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plates and indices the operations are performed with un-

deviating accuracy. The second, and by far the most cu-

rious instrument, is about half the . size of a common

table writing-desk. By this, problems of almost any extent in

multiplication and division, are solved without the possibility

of a mistake. The multiplier and multiplicand in one instance,

and the divisor and dividend in the other, are first properly

arranged; then, by turning a winch, the product or quotient is

found.

CHAP. XXIX.

MATHEMATICS,

Continued.

Introduction Algebra defined, and the term explained History of Alge.

brai Origin of the Name of the Science from whom derived by

whom improved. Modern authors: Ward Jones Newton. Alge-

braical definitions and symbols. Introdnctory treatises: Penning

Bonnycastle Walker Bridge Frend Simpson Maclanrin Saun-

derson Dodson Bland Euler Hales, &c.

A LTHOUGH a competent knowledge of common arith-

metic should be followed by Algebra, yet the student may,

and, perhaps, ought to combine, with his analytical studies,

a diligent* attention to Geometry, of which we shall shortly

speak more at large. Algebraical and Geometrical studies

should go hand in hand, though for convenience, in this work,

we are obliged to speak of them separately; and algebra being

but a more general kind of arithmetic, we begin with that.

Algebra is a general method of resolving mathematical pro-

blems by means of equations : or it is a method of compu-

tation by symbols, which have been invented for expressing

the quantities that are the objects of this science, and shewing

their mutual relation and dependence. It has been thought that

these quantities were, in the infancy of science, denoted by

their names at full length, which being found inconvenient,

VOL. I. , 2 F

434 MATHEMATICS.

\vere succeeded by abbreviations, or by their initials. At length

certain letters of the alphabet were adopted, as general repre-

sentations of all quantities ; 6ther symbols or signs were intro-

duced to prevent circumlocution, and to facilitate the compa-

rison of various quantities with one another ; and in conse-

quence of the use of letters and other general symbols, algebra

obtained the name of literal or universal arithmetic.

The origin of the name algebra is not easily ascertained :

from its prefix Al, it is supposed to be of Arabic original ;

but its etymology has been variously assigned by different au-

thors. Some have assumed that the word comes from an

Arabic term, to restore, to which was added another, to com-

pare, and hence it was formerly denominated the science of

restitution and comparison ; or resolution and equation. Ac-

cordingly, Lucas de Bur^o, the first European author on

Algebra, calls it the rule of restoration and opposition. Others

have derived it from the word Geber, which was either the

name of a celebrated mathematician to whom the invention

of the science is ascribed ; or from the word Geber, which

forms, with the particle al, the term algebra, signifying a

reductioii of broken numbers or fractions into integers. The

science has, been distinguished by other names besides algebra.

Lucas de Burgo calls it Vurte magiore, or greater art, in

opposition to common arithmetic, which is denominated Parte

mitiore, or the lesser art.

Some writers have defined algebra, as the art of resolving

mathematical problems, but this is rather the idea attached to

analysis, or the analytic art in general, than to algebra, which

is only a particular branch of it. Algebra consists of two

parts, viz. the method of calculating magnitudes or quantities

represented by letters or other characters, and the mode of

applying these calculations to the solution of pr(5blems. Whcu

algebra is applied to the solution of problems, all the quanti-

ties that are involved in the problem are expressed by letters ;

and all the conditions that serve their mutual relation, and by

which they are compared with one another, arc signified by

ALGEBRA. 435

their appropriate characters, and they are tlms thrown into

one or more equations, as the case requires : this is called

synthesis, or composition. When this has been done, the

unknown quantity is disengaged, by a variety of analytical

operations, from those that are known, and brought to stand

alone on one side of the equation, while the known quantities

are on the other side ; xind thus the value of the unknown

quantity is investigated and obtahied. This process is called

analysis, or resolution ; hence algebra is a species of the

analytic art, and is called the modern analysis, in contradis-

tinction to the ancient analysis, which chiefly regarded geome-

try and its application.

The origin of the science of algebra, like the derivation of

the term, is not easily determined. The most ancient treatise,

on that part of analytics properly called algebra, now known,

is that of Diophantus, a Greek author, who flourished in the

fourth century of the Christian aera ; and who, it appears,

wrote thirteen books *' Arithmeticorum," though only six of

them are preserved, which .were printed together with a

single book on multangular numbers, in a Latin translation

by Xylander, in 1575, and afterwards in Greek and Latin,

with a commentaiy, in l621 and I67O, by Gaspar Bachet and

M. Fermat. There must, however, have been other works

on the subject, though unknown to the moderns, because these

do not contain the elements of the science ; they are merely

collections of diflicult questions relating to square and cube

numbers, and other curious properties of numbers, with

their solutions. To have been able to form these questions,

and to have arrived at the solutions, required complete ele-

mentary treatises, well known to the author, and something

similar, no doubt, to existing elementary books, as will appear

from the following considerations.

