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PR X A C = C B^, that is, ax y^ \ the Fluxion of which

equation is a x 2 y y ; therefore, x -J^. > w hich substi-

a

tuted for x, makes the general expression for the Subtangent

C T n ^iJ (by substituting a x for - its value,)

a . o

= 2 X. So that, the Subtangent C T is double the absciss

A C ; and consequently, A T is 3: AC.

Of the several works w ritten on this subject, the follow ing

seem to require particular notice.

" The method of Fluxions and infinite Series, with its ap-

plication to the Geometry of curve lines, by the inventor. Sir

Isaac Newton, to which is subjoined, a perpetual Comment

upon the whole work," S(,c. by John Colson. This will al-

ways be regarded as a standard work, but it is not exactly

suited to young persons entering upon the subject.

FLUXIONS. 507

Mr. John Rowe's " Introduction to the Doctrine of

Fluxions," is divided into two parts, the first treats of the

direct method of fluxions, and the second of the inverse

method. This latter, which is unquestionably the most difficult

part of the fluxionary calculus, viz. that of finding the fluent

from the fluxion being given, is introduced with the doctiine

of infinite series. The whole is written in so plain and per-

spicuous a manner, that the learner, who is previously well

acquainted with the arithmetic of rsurds, need not be deterred

from entering upon the study of Mr. Rowe's treatise. Of this

the third edition was printed in 1767.

The same may be said of " The Principles of Fluxions,**

by the Rev. S. Vince, D. D. F. R. S., which is a neat intro-

duction to this branch of science. The third edition of which

was published in 1805.

A more elaborate and more difficult piece is entitled " The

Doctrine and Application of Fluxions, &c., in two parts, by

Thomas Simpson, F. R. S." But the principal work is Mr.

Maclaurin's " Treatise of Fluxions," in two volumes, 4to.

In this the subject is handled agreeably to the method of rea-

soning used by the ancient mathematicians, without having

^course to algebraic solutions. To his demonstrations of this

doctrine he has added many valuable improvements of it, and

has applied it to so many curious and useful inquiries, that

the work has been denominated a store-house of mathematical

learning, rather than a treatise on one branch of it. Through-

out the whole there appears a masterly genius and uncom-

mon address. The biographer of Mr. Maclaurin, speak-

ing of him in connexion with his Treatise on Fluxions, says,

" he had a quick, comprehensive view, taking in, at once, all

the means of investigation, he could select the fittest for his

purpose, and apply them with exquisite art and method.

This is a faculty not to be acquired by exercise only ; we

ought rather to call it a species of that taste, the gift of na-

ture, which in mathematics, as in other things, distinguishes

excellence from mediocrity."

508 MATHEMATICS.

DOCTRINE OF CHANCES.

The doctrine of Chances being of great importance,

when applied to tlie solution of questions in life-annuities,

insurance, reversions, &c. it is necessary briefly to point out

those works in wliich its principles are laid down and in-

vestigated.

This subject, no less useful than interesting and curious,

does not appear to have engaged the attention of mathemati-

cians in former times, so murh as its importance may seem to

have required. Tlie writers upon it, in our own language, are

comparatively few. To M. Huygens we are indebted for

the first regular tract on this subject, which was in the Latin

language, and entitled *' De Ratiociniis in Ludo Aleae;" this

work, however, from the comparatively few problems which it

contains, and the want of demonstration to some of them,

cannot be regarded as an elementary treatise. To this suc-

ceeded a small anonymous tract entitled, " on the Laws of

Chance," published in London in J C)92, and a French publica-

tion Entitled ** L' Analyse des Jeux de Hazard," written by

M. Monmort, and published in 1708. Iliis author, following

tlie mode of M. Huygens in the solution of his problems^

M. de Moivre, who objected to it, published his work " On

the Doctrine of Chances," which was first published in 1717,

but which has been twice or thrice reprinted since. The edi-

tion of 1756, is, it is believed, the best. M. De Moivre pro-

ceeds from the most simple to the most complicated cases ; so

that, by the variety of his problems, as well as by the improve-

ments and additions which he made in two subsequent edi-

tions, he has rendered his work the best and most copious that

has ever been written on the subject. In the year 1 740, Mr.

Thomas Simpson published a very thin and small quarto, on

*' The Nature and Laws of Chance, illustrated wiih a great va-

riety of Examples," which, like his other publications, is not

only clear and concise, but contains some problems, the solu*

tious of which had never before been communicated to the

DOCTRINE OF CHANCES. 509

public. Mr. Dodson, in the year 1753, rendered the subject

still more accessible to persons not far advanced in analytical

studies, by publishing in the second volume of his " Mathema-

tical Repository," a number of questions with their solutions,

but chiefly with the view of applying them to the doctrine of

annuities and survivorships. In addition to these, may be

mentioned a small tract, " De Mensura Sortis," given by M.

