BEZOUT'S ARITHMETIC.

1. In general, every thing is called quantity, which is susceptible of increase and diminution. Extent, duration, weight, etc. are quantities. Every thing that is quantity is the object of the mathematics; but arithmetic, which makes a part of these sciences, only considers quantities when they are expressed in numbers.

2. Arithmetic is then the science of numbers: it considers their nature and properties; and its object is to give easy methods for representing numbers, and for compounding and decompounding them; which is called calculating.

3. To form an exact idea of numbers, we must first know what is understood by a unit.

4. The unit is a quantity taken, in general, arbitrarily, to serve as a term of comparison to all quantities of the same kind: thus when we say, a certain body weighs five pounds, the pound is the unit; it is the quantity to which we compare the weight of the body; we might, with equal propriety, take the ounce for the unit, and then the weight of the same body would be signified by eighty.

5. Number shows of how many units or parts of a unit a quantity is composed.

If the quantity be composed of whole units, the number which expresses it is called a whole number; and if it be composed of whole units and parts of a unit, or simply of parts of a unit, then the number is called fractional or fraction: three and a half is a fractional number; three-fourths is a fraction. Whole units are called integers.

6. A number which we express without signifying the species of the units, as when we say simply three or three times, four or four times, is called an abstract number; and when we express at the same time the species of the units, as when we say four books, three dollars, it is called a concrete number.

OF NUMERATION AND DECIMALS.

7. Numeration is the art of expressing all numbers by a limited number of names and characters: these characters are called figures.

8. The characters made use of in numeration, and the names of the numbers which they represent, are as follows:

cipher, or unit, or nought

one two three

six

four five 01 2 3 4 5 6

seven eight nine 7 8 9

To represent all other numbers with these ten characters, ten units are expressed by a single unit, and we count tens as we count the simple units, that is to say, we count two tens, three tens, etc. as far as 9: to represent these new units, we employ the same figures as for the simple units, but we distinguish them by the place which we make thẻm occupy, in placing them on the left of the simple units.

Thus, to represent fifty-four, which contains five tens and four units, we write 54.

To represent sixty, which contains an exact number of tens and no units, we write 60, putting a cipher in the place of units, which shows that there are no simple units, and makes the figure 6 represent a number of tens. We can, by this means, count as far as ninety-nine inclusive.

9. Let us remark as we go along, this property of numeration; namely, that a figure placed on the left of another figure, or of a cipher, represents a number ten times as great as if it were alone.

10. From 99 we count as far as nine hundred and ninetynine, in a similar manner. Of ten tens, we compose a single unit, which we call hundred; we count hundreds from one to nine, and represent them by the same figures, but in placing these figures on the left of the tens.

Thus, to express eight hundred and fifty-nine, which number contains eight hundreds, five tens, and nine units, we write 859. If it were eight hundred and nine, which contains eight hundreds, no tens, and nine units, we should write 809; that is to say, we should put a cipher in the place of the tens which are wanting. If the units were also wanting, we

should put two ciphers: thus to express eight hundred, we write 800.

11. Let us again remark, that a figure placed on the left of two others, or of two ciphers, expresses a number a hundred times as great as if it were alone.

12. From nine hundred and ninety-nine, we can, by the same artifice, count as far as nine thousand nine hundred and ninety-nine, in forming of ten hundreds a unit called thousand, counting these units as before, and representing them by the same figures placed on the left of the hundreds.

Thus, to represent seven thousand eight hundred and fiftynine, we write 7859; to express seven thousand and nine, we write 7009; and for seven thousand, we write 7000, where we see that a figure on the left of three others, or of three ciphers, expresses a number a thousand times as great as if it were alone.

13. In continuing thus to include ten units of a certain order in a single unit, and to place these new units in ranks more and more advanced towards the left; we are able to express in a uniform manner, and with ten characters only, all the whole numbers imaginable.

* Or, we can represent all possible numbers with the ten following characters, which are called figures: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The nine first represent one, two, three, four, five, six, seven, eight, nine; the character 0, is called nought, and does not represent any number.

To effect this, in proceeding from the right hand towards the left, the first figure is considered as representing units; the second, tens; the third, hundreds; the fourth, thousands; the fifth, tens of thousands; the sixth, hundreds of thousands; the seventh, millions, etc. Also, O is put in the place of the units, the tens, the hundreds, the thousands, the tens of thousands, the hundreds of thousands, the millions, etc. which are wanting in a number.

It is evident, in effect, that with these ten characters we ean represent all the whole numbers possible, since the following numbers, one, two, three, four, five, six, seven, eight, nino ton eleven, etc. which, beginning with a unit, go on in

creasing by a unit to infinity, may be expressed by these ten characters.

14. To read with facility a number expressed by any number of figures at pleasure, we separate them into periods of three figures each, proceeding from the right hand towards the left: we shall give to these periods the following names, beginning at the right hand, units, thousands, millions, billions, quadrillions, quintillions, sextillions, etc. The first figure of each period (beginning always at the right) will have the name of that period; the second, that of tens; and the third, that of hundreds.

Thus, beginning at the left, we read each period as if it stood alone, and we pronounce at the end of each the name of that same period; for example, to read the following number:

quadrillions trillions billions millions thousands units

we say twenty-three quadrillions, four hundred and fifty-six trillions, seven hundred and eighty-nine billions, two hundred and thirty-four millions, five hundred and sixty-five thousands, four hundred and fifty-six units.

From what has been said above, knowing that there are three places of figures, that is to say, units, tens, and hundreds under each name, we can easily write down in figures any number which is expressed in words, by first writing the names of the different periods in succession, beginning with the highest name mentioned in the question, and terminating with the units, and then placing such figures under each name as the nature of the question requires; for example, if it be required to write down in figures the following number, fifty billions, thirty-two millions, five hundred and two thousands, six hundred and fifteen, I write the names of the different periods, beginning with the highest name mentioned in the question, thus,

Billions Millions Thousands Units

then as the question requires fifty billions, I write 50 under billions, putting a cipher in the place of units, and leaving the place of hundreds vacant, because a cipher placed on the left

of the 5 would not at all alter its value; then as the question requires thirty-two millions, I place 32 under millions, but I must not here leave the place of hundreds vacant as I did under billions, for as the value of the 5 which stands under billions depends upon the number of places to which it is removed towards the left, that omission would decrease its value ten-fold, I therefore write 032 under millions; again, as the question requires five hundred and two thousands, I write 502 under thousands; and lastly, for the six hundred and fifteen, I write 615 under units, which completes the proposed number.

Let the following numbers be expressed in figures. 1. Four thousand six hundred and twenty.

2. Fifty-six millions three hundred thousand and ten. 3. Seventeen billions, forty-five millions, three thousand and four.

4. One hundred and six trillions, sixty-seven billions, twenty-four thousand and sixty.

5. Three hundred quadrillions, six trillions, seventy billions, three hundred millions and sixteen.

15. From the numeration which we have explained, and which is purely of agreement, it follows that as we advance from the right hand towards the left, the units of which each' figure is composed are ten times as great as they would be if the figure stood one place farther to the right; and that, consequently, to render a number ten times, a hundred times, a thousand times, etc. greater, we have only to put one, two, three, etc, ciphers on the right of its unit figure: on the contrary, as we retrograde from the left hand towards the right, the units in each figure are only one-tenth of what they would be if the figure stood one place farther towards the left,

16. Such is numeration: it is the basis of every other method of counting, although in many arts we do not always count by tens, by tens of tens, etc.

17. To estimate quantities less than the unit which we have chosen, we divide this into other smaller units. The number of them is indifferent in itself, provided that we can measure