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William Shepherd.

Systematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) online

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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."



Online LibraryWilliam ShepherdSystematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) → online text (page 36 of 44)