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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-

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-

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