
 6 11 
 
 1-72 
 
 0-98500 28 
 
 57 57 
 
 1 96 
 
 1-23 
 
 91805 01 
 
 245 51 
 
 6 02 
 
 1-73 
 
 0-98557 85 
 
 55 61 
 
 1 90 
 
 1-24 
 
 0-92050 52 
 
 239 49 
 
 5 91 
 
 1-74 
 
 0-98613 46 
 
 53 71 
 
 1 85 
 
 1-25 
 
 0-92290 01 
 
 233 58 
 
 5 81 
 
 1-75 
 
 0-98667 17 
 
 51 86 
 
 1 79 
 
 1-26 
 
 0-92523 59 
 
 227 77 
 
 5 71 
 
 1-76 
 
 0-98719 03 
 
 50 07 
 
 1 75 
 
 1-27 
 
 0-92751 36 
 
 222 06 
 
 5 61 
 
 1-77 
 
 0-98769 10 
 
 48 32 
 
 1 68 
 
 1-28 
 
 0-92973 42 
 
 216 45 
 
 5 52 
 
 1-78 
 
 0-98817 42 
 
 46 64 
 
 1 65 
 
 1-29 
 
 0-93189 87 
 
 210 93 
 
 5 41 
 
 1-79 
 
 0-98864 06 
 
 44 99 
 
 I 59 
 
 z 3 
 
XXXVI 
 
 t. 
 
 H. 
 
 A 
 
 A2 
 
 t. 
 
 H. 
 
