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This book, notwithstanding the high reputatiofi which it

had in the beginning of the lart century, and to its intrinsic

merit we have heard several able mathematicians, now no

more, bear most decisive evidence, as having themselves been

almost, or perhaps altogether, indebted to it for their delight

in science, is now seldom inquired for. It has made way for

others, which we shall describe hereafter, and which seem

better adapted to the wants of young persons at this period.

Perhaps, in nothing, is the present age, including the last thirty

years, more distinguished than in the production of elementary

works in useful knowledge and real science.

Very similar to Ward's " Young Mathematician's Guide,"

wa# ^Ir, Jones's " Syi\opsis Palmariorum Matheseos," pub-j

ALGEBRA. 445

lished in 1706. This compendium contains, thongh in a

shorter compass, the same subjects as those treated of by

Mr. Ward, with the addition of the principles of Projection,

the elements of Trigonometry, Mechanics, ayd Optics. Tlie

author, who was father to the late illustrious Sir William

Jones, said he designed his work for the benefit, and adapted

to the capacities of beginners. It appears, however, to have

been rather too brief for the purpose intended. Mr. Jones

published in 1711 a collection of Sir Isaac Newton's papers,

entitled " Analysis per quantitatum series, fluxiones, ac dif-

ferentias ; cum enumeratione linearum tertii ordinis." Sir

Isaac Newton's " Arithmetica Universalis, sive de Compo-

sitione et Resolutione Arithmetica liber," was published in

1707, since which, it has gone through many editions, and is

included in Dr. Horsley's edition of Sir Isaac Newton's works,

in five volumes, 4to., the Arithmetica standing first in the col-

lection. This treatise was ' the text-book of the author at

Cambridge, and though not intended for publication, it con-

tains many very considerable improvements in analytics. Com-

mentaries have been published on this work, by S'Gravesande

and others. Iji the year 1769, Dr. Wilder, mathematical

professor of Trinity College, Dublin, published a translation

of the " Arithmetica Universalis," which had been made by

Mr.- Ralphson, and corrected by Mr. Cunn. This he illus-

trated and explained in a series of notes ; to the right under*-

standing of which work, thus presented to the public, with an

additional treatise upon the measures of ratios, Dr. Wilder

says it is only necessary that the student should be well

versed in the Elements of Euclid, and be master of common

arithmetic, as it is taught in the schools.

Without pretending to enumerate all the introductory works

to this science, that have, of late years, been published, we

shall point out' to our readers a sufficient number to allow

them a choice, and mention some of their chief, or distin-

guishing merits. It is presumed, that the pupil, previously to

his entrance upon a course of algebra, is master of the ele-

446 MATHExMATICS.

mentary rules of common arithmetic ; and if he is conversant

with fractions and decimals, he will enter with more advan-

tage upon the study of algebra.

Previously t(^ enumerating particular treatises on Algebra,

we may, in few words, explain the nature of the subject gene-

rally, and the characters which are used by almost all au-

thors.

Every figure, or common arithmetical character, has a de-

terminate value ; thus the figures 5, 7, 9, always represent the

same number, viz. the collections of five, seven, and nine

units; but algebraical characters must be general, and inde-

pendent of any particular signification, adapted to the repre-

sentation of all sorts of quantities, according to the nature of

the questions to which they are applied. To answer genera!

purposes, they should be simple, and easy to describe, so as

hot to be troublesome in operation, nor difficult to remember.

These advantages meet in the letters of the alphabet, which

are therefore usually adopted to represent magnitudes in al-

gebra ; and we have shewn above, in what way, and by whom

they were introduced.

In algebraical investigations, some quantities are assumed

as known or given, and the value of others is unknown, and to

be found out ; the former are commonly represented by the

leading letters of the alphabet, , h, r, d, &c. ; the latter by

the final letters, er, x, y, z. Though it often tends to relieve

the memory, if the initial letter of the subject under consi-

deration be made use of, whether that be known or unknown:

thus r may denote a radius, b a base, p a perpendicular, s a

side, d density, m mass, &c.

The characters used to denote the operations, are princi-

pally these :

+ signifies addition, and is named plus.

signifies subtraction, and is named minus,

X denotes multiplication, and is named into.

-f- denotes division, and is named "by.

-^ t tlie mark of radicality denotes the square root; with a

ALGEBRA. 447

3 before it, thus \/, tUe cube root; with a 4, thus ^v^j lli

fourth, or biquadrate root ; thus "v^, the th root.

Proportion is commonly denoted by a colon between tlie

antecedent and consequent of each ratio, and a double colon

between the two ratios: thus, if a be to h as c to J, we state

it as follows, a:h: : C'.d.

:=. is the symbol of equality.

Hence, a + 6 denotes the sum of the quantities represented^

by a and b.

a b denotes their difference when b is the less: fe ,

their difference when is the less : a^b, the difference when

it is not known which is the greater, ax 6, or a . &, or abf

represents the product of a multiplied into b.

a-i-b, or J, shews that the number represented by a is

to be divided by that which is represented by b.

T is the reciprocal of - , and - the reciprocal of a.

a:b: :c:d denotes that a is in the same proportion to b^

as c h to d.

xzza b + c IS an equation, shewing that x is equal to the

difference of a and b, added to the quantity c.

