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

For the better understanding of the principles of Notation here explained, the following Table is given.

Hundreds of Millions co Tens of Millions

Thousands

Hundreds of Thousands

Tens of Thousands

5 4 3 2 1 Fifty-four thousand 321.

6 5 4 3 2 1 654 thousand 321.

4 3 2 1 Seven millions 654 thous. 321.

8 7 6 5 4 3 2 1 Eighty-seven millions 654, 321.

9 8 7 6 5 4 3 2 1 987 millions, 654, 321.

The first column on the right hand contains units, and the figure 1 in that column, represents the number One. The second line consists of 1 Unit and two Tens, or twentyone; the third line of 3 hundreds, 2 tens, and 1 unit, or three hundred and twenty-one; and in the same manner the lowest line contains 9 hundreds of millions, 8 tens of millions, and 7 millions; in all 987 millions; also 6 hundreds of thousands, 5 tens of thousands, and 4 thousands; in all 654 thousands; and lastly, 3 hundreds, 2 tens, and 1 unit so that the whole sum expressed by the g figures in the lowest line, is nine hundred and eighty-seven millions, six hundred and fifty-four thousands, three hundred and twenty-one.

In the same manner, Numeration may any extent, as in the following example.

Trillions

Hundreds of thousands

Tens of thousands

Thousands

be carried on to

of Billions

Where 19 figures represent the sum Three Trillions, two hundred and twelve thousand, three hundred and fortyfive Billions, six hundred and seventy-eight thousand, nine hundred and eighty-seven Millions, six hundred and fiftyfour Thousands, three hundred and twenty-one.

OF ADDITION.

THE fundamental operations of Arithmetical calculation are these four; Addition, Subtraction, Multiplication, and

VOL. I.

Z

Division:

Division or rather, as quantities are susceptible of no other modification but augmentation and diminution; the two last operations, Multiplication and Division, are in fact only speedy methods of performing the two first operations, Addition and Subtraction.

By Addition we assemble and express on paper, the aggregate value of a number of separate quantities. When the quantities or the numbers by which they are expressed, consist of only one place of figures, as when 3, 5, 7, and 9,

9

7

5

3

Sum 24

are to be added together, we say thus; three and five are eight, and seven are fifteen, and nine are twenty-four, writing 2 for the number of tens, and 4 for the remaining units, as in the margin: But when the sums to be added together consist of more than one place of figures, the scholar must be careful to place them so as that Units shall be immediately, under units, Tens under tens, Hundreds under hundreds, &c. as in the anannexed example, where the inhabitants of the principal towns of a certain county, being calculated to be 4,386, 2,285, 7,309, 3,025, and 1766; it is required to know the amount of the population of these five towns.

Thousands.

Hundreds.

Tens.

Units.

4386

2285 7309 3025 1766

Write down these several sums, as in the margin; then drawing a line under the whole, say 6 and 5 are 11, and 9 are 20, and 5 are 25, and 6 are 31; that is, 3 tens and 1 unit; then write this 1 in the place of units, and carrying (as it is termed) the three tens to the second column of figures, say, 3 and 6 are 9, and 2 are 11, and (passing over the nought) 8 are 19, and 8 are 27 here are 2 tens and 7 units, which units are to be written under the second column, and the 2 tens carried or added to the third column. Then say 2 and 7 are 9, and 3 are 12, and 2 are 14, and 3 are 17; where the 7 units are to be written under the column now

18771

summed