 A 
 
 A* 
 
 1-80 
 
 0-98909 05 
 
 43 40 
 
 1 54 
 
 2-30 
 
 0-99885 68 
 
 5 56 
 
 25 
 
 1-81 
 
 0-98952 45 
 
 41 86 
 
 1 50 
 
 2-31 
 
 0-99891 24 
 
 5 31 
 
 24 
 
 1-82 
 
 0-98994 31 
 
 40 36 
 
 1 44 
 
 2-32 
 
 0-99896 55 
 
 5 07 
 
 23 
 
 1-83 
 
 0-99034 67 
 
 38 92 
 
 41 
 
 2-33 
 
 0-99901 62 
 
 4 84 
 
 23 
 
 1-84 
 
 0-99073 59 
 
 37 51 
 
 36 
 
 2-34 
 
 0-99906 46 
 
 4 61 
 
 20 
 
 1-85 
 
 0-99111 10 
 
 36 15 
 
 33 
 
 2-35 
 
 0-99911 07 
 
 4 41 
 
 21 
 
 1-86 
 
 0-99147 25 
 
 34 82 
 
 27 
 
 2-36 
 
 0-99915 48 
 
 4 20 
 
 19 
 
 1-87 
 
 0-99182 07 
 
 33 55 
 
 24 
 
 2-37 
 
 0-99919 68 
 
 4 01 
 
 19 
 
 1-88 
 
 0-99215 62 
 
 32 31 
 
 20 
 
 2-38 
 
 0-99923 69 
 
 3 82 
 
 18 
 
 1-89 
 
 0-99247 93 
 
 31 11 
 
 16 
 
 2-39 
 
 0-99927 51 
 
 3 64 
 
 17 
 
 1-90 
 
 0-99279 04 
 
 29 95 
 
 12 
 
 2-40 
 
 0-99931 15 
 
 3 47 
 
 16 
 
 1-91 
 
 0-99308 99 
 
 28 83 
 
 08 
 
 2-41 
 
 0-99934 62 
 
 3 31 
 
 16 
 
 1-92 
 
 0-99337 82 
 
 27 75 
 
 06 
 
 2-42 
 
 0-99937 93 
 
 3 15 
 
 15 
 
 1-93 
 
 0-99365 57 
 
 26 69 
 
 01 
 
 2-43 
 
 0-99941 08 
 
 3 00 
 
 14 
 
 1-94 
 
 0-99392 26 
 
 25 68 
 
 99 
 
 2-44 
 
 0-99944 08 
 
 2 86 
 
 14 
 
 1-95 
 
 0-99417 94 
 
 24 69 
 
 95 
 
 2-45 
 
 0-99946 94 
 
 2 72 
 
 12 
 
 1-96 
 
 0-99442 63 
 
 23 74 
 
 91 
 
 2-46 
 
 0-99949 66 
 
 2 60 
 
 14 
 
 1-97 
 
 0-99466 37 
 
 22 83 
 
 89 
 
 2-47 
 
 0-99952 26 
 
 2 46 
 
 11 
 
 1-98 
 
 0-99489 20 
 
 21 94 
 
 85 
 
 2-48 
 
 0-99954 72 
 
 2 35 
 
 11 
 
 1-99 
 
 0-99511 14 
 
 21 09 
 
 84 
 
 2-49 
 
 0-99957 07 
 
 2 24 
 
 12 
 
 2-00 
 
 0-99532 23 
 
 20 25 
 
 78 
 
 2-50 
 
 0-99959 31 
 
 a 12 
 
 10 
 
 2-01 
 
 0-99552 48 
 
 19 47 
 
 79 
 
 2-51 
 
 0-99961 43 
 
 2 02 
 
 10 
 
 2-02 
 
 0-99571 95 
 
 18 68 
 
 73 
 
 2-52 
 
 0-99963 45 
 
 92 
 
 9 
 
 2-03 
 
 0-99590 63 
 
 17 95 
 
 72 
 
 2-53 
 
 0-99965 37 
 
 83 
 
 10 
 
 2-04 
 
 0-99608 58 
 
 17 23 
 
 69 
 
 2-54 
 
 0-99967 20 
 
 73 
 
 8 
 
 2-05 
 
 0-99625 81 
 
 16 54 
 
 67 
 
 2-55 
 
 0-99968 93 
 
 65 
 
 8 
 
 2-06 
 
 0-99642 35 
 
 15 87 
 
 65 
 
 2-56 
 
 0-99970 58 
 
 57 
 
 7 
 
 2-07 
 
 0-99658 22 
 
 15 22 
 
 61 
 
 2-57 
 
 0-99972 15 
 
 50 
 
 9 
 
 2-08 
 
 0-99673 44 
 
 14 61 
 
 61 
 
 2-58 
 
 0-99973 65 
 
 41 
 
 7 
 
 2-09 
 
 0-99688 05 
 
 14 00 
 
 57 
 
 2-59 
 
 0-99975 06 
 
 34 
 
 7 
 
 2-10 
 
 0-99702 05 
 
 13 43 
 
 55 
 
 2-60 
 
 0-99976 40 
 
 27 
 
 6 
 
 2-11 
 
 0-99715 48 
 
 12 88 
 
 54 
 
 2-61 
 