V^ fl, or a , IS the square root of a ; '\/a, or a , is the

X

cube root of a; and "'\/a, or a , is the wth root of a.

d^ is the square of a,- a^ the cube of a; a^ the fourth

power of a ; and a"^ the wth power of a.

a + bxc, or {a + b) c, is the product of the compound

quantity a + 6 multiplied by the simple quantity c. Using

the bar , or the parenthesis ( ), as a vinculum, to con-

nect several quantities into one..

a^b-i-a b, or t> expressed like a fraction, is the quo-

tient of rt + 6 divided by a b.

5a denotes that the quantity a is to be taken 5 times, and

7. {b + c) is 7 times b + c. And these numbers, 5 or 7,

448 MATHEMATICS.

shewing how often the quantities are 'to be taken, or multi-

plied, are called co-efficients.

Like quantities, are those which consist of the same letters,

and powers. As a and 3a ; or a6 and Aab ; or Sarbc and

5a'bc.

Unlike quantities, are those which consist of different letters,

or different powers. As a and b ; or 2a and a^ ; or 3ab^ and

Sabc.

Simple quantities, or monomials, are those which consist of

one term only. As 3a, or-5ab, or Gabc^.

Compound quantities, are those whrdi consist of two or

more terms. As a-\-b, or a -h 26 3c.

And when the compound quantity consists of two terms, it

is called a binomial ; when of three terms, it is a trinomial ;

when of four terms, a quadrinomial ; more than four terms, a

multinomial, or polynomial.

Positive or affirmative quantities, are those which are to be

added, or have the sign + . As a or + a, or ab ; for when

a quantity is found without a sign, it is understood to be posi-

tive, or to have the sign + prefixed.

Negative quantities, are those which are to be subtracted,

as a, or 2a6, or Sab"^.

Like signs, are either all positive ( + ), or all negative ( ).

Unlike signs, are when some are positive ( + ), and others

negative ( )

In every quantity we may consider two things, its value,

and its manner of existing with regard to other magnitudes

which enter with it into the same calculation. The vahie of

a quantity is expressed by the letter or by the character des-

tined to represent the number of its units. But as to the

xnode of existence, with regard to others, some rn^gnitudes may

affect the calculation either in the same or in opposite senses ;

which renders it necessary to distinguish two sorts of quanti-

ties, positive and negative. Thus whether a man have a

thousand pounds in property or stock, or be a thousand pounds

in debt, may be represented by characters, either arithmetical

ALGEBRA. 449

or algebraical ; but since an actual property is directly oppo-

site in its nature to a debt, the two must be marked by dif-

ferent symbols : so that, if property be reckoned a positive

quantity, and marked + , a debt owed must be estimated as

negative, and marked . Again, if, commencing at the

same point, motion towards the east be considered as a posi-

tive quantity in an investigation, motion towards the west,

which is opposite to the former, must enter the same calcula -

tion as a negative quantity. If the elevations of the sun

above the horizon are considered 'as positive quantities, the

depressions of the sun below the horizon must be treated as

negative quantities. It is the same with all quantities^ which^

when considered together, exist differently with respect to one

another.

A residual quantity is a binomial having one of the terms

negative. As a 2b.

The power of a quantity (a), is its square (a^), or cube (a^),

or biquadrate (a?), &c. ; called also the 2d power, or 3d

power, or 4th power, &c.

Tlie index, or exponent, is the number which denotes the

power or root of a. quantity. So 2 is the exponent of the

square or 2d power a^ ; and 3 is the index of the cube or 3d

power ; and 4. is the index of the square root a or ^a ; and

I

J fk the index of the cube root a or ^\/a.

A rational quantity, is that which has no radical sign (-\/)

or index annexed to it. As a, or Sab.

An irrational quantity, or surd, is that which has not an

exact root, or is expressed by means of the radical sign \/.

I

As \/2, or \/a, or ^\/S cr ^

One of the easiest and most simple Introductions to this

science, is that by Fenning, which, if our recollection serves

us, is introduced with an account of fractions by common

numbers. It is many years since we have seeq this work,

anu, perhaps, it is not now to be met with but on stalls, or

VOL. I. 2 G

4d6 mathkMatics.

in second-liand catalogues. In lieu of this, we may notic*

an excellent little treatise by Mr. Bonnycastle, entitled " An

Introduction to Algebra, with Notes and Observations for the

Use of Schools, &c." Tliis compendium is formed upon the

model of larger works, and is intended as an introduction to

them. It supposes, however, that the person making "use of

it, as a first book, on the -subject, has the advantage of a living

instructor to aid him in difficulties that will inevitably occur

to check his progress. To those who have no instructor, we

would recommend a work, which, from a slight view of it

(for it has but lately come into our hands), appears to obviate

all difficulties, by explaining every thing in a full and familiar

manner as fjjr as it goes : it is entitled " The Philosophy of

Arithmetic, considered as a branch of Mathematical Science,

and the Elements of Algebra, &c. by John Walker." This

volume is divided into twenty-eight chapters, of which thirteen

are devoted to the elucidation of the principles of common

arithmetic. Mr. Walker observes, that " the scientific prin-

ciples of common arithmetic are so coincident with those of

algebra, or universal arithmetic, that to persons acquainted

with the former, the Elements of the latter offer no serious

difficulty. Of the Elements of Algebra, therefore, I have

given such a view, as may open that wide field of science to

the student, and enable him, at his pleasure, to extend his

progress, by the aid of any of the larger works extant on the

subject. Having designed this work for the instruction of

those who come to it most uninitiated in science, I have aimed

at giving a clear and full explanation of the most elementary

principles." From the parts that we have examined, it does

appear that the author has succeeded in the accomplishment

of his object. It must, however, be observed, that Mr.