De Moivre, in his " Miscellanea Analytica," and some pa-

pers published at different times in the Transactions of the

learned Societies on the Continent, and those of the Royal

Society of London." " Among which," says Mr. Morgan, in

the article Chances, in the New Cyclopedia, " may be parti-

cularly mentioned an " Essay on the Method of calculating

the e.xact probability of all conclusions founded on Induction,"

and a " Supplement " to that Essay ; the one preserved from

the papers of the late Rev. Mr. Bayes, and communicated

with an appendix, by Dr. Price, to the Royal Society in the

year 1762, and the other chiefly written by Dr. Price, and

communicated in the following year. " These tracts contain

the investigation of a problem, the converse of which had for-

merly exercised the ingenuity of M. Bernoulli, De Moivre,

and Simpson. Indeed both the problem, and its converse,

may be considered not only as the most difficult, but as the

most important that can be proposed on the subject, having no

less an object in view, than to shew what reason we have for

believing that there are in the constitution of things, fixed laws,

according to which events happen ; and that therefore, the

frame of the world must be the effect of the wisdom and

power of an intelligent cause, and thus to confirm the argu-

ment, taken from final causes, for the existence of Deity."

Besides the above-named works on the Doctrine of Chan-

ces, which is of very great consequence in this country, where

the valuation of an immense property, and the future provision

of many thousands, entirely depend on the right knowledge of

it, we may mention a treatise just published, entitled " Tl^e

510 MATHEMATICS.

Doctrine of Chances, or the Theory of Gaming made easy to

every Person acquainted with common Arithmetic, so as to

enable them to calculate the probabilities of events, in Ix)tte-

ries, Cards, Horse Racing, Dice, &c. with tables on Chance

never before published, by Wm. Rouse." The principles of

the Doctrine of Chances cannot be altered they are invariable ;

but Mr. Rouse may have made them more intelligible to the

unlearned, among whom we presume gamblers are usually to be

reckoned, who require a strong stimulus to engage their atten-

tion, but who have not sufficient energy to make study either the

business or amusement of life. The tables, mentioned in the

title, are intended to shew, at one view, the Chances for and

against winning any assigned number of Games, at any kind of

play, out of a given number, &c. How far this volume has

a claim to the character of accuracy we know not, it being

too recent a publication to admit of its having been read.

If the doctrine of Chances could be applied only to the

principles of Gaming, it would be of comparative little value ;

but as upon that doctrine depends every thing relating to Annu-

ities, to Survivorships, and to Reversions, we shall endea-

Tour, in a few words, to state the connexion of these several

subjects.

Chance is particularly used for the probability of an event,

and is greater or less, according to the number of chances

there are by which it may happen, compared with the number

by which it may either happen or fail. Thus, if an event has

three chances to happen, and two to fail, the probability of its

happening may be estimated at I ths, and of its failing, | ths.

Hence it appears, that if the probability of its happening and

failing be .added together, the sum is equal to unity. The ex-

pectation of obtaining any thing is estimated by the value of

that thing, multiplied by the probability of obtaining it. The

risk of losing any thing is estimated by the value of the thing

multiplied by the probability of losing it. Applying this to

gamingf we say, if from the expectations which the gamesters

ANNUITIES, INSURANCE, &C. 511

have upon the whole snm deposited, the particular sums they

deposit, that is, their own stakes, be subtracted, there will re-

main the gain, if the difference is positive ; or the loss, if the

difference be negative. Again, if from the respective expecta-

tions which either gamester has upon the sum deposited by his

adversary, the risk of losing what he himself deposits be sub-

tracted, there will likewise remain his gain or loss.

ANNUITIES, INSURANCE, &c.

In the application of the subject to insurance and life-

annuities, we say, if there is a certain number of chances by

which the possession of a sum can be secured, and also a

certain number of chances by which it may be lost, that sum

may be insured for that part of it, which shall be to the

whole, as the number of chances there is to lose it, is to the

number of all the chances.

From the bills of mortality in different places, tables have

been constructed, which shew how many persons, upon an

average, out of a certain number born, are lost, or have died at

the end of each year, to the extremity of life : from such tables

\\\e probability of the continuance of a life of any proposed age

is known. Hence we have tables calculated, shewing the

Expectation of human life at every age, according to the

probabilities of life at every age; and from these tables,

founded upon the doctrine of chances, the value of Annuities,

and of the insurance of single and joint lives, &c. is ascertained. .

The present value of a life-annuity is the sum that would

be sufficient, allowing for the chances of life failing, to pay the

annuity without loss : of course, if money bore no interest, the

value of an annuity would be equal to the sum, to be yearly

paid, multiplied by the number of years, equal to the ex-

pectation of life. But since money is capable of being im-

proved at interest, the sum just mentioned, would be more!