 0-99977 67 
 
 21 
 
 6 
 
 2-12 
 
 0-99728 36 
 
 12 34 
 
 51 
 
 2-62 
 
 0-99978 88 
 
 15 
 
 6 
 
 2-13 
 
 0-99740 70 
 
 11 83 
 
 50 
 
 2-63 
 
 0-99980 03 
 
 1 09 
 
 6 
 
 2-14 
 
 0-99752 53 
 
 11 33 
 
 47 
 
 2-64 
 
 0-99981 12 
 
 1 03 
 
 5 
 
 2-15 
 
 0-99763 86 
 
 10 86 
 
 47 
 
 2-G5 
 
 0-99982 15 
 
 98 
 
 5 
 
 2-16 
 
 0-99774 72 
 
 10 39 
 
 43 
 
 2-66 
 
 0-99983 13 
 
 93 
 
 5 
 
 2-17 
 
 0-99785 11 
 
 9 96 
 
 44 
 
 2-67 
 
 0-99984 06 
 
 88 
 
 4 
 
 2-18 
 
 0-99795 07 
 
 9 52 
 
 39 
 
 2-68 
 
 0-99984 94 
 
 84 
 
 5 
 
 ( 2-19 
 
 0-99804 59 
 
 9 13 
 
 41 
 
 2-69 
 
 0-99985 78 
 
 79 
 
 4 
 
 2-20 
 
 0-99813 72 
 
 8 72 
 
 36 
 
 2-70 
 
 0-99986 57 
 
 75 
 
 4 
 
 2-21 
 
 0-99822 44 
 
 8 36 
 
 38 
 
 2-71 
 
 0-99987 32 
 
 71 
 
 4 
 
 2-22 
 
 0-99830 80 
 
 7 98 
 
 34 
 
 2-72 
 
 0-99988 03 
 
 67 
 
 4 
 
 2-23 
 
 0-99838 78 
 
 7 64 
 
 33 
 
 2-73 
 
 0-99988 70 
 
 63 
 
 2 
 
 2-24 
 
 0-99846 42 
 
 7 31 
 
 33 
 
 2-74 
 
 0-99989 33 
 
 61 
 
 4 
 
 2-25 
 
 0-99853 73 
 
 6 98 
 
 30 
 
 2-75 
 
 0-99989 94 
 
 57 
 
 3 
 
 2-26 
 
 0-99860 71 
 
 6 68 
 
 30 
 
 2-76 
 
 0-99990 51 
 
 54 
 
 3 
 
 2-27 
 
 0-99867 39 
 
 6 38 
 
 29 
 
 2-77 
 
 0-99991 05 
 
 51 
 
 3 
 
 2-28 
 
 0-99873 77 
 
 6 09 
 
 27 
 
 2-78 
 
 0-99991 56 
 
 48 
 
 2 
 
 2-29 
 
 0-99879 86 
 
 5 82 
 
 26 
 
 2-79 
 
 0-99992 04 
 
 46 
 
 3 
 
TABLE I, 
 
 t. 
 
 H. 
 
 A 
 
 A* 
 
 t 
 
 H. 
 
 A 
 
 A2 
 
 2-80 
 
 0-99992 50 
 
 43 
 
 2 
 
 2-91 
 
 0-99996 13 
 
 23 
 
 1 
 
 2-81 
 
 0-99992 93 
 
 41 
 
 3 
 
 2-92 
 
 0-99996 36 
 
 22 
 
 1 
 
 2-82 
 
 0-99993 34 
 
 38 
 
 1 
 
 2-93 
 
 99996 58 
 
 21 
 
 2 
 
 2-83 
 
 0-99993 72 
 
 37 
 
 3 
 
 2-94 
 
 0-99996 79 
 
 19 
 
 1 
 
 2-84 
 
 0-99994 09 
 
 34 
 
 1 
 
 
 
 
 
 
 
 
 
 2-95 
 
 0-99996 98 
 
 18 
 
 1 
 
 2-85 
 
 0-99994 43 
 
 33 
 
 2 
 
 2-96 
 
 0-99997 16 
 
 17 
 
 
 
 2-86 
 
 0-99994 76 
 
 31 
 
 2 
 
 2-97 
 
 0-99997 33 
 
 17 
 
 2 
 
 2-87 
 
 0-99995 07 
 
 29 
 
 2 
 
 2-98 
 
 0-99997 50 
 
 15 
 
 1 
 
 2-88 
 
 0-99995 36 
 
 27 
 
 1 
 
 2-99 
 
 0-99997 65 
 
 14 
 
 _ 
 
 2-89 
 
 0-99995 63 
 
 26 
 
 2 
 
 
 
 
 
 
 
 
 
 3-00 
 
 0-99997 79 
 
 
 
 _ 
 
 2-90 
 
 0-99995 89 
 
 24 
 
 1 
 
 
 
 
 
 *** This table has been continued to S'OO from the data in Kramp's 
 Treatise on Astronomical Refractions. The description of it in the work 
 was written before this addition was made. 
 
TABLE II. 
 
 t. 
 
 K 
 
 A 
 
 t. 
 
 K 
 
 A 
 
 t. 
 