Bonnycastle's " Introduction" includes a number of topics

not touched on even by Mr. Walker. Of these, we may

mention the Diophantine Probtems, and the Summation of

Infinite Series, which b a very important part of some of the

practical mathematics.

ALGEBRA. , 451

" Lectures on the Elements of Algebra," &c. by the Rev.

B. Bridge, A. M. may be safely recommended as a valuable

introduction to the science ; we admit the truth of the author's

assertion, " that the substance pf the Lectures is perfectly

within the comprehension of students at the age of fifteen or

6ixteei>." In some of these lectures, the learner's ingenuity

will, however, be tried ; but*the subjects are interesting, and

worthy the exertion he may be called on to make in the in-

vestigation.

In connexion with any of the above-named works, tha

pupil may read the first part of Maclaurin's Treatise of Al-

gebra, which, tliough deficient in the number of its examples,

is written in a remarkably clear, not to say elegant, style. It

proceeds only to Quadratic Equations, and the doctrine of

Surds.

" The Principles of Algebra," by William Frend, may be?

consulted with advantage, but it cannot be recommended, by

itself, as an introductory work to the science, because we feel

no objection to the usual modes of notation and expression,

which Mr. Frend endeavours to exchange for others, as we

apprehend, not at all-more intelligible. Algebra, like every

branch of real science, has, no doubt, its difficulties ; and the

youth who would make real proficiency in it, either with or

without the aid of a tutor, must, in the first instance, be con-

tent to advance slowly, feel every step of the ground on which

he treads, and fully comprehend every term he may meet

with. To such a one, we are sure there caw be no real ob-

stacles in the use of the algebraical terms which are found in

common books ; nor can he be expected to make much pro-

gress in the science, who is frightened with the words plus,

minus, sines, co-efficient, &c. We have, however, said, that

the learner may consult the " Principles of Algebra" with ad-

vantage ; and we regret that the book is become so scarce as

rarely to be met with. It was published in two parts. The

Jirst proceeds to Cardan's rule for the solution of Equations

2 G 3

452 ^ MATHEMATICS.

of the third order; the second part contains tlie theory of

Equations established on mnlhematical demonstration.

Having gone througli the whole, or the introductory parts

of either of the foregoing elementary books, the student may

take in connexion uith his present pursuits, some parts of

IMr. Thomas Simpson's Algebra, which treats on topics not

to be found in any of the othew. Or he may advance to the

second part of Maclaurin's Algebra, " On the Genius and

Resolution of Equations of all Degrees," &c.

Much curious and valuable mathematical knowledge will

be found in Saunderson's ** Elements of Algebra," in two

volumes, 4to. should they fall in the way of the pupil. Dr.

Saunderson, though blind, was one of the ablest mathema-

ticians of the age. He lost his sight when he was only eight

years old ; yet so great were his talents, and so steady his ap-

plication to the classics and maUiematics, that he could, at an

early period of life, take pleasure in hearing the works of

Euclid, Archimedes, and Diophantus, read in their original

Greek. At the age of twenty-tive, he went to Cambridge,

and his fame soon filled the University. Newton's Principia,

Optics, and Universal Arithmetic, were the foundation of the

lectures which he delivered to the students of that seat of

learning, and they afforded hiih a noble field for the display

of his genius. Great numbers came, some, no doubt, through

motives of curiosity, to hear a blind man give lectures on

optics, discourse on the nature of light and colours, ex-

plain the theory of vision, the phenomenon of the rainbow,

and other objects of sight; but none of liis auditors went

away disappointed; and he always interested, as well as in-

structed, those who came for the purpose of gaining know-

ledge. He succeeded Mr. Whiston in the mathematical pro-

fessor's chair, and, from this time, in 1711, he gavc^ up his

whole time to his pupils, for whose use he had composed

something new and important on almost every branch of the

mathematics. But he discovered no intention to publish any

ALGEBRA. 453

thing, till, by the persuasion of his friends, he prepared his

*' Elements of Algebra" for the press.

Dr. Saunderson had a peculiar meOiod of "performing

arithmetical calculations by nn ingenious machine and me-

thod, which has been denominated his " Palpable Arithmetic,"

and which is particularly described in the first volume of the

work to which we are now directing the reader's attentioQ.

An Abridgment, or Select Parts of Dr. Saunderson's " Ele-

ments of Algebra," was published in an octavo volume, in the

year 1755, which has passed through several editions, the

fourth being printed in 1776. This is a judicious compen-

dium of the larger work, but is not better adapted to learners

than Bonnycastle's, Bridge's, or some other introductions to

the science, of more modern date. For the sake of beginners,

the compiler has prefixed to the Select Parts of Dr. Saunder-

son's Elements, an Introduction to Vulgar and Decimal Frac-

tions, and a collection o"f Arithmetical questions, in order that

the learner may try his skill in common arithmetic before he

enters upon the study of Algebra.

The young algebraist may consult with much advantage

some other books not avowedly elementary, but which con-

tain a large number of excellent problems, the solution of

M'hich will exercfee his ingenuity, and invigorate his powers.

Of these, the first is Dodson's Mathematical Repository, three

volumes, 12mo. 1748. We refer particularly to the first,

and part of the second volume^ the other parts will be no-

ticed hereafter. The early problems of this work are adapted

to those who are but just entering on the science; they in-

crease in difficulty as the pupil is supposed to become stronger

in the pursuit.

Another excellent work of this kind is entitled " Select

Exercises for Young Proficients in Mathematics, &c. by

Thomas Simpson, 1752." Of this volume, the first part con-

tains a number of algebraical problems, with their solutions,

designed as proper exercises for young beginners, in which

454 MATHEMATICS.

the art of managing Equations, and the various methods a(

substitution, are taught and illustrated.