In his prefatory remarks, addressed to a person by the name

of Dionysius, the author recites the names and generations of

the powers, as the square, cube, biquadrate, &c. and gives

them names according to the sura of the indices of the pow-

2 F S

436 MATHEMATICS.

ers, and he marks those powers with the Greek initials, an6

the unknown quantity he expressed by the word agiG/xo?, or

the number. In treating on the multiplication and division of

simple species, he shews what the product and quotient will

be, observing that minus, multiplied by minus, produces plus ;

and that minus into plus, gives minus. Supposing his reader

acquainted with the common operations of the first four rules^

viz. addition, subtraction, multiplication, and division of com-

pound species, he proceeds to remark on the preparation of the

equations that are deduced from the questions, which is now cal-

led the reduction of equations, by collecting like quantities,

changing the signs of those that are removed from one side of the

equation to the other, which operation is termed by the

moderns transposition, so as to bring the equation to simple

terms. Then depressing it to a lower degree by equal division,

when the powers of the unknown quantity are in every term ;

which reduction of the complex equation being made, the

author proceeds no farther, but merely states tvhat the root is,

without giving any rules for finding it, or for the resolution of

the equations, thus intimating, that rules for this purpose were

to be found in some other work or works well known at the

time, whether they were the productions of his own or another

mind. The great excellence of Diophantus' collection of

questions, which seems to be a series of exercises for rules

M'hich had been elsewhere given, is the ii^at mode of notation

or substitution, which being once made, the reduction to the

final equation is obvious. Tlie work indicates much accurate

knowledge of the science of algebra, but as the author redu-

ces all his notations either to simple equations or simple quad-

ratics, it does not appear how far his knowledge extended to

the resolution of compound or affected equations.

It has, however, been thought tliat algebra was not wholly

unknown to the ancient mathematicians, long before the age

of Diophantus ; and there have been those who observed, or

who thought they observed, the traces of it in many places,

though it seems as if the authors bad intentionally concealed it^

ALGEBRA. 437

Something of this appear^ in Euclid, or at least in his com-

mentator Theon, who says that Plato had begun to teach it.

There are other instances of it in Pappus, and still more in

Archimedes and Apollonius, but it. must be observed, that the

analysis of these authors is rather geometrical than algebraical ;

and hence Diophantus may be considered as the only author

come down to us among the Greeks, who treated profe/ssedly

of algebra.

Our knowledge of the science of algebra was derived, not

from Diophantus, but from the Moors or Arabians, but

whether the Greeks or Arabians were the inventors of it, has

been a subject of dispute, into which we shall not enter. It

is, at any rate probable, that it was much more ancient than

Diophantus, because, as we have seen, his treatise refers to,

and depends upon works similar and prior to his own existing

treatise. But wherever algebra was invented and first cultl-

vaCfed, the science, and also the name of it were transmitted

to Europe, and particularly to Spain, by the Arabians or

Saracens, about the year 1 100, or perhaps somewhat sooner.

Italy took the lead in the cultivation of this science after its

introduction into Europe, and Lucas de Burgo, to whom we

have already referred, a Franciscan friar, was the first author

on the subject, who wrote several treatises between the years

1475 and 1510, btft his principal work, entitled " Summa

Arithmeticae et Geometriae," &c., was published at Venice in

1494, and afterwards in 1.523. In this work he mentions se-

veral writers, and particularly Leonardus Pisanus, placed by

Vossius about the year 1400, and said to have been the first

of the moderns who wrote of algebra, from whom he derived

his knowledge of that science ; and from the treatise of LeOr

nardus, not now extant, the contents of that of Lucas were

chiefly collected. Leonardus was supposed to have flourished

about the end of the fourteenth century, but it is now ascer-

tained from an ancient manuscript, that he lived two hundred

years before this, or at the very commencement of the thir-

teenth century, and of course, that Italy is indebted to hip)

438 MATHEMATICS.

for its first knowledge of algebra^. His proper name was

Bonacci : he was a merchant, who traded in the sea-ports of

Africa and the Levant. Being anxious to obtain an acquaint-

ance with the sciences that were eagerly cultivated among the

Arabians, and particularly that of Algebra, he travelled into

that country, and it should seem from the authority of the ma-

nuscript above referred to, that Leonardus had penetrated

deeply into the secrets of the algebraic analysis ; that he was

particularly acquainted with the analysis of problems similar

in kind to those of Diophantus, and that he had made long

voyages into Arabia, and other Eastern countries, with the ex-

press view of gaining a deeper knowledge of the mathematics.

According to Lucas de Burgo, the knowledge of the Euro-

peans in his time, or about the year 1 500, extended no farther

than quadratic equations, of which they used only the positive

roots ; that they admitted only one unknow n quantity ; that

they had no marks or signs either for quantities or operatiohs,

excepting a few abbreviations of the wor<ls or names ; and

that the art was merely employed in resolving certain numeral

problems. If the science had been carried farther in Africa

thani quadratic equations, which 'was probably the case, as has

been inferred from an Arabic manuscript, said to be on cubic

equations, deposited in the library of Leyden, by Warner, the

Europeans had, at this period, obtained only an imperfect

knowledge of it.

ThepubUcalion of the works of Lucas de Burgo promoted

the study, and extended the knowledge of algebra, so that

early in the sixteenth century, or about the year 1305, Scipio

Ferreus, professor of mathematics at Bononin, discovered the

first rule for resolving one case of a compound cubic equation.

Cardaiius was the next Italian that distinguished himself, by

the cultivation and improvement of algebra : he published nine

books of his arithmetical writings, in 1539, at Milan, where he

practised physic, and read lectures on the mathematics : six

years afterwards he published a tenth book containing the

whole doctrine of cubic equations. Cardan denominates

ALGEBRA. 439

algebra, after Lucas de Burgo and others, " Ars magna," or

" Regulae Algebraica," and ascribes the mvention of it, to

Mahomet, the son of Moses, an Arabian. Dr. Hutton, io

a curious and very elaborate article, Algebra, in his Dic-

tionary, has given a full account of Cardan's treatise, and of

the methods which he took to obtain the discovery of solving

three cases of cubic equations from Tartalea. To this ar-

ticle we refer our readers, premising, however, that it will not

be intelligible to any but those who have made some progress

in the science.

