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of so many eminent authors, and was the common depository
of every species of human knowledge, was entirely devoted to
the flames by the Arabs. From tliis period, several ages fol-
lowed of the most wretched ignorance and barbarism, so that
it seems wonderful that the sciences ever recovered the deadly
blow ; but, as has been mentioned, some philosophers of
Alexandria escaped the vengeance of their barbarous con-
querors, and these of course carried with them a remnant of
that general learning for which this school was so justly cele-
brated. But, destitute of books, of instruments, and almost
the means of existence without manual labour, very little far-
ther knowledge could have been accumulated, and still less
propagated ; so that philosophy and mathematics must have
become extinct, had not the Arabians themselves, within less
than two centuries from this fatal catastrophe, become the ad-
mirers and supporters of those very sciences which they, aft,
at least, their ancestors, had so nearly annihilated.

Of all the branches of mathematics, astronomy was that



ARITHMETIC. 4gt

which the Arabs held in the greatest esttffiatidn, though they
did not wholly neglect the others. Our present system of
arithmetic is derived from them, though they were probably
not the inventors, but had acquired their knowledge of it from
the Indians. It is not in our power to enumerate all the
great men among the Arabians that appeared from the period
of which we are now speaking, to the beginning of the thir-
teenth century. Among the Arabian princes, who were also
men of science, we may mention Almansor, who flou-
rished about the year 754 ; Al-Maimon, who reigned from
813 to 833, in whose time, in consequence of the great sup-
port and assistance which he afforded the sciences, they made
very considerable progress. Alfragan, Thebit, and Albategni,
were particularly distinguished about this period. Thebit was
a considerable algebraist, geometrician, and astronomer ; Al-
fragan composed elements of the latter branch of science, of
which several editions have been published since the inven-
tion of printing ; and Albategni, in consequence of his nume-
rous and important observations, and very accurate knowledge,
was sumamed the Arabian Ptolemy. About the year 1100,
Alhazen, a very celebrated Arab, settled in Spain, and com^*
posed a treatise on optics ; and to him we are indebted for
the first theory of refraction and twilight.

Thus have we given a brief sketch of the first rise, the de-
cline, and revival of the mathematical sciences. After this
period, they began to be propagated in several European
countries, and have continued to be cultivated and improved
from that to the present time. The reader is referred to Mon-
tucla's History of Mathematics, for a complete account of
almost every thing that is valuable and deserving of notice on
the subject. Montucla's History was first published in two
volumes, 4to. but it was afterwards continued by Lalande,
making, in the whole, four volumes.

ARITHMETIC.
Arithmetic^ or the art of numbering, is that part of ma-



422 MATHEMATICS.

thematics which considers the powers of numbers, and teaches
how to compute or calculate truly, and with ease and expedi-
tion. The great leading rules or operations, after we have
learnt to name the several figures and combinations of figures,
are addition, subtraction, multiplication, and division. Be*
sides these, many other useful rules have been contrived, for
the purpose of facilitating and expediting computations, mer-
cantile and astronomical ; and which afford the most salutary
aid to the philosopher, the man of business, and the political
economist.

We know very little respecting the origin and invention of
arithmetic ; like other branches of knowledge connected with
the interests of society, it probably originated in the wants of
man, and at first proceeded very slowly toward that point of
perfection to which it has long since arrived. History fixe*
neither the author nor the time when its elementary principles
were discovered, or, perhaps, to speak more properly, con-
trived. Some knowledge of numbers must have existed
in the earliest ages of mankind, which was probably sug-
gested to them by tlieir own fingers or toes, by their flocks
and herds, and by a variety of objects that surrounded them.
Their powers of numeration would, at first, have been very
limited, and before the art of writing was invented, it must
have depended either on memory, or on such artificial helps
as might be most easily obtained.