 K 
 
 A 
 
 o-oo 
 
 o-ooooo 
 
 538 
 
 0-45 
 
 0-23851 
 
 513 
 
 0-90 
 
 0-45618 
 
 446 
 
 o-oi 
 
 0-00538 
 
 538 
 
 0-46 
 
 0-24364 
 
 512 
 
 0-91 
 
 0-46064 
 
 445 
 
 0-02 
 
 0-01076 
 
 538 
 
 0-47 
 
 0-24876 
 
 512 
 
 0-92 
 
 0-46509 
 
 443 
 
 0-03 
 
 0-01614 
 
 538 
 
 0-48 
 
 0-25388 
 
 510 
 
 0-93 
 
 0-46952 
 
 441 
 
 0-04 
 
 0-02152 
 
 538 
 
 0-49 
 
 0-25898 
 
 509 
 
 0-94 
 
 0-47393 
 
 439 
 
 005 
 
 0-02690 
 
 538 
 
 0-50 
 
 0-26407 
 
 508 
 
 0-95 
 
 [0-47832 
 
 438- 
 
 0-06 
 
 0-03228 
 
 538 
 
 0-51 
 
 0-26915 
 
 506 
 
 0-96 
 
 0-48270 
 
 435 
 
 0-07 
 
 0-03766 
 
 537 
 
 0-52 
 
 0-27421 
 
 506 
 
 0-97 
 
 0-48705 
 
 434 
 
 0-08 
 
 0-04303 
 
 537 
 
 0-53 
 
 0-27927 
 
 504 
 
 0-98 
 
 0-49139 
 
 431 
 
 0-09 
 
 0-04840 
 
 538 
 
 0-54 
 
 0-28431 
 
 503 
 
 0-99 
 
 0-49570 
 
 430 
 
 0-10 
 
 0-05378 
 
 536 
 
 0-55 
 
 0-28934 
 
 502 
 
 1-00 
 
 0-50000 
 
 428 
 
 0-11 
 
 0-05914 
 
 537 
 
 0-56 
 
 0-29436 
 
 500 
 
 1-01 
 
 0-50428 
 
 425 
 
 0-12 
 
 0-06451 
 
 536 
 
 0-57 
 
 0-29936 
 
 499 
 
 1-02 
 
 0-50853 
 
 424 
 
 0-13 
 
 0-06987 
 
 536 
 
 0-58 
 
 0-30435 
 
 498 
 
 1-03 
 
 0-51277 
 
 422 
 
 0-14 
 
 0-07523 
 
 536 
 
 0-59 
 
 0-30933 
 
 497 
 
 1-04 
 
 0-51699 
 
 420 
 
 0-15 
 
 0-08059 
 
 535 
 
 0-60 
 
 0-31430 
 
 495 
 
 1-05 
 
 0-52119 
 
 418 
 
 0-16 
 
 0-08594 
 
 535 
 
 0-61 
 
 0-31925 
 
 494 
 
 1-06 
 
 0-52537 
 
 415 
 
 0-17 
 
 0-09129 
 
 534 
 
 0-62 
 
 0-32419 
 
 492 
 
 1-07 
 
 0-52952 
 
 414 
 
 0-18 
 
 0-09663 
 
 534 
 
 0-63 
 
 0-32911 
 
 491 
 
 1-08 
 
 0-53366 
 
 412 
 
 0-19 
 
 0-10197 
 
 534 
 
 0-64 
 
 0-33402 
 
 490 
 
 1-09 
 
 0-53778 
 
 410 
 
 0-20 
 
 0-10731 
 
 533 
 
 0-65 
 
 0-33892 
 
 488 
 
 1-10 
 
 0-54188 
 
 407 
 
 0-21 
 
 0-11264 
 
 532 
 
 0-66 
 
 0-34380 
 
 486 
 
 1-11 
 
 0-54595 
 
 406 
 
 0-22 
 
 0-11796 
 
 532 
 
 0-67 
 
 0-34866 
 
 486 
 