than sufficient to pay the annuity, and it will be as mucll-

more than sufficient, according as the interest is greater. Thus

it is found; that the expectation of a life of twenty, is equal to

512 MATHEMATICS.

thirty three years and a half, therefore to purchase an annuity

of of 1 per annum, for such a life, a person must pay, if there

Mere no interest on money, the sum of 33. 10s. but money

improved at 5 per cent, interest, doubles itself in about fourteen

years ; and from this, and other circumstances which we cannot

here enter into, the sum to be paid, will not be more than

about o 14. From the principles now described, methods are

obtained for calculating the value of an annuity for a person of

any given age for the longest of two or more lives, &c. Or

the value of a given sum payable at the decease of a person,

whenever tliat shall happen, that is, the value of an assurance

of any given sum on the whole duration of life, which is the

principal object of several most respectable insurance offices

in London. On the same principles as those already described,

tlie value of reversions is found, and the sum necessary to be

paid to insure property from the risk of fire, and loss by sea,

&c. This may at first sight appear very strange, and on that

account we shall transcribe the following paragraphs from

an admirable critique, in the Edinburgh Review, on a work

entitled " Essai Philosophique sur les Probabihtes. Par

M. Le Compte Laplace." " The way in which probability

is affected by the indefinite multiplication of events, is a

remarkable part of this theory. If out of a system of events

governed by chance (or by no perceivable law) you take a

small number, you will find great irregularity, and nothing

that looks like order, or obedience to a general rule. Increase

the number of events, or take in a larger extent of the ^boiain

over which you suppose chance to preside, you will find the

irregularities bear a much less proportion to the whole ; they

vijl in a certain degree compensate for one another; and

something like order and regularity will begin to emerge. In

proportion as the events are farther multiplied, tliis convergency

will become more apparent; and in summing up the total

amount, the events will appear adjusted to one another,, by

rules, from which hardly any deviation can be perceived.

" Thus, in considering the subject of life and death ; if W6

ANNUITIES, INSURANCE, SiC. 513

take a small extent of country, or a few people, a single

parish for instance, nothing like a general rule will be dis-

covered. The proportion of the deaths to the numbers alive,

or to the numbers born ; of those living at any age to those

above or below that age, all this will appear the most dif-

ferent in one year, compared with the next ; or in one district

compared with another. But subject to your examination the

parish registeYs of a great country, or a populous city, and

the facts will appear quite different. You will find the pro-

portion of those that die annually out of a given number of

inhabitants, fixed with great precision, as well as of those that

are born, and that have reached to the different periods of

life. In the first case, the irregularities bear a great pro-

portion to the whole ; in the second, they compensate for

one another ; and a rule emerges, from which the deviations

on opposite sides appear almost equal.

" This is true not only of natural events, but of those that

arise from the institutions of society, and the transactions of

men with one another Hence insurance against fire, and the

dangers of the sea. Nothing is less subject to calculation,

than the fate of a particular ship, or a particular house,

though under given circumstances. But let a vast nnmber of

ships, in these circumstances, or of houses be included, and

the chance of their perishing, to that of their being preserved,

is matter of calculation founded on experience, and reduced

to such certainty, that men daily stake their fortunes on the

accuracy of the results.

" This is true, even where chance might be supposed to

predominate the most ; and where the causes that produce

particular effects, are the most independent of one an-

other.

" Laplace observes, that at Paris, in ordinary times, the

number of letters returned to the Post Office, the persons to

whom they were directed not being foundj was nearly the

same from one year to another. We have heard the same

VOL. I. 2 L

.514 MATHEMATICS.

remark stated of the Dead Letter Office, as it is called ia

Loudon.

" Such is the consequence of the multiplication of the

events least under the controul of faxt causes: And the

instances just given, are sufficient to illustrate the truth of the

general proposition; which Laplace has thus stated.

" The recunences of events that depend on chance,

approach to fixt ratios as the events become more numerous,

in such a manner, that the probability of the mean results not

differing from those ratios by any given quantity, may come

nearer to certainty than the smallest limit that can be as-

signed."

One of the first treatises on the subject of interest and

annuities, that requires to be mentioned, is Ward's " Clavis

Usurae," published in 1709, and which, on account of its

scarcity, is published as the last tract in the fifth volume of

the " Scriptores Logarithmici" of Mr. Baron Maseres, with

an appendix by the Baron himself, who considers Ward's

tract as the most complete treatise that has ever been published

on the subject. We may farther observe, that in this fifth

volume of the Scriptores Logarithmici, there are several other

pieces on the same subject, by De Moivre, Robertson, Dr.

Halley, and others.

Mr. Smart, in 1727, published some very valuable tables

on interest and annuities, which have been in use from that

time to the present, and which Mr. Francis Baily has inserted

in a work of his, shortly to be noticed. The late learned and

very excellent Dr. Price, besides some papers in the Trans-

actions of the Royal Society, published two volumes, though

at different times, entitled " Observations on Reversionary

Payments on schemes for providing annuities for widows,

and for persons in old age on the method of calculating the

values of assurances on lives, with Essays on different subjects,

on the Doctrine of Life Annuities, and Political Arithmetic,

Tables," See. Uc, Although this cannot be deemed an elemeu-

ANNUITIES, INSURANCE, &C. 515

tary work, yet it contains a vast fund of valuable matter,

connected with all those subjects which have their foundation

on the Doctrine of Chances and Annuities. The tables are

numerous, and methods of forming are clearly given.

^' Tlie doctrine of Annuities and Assurances on Lives and

survivorships," was explained by Mr. William Morgan, in a

8vo volume, published in 1779. Besides these, Mr. Dodson,

in his Mathematical Repository, Mr. Thomas Simpson,

M. De Moivre, and Baron Maseres, have all treated on the

subject; and it must be observed, that those who would

enter deeply into it, should be well acquainted with algebra

and fluxions.

