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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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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



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 37 of 44)