A long hundred is 120; a great gross is 144 dozen : but these, and many other denominations of a similar kind, are rapidly going out of use.

The Student will consult his own advantage and convenience by committing these ten tables to memory, omitting such of the observations as may depend upon principles beyond the extent of his progress in the subject.

THE

PRINCIPLES AND PRACTICE

OF

ARITHMETIC.

CHAPTER I.

DEFINITIONS, PRELIMINARY NOTIONS, NOTATION, NUMERATION, AND FUNDAMENTAL OPERATIONS.

ARTICLE I. DEFINITION I.

ARITHMETIC is the Science which treats of magnitudes, with reference to the consideration of how many or how few.

2. DEF. 2. An Unit, or, as it is generally called, Unity, is the representation of a thing considered in its individual capacity, without regard to the parts of which it may be made up, and it is the Base or Element of all arithmetical computations.

Thus, each of the terms, a man, a house, a pound, &c., denotes one individual of its kind, being the same as one man, one house, one pound, &c., respectively; and these are the bases or elements by means of which several men, several houses, several pounds, &c., may be computed.

3. DEF. 3. Number signifies a multitude or collection of two or more units, or denotes an assemblage of two or more distinct objects of the same kind.

Thus, two men, three houses, four pounds, &c., which are represented by the numbers, two, three, four, &c. denote more than one individual of the same kind, the single individuals being repeated twice, thrice, four times, &c., respectively. Numbers thus viewed are termed Whole Numbers or Integers; and for the sake of uniformity, the Unit is considered the first or least integer.

H. A.

1

4. DEF. 4. Numbers used to express one or more individuals of specified kinds, are called applicate or concrete numbers; whereas two, three, four, &c., by themselves, not particularizing the kinds of individuals, are termed abstract numbers.

NOTATION.

5. DEF. 1. Notation is the method of expressing by certain symbols, or characters, any proposed number, or quantity arithmetically considered.

6. DEF. 2. The Symbol or Representative of unit or unity, is 1; but instead of other numbers being expressed by assemblages or multitudes of units placed together, which would soon become embarrassing, other characters or symbols have been invented, by means of which every number however great, may be expressed; and instead of a different symbol being adopted for every different number, which would soon become equally inconvenient, all numbers are expressed by means of the following ten symbols, or as they are usually termed Figures, and sometimes Digits, which have their names respectively annexed:

1, 2, 3, 4, 5, 6, 7, 8, 9, 0: one, two, three, four, five, six, seven, eight, nine, zero: the first nine of which are all defined by their names; and the last which is variously denominated Nought, Cipher, or Zero, when standing by itself has no signification, or at most, denotes the absence of number, and is to be regarded merely as an auxiliary digit, for the purposes hereafter to be explained.

7. DEF. 3. Whenever a figure is placed on the right of the same or any other figure, it has, by universal agreement, the effect of increasing the value of the lastmentioned figure tenfold, at the same time that it retains its own value.

Thus, beginning with the auxiliary digit 0, we have the following numbers and their representations:

and it is obvious that by means of two figures, this kind of notation may be continued till we arrive at ninetynine, whose symbol will be 99.

8. DEF. 4. Beyond this number, the use of two, either the same or different figures, will not enable us to go, but a repetition of the contrivance in the last Article, will by means of more figures supply the defect.

Thus, supposing the effect of any figures being placed on the right of symbols formed as above, to be to increase all their values tenfold, we shall have

so likewise of succeeding numbers; thus, we have

345,

three hundred and forty-five,

586:

five hundred and eighty-six :

and again, 999 will be nine hundred and ninety-nine, which is the largest number capable of being expressed by three figures.

Here, the first figure on the right hand is said to occupy the units' place, the second the place of tens, and the third that of hundreds.

Of the auxiliary digit 0, the sole use is in the effect specified in the last two Articles; and all figures to the right of it will therefore be unaffected by it.

9. DEF. 5. In estimating numerical magnitudes, we proceed in order from hundreds, to thousands, tens of thousands, and hundreds of thousands; millions, tens of millions, and hundreds of millions; in precisely the same manner as we have done above from units to tens, and from tens to hundreds.

10. DEF. 6. Agreeably to the principle of Article (7), it is assumed that " any figure placed on the right of one or more others, has the effect of increasing every one of them tenfold without altering its own value;" and this enables us to express with facility any number whatever.

Thus,

(1) 1000 will represent One Thousand.

(2) 5493 will represent Five Thousands, four Hundreds, and ninety-three.

(3) 23456 will represent Twenty-three Thousands, four Hundreds, and fifty-six.

(4) 729054 will represent Seven hundred and twenty-nine Thousands, and fifty-four.

(5) 1803205 will represent One Million, eight hundred and three Thousands, two Hundreds and five.

(6) 32754081 will represent Thirty-two Millions, seven hundred and fifty-four Thousands, and eighty-one. (7) 473025004 will represent Four Hundred and seventy-three Millions, twenty-five Thousands, and four.

11. If the first three figures beginning from the right hand be denominated so many units, tens of units and hundreds of units, it follows that the next three figures taken the same way will be thousands, tens of thousands, and hundreds of thousands: the next three in order will be millions, tens of millions, and hundreds of millions and so on.

Whence, to express in figures any number proposed, we have only to consider in which of these divisions each part of it ought to be found, observing that three figures from the right must be taken to make each division complete, before we proceed to the next.

Ex. 1. Express by means of figures; Thirty-five thousand, eight hundred and nineteen.

Here, eight hundred and nineteen belongs to the first division on the right, and is written 819:

also, thirty-five thousand must be found in the second division from the right, and is 35:

whence, the proposed number will be expressed by 3 5 8 19.

Ex. 2. Write down in figures the number; Five million, twenty-five thousand, six hundred and seven.

In this case, the first division on the right will be 607; the second will be 025, the digit 0 being affixed to the left of the others without altering their values, to make up the required number of three; and the third is 5: so that the expression required will be

50256 07.

Ex. 3. Express by figures the following number;