 1-12 
 
 0-55001 
 
 403 
 
 0-23 
 
 0-12328 
 
 532 
 
 0-68 
 
 0-35352 
 
 483 
 
 1-13 
 
 0-55404 
 
 402 
 
 0-24 
 
 0-12860 
 
 531 
 
 0-69 
 
 0-35835 
 
 482 
 
 1-14 
 
 0-55806 
 
 399 
 
 0-25 
 
 0-13391 
 
 530 
 
 0-70 
 
 0-36317 
 
 481 
 
 1-15 
 
 0-56205 
 
 397 
 
 0-26 
 
 0-13921 
 
 530 
 
 0-71 
 
 0-36798 
 
 479 
 
 1-16 
 
 0-56602 
 
 396 
 
 0-27 
 
 0-14451 
 
 529 
 
 0-72 
 
 0-37277 
 
 478 
 
 1-17 
 
 0-56998 
 
 393 
 
 0-28 
 
 0-14980 
 
 523 
 
 0-73 
 
 0-37755 
 
 476 
 
 1-18 
 
 0-57391 
 
 391 
 
 0-29 
 
 0-15508 
 
 527 
 
 0-74 
 
 0-38231 
 
 474 
 
 1-19 
 
 0-57782 
 
 389 
 
 0-30 
 
 0-16035 
 
 527 
 
 0-75 
 
 0-38705 
 
 473 
 
 1-20 
 
 0-58171 
 
 387 
 
 0-31 
 
 0-16562 
 
 526 
 
 0-76 
 
 0-39178 
 
 471 
 
 "1-21 
 
 0-58558 
 
 384 
 
 0-32 
 
 0-17088 
 
 526 
 
 0-77 
 
 0-39649 
 
 469 
 
 1-22 
 
 0-58942 
 
 383 
 
 0-.33 
 
 0-17614 
 
 524 
 
 0-78 
 
 0-40118 
 
 468 
 
 1-23 
 
 0-59325 
 
 380 
 
 0-34 
 
 0-18138 
 
 524 
 
 0-79 
 
 0-40586 
 
 466 
 
 1-24 
 
 0-59705 
 
 378 
 
 0-35 
 
 0-18662 
 
 523 
 
 0-80 
 
 0-41052 
 
 465 
 
 1-25 
 
 0-60083 
 
 377 
 
 0-36 
 
 0-19185 
 
 522 
 
 0-81 
 
 0-41517 
 
 462 
 
 1-26 
 
 0-60460 
 
 373 
 
 0-37 
 
 0-19707 
 
 522 
 
 0-82 
 
 0-41979 
 
 461 
 
 1-27 
 
 0-60833 
 
 372 
 
 0-38 
 
 0-20229 
 
 520 
 
 0-83 
 
 0-42440 
 
 459 
 
 1-28 
 
 0-61205 
 
 370 
 
 0-39 
 
 0-20749 
 
 519 
 
 0-84 
 
 0-42899 
 
 458 
 
 1-29 
 
 0-61575 
 
 367 
 
 0-40 
 
 0-21268 
 
 519 
 
 0-85 
 
 0-43357 
 
 456 
 
 1-30 
 
 0-61942 
 
 366 
 
 0-41 
 
 0-21787 
 
 517 
 
 0-86 
 
 0-43813 
 
 454 
 
 1-31 
 
 0-62308 
 
 363 
 
 0-42 
 
 0-22304 
 
 517 
 
 0-87 
 
 0-44267 
 
 452 
 
 1-32 
 
 0-62671 
 
 361 
 
 0-43 
 
 0-22821 
 
 515 
 
 0-88 
 
 0-44719 
 
 450 
 
 1-33 
 
 0-63032 
 
 359 
 
 0-44 
 
 0-23336 
 
 515 
 
 0-89 
 
 0-45169 
 
 449 
 
 1-34 
 
 0-63391 
 
 356 
 
' t. 
 
 K 
 
 A 
 
 t. 
 
 K 
 
 A 
 
 t. 
 