Tartalea, or Tartaglia, of Bressia, was, as we have seen, the

contemporary of Cardan, and published his book of Algebra,

entitled " Quesiti h Invenzioni diverse," in 1546, at Venice,

where he resided as public lecturer in mathematics. This

work was dedicated to Henry VIII. of England, and consists

of nine books, the last of which contains all those questions

that relate to arithmetic and algebra. They comprehend ex-

ercises of simple and quadratic equations, and evince the great

skill of the author in the science of algebra. In 1556, Tartalea

published at Venice, a very large work, in folio, on arithmetic,

geometry, and algebra, the latter of which was incomplete,

going no farther than quadratic equations. Michael Stifelius,

and John Scheubelius, were contemporaries of Tartalea and

Cardan. 1^ " Arithmetica Integra" of the former, is deem-

ed by Dr. Hutton an excellent treatise on arithmetic and al-

gebra. It was printed at Norimberg, in 1544. Stifelius

ascribes the invention of the science to Geber, an Arabian

astronomer.

Scheubelius, professor of mathematics at Tubingen, pub-

lished several treatises on arithmetic and algebra. He is the

first modern algebraist who mentions Diophantus, as the per-

son to whom writers ascribe this art. This is recorded in his

work, entitled " Algebrae Compendiosa facilisque Descriptio,

qua depromuntur magna Arithmetices miracula." Dr. Hutton

has analysed the work, and has ascribed to this, and the

other jiuthors already mentioned, the chief inventions due to

U& MATHEMATICS.

them in perfecting the science, into the minutiae of which ouf

limits do not permit us to enter. Schcubelius treated of only

two orders of tqiiations, viz. the simple and quadratic ; but he

gives the four fundamental rules in the arithmetic of surds.

As he takes no notice of cubic equations, it is probable, that,

though they were known in Italy, this author had not heard

of them in Germany.

Robert Recorde, in England, published the first part of

his arithmetic in 1562, and the second part in 1557, under

the title of the " Whetstone of Witte," &c. The algebra of

Peletarius was printed in Paris in 1558, with the following

title,- " Jacobi Peletarii Cenomani, de Occulte P%rte Nume-

rorum, quam Algebram vocant." Peter Ramus published

his Arithmetic and Algebra about the year 1560. He ex-

presses the powers by /, q, c, b, being the initials of latus,

quadratus, cubus, and biquadratus ; nevertheless, he only treats

of simple and quadratic equations.

In 1567, Nonius, or, as it is often spelt, Nunez, a Portu-

guese, published his Algebra in Spanish, though he says it

had been written in the Portuguese language thirty years be-

fore. Omitting some others, we njay observe, that Simon

Stevinus, of Bruges, published his Algebra very soon after his

Arithmetic, which appeared in 1585, and both were printed

lu an edition of his works in 1634, with notes <ihd additions

bv Albert Girard. About the same time with Stevinus, ap-

peared Francis Vieta, who contributed more to the improve-

ment of algebraic equations than any former author. His

algebraical works were Mritten about the year l6(J(); some of

them were not published tilt after his death in 1603. In

1646, all his mathematical works were collected by Francis

Schooten. Vieta's improvements comprehend, among others,

the following particulars. He introduced the general use of

the letters of the alphabet, to denote indefinite given quanti-

ties, and he expresses the unknown quantities by the vowels,

in capitals. A, E, I, O, II, Y, and the known ones by the

50nsonant<<, B, C, D, &.c. He also invented many terms and

ALGEBRA. 441

forms of expression, which are still in use ; as co-efficient,

affirmative, and negative, pure and adfected, or affected, and

the line, or vinculum, over compound quantities, thus:

A+B+C, - r^ipr*.

Albert Girard, an ingenious Dutch mathematician, the

editor of Sievinus's arithmetic, who died in 1633, deserves

notice on account of his work entitled "Invention Nouvelle

en I'Algebra, tant pour la Solution des Equations, que pour

recoignostre le nombre des Solutions qu'elles regoivent, avec

plusieurs choses qui sont necessaires k la perfection de ceste

divine Science." The next person who claims particular no-

tice, is Thomas Harriot, who died at the age of sixty years, in

3621, and whose Algebra was published by Walter Warner,

in 1631, under the title of " Artis Analytics Praxis, ad

iEquationes Algebraicas nova, expedita, et generali methodo,

resolvendas," a work which, according to Dr. Hutton, shews,

in all its parts, marks of great genius and originality. On the

foundation laid by Harriot, says Dr. AYallis, Pes Cartes,

without naming him, hath built the greatest part, if not the

whole, of his algebra or geometry, without which, he adds,

" that whole superstructure of Des Cartes had never been."

Harriot introduced the uniform use of the small letters, a, b,

c, d, &c. expressing the unknown quantities by the vowels,

and the known ones by the consonants, joining them together

in the form^of a word, to represent the product of any num-

ber of these literal quantities, thus a b c signified that a, A,

and c, were multiplied together. Oughtred, contemporary

with Harlot, was born in the year 1573, and died in l660.