A much more modem work, but one of considerable utility

to the student in Algebra, is the following, " Algebraical Pro-

blems, producing Simple and Quadratic equations, withtheir

solutions, designed as an introduction to the higher branches

of analytics: by the Rev. M. Bland, A. M. \S\2" These

problems, of \\hich there are several hundreds, are designed

solely to point out the various methods employed by Analysts

in the solution of Equations. They are arranged in the usual

manner : the Jirst part containing simple Equations ; the se-

cond, pure Quadratics, and others, that may be solved without

completing the square ; and third/i/, adfected Quadratics.

The author, who has employed much industry and skill in the

compilation of the volume, tells his readers that he has con-

sulted many books, and as utility was the sole object which he

had in view, he has taken his examples from every source,

and has altered them to suit his purpose. At the head of

each section he has given the common rules, so that if the

reader is acquainted with the practice previously to Equations,

Mr. Bland's volume may be considered as a good introduction

to the science at that point. Of the Lady's Diary and Ley-

bourn's Mathematical Repository, we shall s]>eak in our next

chapter.

Mr. Bonnycastle, in 1813, published a much larger work

on this subject, than that which we have already noticed, it is

entitled " A Treatise on Algebra in Practice and Theory," iu

two vols. 8vo. The first volume is devoted chiefly, though in

an extended form, to the same subjects as he had already dis-

cussed in the smaller work ; the second, denominated by the au-

thor " the Theoretical part," will afford much exercise to the

talents and ingenuity of the student, who has already inade

considerable progress in the analytic science, aiid will probably

open to him a new field of speculation. To the first volume

19 prefixed an excellent historical introduction : in the latter

ALGEBRA. 455

jmrt of the second volume is shewn the application of Algebra

to Geometry, and the doctrine of curves.

After, or in connexion with, this work pf Mr. Bonnjcastlc,

may be taken in hand the second volume of the " Elements

of Algebra," by Leonard Euler. This work V4;as pub-

lished in the German language, in 1770, and has since been

translated into the French and English, with notes and addi-

tions by the editors. Among the latter there is a very learned

and copious tract of tlie celebrated La Grange, oit " conti-

nued fractions," anfl such parts of the indeterminate analysis,

as had not been sufficiently treated of by the author; " the ,

whole," says Mr. Bonnycastle, ," formii^ one of the most

profound treatises on this branch of the' subject that has ever

yet appeared."

Before the pupil has arrived at this period of his studies, he

will necessarily be acquainted with almost every tiling that hat

been written on the subject, and will of himself know where

to look for subjects \\hich may engage his attention. We

shall, therefore, only observe, that there is a valuable work in

Xhe Latin language, published at Dublin in 1784, entitled

" Analysis ^quationum, Auctore Guil. Hales. D. D." The

author of which says, that he has endeavoured to follow the

tract of Wallis, Maclaurin, Saunderson, De Moirre, Simp-

son, Clairaut, D'Alerabert, Euler, La Grange, Waring,

Bertrand, Landen, Hutton, &c. " qui aut scriptis Newtoui

illustrandis, aut algebrae limitibus latins proferendis feUcis-

;5ime operam dederunt."

With respect to some of the authors above enumerated, we

may observe, in addition to what we have already noticed, that

M. Clairaut published his " Elemens d'Algebre," in 1746,

in which he made many improvements with reference to the

irreducible case in cubic equations. A fifth edition of this

work was published at Paris, with notes and large additions,

in 1797. M. Landen published his " Residual Analysis" in

1764, his " Mathematical Lucubrations" in 1765, and his

'^ Mathematical Memoirs " in 1 780. The Memoirs of the .

456 MATHEMATICS.

Berlin and Petersburgh Academies abound with improvements

on series and oilier branches of analysis by Euler, La Grange,

and other illustrious mathematicians. Dr. Waring, late of

Cambridge, communicated many valuable papers to the Royal

Society, .who have caused them to be printed in their Trans-

actions, and many of his improvements are contained in his

separate publications, which, it must be acknowledged, are

too abstruse for ordinary mathematicians. These are entitled

** Meditationes Algebraicae," " Proprietates Algebraicarum

Curvarum," and " Meditationes Analyticae." Mr. Baron

Maseres claims to be mentioned not only as an 'original writer

on the analytical branch of Science, but also on account of the

labour and expense which he has bestowed on the publication

of the " Scriptores Logarithmici," in six large vols. 4to.

between the year 1791 and 1807, containing many curious

and useful tracts, which are thus preserved from being lost,

and many valuable papers of his own on the binomial theo-

rem, series, &c. The Baron's separate publications oo

Algebra, are, (1). "A Dissertation on the Negativie Sign in

Algebra." (2). ** Principles of the Doctrine of Life Annu-

ities.'* (3). " Tracts on the Resolution of Affected Algebraic

Equations," &c

CHAP. XXX.

MATHEMATICS,

Continued.

Advantages, History and Province of Geometry Principles of Geometry.

Elementary treatises, Simson's ".Elements" Cunn Tacquet De Chales

Whiston Barrow Simpson's Bonnycastle's Payne's and Cowley'^

Geometry. Matton's Playfair's Leslie's Reynard's ; and Keith's. Ap-

plication of Algebra to Geometry. Simpson Frend Boni)ycastle-^

Lady's Diary, and Leyboum's Mathematical Repository.