 K 
 
 A 
 
 1-35 
 
 0-63747 
 
 355 
 
 1'85 
 
 0-78790 
 
 246 
 
 2-35 
 
 0-88705 
 
 152 
 
 1-36 
 
 0-64102 
 
 352 
 
 1-86 
 
 0-79036 
 
 244 
 
 2-36 
 
 0-88857 
 
 151 
 
 1-37 
 
 0-64454 
 
 350 
 
 1-87 
 
 0-79280 
 
 242 
 
 2-37 
 
 0-89008 
 
 149 
 
 1-38 
 
 0-64804 
 
 348 
 
 1-88 
 
 0-79522 
 
 239 
 
 2-38 
 
 0-89157 
 
 147 
 
 1-39 
 
 0-65152 
 
 346 
 
 1-89 
 
 0-79761 
 
 238 
 
 2-39 
 
 0-89304 
 
 146 
 
 1-40 
 
 0-65498 
 
 343 
 
 1-90 
 
 0-79999 
 
 236 
 
 2-40 
 
 0-89450 
 
 145 
 
 1-41 
 
 0-65841 
 
 341 
 
 1-91 
 
 0-80235 
 
 234 
 
 2-41 
 
 0-89595 
 
 143 
 
 1-42 
 
 0-66182 
 
 339 
 
 1-92 
 
 0-80469 
 
 231 
 
 2-42 
 
 0-89738 
 
 141 
 
 1-43 
 
 0-66521 
 
 337 
 
 1-93 
 
 0-80700 
 
 230 
 
 2-43 
 
 0-89879 
 
 140 
 
 1-44 
 
 0-66858 
 
 335 
 
 1-94 
 
 0-80930 
 
 228 
 
 2-44 
 
 0-90019 
 
 138 
 
 1-45 
 
 0-67193 
 
 333 
 
 1-95 
 
 0-81158 
 
 225 
 
 2-45 
 
 0-90157 
 
 136 
 
 1-46 
 
 0-67526 
 
 330 
 
 1-96 
 
 0-81383 
 
 224 
 
 2-46 
 
 0-90293 
 
 135 
 
 1-47 
 
 0-67856 
 
 328 
 
 1-97 
 
 0-81607 
 
 221 
 
 2-47 
 
 0-90428 
 
 134 
 
 1-48 
 
 0-68184 
 
 326 
 
 1-98 
 
 0-81828 
 
 220 
 
 2-48 
 
 0-90562 
 
 132 
 
 1-49 
 
 0-68510 
 
 323 
 
 1-99 
 
 0-82048 
 
 218 
 
 2-49 
 
 0-90694 
 
 131 
 
 1-50 
 
 0-68833 
 
 322 
 
 2-00 
 
 0-82266 
 
 215 
 
 2-50 
 
 0-90825 
 
 129 
 
 1-51 
 
 0-69155 
 
 319 
 
 2-01 
 
 0-82481 
 
 214 
 
 2-51 
 
 0-90954 
 
 128 
 
 1-52 
 
 0-69474 
 
 317 
 
 2-02 
 
 0-82695 
 
 212 
 
 2-52 
 
 0-91082 
 
 126 
 
 1-53 
 
 0-69791 
 
 315 
 
 2-03 
 
 0-82907 
 
 210 
 
 253 
 
 0-91208 
 
 124 
 
 1-54 
 
 0-70106 
 
 313 
 
 2-04 
 
 0-83117 
 
 207 
 
 2-54 
 
 0-91332 
 
 124 
 
 1-55 
 
 0-70419 
 
 310 
 
 . 2-05 
 
 0-83324 
 
 206 
 
 2-55 
 
 0-91456 
 
 122 
 
 1-56 
 
 0-70729 
 
 309 
 
 2-06 
 
 0-83530 
 
 204 
 
 2-56 
 
 0-91578 
 
 120 
 
 1-57 
 
 0-71038 
 
 306 
 
 2-07 
 
 0-83734 
 
 202 
 
 2-57 
 
 0-91698 
 
 119 
 
 1-58 
 
 0-71344 
 
 304 
 
 2-08 
 
 0-83936 
 
 201 
 
 2-58 
 
 0-91817 
 
 118 
 
 1-59 
 
 0-71648 
 
 301 
 
 2-09 
 
 0-84137 
 
 198 
 
 2-59 
 
 0-91935 
 
 116 
 
 1-60 
 
