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William Shepherd.

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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



Online LibraryWilliam ShepherdSystematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) → online text (page 42 of 44)