The introduction of arithmetic as a science, and the im-
provements it underwent, must, in a great degree, have de-
pended upon the introduction and establishment of commerce;
and as that was extended and improved, and other sciences
were discovered and cultivated, arithmetic would be improved
likewise. Hence it has been inferred, that arithmetic was
a Tyrian invention, or, at least, that it was much in-
debted to the Tyrians or Phoenicians for its progress in the
world. Proclus, in his Commentary on the first book of
Euclid, says, the Phoenicians, by reason of their great traffic
and commerce, were the inventors of arithmetic. This also



ARITHMETIC. 423

M'as the opmion of Strabo. Others, however, have traced the
origin of this art to Eg\'pt, and they say tliat Theut, or Thot,
was the inventor of numbers, that from thence the Greeks
adopted the idea of ascribing to their Mercury, correspond-
ing to the Egyptian Theut, or Hermes, the superintendence
of commerce and arithmetic.

Josephus affirms, that arithmetic passed from Asia into
Egypt, where it was cultivated and improved, so much so,
that a considerahie part of Egyptian philosophy, and dieology
likewise, seems to have turned altogether upon numbers. Atha-
nasius Kircher shews that the Egyptians explained every thing
by numbers. So highly was the art of numbering or arith-
metic esteemed by the ancients, that Pythagoras affirmed
that the nature of numbers pervades the whole universe, and
that the knowledge of numbers is the knowledge of tiie Deity
himself.

From Egypt, arithmetic was transmitted to the Greeks by
Pytliagoras and his followers, and among them it was the sub-
ject of particular attention, as is evident from the writings of
Archimedes and other illustrious philosophers ; and with the
improvements which it derived from them, it passed to the
Romans, and from rfiem it came to us.

"Rie ancient arithmetic was very different from that of the
modems, in several respects, and particularly in their method
of notation. It has been thought that the Greeks and Ro-
mans at first used pebbles in their calculations, and, in proof
of this, we find the Greek and Latin words signifying calcu-
lation, are derived from the words ^fi(^a^ and calculus, a little
stone; and the Greek verb ^^(pi^u signifies, among other
meanings, to calculate. In confirmation of this theory, it is
observed, that the Indians are at this time very e'>cpert in com-
puting by means of their fingers, without the use of pen and
ink ; and that the natives of Peru, by different arrangements of
their maize, surpass the European, aided by all his rules, both
in accuracy and despatch.

" It is easy to imagine," says the author of the " Origin of



424 MATHEMATICS.

Laws, Arts, and Sciences," p. 218, Sec. vol.i. "how, by the
help of tlieir fingers and a few little stones, men might per-
form considerable calculations. If it is demanded how these
primitive arithmeticians managed when they were to count a
great number of objects, which obliged them several times to
recommence the decimal numeration, I answer, that it seems
probable they marked tens of units by one symbol, and hun-
dreds by another. Perhaps they expressed tens by white
stones, and hundreds by stones of another colour. After this
discovery, it would not be difficult to contrive symbols for ex-
pressing tens of hundreds, or thousands, &,c. &c.

" Perhaps the first arithmeticians might make use of sym-
bols of the same colour to express tens, hundreds, &c. only
observing to place them so with regard to one another, as to
determine their relative value, as we do with our cyphers,
which have different values, according to the rank they hold,
and the place they occupy. By such means, mankind might
carry the practice of numeration further than their necessities
and way of life required. The invention of these methods of
numeration would naturally lead men to the knowledge of ad-
dition. As soon as they knew how to number with facility
a collection of objects, however great, it would require no
great effort to number several of these together, that is, to
add them. They had nothing to do but to place the symbols
of their several numbers under one another, so as to have
their units, their tens, and their hundreds, 8cc. under their eye
at once, and then to reduce all these several symbols into one.
They would not be long in discovering the art of performing
this reduction. They had only to sum up separately first
their units, then their tens, and then their hundreds, and to
form the symbol of each of these sums as they discovered
them ; to do that, in a word, by parts, which the weakness of
our faculties will not permit us to do at once.