JN EXT to Arithmetic should follow Geometry, in a course

had in the beginning of the lart century, and to its intrinsic

merit we have heard several able mathematicians, now no

more, bear most decisive evidence, as having themselves been

almost, or perhaps altogether, indebted to it for their delight

in science, is now seldom inquired for. It has made way for

others, which we shall describe hereafter, and which seem

better adapted to the wants of young persons at this period.

Perhaps, in nothing, is the present age, including the last thirty

years, more distinguished than in the production of elementary

works in useful knowledge and real science.

Very similar to Ward's " Young Mathematician's Guide,"

wa# ^Ir, Jones's " Syi\opsis Palmariorum Matheseos," pub-j

ALGEBRA. 445

lished in 1706. This compendium contains, thongh in a

shorter compass, the same subjects as those treated of by

Mr. Ward, with the addition of the principles of Projection,

the elements of Trigonometry, Mechanics, ayd Optics. Tlie

author, who was father to the late illustrious Sir William

Jones, said he designed his work for the benefit, and adapted

to the capacities of beginners. It appears, however, to have

been rather too brief for the purpose intended. Mr. Jones

published in 1711 a collection of Sir Isaac Newton's papers,

entitled " Analysis per quantitatum series, fluxiones, ac dif-

ferentias ; cum enumeratione linearum tertii ordinis." Sir

Isaac Newton's " Arithmetica Universalis, sive de Compo-

sitione et Resolutione Arithmetica liber," was published in

1707, since which, it has gone through many editions, and is

included in Dr. Horsley's edition of Sir Isaac Newton's works,

in five volumes, 4to., the Arithmetica standing first in the col-

lection. This treatise was ' the text-book of the author at

Cambridge, and though not intended for publication, it con-

tains many very considerable improvements in analytics. Com-

mentaries have been published on this work, by S'Gravesande

and others. Iji the year 1769, Dr. Wilder, mathematical

professor of Trinity College, Dublin, published a translation

of the " Arithmetica Universalis," which had been made by

Mr.- Ralphson, and corrected by Mr. Cunn. This he illus-

trated and explained in a series of notes ; to the right under*-

standing of which work, thus presented to the public, with an

additional treatise upon the measures of ratios, Dr. Wilder

says it is only necessary that the student should be well

versed in the Elements of Euclid, and be master of common

arithmetic, as it is taught in the schools.

Without pretending to enumerate all the introductory works

to this science, that have, of late years, been published, we

shall point out' to our readers a sufficient number to allow

them a choice, and mention some of their chief, or distin-

guishing merits. It is presumed, that the pupil, previously to

his entrance upon a course of algebra, is master of the ele-

446 MATHExMATICS.

mentary rules of common arithmetic ; and if he is conversant

with fractions and decimals, he will enter with more advan-

tage upon the study of algebra.

Previously t(^ enumerating particular treatises on Algebra,

we may, in few words, explain the nature of the subject gene-

rally, and the characters which are used by almost all au-

thors.

Every figure, or common arithmetical character, has a de-

terminate value ; thus the figures 5, 7, 9, always represent the

same number, viz. the collections of five, seven, and nine

units; but algebraical characters must be general, and inde-

pendent of any particular signification, adapted to the repre-

sentation of all sorts of quantities, according to the nature of

the questions to which they are applied. To answer genera!

purposes, they should be simple, and easy to describe, so as

hot to be troublesome in operation, nor difficult to remember.

These advantages meet in the letters of the alphabet, which

are therefore usually adopted to represent magnitudes in al-

gebra ; and we have shewn above, in what way, and by whom

they were introduced.

In algebraical investigations, some quantities are assumed

as known or given, and the value of others is unknown, and to

be found out ; the former are commonly represented by the

leading letters of the alphabet, , h, r, d, &c. ; the latter by

the final letters, er, x, y, z. Though it often tends to relieve

the memory, if the initial letter of the subject under consi-

deration be made use of, whether that be known or unknown:

thus r may denote a radius, b a base, p a perpendicular, s a

side, d density, m mass, &c.

The characters used to denote the operations, are princi-

pally these :

+ signifies addition, and is named plus.

signifies subtraction, and is named minus,

X denotes multiplication, and is named into.

-f- denotes division, and is named "by.

-^ t tlie mark of radicality denotes the square root; with a

ALGEBRA. 447

3 before it, thus \/, tUe cube root; with a 4, thus ^v^j lli

fourth, or biquadrate root ; thus "v^, the th root.

Proportion is commonly denoted by a colon between tlie

antecedent and consequent of each ratio, and a double colon

between the two ratios: thus, if a be to h as c to J, we state

it as follows, a:h: : C'.d.

:=. is the symbol of equality.

Hence, a + 6 denotes the sum of the quantities represented^

by a and b.

a b denotes their difference when b is the less: fe ,

their difference when is the less : a^b, the difference when

it is not known which is the greater, ax 6, or a . &, or abf

represents the product of a multiplied into b.

a-i-b, or J, shews that the number represented by a is

to be divided by that which is represented by b.

T is the reciprocal of - , and - the reciprocal of a.

a:b: :c:d denotes that a is in the same proportion to b^

as c h to d.

xzza b + c IS an equation, shewing that x is equal to the

difference of a and b, added to the quantity c.

V^ fl, or a , IS the square root of a ; '\/a, or a , is the

X

cube root of a; and "'\/a, or a , is the wth root of a.

d^ is the square of a,- a^ the cube of a; a^ the fourth

power of a ; and a"^ the wth power of a.

a + bxc, or {a + b) c, is the product of the compound

quantity a + 6 multiplied by the simple quantity c. Using

the bar , or the parenthesis ( ), as a vinculum, to con-

nect several quantities into one..

a^b-i-a b, or t> expressed like a fraction, is the quo-

tient of rt + 6 divided by a b.