His " Clavis" was published in 1631; he chiefly follows

Vieta, in the notation by capitals. This author, in algebraical

multiplication, either joins the letters in the form of a word,

or connects them with the sign x , introducing, for the first

time, this character of multiplication ; thus the three terms

A X A, or A A, or Aq, meant the same quantity. He intro-

duced many useful contractions in the multiplication and di-

vision of decimals, and he used the following form for the

442 MATHEMATICS.

terms of proportion 7 . 9 : : 28 . 36, which is nearly the same

as that now in use.

Des Cartes published his Geometry in lC37, which may

be considered as an appHcation of algebra to geometry, and

not as a separate treatise on either of these sciences. His

inventions and discoveries comprehend the application of al-

gebra to the geometry of curve lines, the construction of equa-

tions of the higher orders, and a rule for resolving biquadratic

equations, by means of a cubic and two quadratics. With a

view to the more easy application of equations to the con-

struction of problems, Des Cartes mentions many particulars

concerning the nature and reduction of equations, states

them in his own language and manner, and frequently ac-

companied with his own improvements. Here he chiefly fol-

lowed Cardan, Viefa, and Harriot, and especially the last,

explaining some of their rules aiid^ discoveries more distinctly,

with variations in the notation, in which he puts the first

letters of the alphabet, a, b, c, d, &c. for known quantities,

and the latter lettei-s, m, x, y, z, for unknown quantities.

Fermat, who published Diophantus' Arithmetic, with va-

luable notes, was a contemporary of Des Cartes, and a com-

petitor for some of his most valuable discoveries. He had,

before the publication of Des Cartes' Geometry, applied al-

gebra to curve lines, and had discovered a method of tangents,

and a method of maximis et minimis, approaching very nearly

to the method of fluxions; to be noticed hereafter. At this

period, Algebra had acquired a regular and permanent form,

and from this time, the writers on the whole, or detached

parts of the science, became so numerous, that it would be

impossible in a short article to enumerate all their works ;

we shall accordingly only mention those that appear the niost

prominent.

The Geometry of Des Cartes engaged the attention of

several mathematicians in Holland, where it was first pub-

lished, and also in France and England. Francis Schooten,

professor at Leyden, was one of tbfe earliest cultivators of the

ALGEBRA. 443

new geometry; and, in 1649, le published a translation of

Des Cartes' work into the Latin language. Huygens, illus-

trious for his many discoveries in mechanics, directed his at-

tention to the algebraic analysis ; and, among other works, he

published a short piece, entitled " De Ratiociniis in Ludo

Alae," in order to shew the usefulness of algebra. Herigone,

in 1634, published at Paris the first course of mathematics,

in five volumes, 8vo. containing a treatise on algebra, which,

according to Dr. Hutton, bears evident marks of originality

and ingenuity. Cavalerius, in the following year, published

his " lodivisibles," which introduced a new aera into analytical

science, and new modes of computation. In 1655, Dr.

Wallis published his. " Arithmetica Infinitorum," which was

a great improvement on the Indivisibles of Cavalerius, and

led the way to infinite series, the binomial theorem, and the

method of fluxions.

Mr. Kinckhuysen, in I66I, published a treatise of algebra

in the Dutch language, which Sir Isaac Newton, when pro-

fessor of mathematics at Cambridge, used and improved, and

which he designed to republish, with his method of fluxions

and infinite series, but was prevented by the accidental burn-

ing of some of his papers. In 1665, or I666, Sir Isaac

Newton made several of his most valuable discoveries, though

they were not published till a later period ; such as the binor

mial theorem, the method of fluxions and infinite series, the

Quadrature, rectification, &c. of curves, the investigation of

the roots of all sorts of equations, both numeral and literal, in

infinite converging series, the reversion of series, &g.

The " Elements of Algebra" were published by John

Kersey, in 1675, in two volumes, folio, containing the illus-

tration of the science, and tlie nature of equations, the expli-

cation of Diophantus' problems, and many additions concern-

ing mathematical composition and resolution. This work is

thought by Dr. Hutton to be ample and complete. Dr.

Wallis's Treatise of Algebra, both historical and practical,

shewing the origin, progress, and advancement of it $rom

444 MATHEMATICS.

time to time, was published in 1685, is folio. In 1687, Dr,

Halley communicated in the Philosophical Transactions the

construction of cubic and biquadratic equations, by a parabola

nd circle, with improvements of the methods of Des Cartes,

Baker, &c. ; and a memoir on the number of the roots of

equations, with their limits and signs.

Mr. John Ward published in 1695 " A Compendium of

Algebra;" and, in 1706, the first edition of ".The Young

Mathematician's Guide," which includes Arithmetic, vulgar

and decimal; Algebra; the Elements of Geometry; Conic

Sections; the Arithmetic of Infinites; and an Appeofiix of

Practical Gauging. This work was, a century ago, extremely

popular, and passed through several editions ; in the Preface

to tlie fifth edition, in 1722, the author says, that his " book

has answered to its title so well, that I believe I may truly

say, without vanity, that this treatise hath proved a very help>

fjul guide to nearly five thousand persons, and perhaps most

of them such as would never have looked into the Mathe-

matics at all but for it." In describing his plan, Mr. Ward

says, " I began with an unit in Arithmetic, and a point in

Geometry; and, from these foundations, proceeded gradually

on, leading the young learner, step by step, with all the plain-

ness possible."

deviating accuracy. The second, and by far the most cu-

rious instrument, is about half the . size of a common

table writing-desk. By this, problems of almost any extent in

multiplication and division, are solved without the possibility

of a mistake. The multiplier and multiplicand in one instance,

and the divisor and dividend in the other, are first properly

arranged; then, by turning a winch, the product or quotient is

found.