Mr. Francis Baily, in 1809, published, as an introductory

book, " The Doctrine of Interest and Annuities analytically

explained," &c. The reasons which Mr. Baily assigns for the

publication of his work, which is a good and useful perform-

ance, may be applied generally as a recommendation of the

science itself to the attention of the mathematical student:

" At this time," he says, " a more than ordinary degree of

attention has been given to it, arising from the great variety

and extent of property, which is affected by circumstances,

involving the consideration of this subject. For, when we

consider the numerous cases of daily occurrence, in which the

question of interest is unavoidably concerned, both simple and

compound ; when we consider the great and extensive business

which is constantly transacting, in the purchase and sale of

annuities of various kinds, immediate and reversionary, tena-

porary and perpetual, increased no doubt by the numerous

offices which have sprung up within these few years, for that

very purpose ; -when we consider also the immense quantity

of lands which are held on leases, for different terms of years,

and which leases are continually required to be exchanged,

sold, or renewed ; but above all, when we consider its ap-

plication in regard to our finances, and the state of the Na

tioual Debt, with the help it affords us in pointing out tlje

2 L 2

516 MATHEMATICS.

easiest and most effectual method of alleviating our present

incumbrances the subject assuredly acquires a degiee of im-

portance which it never before aspired to."

Mr. Daily's work treats of the doctrine of Interest, Annui-

ties, Reversions, and the Renewal of Leases, and of various

subjects of Finance.

Mr. Baily, in the year 1810, published a much larger and

more comprehensive work on this subject, to which, in 1813,

he added an appendix, making together two volumes, 8vo.

It is entitled " The doctrine of Life Annuities and Assurances,

analytically investigated and practically explained," &c. Of

this treatise, he says, it must be considered as a continuation

of the other work, and will, he believes, contain all that is

useful or interesting on the science, and he adds, that he has

taken care to comprehend in it '' such additional information

as a more improved analysis, and more recent discoveries in

the science have been able to afford." It contains the ele-

mentary principles of the Laws of Chance, with remarks on

the probabilities of life Methods for determining the value

of annuities on single or joint lives ; and on the longest of any

number of lives problems for the solution of cases of abso-

lute and contingent fleversioDary annuities -cases of annuities

depeiKling on survivorships, between two and three lives a

full explanation of the doctrine of Assurances, and other in-

teresting matter.

We have, by the same author, a smaller work, entitled

" Tables for the purchasing and renewing of leases, with

rules for determining the value of the reversion of estates,"

&c. Sec.

CHAP. XXXIII.

MATHEMATICS,

Continued.

Navigation What the art comprises History of Ancient writers on,

vii. Medina Cortes Frizins Nunez Coignet Norman Mercator

Wright Napier Norwood Bond. Modern writers recommended

Mackay Mendoza Robertson ; and Maskelyne. Mensuration His-

tory and Practice of Writers on Hawney Robertson Hutton; and

Bonnycastle. Surveying ^writers Leslie Hutton ^Crocker Davis

and others. Marine Swreying. Levelling, practice of. Dialling,

practice of, and writers on.

1 HE word Navigation is derived from two Latin words,

viz. navis, a ship, and ago, to manage, and the great end of

the art of Navigation, is to instruct the mariner how to con-

duct a ship through the wide and pathless ocean, to any given

country, by the safest and shortest way. This art comprises

the method of giving to the vessel the desired direction, by

means of the sails and a rudder, which has been denominated

seamanship, as well as the mode of ascertaining the relative

situation of the ship at any assigned time, and its future course.

For the performance of this, much previous instruction i^

^1$. MATHEMATICS.

necessary; sucb, for instance, as will give a perfect Inow-^

ledge of the figure of the earth, with the various real or

imaginary lines upon it, so as to be able to ascertain the

distance and situation of places, with respect to one an-

other. The mariner must also know the use of the several

instruments employed in measuring the ship's way, as the log

and half-minute glass ; also of the quadrant, to take the altitude

of the sun and stars, the compass to represent the sensible

horizon, and the azimuth compass, to take the azimuth or

amplitude of the sun, in order to ascertain the variation of

the needle. He must be conversant with maps and sea-charts,

and should know the depths of water in particular places, the

times of the ebbing and flowing of tides upon the coasts that

he may have occasion to approach. These and other things

will, perhaps, be best learnt by practice, but an able sailor

should be well grounded in the elementary principles of ma-

thematics, which will teach him the reason and foundation of

bis wh6le art

The antiquity and general utility of the art of Navigation

will not be disputed : its very early history it is impossible to

ascertain, but it is known that the Pha^nicians were skilful

navigators, at a period when other nations had scarcely ven-

tured from the shore. From Phoenicia it may be traced to

Egypt and Greece, but the Carthaginians soon eclipsed the

inhabitants of those countries in nautical knowledge and ex-

perienee, and by this means acquired those vast resources that

enabled them so long to withstand the Roman power. It was

during the struggle between Carthage and Rome, that the

first traces of the art may be discovered in Italy. The subse-

quent destruction of Carthage consigned navigation, and with

it commerce, to the Romans : the latter was left to the ex-

ertions of the conquered provinces, and the former was chiefly

cultivated to aid their favourite views of universal dominion,

and upon the ruin of that empire, navigation shared the fate

of literature and the other arts. When Attila destroyed

Mantua, VeroDa, and other adjacent cities, such of tlie in-

NAVIGATION. 519

habitants as escaped the bloody slaughter, fled (o the islands

equation is a x 2 y y ; therefore, x -J^. > w hich substi-

a

tuted for x, makes the general expression for the Subtangent

C T n ^iJ (by substituting a x for - its value,)

a . o

= 2 X. So that, the Subtangent C T is double the absciss

A C ; and consequently, A T is 3: AC.