" It was not difficult to proceed from the practice of nu-
meration to addition, it was still more easy to find out the art
of multiplying one number by another. Tnere is reason to



ARITHMETIC. ^

think, that multiplication was at first performed by means of
addition. The steps of the human mind are naturally slow.
It requires no little time and labour to pass the medium
which divides one part of science from another, however ana-
logous they may seem to be. At first, therefore, it is prob-
able multiplication and addition made but one operation. Had
they, for example, to multiply 12 by 4, they formed the sym-
bol of 12 four times, and then reduced these four symbols
into one, by the rules just laid down. But this method of
multiplication by ^dition must have been very tedious and
perplexing, when either of the numbers was considerable. If
they were to multiply 15 by 13, they had to make the symbol
of 15 thirteen times, and then to sum up these thirteen sym-
bols. Those who were most practised in calculation, would
soon discover that they might abridge this operation, by form-
ing the symbol of 15 three times, and once that of 150, which
is the product of 1 by 10; and then sum up these four sym-
bols. Such was probably the first step of the hum'an mind
towards multiplication, properly so called, or the art of adding
equal numbers with greater facility and expedition. This
operation, however, could never be performed with ease, till
those who practised calculation had by heart the product of
all the numbers under ten."

The Hebrews and Greeks at a very early period, it is well
known, had recourse to the letters of their alphabet for the
representation of their numbers, and the Romans followed
their example. The Greeks, in particular, had two methods ;
the first resembled that of the Romans, with which we are
well acquainted, and is still used in numbering the chapters,
and other divisions of books, dates of years, &c. They after-
wards had recourse to another method, in which the first nine
letters of their alphabet represented the first numbers from
1 to 9, and the next nine letters represented any number of
tens, from 1 to 9, that is, 10, 20, &c. to 90. Any number
of hundreds they expressed by other letters, in the same, or
a similar way, as has been explained above with regard to



426 MATHEMATICS.

pebbles, thus approaching to the more perfect decuple or de-
cimal scale of progression used by the Arabians, who, as wc
have seen, probably received it from the Indians, to whom it
was perhaps carried by Pythagoras.

The introduction of the Arabian or Indian notation into
Europe, about the tenth century, made a material alteration
in the state of arithmetic ; and this is thought to have been
one of the greatest improvements which this science had re-
ceived since the first discovery of it. The method of notation
was certainly brought from the Arabians Jnto Spain, by the
Moors or Saracens in the tenth century. Gerbert, after-
wards Pope Silvester II., v.ho died in the year 1003, carried
this notation from the Moors of Spain into France, it is be-
lieved about the year 960, and it was known in this country
early in the eleventh century, or perhaps sooner. As science
and literature advanced in Europe the knowledge of numbers
was also extended, and the writers in this art were very much
multiplie'd.

The next considerable improvement in this branch of
science, after the introduction of the numeral figures of the
Arabians, was. that of decimal parts, for which we are in-
debted to Regiomontamis ; who, about the year 14G4, divided
the radius of a circle (see Trigonometry) into 10,000,(X)0, so
that if the radius be denoted by l,the sines, co-sines, See. will
be expressed by so many places of decimal fractions, as there are
cyphers following 1 . This seems to have been the introductiorr
of decimal parts. The first person who wrote professedly on the
subject, and introduced the name " disme" or " decimals" was
Simon Sterinus, in a treatise intitled " Disme," subjoined to
his arithmetic, published in French, and printed at Leyden
in 158.5 ; since which, the method of decimals has been prac-
tised by many others, and is now become universal.

Arithmetic, in its present state, is divided into different kinds;
our concern with it is only as it relates to theory zwA practice.
Theoretical or speculative arithmetic is the science of the pro-
perties, relations, &c. of numbers, considered abstractedly, with



ARITHMETIC. 427

the reasons and demonstrations of the several mles. Euclid,
to whom we shall shortly have occasion to refer more parti-
cularly, furnishes a theoretical arithmetic, in the seventh,
eighth and ninth books of his Elements. The practical part
consists in the application of the rules resulting from the theory
to the solution of questions or problems, and arithmetic in
this respect has Been called an art : viz. the art of numbering
or computing ; that is, from certain numbers given, of finding
certain others, whose relation to the former becomes known.