5a denotes that the quantity a is to be taken 5 times, and

7. {b + c) is 7 times b + c. And these numbers, 5 or 7,

448 MATHEMATICS.

shewing how often the quantities are 'to be taken, or multi-

plied, are called co-efficients.

Like quantities, are those which consist of the same letters,

and powers. As a and 3a ; or a6 and Aab ; or Sarbc and

5a'bc.

Unlike quantities, are those which consist of different letters,

or different powers. As a and b ; or 2a and a^ ; or 3ab^ and

Sabc.

Simple quantities, or monomials, are those which consist of

one term only. As 3a, or-5ab, or Gabc^.

Compound quantities, are those whrdi consist of two or

more terms. As a-\-b, or a -h 26 3c.

And when the compound quantity consists of two terms, it

is called a binomial ; when of three terms, it is a trinomial ;

when of four terms, a quadrinomial ; more than four terms, a

multinomial, or polynomial.

Positive or affirmative quantities, are those which are to be

added, or have the sign + . As a or + a, or ab ; for when

a quantity is found without a sign, it is understood to be posi-

tive, or to have the sign + prefixed.

Negative quantities, are those which are to be subtracted,

as a, or 2a6, or Sab"^.

Like signs, are either all positive ( + ), or all negative ( ).

Unlike signs, are when some are positive ( + ), and others

negative ( )

In every quantity we may consider two things, its value,

and its manner of existing with regard to other magnitudes

which enter with it into the same calculation. The vahie of

a quantity is expressed by the letter or by the character des-

tined to represent the number of its units. But as to the

xnode of existence, with regard to others, some rn^gnitudes may

affect the calculation either in the same or in opposite senses ;

which renders it necessary to distinguish two sorts of quanti-

ties, positive and negative. Thus whether a man have a

thousand pounds in property or stock, or be a thousand pounds

in debt, may be represented by characters, either arithmetical

ALGEBRA. 449

or algebraical ; but since an actual property is directly oppo-

site in its nature to a debt, the two must be marked by dif-

ferent symbols : so that, if property be reckoned a positive

quantity, and marked + , a debt owed must be estimated as

negative, and marked . Again, if, commencing at the

same point, motion towards the east be considered as a posi-

tive quantity in an investigation, motion towards the west,

which is opposite to the former, must enter the same calcula -

tion as a negative quantity. If the elevations of the sun

above the horizon are considered 'as positive quantities, the

depressions of the sun below the horizon must be treated as

negative quantities. It is the same with all quantities^ which^

when considered together, exist differently with respect to one

another.

A residual quantity is a binomial having one of the terms

negative. As a 2b.

The power of a quantity (a), is its square (a^), or cube (a^),

or biquadrate (a?), &c. ; called also the 2d power, or 3d

power, or 4th power, &c.

Tlie index, or exponent, is the number which denotes the

power or root of a. quantity. So 2 is the exponent of the

square or 2d power a^ ; and 3 is the index of the cube or 3d

power ; and 4. is the index of the square root a or ^a ; and

I

J fk the index of the cube root a or ^\/a.

A rational quantity, is that which has no radical sign (-\/)

or index annexed to it. As a, or Sab.

An irrational quantity, or surd, is that which has not an

exact root, or is expressed by means of the radical sign \/.

I

As \/2, or \/a, or ^\/S cr ^

One of the easiest and most simple Introductions to this

science, is that by Fenning, which, if our recollection serves

us, is introduced with an account of fractions by common

numbers. It is many years since we have seeq this work,

anu, perhaps, it is not now to be met with but on stalls, or

VOL. I. 2 G

4d6 mathkMatics.

in second-liand catalogues. In lieu of this, we may notic*

an excellent little treatise by Mr. Bonnycastle, entitled " An

Introduction to Algebra, with Notes and Observations for the

Use of Schools, &c." Tliis compendium is formed upon the

model of larger works, and is intended as an introduction to

them. It supposes, however, that the person making "use of

it, as a first book, on the -subject, has the advantage of a living

instructor to aid him in difficulties that will inevitably occur

to check his progress. To those who have no instructor, we

would recommend a work, which, from a slight view of it

(for it has but lately come into our hands), appears to obviate

all difficulties, by explaining every thing in a full and familiar

manner as fjjr as it goes : it is entitled " The Philosophy of

Arithmetic, considered as a branch of Mathematical Science,

and the Elements of Algebra, &c. by John Walker." This

volume is divided into twenty-eight chapters, of which thirteen

are devoted to the elucidation of the principles of common

arithmetic. Mr. Walker observes, that " the scientific prin-

ciples of common arithmetic are so coincident with those of

algebra, or universal arithmetic, that to persons acquainted

with the former, the Elements of the latter offer no serious

difficulty. Of the Elements of Algebra, therefore, I have

given such a view, as may open that wide field of science to

the student, and enable him, at his pleasure, to extend his

progress, by the aid of any of the larger works extant on the

subject. Having designed this work for the instruction of

those who come to it most uninitiated in science, I have aimed

at giving a clear and full explanation of the most elementary

principles." From the parts that we have examined, it does

appear that the author has succeeded in the accomplishment

of his object. It must, however, be observed, that Mr.