CHAP. XXIX.

MATHEMATICS,

Continued.

Introduction Algebra defined, and the term explained History of Alge.

brai Origin of the Name of the Science from whom derived by

whom improved. Modern authors: Ward Jones Newton. Alge-

braical definitions and symbols. Introdnctory treatises: Penning

Bonnycastle Walker Bridge Frend Simpson Maclanrin Saun-

derson Dodson Bland Euler Hales, &c.

A LTHOUGH a competent knowledge of common arith-

metic should be followed by Algebra, yet the student may,

and, perhaps, ought to combine, with his analytical studies,

a diligent* attention to Geometry, of which we shall shortly

speak more at large. Algebraical and Geometrical studies

should go hand in hand, though for convenience, in this work,

we are obliged to speak of them separately; and algebra being

but a more general kind of arithmetic, we begin with that.

Algebra is a general method of resolving mathematical pro-

blems by means of equations : or it is a method of compu-

tation by symbols, which have been invented for expressing

the quantities that are the objects of this science, and shewing

their mutual relation and dependence. It has been thought that

these quantities were, in the infancy of science, denoted by

their names at full length, which being found inconvenient,

VOL. I. , 2 F

434 MATHEMATICS.

\vere succeeded by abbreviations, or by their initials. At length

certain letters of the alphabet were adopted, as general repre-

sentations of all quantities ; 6ther symbols or signs were intro-

duced to prevent circumlocution, and to facilitate the compa-

rison of various quantities with one another ; and in conse-

quence of the use of letters and other general symbols, algebra

obtained the name of literal or universal arithmetic.

The origin of the name algebra is not easily ascertained :

from its prefix Al, it is supposed to be of Arabic original ;

but its etymology has been variously assigned by different au-

thors. Some have assumed that the word comes from an

Arabic term, to restore, to which was added another, to com-

pare, and hence it was formerly denominated the science of

restitution and comparison ; or resolution and equation. Ac-

cordingly, Lucas de Bur^o, the first European author on

Algebra, calls it the rule of restoration and opposition. Others

have derived it from the word Geber, which was either the

name of a celebrated mathematician to whom the invention

of the science is ascribed ; or from the word Geber, which

forms, with the particle al, the term algebra, signifying a

reductioii of broken numbers or fractions into integers. The

science has, been distinguished by other names besides algebra.

Lucas de Burgo calls it Vurte magiore, or greater art, in

opposition to common arithmetic, which is denominated Parte

mitiore, or the lesser art.

Some writers have defined algebra, as the art of resolving

mathematical problems, but this is rather the idea attached to

analysis, or the analytic art in general, than to algebra, which

is only a particular branch of it. Algebra consists of two

parts, viz. the method of calculating magnitudes or quantities

represented by letters or other characters, and the mode of

applying these calculations to the solution of pr(5blems. Whcu

algebra is applied to the solution of problems, all the quanti-

ties that are involved in the problem are expressed by letters ;

and all the conditions that serve their mutual relation, and by

which they are compared with one another, arc signified by

ALGEBRA. 435

their appropriate characters, and they are tlms thrown into

one or more equations, as the case requires : this is called

synthesis, or composition. When this has been done, the

unknown quantity is disengaged, by a variety of analytical

operations, from those that are known, and brought to stand

alone on one side of the equation, while the known quantities

are on the other side ; xind thus the value of the unknown

quantity is investigated and obtahied. This process is called

analysis, or resolution ; hence algebra is a species of the

analytic art, and is called the modern analysis, in contradis-

tinction to the ancient analysis, which chiefly regarded geome-

try and its application.

The origin of the science of algebra, like the derivation of

the term, is not easily determined. The most ancient treatise,

on that part of analytics properly called algebra, now known,

is that of Diophantus, a Greek author, who flourished in the

fourth century of the Christian aera ; and who, it appears,

wrote thirteen books *' Arithmeticorum," though only six of

them are preserved, which .were printed together with a

single book on multangular numbers, in a Latin translation

by Xylander, in 1575, and afterwards in Greek and Latin,

with a commentaiy, in l621 and I67O, by Gaspar Bachet and

M. Fermat. There must, however, have been other works

on the subject, though unknown to the moderns, because these

do not contain the elements of the science ; they are merely

collections of diflicult questions relating to square and cube

numbers, and other curious properties of numbers, with

their solutions. To have been able to form these questions,

and to have arrived at the solutions, required complete ele-

mentary treatises, well known to the author, and something

similar, no doubt, to existing elementary books, as will appear

from the following considerations.