 0-71949 
 
 300 
 
 2-10 
 
 0-84335 
 
 196 
 
 2-60 
 
 0-92051 
 
 115 
 
 1-61 
 
 0-72249 
 
 297 
 
 2-11 
 
 0-84531 
 
 195 
 
 2-61 
 
 0-92166 
 
 114 
 
 1-62 
 
 072546 
 
 295 
 
 2-12 
 
 0-84726 
 
 193 
 
 2-62 
 
 0-92280 
 
 112 
 
 1-63 
 
 0-72841 
 
 293 
 
 2-13 
 
 0-84919 
 
 190 
 
 2-63 
 
 0-92392 
 
 111 
 
 1-64 
 
 0-73134 
 
 291 
 
 2-14 
 
 0-85109 
 
 189 
 
 2-64 
 
 0-92503 
 
 110 
 
 1-65 
 
 0-73425 
 
 289 
 
 2-15 
 
 0-85298 
 
 188 
 
 2-65 
 
 0-92613 
 
 108 
 
 1-66 
 
 073714 
 
 286 
 
 2-16 
 
 0-85486 
 
 185 
 
 2-66 
 
 0-92721 
 
 107 
 
 1-67 
 
 0-74000 
 
 285 
 
 2-17 
 
 0-85671 
 
 183 
 
 2-67 
 
 0-92828 
 
 106 
 
 1-68 
 
 0-74285 
 
 282 
 
 2-18 
 
 0-85854 
 
 182 
 
 2-68 
 
 0-92934 
 
 104 
 
 1-69 
 
 0-74567 
 
 280 
 
 2-19 
 
 0-86036 
 
 180 
 
 2-69 
 
 0-93038 
 
 103 
 
 1-70 
 
 0-74847 
 
 277 
 
 2-20 
 
 0-86216 
 
 178 
 
 2-70 
 
 0-93141 
 
 102 
 
 1-71 
 
 0-75124 
 
 276 
 
 2-21 
 
 0-86394 
 
 176 
 
 2-71 
 
 093243 
 
 101 
 
 1-72 
 
 0-75400 
 
 274 
 
 2-22 
 
 0-86570 
 
 175 
 
 2-72 
 
 0-93344 
 
 99 
 
 1-73 
 
 0-75674 
 
 271 
 
 2-23 
 
 0-86745 
 
 172 
 
 2-73 
 
 0-93443 
 
 98 
 
 1-74 
 
 0-75945 
 
 269 
 
 2-24 
 
 0'86917 
 
 171 
 
 2-74 
 
 0-93541 
 
 97 
 
 1-75 
 
 0-76214 
 
 267 
 
 2-25 
 
 0-87088 
 
 170 
 
 2-75 
 
 0-93638 
 
 96 
 
 1-76 
 
 0-76481 
 
 265 
 
 2-26 
 
 0-87258 
 
 167 
 
 2-76 
 
 0-93734 
 
 94 
 
 1-77 
 
 076746 
 
 263 
 
 2-27 
 
 0-87425 
 
 166 
 
 2-77 
 
 0-93828 
 
 94 
 
 1-78 
 
 0-77009 
 
 261 
 
 2-28 
 
 0-87591 
 
 164 
 
 2-78 
 
 0-93922 
 
 92 
 
 179 
 
 0-77270 
 
 258 
 
 2-29 
 
 0-87755 
 
 163 
 
 2-79 
 
 0-94014 
 
 91 
 
 1-80 
 
 0-77528 
 
 257 
 
 2-30 
 
 0-87918 
 
 160 
 
 2-80 
 
 0-94105 
 
 90 
 
 1-81 
 
 0-77785 
 
 254 
 
 231 
 
 0-88078 
 
 159 
 
 2-81 
 
 0-94195 
 
 89 
 
 1-82 
 
 0-78039 
 
 252 
 
 2-32 
 
 0-88237 
 
 158 
 
 2-82 
 
 0-94284 
 
 87 
 
 1-83 
 
 0-78291 
 
 251 
 
 2-33 
 
 0-88395 
 
 155 
 
 2-83 
 
 0-94371 
 
 87 
 
 1-84 
 
 0-78542 
 
 248 
 
 2-34 
 
 0-88550 
 
 155 
 
 2-84 
 
 0-94458 
 
 85 
 
xl 
 
 t. 
 
 K 
 
 A 
 
 t. 
 
 K 
 
 A 
 
 t. 
 