Bonnycastle's " Introduction" includes a number of topics

not touched on even by Mr. Walker. Of these, we may

mention the Diophantine Probtems, and the Summation of

Infinite Series, which b a very important part of some of the

practical mathematics.

ALGEBRA. , 451

" Lectures on the Elements of Algebra," &c. by the Rev.

B. Bridge, A. M. may be safely recommended as a valuable

introduction to the science ; we admit the truth of the author's

assertion, " that the substance pf the Lectures is perfectly

within the comprehension of students at the age of fifteen or

6ixteei>." In some of these lectures, the learner's ingenuity

will, however, be tried ; but*the subjects are interesting, and

worthy the exertion he may be called on to make in the in-

vestigation.

In connexion with any of the above-named works, tha

pupil may read the first part of Maclaurin's Treatise of Al-

gebra, which, tliough deficient in the number of its examples,

is written in a remarkably clear, not to say elegant, style. It

proceeds only to Quadratic Equations, and the doctrine of

Surds.

" The Principles of Algebra," by William Frend, may be?

consulted with advantage, but it cannot be recommended, by

itself, as an introductory work to the science, because we feel

no objection to the usual modes of notation and expression,

which Mr. Frend endeavours to exchange for others, as we

apprehend, not at all-more intelligible. Algebra, like every

branch of real science, has, no doubt, its difficulties ; and the

youth who would make real proficiency in it, either with or

without the aid of a tutor, must, in the first instance, be con-

tent to advance slowly, feel every step of the ground on which

he treads, and fully comprehend every term he may meet

with. To such a one, we are sure there caw be no real ob-

stacles in the use of the algebraical terms which are found in

common books ; nor can he be expected to make much pro-

gress in the science, who is frightened with the words plus,

minus, sines, co-efficient, &c. We have, however, said, that

the learner may consult the " Principles of Algebra" with ad-

vantage ; and we regret that the book is become so scarce as

rarely to be met with. It was published in two parts. The

Jirst proceeds to Cardan's rule for the solution of Equations

2 G 3

452 ^ MATHEMATICS.

of the third order; the second part contains tlie theory of

Equations established on mnlhematical demonstration.

Having gone througli the whole, or the introductory parts

of either of the foregoing elementary books, the student may

take in connexion uith his present pursuits, some parts of

IMr. Thomas Simpson's Algebra, which treats on topics not

to be found in any of the othew. Or he may advance to the

second part of Maclaurin's Algebra, " On the Genius and

Resolution of Equations of all Degrees," &c.

Much curious and valuable mathematical knowledge will

be found in Saunderson's ** Elements of Algebra," in two

volumes, 4to. should they fall in the way of the pupil. Dr.

Saunderson, though blind, was one of the ablest mathema-

ticians of the age. He lost his sight when he was only eight

years old ; yet so great were his talents, and so steady his ap-

plication to the classics and maUiematics, that he could, at an

early period of life, take pleasure in hearing the works of

Euclid, Archimedes, and Diophantus, read in their original

Greek. At the age of twenty-tive, he went to Cambridge,

and his fame soon filled the University. Newton's Principia,

Optics, and Universal Arithmetic, were the foundation of the

lectures which he delivered to the students of that seat of

learning, and they afforded hiih a noble field for the display

of his genius. Great numbers came, some, no doubt, through

motives of curiosity, to hear a blind man give lectures on

optics, discourse on the nature of light and colours, ex-

plain the theory of vision, the phenomenon of the rainbow,

and other objects of sight; but none of liis auditors went

away disappointed; and he always interested, as well as in-

structed, those who came for the purpose of gaining know-

ledge. He succeeded Mr. Whiston in the mathematical pro-

fessor's chair, and, from this time, in 1711, he gavc^ up his

whole time to his pupils, for whose use he had composed

something new and important on almost every branch of the

mathematics. But he discovered no intention to publish any

ALGEBRA. 453

thing, till, by the persuasion of his friends, he prepared his

*' Elements of Algebra" for the press.

Dr. Saunderson had a peculiar meOiod of "performing

arithmetical calculations by nn ingenious machine and me-

thod, which has been denominated his " Palpable Arithmetic,"

and which is particularly described in the first volume of the

work to which we are now directing the reader's attentioQ.

An Abridgment, or Select Parts of Dr. Saunderson's " Ele-

ments of Algebra," was published in an octavo volume, in the

year 1755, which has passed through several editions, the

fourth being printed in 1776. This is a judicious compen-

dium of the larger work, but is not better adapted to learners

than Bonnycastle's, Bridge's, or some other introductions to

the science, of more modern date. For the sake of beginners,

the compiler has prefixed to the Select Parts of Dr. Saunder-

son's Elements, an Introduction to Vulgar and Decimal Frac-

tions, and a collection o"f Arithmetical questions, in order that

the learner may try his skill in common arithmetic before he

enters upon the study of Algebra.

The young algebraist may consult with much advantage

some other books not avowedly elementary, but which con-

tain a large number of excellent problems, the solution of

M'hich will exercfee his ingenuity, and invigorate his powers.

Of these, the first is Dodson's Mathematical Repository, three

volumes, 12mo. 1748. We refer particularly to the first,

and part of the second volume^ the other parts will be no-

ticed hereafter. The early problems of this work are adapted

to those who are but just entering on the science; they in-

crease in difficulty as the pupil is supposed to become stronger

in the pursuit.

Another excellent work of this kind is entitled " Select

Exercises for Young Proficients in Mathematics, &c. by

Thomas Simpson, 1752." Of this volume, the first part con-

tains a number of algebraical problems, with their solutions,

designed as proper exercises for young beginners, in which

454 MATHEMATICS.

the art of managing Equations, and the various methods a(

substitution, are taught and illustrated.