In his prefatory remarks, addressed to a person by the name

of Dionysius, the author recites the names and generations of

the powers, as the square, cube, biquadrate, &c. and gives

them names according to the sura of the indices of the pow-

2 F S

436 MATHEMATICS.

ers, and he marks those powers with the Greek initials, an6

the unknown quantity he expressed by the word agiG/xo?, or

the number. In treating on the multiplication and division of

simple species, he shews what the product and quotient will

be, observing that minus, multiplied by minus, produces plus ;

and that minus into plus, gives minus. Supposing his reader

acquainted with the common operations of the first four rules^

viz. addition, subtraction, multiplication, and division of com-

pound species, he proceeds to remark on the preparation of the

equations that are deduced from the questions, which is now cal-

led the reduction of equations, by collecting like quantities,

changing the signs of those that are removed from one side of the

equation to the other, which operation is termed by the

moderns transposition, so as to bring the equation to simple

terms. Then depressing it to a lower degree by equal division,

when the powers of the unknown quantity are in every term ;

which reduction of the complex equation being made, the

author proceeds no farther, but merely states tvhat the root is,

without giving any rules for finding it, or for the resolution of

the equations, thus intimating, that rules for this purpose were

to be found in some other work or works well known at the

time, whether they were the productions of his own or another

mind. The great excellence of Diophantus' collection of

questions, which seems to be a series of exercises for rules

M'hich had been elsewhere given, is the ii^at mode of notation

or substitution, which being once made, the reduction to the

final equation is obvious. Tlie work indicates much accurate

knowledge of the science of algebra, but as the author redu-

ces all his notations either to simple equations or simple quad-

ratics, it does not appear how far his knowledge extended to

the resolution of compound or affected equations.

It has, however, been thought tliat algebra was not wholly

unknown to the ancient mathematicians, long before the age

of Diophantus ; and there have been those who observed, or

who thought they observed, the traces of it in many places,

though it seems as if the authors bad intentionally concealed it^

ALGEBRA. 437

Something of this appear^ in Euclid, or at least in his com-

mentator Theon, who says that Plato had begun to teach it.

There are other instances of it in Pappus, and still more in

Archimedes and Apollonius, but it. must be observed, that the

analysis of these authors is rather geometrical than algebraical ;

and hence Diophantus may be considered as the only author

come down to us among the Greeks, who treated profe/ssedly

of algebra.

Our knowledge of the science of algebra was derived, not

from Diophantus, but from the Moors or Arabians, but

whether the Greeks or Arabians were the inventors of it, has

been a subject of dispute, into which we shall not enter. It

is, at any rate probable, that it was much more ancient than

Diophantus, because, as we have seen, his treatise refers to,

and depends upon works similar and prior to his own existing

treatise. But wherever algebra was invented and first cultl-

vaCfed, the science, and also the name of it were transmitted

to Europe, and particularly to Spain, by the Arabians or

Saracens, about the year 1 100, or perhaps somewhat sooner.

Italy took the lead in the cultivation of this science after its

introduction into Europe, and Lucas de Burgo, to whom we

have already referred, a Franciscan friar, was the first author

on the subject, who wrote several treatises between the years

1475 and 1510, btft his principal work, entitled " Summa

Arithmeticae et Geometriae," &c., was published at Venice in

1494, and afterwards in 1.523. In this work he mentions se-

veral writers, and particularly Leonardus Pisanus, placed by

Vossius about the year 1400, and said to have been the first

of the moderns who wrote of algebra, from whom he derived

his knowledge of that science ; and from the treatise of LeOr

nardus, not now extant, the contents of that of Lucas were

chiefly collected. Leonardus was supposed to have flourished

about the end of the fourteenth century, but it is now ascer-

tained from an ancient manuscript, that he lived two hundred

years before this, or at the very commencement of the thir-

teenth century, and of course, that Italy is indebted to hip)

438 MATHEMATICS.

for its first knowledge of algebra^. His proper name was

Bonacci : he was a merchant, who traded in the sea-ports of

Africa and the Levant. Being anxious to obtain an acquaint-

ance with the sciences that were eagerly cultivated among the

Arabians, and particularly that of Algebra, he travelled into

that country, and it should seem from the authority of the ma-

nuscript above referred to, that Leonardus had penetrated

deeply into the secrets of the algebraic analysis ; that he was

particularly acquainted with the analysis of problems similar

in kind to those of Diophantus, and that he had made long

voyages into Arabia, and other Eastern countries, with the ex-

press view of gaining a deeper knowledge of the mathematics.

According to Lucas de Burgo, the knowledge of the Euro-

peans in his time, or about the year 1 500, extended no farther

than quadratic equations, of which they used only the positive

roots ; that they admitted only one unknow n quantity ; that

they had no marks or signs either for quantities or operatiohs,

excepting a few abbreviations of the wor<ls or names ; and

that the art was merely employed in resolving certain numeral

problems. If the science had been carried farther in Africa

thani quadratic equations, which 'was probably the case, as has

been inferred from an Arabic manuscript, said to be on cubic

equations, deposited in the library of Leyden, by Warner, the

Europeans had, at this period, obtained only an imperfect

knowledge of it.

ThepubUcalion of the works of Lucas de Burgo promoted

the study, and extended the knowledge of algebra, so that

early in the sixteenth century, or about the year 1305, Scipio

Ferreus, professor of mathematics at Bononin, discovered the

first rule for resolving one case of a compound cubic equation.

Cardaiius was the next Italian that distinguished himself, by

the cultivation and improvement of algebra : he published nine

books of his arithmetical writings, in 1539, at Milan, where he

practised physic, and read lectures on the mathematics : six

years afterwards he published a tenth book containing the

whole doctrine of cubic equations. Cardan denominates

ALGEBRA. 439

algebra, after Lucas de Burgo and others, " Ars magna," or

" Regulae Algebraica," and ascribes the mvention of it, to

Mahomet, the son of Moses, an Arabian. Dr. Hutton, io

a curious and very elaborate article, Algebra, in his Dic-

tionary, has given a full account of Cardan's treatise, and of

the methods which he took to obtain the discovery of solving

three cases of cubic equations from Tartalea. To this ar-

ticle we refer our readers, premising, however, that it will not

be intelligible to any but those who have made some progress

in the science.