 K 
 
 A 
 
 2-85 
 
 0-94543 
 
 84 
 
 3-10 
 
 0-96346 
 
 60 
 
 3-35 
 
 0-97615 
 
 42 
 
 2-86 
 
 0-94627 
 
 84 
 
 3-11 
 
 0-96406 
 
 60 
 
 3-36 
 
 0-97657 
 
 41 
 
 2-87 
 
 0-947H 
 
 82 
 
 3-12 
 
 0-96466 
 
 58 
 
 3-37 
 
 0-97698 
 
 40 
 
 2-88 
 
 0-94793 
 
 81 
 
 3-13 
 
 0-96524 
 
 58 
 
 3-38 
 
 0-97738 
 
 40 
 
 2-89 
 
 0-94874 
 
 80 
 
 3-14 
 
 0-96582 
 
 56 
 
 3-39 
 
 0-97778 
 
 39 
 
 2-90 
 
 0-94954 
 
 79 
 
 3-15 
 
 0-96638 
 
 56 
 
 3-40 
 
 0-97817 
 
 359 
 
 2-91 
 
 0-95033 
 
 78 
 
 3-16 
 
 0-96694 
 
 55 
 
 3-50 
 
 0-98176 
 
 306 
 
 2-92 
 
 0-95111 
 
 76 
 
 3-17 
 
 0-96749 
 
 55 
 
 3-60 
 
 0-98482 
 
 261 
 
 2-93 
 
 0-95187 
 
 76 
 
 3-18 
 
 0-96804 
 
 53 
 
 3-70 
 
 0-98743 
 
 219 
 
 2-94 
 
 0-95263 
 
 75 
 
 3-19 
 
 0-96857 
 
 53 
 
 3-80 
 
 0-98962 
 
 185 
 
 2-95 
 
 0-95338 
 
 74 
 
 3-20 
 
 0-96910 
 
 52 
 
 3-90 
 
 0-99147 
 
 155 
 
 2-96 
 
 0-95412 
 
 73 
 
 3-21 
 
 0-96962 
 
 51 
 
 4-00 
 
 0-99302 
 
 129 
 
 2-97 
 
 0-95485 
 
 72 
 
 3-22 
 
 0-97013 
 
 51 
 
 4-10 
 
 0-99431 
 
 108 
 
 2-98 
 
 0-95557 
 
 71 
 
 3-23 
 
 0-97064 
 
 50 
 
 4-20 
 
 0-99539 
 
 88 
 
 2-99 
 
 0-95628 
 
 70 
 
 3-24 
 
 0-97114 
 
 49 
 
 4-30 
 
 0-99627 
 
 73 
 
 3-00 
 
 0-95698 
 
 69 
 
 3-25 
 
 0-97163 
 
 48 
 
 4-40 
 
 0-99700 
 
 60 
 
 3-01 
 
 0-95767 
 
 68 
 
 3-26 
 
 0-97211 
 
 48 
 
 4-50 
 
 0-99760 
 
 48 
 
 3-02 
 
 0-95835 
 
 67 
 
 3-27 
 
 0-97259 
 
 47 
 
 4-60 
 
 0-99808 
 
 40 
 
 3-03 
 
 0-95902 
 
 66 
 
 3-28 
 
 0-97306 
 
 46 
 
 4-70 
 
 0-99848 
 
 31 
 
 3-04 
 
 0-95968 
 
 65 
 
 3-29 
 
 0-97352 
 
 45 
 
 4-80 
 
 0-99879 
 
 26 
 
 3'05 
 
 0-96033 
 
 65 
 
 3-30 
 
 0-97397 
 
 45 
 
 4-90 
 
 0-99905 
 
 21 
 
 3-06 
 
 0-96098 
 
 63 
 
 3-31 
 
 0-97442 
 
 44 
 
 5-00 
 
 0-99926 
 
 
 3-07 
 
 0-96161 
 
 63 
 
 3-32 
 
 0-97486 
 
 44 
 
 
 
 
 3-08 
 
 0-96224 
 
 62 
 
 3-33 
 
 0-97530 
 
 43 
 
 
 
 
 3-09 
 
 0-96286 
 
 60 
 
 3-34 
 
 0-97573 
 
 42 
 
 
 
 
 THE END. 
 
 LONDON: 
 
 Printed by A. SPOTTISWOODE, 
 NevvStreet-Square. 
 
TTTTS "RflOTT TS_ TVTTT1 O"NT TTTT! J" A QL I P T\ A TT! 
 
 RETURN CIRCULATION DEPARTMENT 
 
 TO ^ 202 Main Library 
 
 LOAN PERIOD 1 
 HOME USE 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 
 
 1 -month toana may be renewed by calling 642-3405 
 Vyear loans may be recharged by bringing the books to the Circulation Dt 
 Renewals and recharges may be made 4 days prtor to due date 
 
 DUE AS STAMPED BELOW 
 
 M U'f & Y 
 
 5C 
 
 
 .LJ*. i 
 
 ? 
 
 
 RfX-Eivm** 
 
 AUb & o i-<i" 
 
 
 
 PRCULATION D 
 
 IPT, 
 
 
 AUG IB 1991 
 
 
 
 nro A ft 1001 
 
 
 
 utu u D lyy] 
 Mil list DEC 1 2 W 
 
 9T 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 UNIVERSITY OF CALIFORNIA, B 
 FORM NO. DD6, 60m, 1/83 BERKELEY, CA 94720 
 

 
 UNIVERSITY OF CALIFORNIA LIBRARY