To a person unacquainted with the rules of arithmetic, but
who has in his mind an idea of numbers, and at the same
time knows how to express those numbers, it may naturally
occur to him to ask what is the sum of two or more numbers ?
hence appears the necessity of the rule addition. To obtain
the diflference of two given numbers, subtraction will be re-
quired : to find the product of two or more numbers, we want
another rule called multiplication y and to find how often one
number is contained in another, requires division. These four
rules, viz. addition, subtraction, multiplication, and division,
are, in fact, the whole of arithmetic. For every arithmetical
operation requires the use of some or all of these. These
have been sometimes compared to the simple mechanical
powers, which, variously combined, produce engines of differ-
ent forms and of indefinite force ; which, however they may
at first strike us with surprise and astonishment, will, upon
examination, appear evidently to arise from a proper combin-
ation of the simple powers. In the same manner the most
complex operations in arithmetic are all effected by the appli-
cation of the simple rules just described.

Although it may at first appear, that the student, in
pursuit of general knowledge, has little to do with arithmetic,
excepting, as a branch of science, and that if he understand
the theoretical part it will be amply sufficient, experience will
teach him a different lesson. To be ready in science, pro-
perly, so called, he ought to be an expert practical arith-



428 MATHEMATICS.

metician, and he cannot lay too broad and solid a foundation
in this introductory part of knowledge.

Numerical computations are necessary in almost every step
of useful knowledge, though the manner of the operations
depends upon the nature of the subject to which they are ap-
plied. Arithmetic may, therefore, be regarded as the hand-
maid to her sister arts and sciences ; and, while she is em-
ployed in their service, she must be under their direction, be
ready at their call, and observe the laws they prescribe. Thus,
the calculations of eclipses, the time of the tides, and of the
several changes of the moon, are performed by numbers ; but
the operations must be directed by the precepts of astronomy,
which are the results of an extensive knowledge in that branch
of science. Reckonings at sea, so useful and necessary to
the navigator, are wrought by arithmetic ; but the steps of
the work are conducted by tlie rules of navigation. The
contents of casks, in the art of guaging, are found by num-
bers ; but the principles of the art depend on plane and solid
Geometrju The perpendicular heights of objects are com-
puted by numbers ; but trigonometry, which depends on geo-
metry, directs the manner of their appUcation. The Specific
Gravities of bodies are obtained by arithmetic ; but they could
not have been investigated without the aid of hydrostatics. Simi-
lar observations would apply to the other branches of science,
which shew how important it is, that a person in pursuit of
the general principles of knowledge, should be well grounded
in arithmetic. It remains now to describe those books from
which such knowledge can be readily obtained.

The number of books containing the practice and first
principles of arithmetic, is very great ; almost any of which,
with the assistance of a teacher, will answer the end proposed.
We shall mention but few of these, without, however, mean-
ing, by our silence, to depreciate the others.

The principal school-books in Arithmetic, were formerly
the " School-master's Assistaot/' by Dilworth, and the



ARITHMETIC. 429

*' Tutor's Assistant/' by Walkinghame ; these, for many years,
kept possession of the majority of schools in this country, and
were unquestionably very useful books ; and might at that time
have been recommended as well (o young persons who had no
guides, as to tutors, for whose express use the titles seem to
denote they were, by the authors, designed. Since which the
" Tutor's Guide," by Charles Vyse, became a popular school-
book: this is a much more elaborate work than the other two}
it includes a number of subjects not necessary as a mere intro
duction to Arithmetic ; such as the mensuration.of solids, as
well as superfices, the method of obtaining the specific gra-
vities of bodies, the application of the rules of arithmetic to
the principles of Chronology, &c. Mr. Vyse published a key
to his work, containing solutions of all the questions in the
" Guide."

Dr. Hutton's Treatise is used in many schools, to this also
there is a key. The same may be said of Keith's, of Good-
acre's, Molyneux's, and others. The first part of Molyneux's
is excellent, *as far as it goes, which embraces the four
first rules. Reduction, the Rule of Three, and Practice. We
think it not a judicious plan to throw the doctrine and prac-
tice of Vulgar Fractions, so far forward as many authors do,
because the knowledge of them is required at an early period.
The four rules in Vulgar Fractions, are as easily learnt as any
other part of arithmetic, when properly taught, and should
perhaps be immediately learnt after the same rules in whole
numbers.