Tartalea, or Tartaglia, of Bressia, was, as we have seen, the

contemporary of Cardan, and published his book of Algebra,

entitled " Quesiti h Invenzioni diverse," in 1546, at Venice,

where he resided as public lecturer in mathematics. This

work was dedicated to Henry VIII. of England, and consists

of nine books, the last of which contains all those questions

that relate to arithmetic and algebra. They comprehend ex-

ercises of simple and quadratic equations, and evince the great

skill of the author in the science of algebra. In 1556, Tartalea

published at Venice, a very large work, in folio, on arithmetic,

geometry, and algebra, the latter of which was incomplete,

going no farther than quadratic equations. Michael Stifelius,

and John Scheubelius, were contemporaries of Tartalea and

Cardan. 1^ " Arithmetica Integra" of the former, is deem-

ed by Dr. Hutton an excellent treatise on arithmetic and al-

gebra. It was printed at Norimberg, in 1544. Stifelius

ascribes the invention of the science to Geber, an Arabian

astronomer.

Scheubelius, professor of mathematics at Tubingen, pub-

lished several treatises on arithmetic and algebra. He is the

first modern algebraist who mentions Diophantus, as the per-

son to whom writers ascribe this art. This is recorded in his

work, entitled " Algebrae Compendiosa facilisque Descriptio,

qua depromuntur magna Arithmetices miracula." Dr. Hutton

has analysed the work, and has ascribed to this, and the

other jiuthors already mentioned, the chief inventions due to

U& MATHEMATICS.

them in perfecting the science, into the minutiae of which ouf

limits do not permit us to enter. Schcubelius treated of only

two orders of tqiiations, viz. the simple and quadratic ; but he

gives the four fundamental rules in the arithmetic of surds.

As he takes no notice of cubic equations, it is probable, that,

though they were known in Italy, this author had not heard

of them in Germany.

Robert Recorde, in England, published the first part of

his arithmetic in 1562, and the second part in 1557, under

the title of the " Whetstone of Witte," &c. The algebra of

Peletarius was printed in Paris in 1558, with the following

title,- " Jacobi Peletarii Cenomani, de Occulte P%rte Nume-

rorum, quam Algebram vocant." Peter Ramus published

his Arithmetic and Algebra about the year 1560. He ex-

presses the powers by /, q, c, b, being the initials of latus,

quadratus, cubus, and biquadratus ; nevertheless, he only treats

of simple and quadratic equations.

In 1567, Nonius, or, as it is often spelt, Nunez, a Portu-

guese, published his Algebra in Spanish, though he says it

had been written in the Portuguese language thirty years be-

fore. Omitting some others, we njay observe, that Simon

Stevinus, of Bruges, published his Algebra very soon after his

Arithmetic, which appeared in 1585, and both were printed

lu an edition of his works in 1634, with notes <ihd additions

bv Albert Girard. About the same time with Stevinus, ap-

peared Francis Vieta, who contributed more to the improve-

ment of algebraic equations than any former author. His

algebraical works were Mritten about the year l6(J(); some of

them were not published tilt after his death in 1603. In

1646, all his mathematical works were collected by Francis

Schooten. Vieta's improvements comprehend, among others,

the following particulars. He introduced the general use of

the letters of the alphabet, to denote indefinite given quanti-

ties, and he expresses the unknown quantities by the vowels,

in capitals. A, E, I, O, II, Y, and the known ones by the

50nsonant<<, B, C, D, &.c. He also invented many terms and

ALGEBRA. 441

forms of expression, which are still in use ; as co-efficient,

affirmative, and negative, pure and adfected, or affected, and

the line, or vinculum, over compound quantities, thus:

A+B+C, - r^ipr*.

Albert Girard, an ingenious Dutch mathematician, the

editor of Sievinus's arithmetic, who died in 1633, deserves

notice on account of his work entitled "Invention Nouvelle

en I'Algebra, tant pour la Solution des Equations, que pour

recoignostre le nombre des Solutions qu'elles regoivent, avec

plusieurs choses qui sont necessaires k la perfection de ceste

divine Science." The next person who claims particular no-

tice, is Thomas Harriot, who died at the age of sixty years, in

3621, and whose Algebra was published by Walter Warner,

in 1631, under the title of " Artis Analytics Praxis, ad

iEquationes Algebraicas nova, expedita, et generali methodo,

resolvendas," a work which, according to Dr. Hutton, shews,

in all its parts, marks of great genius and originality. On the

foundation laid by Harriot, says Dr. AYallis, Pes Cartes,

without naming him, hath built the greatest part, if not the

whole, of his algebra or geometry, without which, he adds,

" that whole superstructure of Des Cartes had never been."

Harriot introduced the uniform use of the small letters, a, b,

c, d, &c. expressing the unknown quantities by the vowels,

and the known ones by the consonants, joining them together

in the form^of a word, to represent the product of any num-

ber of these literal quantities, thus a b c signified that a, A,

and c, were multiplied together. Oughtred, contemporary

with Harlot, was born in the year 1573, and died in l660.