Of the several works w ritten on this subject, the follow ing

seem to require particular notice.

" The method of Fluxions and infinite Series, with its ap-

plication to the Geometry of curve lines, by the inventor. Sir

Isaac Newton, to which is subjoined, a perpetual Comment

upon the whole work," S(,c. by John Colson. This will al-

ways be regarded as a standard work, but it is not exactly

suited to young persons entering upon the subject.

FLUXIONS. 507

Mr. John Rowe's " Introduction to the Doctrine of

Fluxions," is divided into two parts, the first treats of the

direct method of fluxions, and the second of the inverse

method. This latter, which is unquestionably the most difficult

part of the fluxionary calculus, viz. that of finding the fluent

from the fluxion being given, is introduced with the doctiine

of infinite series. The whole is written in so plain and per-

spicuous a manner, that the learner, who is previously well

acquainted with the arithmetic of rsurds, need not be deterred

from entering upon the study of Mr. Rowe's treatise. Of this

the third edition was printed in 1767.

The same may be said of " The Principles of Fluxions,**

by the Rev. S. Vince, D. D. F. R. S., which is a neat intro-

duction to this branch of science. The third edition of which

was published in 1805.

A more elaborate and more difficult piece is entitled " The

Doctrine and Application of Fluxions, &c., in two parts, by

Thomas Simpson, F. R. S." But the principal work is Mr.

Maclaurin's " Treatise of Fluxions," in two volumes, 4to.

In this the subject is handled agreeably to the method of rea-

soning used by the ancient mathematicians, without having

^course to algebraic solutions. To his demonstrations of this

doctrine he has added many valuable improvements of it, and

has applied it to so many curious and useful inquiries, that

the work has been denominated a store-house of mathematical

learning, rather than a treatise on one branch of it. Through-

out the whole there appears a masterly genius and uncom-

mon address. The biographer of Mr. Maclaurin, speak-

ing of him in connexion with his Treatise on Fluxions, says,

" he had a quick, comprehensive view, taking in, at once, all

the means of investigation, he could select the fittest for his

purpose, and apply them with exquisite art and method.

This is a faculty not to be acquired by exercise only ; we

ought rather to call it a species of that taste, the gift of na-

ture, which in mathematics, as in other things, distinguishes

excellence from mediocrity."

508 MATHEMATICS.

DOCTRINE OF CHANCES.

The doctrine of Chances being of great importance,

when applied to tlie solution of questions in life-annuities,

insurance, reversions, &c. it is necessary briefly to point out

those works in wliich its principles are laid down and in-

vestigated.

This subject, no less useful than interesting and curious,

does not appear to have engaged the attention of mathemati-

cians in former times, so murh as its importance may seem to

have required. Tlie writers upon it, in our own language, are

comparatively few. To M. Huygens we are indebted for

the first regular tract on this subject, which was in the Latin

language, and entitled *' De Ratiociniis in Ludo Aleae;" this

work, however, from the comparatively few problems which it

contains, and the want of demonstration to some of them,

cannot be regarded as an elementary treatise. To this suc-

ceeded a small anonymous tract entitled, " on the Laws of

Chance," published in London in J C)92, and a French publica-

tion Entitled ** L' Analyse des Jeux de Hazard," written by

M. Monmort, and published in 1708. Iliis author, following

tlie mode of M. Huygens in the solution of his problems^

M. de Moivre, who objected to it, published his work " On

the Doctrine of Chances," which was first published in 1717,

but which has been twice or thrice reprinted since. The edi-

tion of 1756, is, it is believed, the best. M. De Moivre pro-

ceeds from the most simple to the most complicated cases ; so

that, by the variety of his problems, as well as by the improve-

ments and additions which he made in two subsequent edi-

tions, he has rendered his work the best and most copious that

has ever been written on the subject. In the year 1 740, Mr.

Thomas Simpson published a very thin and small quarto, on

*' The Nature and Laws of Chance, illustrated wiih a great va-

riety of Examples," which, like his other publications, is not

only clear and concise, but contains some problems, the solu*

tious of which had never before been communicated to the

DOCTRINE OF CHANCES. 509

public. Mr. Dodson, in the year 1753, rendered the subject

still more accessible to persons not far advanced in analytical

studies, by publishing in the second volume of his " Mathema-

tical Repository," a number of questions with their solutions,

but chiefly with the view of applying them to the doctrine of

annuities and survivorships. In addition to these, may be

mentioned a small tract, " De Mensura Sortis," given by M.

