as the tenth, hundredth, thousandth, &c. part of the principal unit.

To distinguish the decimal parts from the principal units, we separate the ones from the others by a comma, thus the number 53,4, expresses fifty three units, and four tenths of a unit; if we put another fig. at the right of the 4, such as 53,47, each unit of the 7, will express tenths of tenths, or hundredths.—If we had the number 53,476, the 6 would express tenths of hundredths, or thousandths; a fourth fig. would be tenths of thousandths ; a fifth, hundredths of thousandths, and so on infinitely. We may then call this last number in the following manner; fifty three units, four tenths, seven hundredths, six thousandths.But since a hundredth is worth ten thousandths, and consequently the seven hundredths amount to 70 thousandths; since, likewise, a tenth is worth ten hundredths or one hundred thousandths, the four tenths will make four hundred thousandths: We can read this number in a more abridged manner, and say, fifty three units, four hundred and seventy six thousandths. Such is the usual way of reading a number composed of principal units and decimal parts. might also read it thus; fifty three thousand, four hundred and seventy six thousandths; which will be easily understood, since every principal unit is worth ten tenths, a hundred hundredths, a thousand thousandths, and consequently every ten of units is worth ten thousands of thousandths. What we have here explained, is founded upon the general principle of Numeration, which is absolutely the same for decimal parts and for principal units.

We

If we had to read a number, containing no princi pal units, such as 0,4657, we should say, four thousand six hundred and fifty seven ten thousandths, because the units, expressed by the 7, are ten thousandth parts.

The O or cypher, amidst decimal parts, answers two purposes. 1st. It fills the place of the units wanting in the order where it is placed; then it serves to give a value ten times less to the figures at its right. In fact, suppose the number 45,308; the cypher indicates that there are no hundredths, and it gives a value ten times smaller to the 8; for if, omitting the 0, we wrote 45,38, the 8, which expressed thousandths, would now be hundredths.

It is plain, that the value of a number, containing decimal parts, would not be altered by the addition, at its right, of ever so many cyphers; and, conse➡ quently, if, at the right of decimal figures, there be cyphers, they may be safely suppressed.

REMARK. Since the comma serves to discern the decimal parts from the principal units, it is obvious that by moving this comma one fig. lower to the right, or higher to the left, we make this number ten times greater or smaller. Let us take, for instance, the number 534,728; if we move the comma one fig.. to the right, and write 5347,28, the 7, which was of tenths, will become units; the 4, which expressed units, now expresses tens; the 3, which was of tens, is hundreds; the 5, which was hundreds, becomes thousands and, on another hand, the 2, which expressed hundredths, is now of tenths; the 8, from thousandths, becomes hundredths. The moving of the comma, has then rendered every part of the num ber,, and consequently the whole number, ten times greater than it was.-On the contrary, by placing the comma one fig. higher to the left, and writing 53,4728, we make the whole number ten times less, since every one of its parts, decreases in a tenfold proportion.

By a similar reasoning, it is easy to see, that, by moving the comma two, three, four, &c. fig. lower

to the right, we should make the number, 100, 1000, 10000, &c. times greater, and inversely if we moved it to the left.

It is to be wished that the pound, fathom, &c. had been divided into decimal parts; arithmetical calcula tions would thereby have been greatly simplified.

A number is called abstract, which designates no kind of things, such as twice, three times, four times, &c. we call concrete a number applied to things, such as 25 fathoms, 50 pounds. This last kind of number is also called incomplex, when, as in the former example, it contains but units of the same denomination, and complex, when it contains units of various denominations, but reducible to the same, such as 15 fathoms 4 feet 6 inches-54£. 12s. 6d.

All the operations performed upon numbers, are reduced to four, viz. adding, subtracting, multiplying, and dividing. As the process of these operations is not the same for complex and incomplex numbers, we shall treat of them separately.

ANNOTATION.

For brevity's sake, it has been agreed to make use of certain signs; viz.

which signifies more: 3+2, signifies 3 more 2.

less: 9-4-1, means 9 less 4 less 1. × signifies multiplied by: 3×4, or 3 multiplied by 4. signifies equal to: 4+3=5+2, or 4 more 3, is equal to 5 more 2.

ADDITION

OF INCOMPLEX NUMBERS.

ADDITION is an operation directed to find out one number equal to several other numbers taken together, and the number, resulting from this operation, is called the sum.

The process consists, 1st. in placing the numbers one under another, in such a manner that the units be exactly under the units, the tens under the tens, hundreds under hundreds, and so on; and thus all the figures of the same kind of units form a vertical column. 2dly. Adding the figures of each column, beginning by that of simple units; and, if in the sum of any of the columns you find tens, you only write under that column the units belonging to it, and keep the tens to add them to the following column. An example will illustrate this theory.

48073

Numbers to be 259070 added together. 643

Adding up the figure of 3794206) the first column, I find 12 for the sum of the units in the four numbers; then I have written 2, and kept

the one ten to join it to the 4101992 Sum. column of tens; the sum of which having been found to be 19, I write 9 under it, and keep one ten of tens or a hundred, to add it up to the column of hundreds; proceeding on in the same manner, I have found 4101992 for the sum of the four numbers, which must evidently be right, since it is composed of the sums of all the units, tens, hundreds, &c. of those four numbers.

The proof of Addition is done by another addition executed in a contrary direction to the former; that

is to

say, if you have proceeded downwards in the first operation, you must go upwards in the proof. All that we have said, applies, without any exception, to nuinbers containing decimals. Here you see that the disposition of numbers is always such, that the units

of the same kind be placed the ones beneath the others.

numbers

to be

563,...

8621,05..

4,8934

0 236.

cast up.

Sum 9189,1794

Instead of the dots which mark the vacant places at the right of the numbers, we might have put cyphers without changing the value, the comma shewing that these are decimals.

OF SUBTRACTION.

SUBTRACTION is an operation, by which we find out the excess of one number above another; this excess is called the difference.

To proceed to the operation, you write first the greatest number, then place under it the smaller, so that the units correspond to the units, the tens to the tens, &c. Now, nothing would be easier than to find the excess of the larger over the inferior number, if all the fig. of the former were greater than the corresponding ones in the latter, but commonly it being not so, we shall see, by an example, how to perform every kind of subtraction.