His " Clavis" was published in 1631; he chiefly follows

Vieta, in the notation by capitals. This author, in algebraical

multiplication, either joins the letters in the form of a word,

or connects them with the sign x , introducing, for the first

time, this character of multiplication ; thus the three terms

A X A, or A A, or Aq, meant the same quantity. He intro-

duced many useful contractions in the multiplication and di-

vision of decimals, and he used the following form for the

442 MATHEMATICS.

terms of proportion 7 . 9 : : 28 . 36, which is nearly the same

as that now in use.

Des Cartes published his Geometry in lC37, which may

be considered as an appHcation of algebra to geometry, and

not as a separate treatise on either of these sciences. His

inventions and discoveries comprehend the application of al-

gebra to the geometry of curve lines, the construction of equa-

tions of the higher orders, and a rule for resolving biquadratic

equations, by means of a cubic and two quadratics. With a

view to the more easy application of equations to the con-

struction of problems, Des Cartes mentions many particulars

concerning the nature and reduction of equations, states

them in his own language and manner, and frequently ac-

companied with his own improvements. Here he chiefly fol-

lowed Cardan, Viefa, and Harriot, and especially the last,

explaining some of their rules aiid^ discoveries more distinctly,

with variations in the notation, in which he puts the first

letters of the alphabet, a, b, c, d, &c. for known quantities,

and the latter lettei-s, m, x, y, z, for unknown quantities.

Fermat, who published Diophantus' Arithmetic, with va-

luable notes, was a contemporary of Des Cartes, and a com-

petitor for some of his most valuable discoveries. He had,

before the publication of Des Cartes' Geometry, applied al-

gebra to curve lines, and had discovered a method of tangents,

and a method of maximis et minimis, approaching very nearly

to the method of fluxions; to be noticed hereafter. At this

period, Algebra had acquired a regular and permanent form,

and from this time, the writers on the whole, or detached

parts of the science, became so numerous, that it would be

impossible in a short article to enumerate all their works ;

we shall accordingly only mention those that appear the niost

prominent.

The Geometry of Des Cartes engaged the attention of

several mathematicians in Holland, where it was first pub-

lished, and also in France and England. Francis Schooten,

professor at Leyden, was one of tbfe earliest cultivators of the

ALGEBRA. 443

new geometry; and, in 1649, le published a translation of

Des Cartes' work into the Latin language. Huygens, illus-

trious for his many discoveries in mechanics, directed his at-

tention to the algebraic analysis ; and, among other works, he

published a short piece, entitled " De Ratiociniis in Ludo

Alae," in order to shew the usefulness of algebra. Herigone,

in 1634, published at Paris the first course of mathematics,

in five volumes, 8vo. containing a treatise on algebra, which,

according to Dr. Hutton, bears evident marks of originality

and ingenuity. Cavalerius, in the following year, published

his " lodivisibles," which introduced a new aera into analytical

science, and new modes of computation. In 1655, Dr.

Wallis published his. " Arithmetica Infinitorum," which was

a great improvement on the Indivisibles of Cavalerius, and

led the way to infinite series, the binomial theorem, and the

method of fluxions.

Mr. Kinckhuysen, in I66I, published a treatise of algebra

in the Dutch language, which Sir Isaac Newton, when pro-

fessor of mathematics at Cambridge, used and improved, and

which he designed to republish, with his method of fluxions

and infinite series, but was prevented by the accidental burn-

ing of some of his papers. In 1665, or I666, Sir Isaac

Newton made several of his most valuable discoveries, though

they were not published till a later period ; such as the binor

mial theorem, the method of fluxions and infinite series, the

Quadrature, rectification, &c. of curves, the investigation of

the roots of all sorts of equations, both numeral and literal, in

infinite converging series, the reversion of series, &g.

The " Elements of Algebra" were published by John

Kersey, in 1675, in two volumes, folio, containing the illus-

tration of the science, and tlie nature of equations, the expli-

cation of Diophantus' problems, and many additions concern-

ing mathematical composition and resolution. This work is

thought by Dr. Hutton to be ample and complete. Dr.

Wallis's Treatise of Algebra, both historical and practical,

shewing the origin, progress, and advancement of it $rom

444 MATHEMATICS.

time to time, was published in 1685, is folio. In 1687, Dr,

Halley communicated in the Philosophical Transactions the

construction of cubic and biquadratic equations, by a parabola

nd circle, with improvements of the methods of Des Cartes,

Baker, &c. ; and a memoir on the number of the roots of

equations, with their limits and signs.

Mr. John Ward published in 1695 " A Compendium of

Algebra;" and, in 1706, the first edition of ".The Young

Mathematician's Guide," which includes Arithmetic, vulgar

and decimal; Algebra; the Elements of Geometry; Conic

Sections; the Arithmetic of Infinites; and an Appeofiix of

Practical Gauging. This work was, a century ago, extremely

popular, and passed through several editions ; in the Preface

to tlie fifth edition, in 1722, the author says, that his " book

has answered to its title so well, that I believe I may truly

say, without vanity, that this treatise hath proved a very help>

fjul guide to nearly five thousand persons, and perhaps most

of them such as would never have looked into the Mathe-

matics at all but for it." In describing his plan, Mr. Ward

says, " I began with an unit in Arithmetic, and a point in

Geometry; and, from these foundations, proceeded gradually

on, leading the young learner, step by step, with all the plain-

ness possible."

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