Mr. Bonnycastle's ** Scholar's Guide " is an excellent
elementary treatise, in the notes are neat demonstrations of
each rule. A few years since the same author published an
octavo edition of his work, with various alterations, additions,
and improvements ; to which he has prefixed a neat histori-
cal introduction, and has added a Table of the Squares and
Cubes of all the numbers, up to 2000 inclusive, and a table
of the square and cube roots of the same numbers. He has
likewise given a table of all the prime numbers, from 1 to



430 MATHEMATICS.

100,000, that is, of all those numbers, which cannot be divid-'
ed into any number of equal integral parts greater than unity }
such are the numbers, 3, 5, 7, 13, 17, &c. A much larger
table extending to 217,219 is given in Dr. Rees's great work,
the New Cyclopedia, under the article Prime Numbers,
which, says the author of that article, is double the extent to
which they are carried in any other English work, and which
it is presumed may be found useful in a great variety of cases
connected with arithmetical and numeral problems.

Mr. Bonnycastle's " Scholar's Guide" seems deficient in its
examples, which cannot well be too numerous for young be-
ginners : the rules are easily understood, and nothing but con-
siderable practice can give facility in the operations. To ob-
viate this objection, the following work was published, " A
System of Practical Arithmetic, applicable to the present
state of trade and money^transactions, illustrated witli nume-
rous examples under each rule," by the Rev. J. Joyce. A
large portion of this work is devoted to examples of all kinds,
intended to illustrate the rules, which contain much useful
information applicable to the advancing stages of life. The
author of this work, after giving reasons for introducing
Logarithms, the doctrine of Annuities, Reversions, Leases,
&.C. into his book, says, " A book of Arithmetic, for schools,
should contain every thing necessary to be known, previously
to the study of Algebra ; and for the sake of those who wish
to proceed to that science, it should be introductory to it.
This, it is believed, will be found to be one of the character-
istics of the work now offered to the public. There are,
however, thousands who never trouble themselves to learn
beyond the elements of the arithmetic which they acquire at
school ; who, looking to trade and commerce as the objects of
their future lives, seek only for that knowledge which in some
way or other is applicable to those objects. These, in almost
every rank and situation, have frequent occasion to calculate
the interest of money and the discount of bills ; and to as-
certain the value of annuities on single and joint lives, of



ARITHMETIC. 431

survivorships, of leases, of reversionary interests, of funded
property, and of freehold estates." To render these subjects
fuiuiliar to young people was one chief object of the work
last mentioned, and it is presumed that it has been at-
tained.

A key to this system of practical arithmetic has been
published, into which, unfortunately, several errors crept,
owing to the proprietors having employed a person to correct
the proofs who was incompetent to the task, instead of con-
signing the business, as they ought, to the author.

There are besides the introductory books already mentioned,
some on a much larger and more expensive scale ; among
these may be mentioned, one by Malcolm, and another in-
titled " Arithmetic, Rational, and Practical," by John Mair.
We may add, that the most considerable madiematicians have
in former times, as well as in modern days, employed their
talents in illustrating the rules of arithmetic, and in investigat-
ing the properties of numbers.

It may be farther added, that arithmetical operations have
been performed by mechanism, and machines of various kinds
have been invented to perform the fundamental rules. These,
however, can go no fartlier : in cases where any degree of
invention is necessary, the aid of the mind is required. Na-
pier's rods are familiar to every one who has attended to the
subject in hand. In the reign of Charles II. Sir Samuel
Morland invented two arithmetical machines, of which he
published an account under the title " The Description and
Use of two Arithmetic Instruments, together with a short trea-
tise, explaining the ordinary operations of arithmetic." This
work, which is no\^^ very rare, is illustrated with twelve
plates, in which the different parts of the machines are ex-
hibited ; and from these it should seem that the four funda-
mental rules in arithmetic are readily worked, and, to use the
author's own words, " without charging the memory, disturb-
ing the mind, or exposing the operations to any uncertainty."
The present Earl Stanhope about thirty years since invented



432 MATHEMATICS.

and caused to be manufactured two raachines for the like
purposes as those of Mr. Morland. The smallest of these
instruments, intended for addition and subtraction, is not
much larger than an octavo volume, and by means of dial-



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 35 of 44)