De Moivre, in his " Miscellanea Analytica," and some pa-

pers published at different times in the Transactions of the

learned Societies on the Continent, and those of the Royal

Society of London." " Among which," says Mr. Morgan, in

the article Chances, in the New Cyclopedia, " may be parti-

cularly mentioned an " Essay on the Method of calculating

the e.xact probability of all conclusions founded on Induction,"

and a " Supplement " to that Essay ; the one preserved from

the papers of the late Rev. Mr. Bayes, and communicated

with an appendix, by Dr. Price, to the Royal Society in the

year 1762, and the other chiefly written by Dr. Price, and

communicated in the following year. " These tracts contain

the investigation of a problem, the converse of which had for-

merly exercised the ingenuity of M. Bernoulli, De Moivre,

and Simpson. Indeed both the problem, and its converse,

may be considered not only as the most difficult, but as the

most important that can be proposed on the subject, having no

less an object in view, than to shew what reason we have for

believing that there are in the constitution of things, fixed laws,

according to which events happen ; and that therefore, the

frame of the world must be the effect of the wisdom and

power of an intelligent cause, and thus to confirm the argu-

ment, taken from final causes, for the existence of Deity."

Besides the above-named works on the Doctrine of Chan-

ces, which is of very great consequence in this country, where

the valuation of an immense property, and the future provision

of many thousands, entirely depend on the right knowledge of

it, we may mention a treatise just published, entitled " Tl^e

510 MATHEMATICS.

Doctrine of Chances, or the Theory of Gaming made easy to

every Person acquainted with common Arithmetic, so as to

enable them to calculate the probabilities of events, in Ix)tte-

ries, Cards, Horse Racing, Dice, &c. with tables on Chance

never before published, by Wm. Rouse." The principles of

the Doctrine of Chances cannot be altered they are invariable ;

but Mr. Rouse may have made them more intelligible to the

unlearned, among whom we presume gamblers are usually to be

reckoned, who require a strong stimulus to engage their atten-

tion, but who have not sufficient energy to make study either the

business or amusement of life. The tables, mentioned in the

title, are intended to shew, at one view, the Chances for and

against winning any assigned number of Games, at any kind of

play, out of a given number, &c. How far this volume has

a claim to the character of accuracy we know not, it being

too recent a publication to admit of its having been read.

If the doctrine of Chances could be applied only to the

principles of Gaming, it would be of comparative little value ;

but as upon that doctrine depends every thing relating to Annu-

ities, to Survivorships, and to Reversions, we shall endea-

Tour, in a few words, to state the connexion of these several

subjects.

Chance is particularly used for the probability of an event,

and is greater or less, according to the number of chances

there are by which it may happen, compared with the number

by which it may either happen or fail. Thus, if an event has

three chances to happen, and two to fail, the probability of its

happening may be estimated at I ths, and of its failing, | ths.

Hence it appears, that if the probability of its happening and

failing be .added together, the sum is equal to unity. The ex-

pectation of obtaining any thing is estimated by the value of

that thing, multiplied by the probability of obtaining it. The

risk of losing any thing is estimated by the value of the thing

multiplied by the probability of losing it. Applying this to

gamingf we say, if from the expectations which the gamesters

ANNUITIES, INSURANCE, &C. 511

have upon the whole snm deposited, the particular sums they

deposit, that is, their own stakes, be subtracted, there will re-

main the gain, if the difference is positive ; or the loss, if the

difference be negative. Again, if from the respective expecta-

tions which either gamester has upon the sum deposited by his

adversary, the risk of losing what he himself deposits be sub-

tracted, there will likewise remain his gain or loss.

ANNUITIES, INSURANCE, &c.

In the application of the subject to insurance and life-

annuities, we say, if there is a certain number of chances by

which the possession of a sum can be secured, and also a

certain number of chances by which it may be lost, that sum

may be insured for that part of it, which shall be to the

whole, as the number of chances there is to lose it, is to the

number of all the chances.

From the bills of mortality in different places, tables have

been constructed, which shew how many persons, upon an

average, out of a certain number born, are lost, or have died at

the end of each year, to the extremity of life : from such tables

\\\e probability of the continuance of a life of any proposed age

is known. Hence we have tables calculated, shewing the

Expectation of human life at every age, according to the

probabilities of life at every age; and from these tables,

founded upon the doctrine of chances, the value of Annuities,

and of the insurance of single and joint lives, &c. is ascertained. .

The present value of a life-annuity is the sum that would

be sufficient, allowing for the chances of life failing, to pay the

annuity without loss : of course, if money bore no interest, the

value of an annuity would be equal to the sum, to be yearly

paid, multiplied by the number of years, equal to the ex-

pectation of life. But since money is capable of being im-

proved at interest, the sum just mentioned, would be more!

