W

ON ARITHMETIC.

HATEVER is in its nature capable of augmentation and diminution, is termed, quantity: extent, duration, weight, &c., are all quantities: and whatever constitutes quantity, becomes an object of mathematical investigation. That branch of mathematics which considers quantity, as expressed by numbers, is called Arithmetic, from a Greek term signifying number; and may hence be considered as the science of the nature and properties of numbers: its object is to discover sure and easy methods of representing, compounding, and decompounding numbers; by certain operations, constituting calculation.

As all calculation is founded on a knowledge of what is called Unity, it must be observed, that an Unit is a quantity assumed at pleasure, to serve as the medium or standard of comparison, between quantities of the same sort. Thus, when we affirm of two bodies, that the one weighs three pounds, and the other five pounds, we make a pound the standard of comparison, or the Unit: but if we say that the first body weighs forty-eight ounces, and the other eighty ounces, we consider the ounce to be the standard or unit.

By numbers we express how many units, or parts

Y 2

of

an

unit,

unit, are contained in any given quantity. If the quantity consists of entire units, the number by which it is expresed, is called a whole number, as for example, sixteen, fiftynine, two-hundred and four, &c.: but if the quantity contain only parts of any given unit, as three quarters of a pound, the number is called a fraction; and when the quantity consists of entire units and parts of an unit, the number expressing it is said to be fractional, as nineteen and nine-tenths.

Arithmetic must have been known from the earliest period of society: but, although, we cannot conceive a nation, nor even a rational individual to have subsisted without a knowledge of numbers, in their most simple application and uses, yet men may have continued for many ages ignorant of the wonderful extent of their powers. The Greeks were the first European nation who cultivated the art of numbers; and some have imagined, from the terms employed by them, and by the Romans after them, that in their Arithmetical operations, they made use of small stones or pebbles: for both the Greek term Psephizo, and the Latin Calculo, (from which we have our Calculation), are derived from the words in those languages, signifying a pebble, or small stone.

However this may have been, we find the Greeks very early making use of the letters of their alphabet, to represent numbers. Thus the twenty-four letters taken as they stand in the alphabet, with three other characters, introduced in certain places, were made to represent the nine digits, the nine tens, and the nine hundreds.

But the difficulty of carrying on arithmetical operations, to much extent, with such characters, may be easily imagined, and is very evident from calculations still remaining in the works of some ancient Greek geometricians.

The Romans who drew from the Greeks the chief part of heir skill in the Sciences; imitated them also in this mode

of

of expressing numbers; but adopting a different arrangement of the alphabetical characters, as here shown,

I. V. X.

L.

C.

D.

One, Five, Ten, Fifty, One Hundred, Five Hundred,

M.

One Thousand. &c

By the repetition and combination of these numeral characters, any number may be expressed. 1st. By the repetition of a character, the value is also repeated, as, III. repre sent three; XX. twenty; CC. two hundred,

2d. When a character is followed by one of inferior va lue, their values are to be added together, as XII. twelve, LV. fifty five. MDCCCVIII. one thousand eight hundred and eight.

3d. But when a numeral letter of small value is placed before one of a greater, the less is to be subtracted from the greater, in order to have the value of the expression: thus IV. represent four, IX. nine, XL. forty, XC. minety. CD. four hundred, &c.

In old Books we meet with 1. instead of D. for 500. and CIƆ for 1000, but these characters may perhaps have been only inaccurate representations of D and M.

Thousands are also represented by drawing a short line over the numeral character as V. for 5,000. L. for 30,000, CCC. for 300,000.

About the end of the 2d century, a new species of arithmetic was invented, as is supposed, by the great geometrician and geographer Ptolomy. Its object was to avoid the difficulties occasioned by fractions in the common Arithmetic and in it the unit was divided into 60 equal parts; each of these into 60 others; each of these last again into 60 other parts, and so on and from these divisions this kind of Arithmetic was called Sexagesimal, or by Sixties.

The excellent mode of expressing Numbers now used, came into Europe from the Arabians, by way of Spain:

but

but those Arabians did not pretend to be the inventors of these symbols, on the contrary they owned they were derived from the Indians. The period when these Arabic Symbols were introduced into England is uncertain: but inscriptions have been found as far back as in 1090, where they are employed.

The Introduction of these new characters did not immediately put an end to the Sexagesimal Arithmetic, which having been employed in all Astronomical tables, was on this account still retained, at least in the fractions, until Decimal arithmetic came into use.

The most ancient treatises on Arithmetic are certain Books of the Elements of Euclid, who flourished about 280 years before Christ. About the year 1460 Regiomontanus (or Muller of Koningsberg) in his Tables, divided the Radius into 10,000 instead of 60,000 parts; and so far abolished the former Sexagesimal Arithmetic, of which however a vestige still exists, in the division of time, and of a Degree of a Great Circle; for an hour is divided into 60 Minutes, a Minute into 60 Seconds, a Second into 60 Thirds, and so on; and a Degree is divided and subdivided in the same manner, into parts of the same denominations. The greatest improvement however which any age has produced, in Arithmetical operation, is by the invention of Logarithms; a discovery for which the world was indebted to Baron Napier of Merchiston in Scotland, towards the beginning of the 17th century. By these and other means, Arithmetic may now be considered as the science which has attained the nearest to perfection; and in which very important improvements can scarcely be looked for.

NOTATION.

By Notation is meant the art of expressing numbers, by a limited set of characters, called Cyphers or Figures.

The

The Figures now used, and their powers, are the following viz.

1. 2. 3. 4. 5. 6. 7.

8. 9.

one two three four five six seven eight nine To these is added 0 to represent nought, or the absence or negation of all number or quantity.

To represent all other numbers by means of these figures, it has been agreed on, that Ten units should be formed into one aggregate sum, to be called Ten, with which calculation may be carried on, as by a simple unit; as two tens, three tens, six tens, &c. on to nine tens. To represent these new units the former figures are employed, but placed in a different position, to the left hand of their original place. Thus to represent twenty-four, containing two Tens and four units, we write 24: for Sixty, or six tens, without any simple units, we write 60: for ninety-nine, 99.

For Numbers above ninety-nine, on to, and including nine hundred and ninety-nine, another series of Units is formed in the same way, each of which contains Ten of the preceding series, and one hundred of the simple units. This last series is termed hundreds; and by it we express any number, as five hundred and sixty-three; thus, 563: Nine hundred and nine thus, 909: that is, nine hundreds, no odd tens, and nine units. Seven hundred would be, 700, without either tens or units.

Again from nine hundred and ninety-nine, by a similiar process, we can count to nine thousand nine hundred and ninety-nine: forming a fresh series of Units called Thousands, each containing ten hundreds. Thus, seven thousand four hundred and thirty-five, will be written 7435; eight thousand and six, that is eight thousands, no hundreds nor tens, and six simple units, 8006. The year One thousand eight hundred and eight, 1808.

For