A much more modem work, but one of considerable utility

to the student in Algebra, is the following, " Algebraical Pro-

blems, producing Simple and Quadratic equations, withtheir

solutions, designed as an introduction to the higher branches

of analytics: by the Rev. M. Bland, A. M. \S\2" These

problems, of \\hich there are several hundreds, are designed

solely to point out the various methods employed by Analysts

in the solution of Equations. They are arranged in the usual

manner : the Jirst part containing simple Equations ; the se-

cond, pure Quadratics, and others, that may be solved without

completing the square ; and third/i/, adfected Quadratics.

The author, who has employed much industry and skill in the

compilation of the volume, tells his readers that he has con-

sulted many books, and as utility was the sole object which he

had in view, he has taken his examples from every source,

and has altered them to suit his purpose. At the head of

each section he has given the common rules, so that if the

reader is acquainted with the practice previously to Equations,

Mr. Bland's volume may be considered as a good introduction

to the science at that point. Of the Lady's Diary and Ley-

bourn's Mathematical Repository, we shall s]>eak in our next

chapter.

Mr. Bonnycastle, in 1813, published a much larger work

on this subject, than that which we have already noticed, it is

entitled " A Treatise on Algebra in Practice and Theory," iu

two vols. 8vo. The first volume is devoted chiefly, though in

an extended form, to the same subjects as he had already dis-

cussed in the smaller work ; the second, denominated by the au-

thor " the Theoretical part," will afford much exercise to the

talents and ingenuity of the student, who has already inade

considerable progress in the analytic science, aiid will probably

open to him a new field of speculation. To the first volume

19 prefixed an excellent historical introduction : in the latter

ALGEBRA. 455

jmrt of the second volume is shewn the application of Algebra

to Geometry, and the doctrine of curves.

After, or in connexion with, this work pf Mr. Bonnjcastlc,

may be taken in hand the second volume of the " Elements

of Algebra," by Leonard Euler. This work V4;as pub-

lished in the German language, in 1770, and has since been

translated into the French and English, with notes and addi-

tions by the editors. Among the latter there is a very learned

and copious tract of tlie celebrated La Grange, oit " conti-

nued fractions," anfl such parts of the indeterminate analysis,

as had not been sufficiently treated of by the author; " the ,

whole," says Mr. Bonnycastle, ," formii^ one of the most

profound treatises on this branch of the' subject that has ever

yet appeared."

Before the pupil has arrived at this period of his studies, he

will necessarily be acquainted with almost every tiling that hat

been written on the subject, and will of himself know where

to look for subjects \\hich may engage his attention. We

shall, therefore, only observe, that there is a valuable work in

Xhe Latin language, published at Dublin in 1784, entitled

" Analysis ^quationum, Auctore Guil. Hales. D. D." The

author of which says, that he has endeavoured to follow the

tract of Wallis, Maclaurin, Saunderson, De Moirre, Simp-

son, Clairaut, D'Alerabert, Euler, La Grange, Waring,

Bertrand, Landen, Hutton, &c. " qui aut scriptis Newtoui

illustrandis, aut algebrae limitibus latins proferendis feUcis-

;5ime operam dederunt."

With respect to some of the authors above enumerated, we

may observe, in addition to what we have already noticed, that

M. Clairaut published his " Elemens d'Algebre," in 1746,

in which he made many improvements with reference to the

irreducible case in cubic equations. A fifth edition of this

work was published at Paris, with notes and large additions,

in 1797. M. Landen published his " Residual Analysis" in

1764, his " Mathematical Lucubrations" in 1765, and his

'^ Mathematical Memoirs " in 1 780. The Memoirs of the .

456 MATHEMATICS.

Berlin and Petersburgh Academies abound with improvements

on series and oilier branches of analysis by Euler, La Grange,

and other illustrious mathematicians. Dr. Waring, late of

Cambridge, communicated many valuable papers to the Royal

Society, .who have caused them to be printed in their Trans-

actions, and many of his improvements are contained in his

separate publications, which, it must be acknowledged, are

too abstruse for ordinary mathematicians. These are entitled

** Meditationes Algebraicae," " Proprietates Algebraicarum

Curvarum," and " Meditationes Analyticae." Mr. Baron

Maseres claims to be mentioned not only as an 'original writer

on the analytical branch of Science, but also on account of the

labour and expense which he has bestowed on the publication

of the " Scriptores Logarithmici," in six large vols. 4to.

between the year 1791 and 1807, containing many curious

and useful tracts, which are thus preserved from being lost,

and many valuable papers of his own on the binomial theo-

rem, series, &c. The Baron's separate publications oo

Algebra, are, (1). "A Dissertation on the Negativie Sign in

Algebra." (2). ** Principles of the Doctrine of Life Annu-

ities.'* (3). " Tracts on the Resolution of Affected Algebraic

Equations," &c

CHAP. XXX.

MATHEMATICS,

Continued.

Advantages, History and Province of Geometry Principles of Geometry.

Elementary treatises, Simson's ".Elements" Cunn Tacquet De Chales

Whiston Barrow Simpson's Bonnycastle's Payne's and Cowley'^

Geometry. Matton's Playfair's Leslie's Reynard's ; and Keith's. Ap-

plication of Algebra to Geometry. Simpson Frend Boni)ycastle-^

Lady's Diary, and Leyboum's Mathematical Repository.

JN EXT to Arithmetic should follow Geometry, in a course

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44