than sufficient to pay the annuity, and it will be as mucll-

more than sufficient, according as the interest is greater. Thus

it is found; that the expectation of a life of twenty, is equal to

512 MATHEMATICS.

thirty three years and a half, therefore to purchase an annuity

of of 1 per annum, for such a life, a person must pay, if there

Mere no interest on money, the sum of 33. 10s. but money

improved at 5 per cent, interest, doubles itself in about fourteen

years ; and from this, and other circumstances which we cannot

here enter into, the sum to be paid, will not be more than

about o 14. From the principles now described, methods are

obtained for calculating the value of an annuity for a person of

any given age for the longest of two or more lives, &c. Or

the value of a given sum payable at the decease of a person,

whenever tliat shall happen, that is, the value of an assurance

of any given sum on the whole duration of life, which is the

principal object of several most respectable insurance offices

in London. On the same principles as those already described,

tlie value of reversions is found, and the sum necessary to be

paid to insure property from the risk of fire, and loss by sea,

&c. This may at first sight appear very strange, and on that

account we shall transcribe the following paragraphs from

an admirable critique, in the Edinburgh Review, on a work

entitled " Essai Philosophique sur les Probabihtes. Par

M. Le Compte Laplace." " The way in which probability

is affected by the indefinite multiplication of events, is a

remarkable part of this theory. If out of a system of events

governed by chance (or by no perceivable law) you take a

small number, you will find great irregularity, and nothing

that looks like order, or obedience to a general rule. Increase

the number of events, or take in a larger extent of the ^boiain

over which you suppose chance to preside, you will find the

irregularities bear a much less proportion to the whole ; they

vijl in a certain degree compensate for one another; and

something like order and regularity will begin to emerge. In

proportion as the events are farther multiplied, tliis convergency

will become more apparent; and in summing up the total

amount, the events will appear adjusted to one another,, by

rules, from which hardly any deviation can be perceived.

" Thus, in considering the subject of life and death ; if W6

ANNUITIES, INSURANCE, SiC. 513

take a small extent of country, or a few people, a single

parish for instance, nothing like a general rule will be dis-

covered. The proportion of the deaths to the numbers alive,

or to the numbers born ; of those living at any age to those

above or below that age, all this will appear the most dif-

ferent in one year, compared with the next ; or in one district

compared with another. But subject to your examination the

parish registeYs of a great country, or a populous city, and

the facts will appear quite different. You will find the pro-

portion of those that die annually out of a given number of

inhabitants, fixed with great precision, as well as of those that

are born, and that have reached to the different periods of

life. In the first case, the irregularities bear a great pro-

portion to the whole ; in the second, they compensate for

one another ; and a rule emerges, from which the deviations

on opposite sides appear almost equal.

" This is true not only of natural events, but of those that

arise from the institutions of society, and the transactions of

men with one another Hence insurance against fire, and the

dangers of the sea. Nothing is less subject to calculation,

than the fate of a particular ship, or a particular house,

though under given circumstances. But let a vast nnmber of

ships, in these circumstances, or of houses be included, and

the chance of their perishing, to that of their being preserved,

is matter of calculation founded on experience, and reduced

to such certainty, that men daily stake their fortunes on the

accuracy of the results.

" This is true, even where chance might be supposed to

predominate the most ; and where the causes that produce

particular effects, are the most independent of one an-

other.

" Laplace observes, that at Paris, in ordinary times, the

number of letters returned to the Post Office, the persons to

whom they were directed not being foundj was nearly the

same from one year to another. We have heard the same

VOL. I. 2 L

.514 MATHEMATICS.

remark stated of the Dead Letter Office, as it is called ia

Loudon.

" Such is the consequence of the multiplication of the

events least under the controul of faxt causes: And the

instances just given, are sufficient to illustrate the truth of the

general proposition; which Laplace has thus stated.

" The recunences of events that depend on chance,

approach to fixt ratios as the events become more numerous,

in such a manner, that the probability of the mean results not

differing from those ratios by any given quantity, may come

nearer to certainty than the smallest limit that can be as-

signed."

One of the first treatises on the subject of interest and

annuities, that requires to be mentioned, is Ward's " Clavis

Usurae," published in 1709, and which, on account of its

scarcity, is published as the last tract in the fifth volume of

the " Scriptores Logarithmici" of Mr. Baron Maseres, with

an appendix by the Baron himself, who considers Ward's

tract as the most complete treatise that has ever been published

on the subject. We may farther observe, that in this fifth

volume of the Scriptores Logarithmici, there are several other

pieces on the same subject, by De Moivre, Robertson, Dr.

Halley, and others.

Mr. Smart, in 1727, published some very valuable tables

on interest and annuities, which have been in use from that

time to the present, and which Mr. Francis Baily has inserted

in a work of his, shortly to be noticed. The late learned and

very excellent Dr. Price, besides some papers in the Trans-

actions of the Royal Society, published two volumes, though

at different times, entitled " Observations on Reversionary

Payments on schemes for providing annuities for widows,

and for persons in old age on the method of calculating the

values of assurances on lives, with Essays on different subjects,

on the Doctrine of Life Annuities, and Political Arithmetic,

Tables," See. Uc, Although this cannot be deemed an elemeu-

ANNUITIES, INSURANCE, &C. 515

tary work, yet it contains a vast fund of valuable matter,

connected with all those subjects which have their foundation

on the Doctrine of Chances and Annuities. The tables are

numerous, and methods of forming are clearly given.

^' Tlie doctrine of Annuities and Assurances on Lives and

survivorships," was explained by Mr. William Morgan, in a

8vo volume, published in 1779. Besides these, Mr. Dodson,

in his Mathematical Repository, Mr. Thomas Simpson,

M. De Moivre, and Baron Maseres, have all treated on the

subject; and it must be observed, that those who would

enter deeply into it, should be well acquainted with algebra

and fluxions.

