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Secretary of the Royal Astronomical Society

LONGMAN S, PATERNOSTE

SE eee Swe te ee ee ee eS

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 probléme relatif aux jeux de hasard, proposé @ un austére janse- niste par un homme du monde, a été l’origine du calcul des aegeag a Ge oisson.

A

CMQGROYD

vi PREFACE.

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;

PREFACE Vil

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- iedge 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

wp The same may be said of several propositions given by Newton.

(|

Vill PREFACE. | }

all 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. JI 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,

PREFACE. ix

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 aclear 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

x PREFACE.

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 be 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 acloser 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 ccmmence at the death of an individual ; but refused to insure lives: the consequence was, that, by a very obvious evasion, insur-

PREFACE, xi

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.

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

Xli PREFACE.

it must be remembered that an annuity of, say £3 a year, diminishing by £1 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, £1, £0, £(—1), £(—2), £(—3), &e.

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-

i ae

PREFACE. Xili

custom him to make those efforts of mind, and to bear 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

XVi PREFACE,

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.

eg eS en

ADEE SCG Sees Be a

CONTENTS.

CHAPTER IL

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 - - . ax = - = 30

CHAPTER III. On Inverse Probabilities - S ~ fe e

Cr 09

CHAPTER IV.

Use of the Tables at the end of this Work - » ~ - 69 CHAPTER V.

On the Risks of Loss or Gain - oer o s - ¥8 CHAPTER VL

On common Notions with regard to Probability - . - IR

CHAPTER VIL On Errors of Observation, and Risks of Mistake - - - 18

CHAPTER VIIL

| On the Application of Probabilities to Life Contingencies - - 158 CHAPTER IX.

: On Annuities and other Money Contingencies - ~ - 181 CHAPTER X,

ro coy b>

On the Value of Reversions and Insurances « = ie

XVill OONTENTS.

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 vf tne Interests of the different Members in an Insurance Office - . - e - 267

CHAPTER XIII. Miscellaneous Subjects connected with Insurance, &c. a - 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 - . ~ eas |

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 Survivorshir - 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 - - - - - - XXvVi

APPENDIX THE SIXTH. On a Question connected with the Valuation of the Assets of an In- surance Office - . a “ - - XXX1

Table I. ” ° m » S<Xxtv Table II. . ie . ~ XXXvVili .

ON

PROBABILITIES.

CHAPTER I.

ON THE NOTION OF PROBABILITY AND ITS MEASURE- MENT 5 ON THE PROVINCE OF MATHEMATICS WITH REGARD TO IT, AND REPLY TO OBJECTIONS.

~“Wuen 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 eases 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 B

of ee . , A - , é “ s « <

Bis ESSAY ON PRVBABILITIES. aiccidlents of life, and contains the principles on which jt 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

SS ore

}

PE eR aA

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 wild 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 ess, we feel the possibility of an answer to the question, ‘‘ 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 Romanem- | Two straight | Fire will | Of 1,000,000 of

pire was over- | lines cannot burn. | children born,some turned bynorth- | enclose a will die aged fifty, ern barbarians. space. 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 apswer, unques- tionably, the first, for the second will certainly not hap-

es ste i

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 @ Oe, OF Cy 08. or 2, or y, or 2, 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 0.

He will draw either a, or b, or c.

ie will draw either a or b OF seccsseee OF ry. He will draw either a or 5 or .,...006. OF Y OF %

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

an

6 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 “ there is a remote chance of it” is found to give that which is conveyed in “ 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. | a must be drawn. Of two children born, one or | a or 6 must be drawn. other will die aged fifty. Of three children born, one or | a, or 6, or c must be drawn. other will die aged fifty. &e. &e. &c. &e. &e. &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. |

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 beiongs 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 6, 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 1/. 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 1/. 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 1/.? 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,