Augustus De Morgan.

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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

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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