Mr. Francis Baily, in 1809, published, as an introductory

book, " The Doctrine of Interest and Annuities analytically

explained," &c. The reasons which Mr. Baily assigns for the

publication of his work, which is a good and useful perform-

ance, may be applied generally as a recommendation of the

science itself to the attention of the mathematical student:

" At this time," he says, " a more than ordinary degree of

attention has been given to it, arising from the great variety

and extent of property, which is affected by circumstances,

involving the consideration of this subject. For, when we

consider the numerous cases of daily occurrence, in which the

question of interest is unavoidably concerned, both simple and

compound ; when we consider the great and extensive business

which is constantly transacting, in the purchase and sale of

annuities of various kinds, immediate and reversionary, tena-

porary and perpetual, increased no doubt by the numerous

offices which have sprung up within these few years, for that

very purpose ; -when we consider also the immense quantity

of lands which are held on leases, for different terms of years,

and which leases are continually required to be exchanged,

sold, or renewed ; but above all, when we consider its ap-

plication in regard to our finances, and the state of the Na

tioual Debt, with the help it affords us in pointing out tlje

2 L 2

516 MATHEMATICS.

easiest and most effectual method of alleviating our present

incumbrances the subject assuredly acquires a degiee of im-

portance which it never before aspired to."

Mr. Daily's work treats of the doctrine of Interest, Annui-

ties, Reversions, and the Renewal of Leases, and of various

subjects of Finance.

Mr. Baily, in the year 1810, published a much larger and

more comprehensive work on this subject, to which, in 1813,

he added an appendix, making together two volumes, 8vo.

It is entitled " The doctrine of Life Annuities and Assurances,

analytically investigated and practically explained," &c. Of

this treatise, he says, it must be considered as a continuation

of the other work, and will, he believes, contain all that is

useful or interesting on the science, and he adds, that he has

taken care to comprehend in it '' such additional information

as a more improved analysis, and more recent discoveries in

the science have been able to afford." It contains the ele-

mentary principles of the Laws of Chance, with remarks on

the probabilities of life Methods for determining the value

of annuities on single or joint lives ; and on the longest of any

number of lives problems for the solution of cases of abso-

lute and contingent fleversioDary annuities -cases of annuities

depeiKling on survivorships, between two and three lives a

full explanation of the doctrine of Assurances, and other in-

teresting matter.

We have, by the same author, a smaller work, entitled

" Tables for the purchasing and renewing of leases, with

rules for determining the value of the reversion of estates,"

&c. Sec.

CHAP. XXXIII.

MATHEMATICS,

Continued.

Navigation What the art comprises History of Ancient writers on,

vii. Medina Cortes Frizins Nunez Coignet Norman Mercator

Wright Napier Norwood Bond. Modern writers recommended

Mackay Mendoza Robertson ; and Maskelyne. Mensuration His-

tory and Practice of Writers on Hawney Robertson Hutton; and

Bonnycastle. Surveying ^writers Leslie Hutton ^Crocker Davis

and others. Marine Swreying. Levelling, practice of. Dialling,

practice of, and writers on.

1 HE word Navigation is derived from two Latin words,

viz. navis, a ship, and ago, to manage, and the great end of

the art of Navigation, is to instruct the mariner how to con-

duct a ship through the wide and pathless ocean, to any given

country, by the safest and shortest way. This art comprises

the method of giving to the vessel the desired direction, by

means of the sails and a rudder, which has been denominated

seamanship, as well as the mode of ascertaining the relative

situation of the ship at any assigned time, and its future course.

For the performance of this, much previous instruction i^

^1$. MATHEMATICS.

necessary; sucb, for instance, as will give a perfect Inow-^

ledge of the figure of the earth, with the various real or

imaginary lines upon it, so as to be able to ascertain the

distance and situation of places, with respect to one an-

other. The mariner must also know the use of the several

instruments employed in measuring the ship's way, as the log

and half-minute glass ; also of the quadrant, to take the altitude

of the sun and stars, the compass to represent the sensible

horizon, and the azimuth compass, to take the azimuth or

amplitude of the sun, in order to ascertain the variation of

the needle. He must be conversant with maps and sea-charts,

and should know the depths of water in particular places, the

times of the ebbing and flowing of tides upon the coasts that

he may have occasion to approach. These and other things

will, perhaps, be best learnt by practice, but an able sailor

should be well grounded in the elementary principles of ma-

thematics, which will teach him the reason and foundation of

bis wh6le art

The antiquity and general utility of the art of Navigation

will not be disputed : its very early history it is impossible to

ascertain, but it is known that the Pha^nicians were skilful

navigators, at a period when other nations had scarcely ven-

tured from the shore. From Phoenicia it may be traced to

Egypt and Greece, but the Carthaginians soon eclipsed the

inhabitants of those countries in nautical knowledge and ex-

perienee, and by this means acquired those vast resources that

enabled them so long to withstand the Roman power. It was

during the struggle between Carthage and Rome, that the

first traces of the art may be discovered in Italy. The subse-

quent destruction of Carthage consigned navigation, and with

it commerce, to the Romans : the latter was left to the ex-

ertions of the conquered provinces, and the former was chiefly

cultivated to aid their favourite views of universal dominion,

and upon the ruin of that empire, navigation shared the fate

of literature and the other arts. When Attila destroyed

Mantua, VeroDa, and other adjacent cities, such of tlie in-

NAVIGATION. 519

habitants as escaped the bloody slaughter, fled (o the islands

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