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of the demonstrations are original; and some of them, in which the employment of figures is supplemented by that of general symbols, may be acceptable to the Algebraist, as well as to the Arithmetician.

Although written with especial reference to the wants of the National Teachers, the PRINCIPLES OF ARITHMETIC will be found to embrace everything upon which, under the head of Arithmetic, candidates for Civil Service appointments are examined.

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D. O'SULLIVAN.

DUBLIN,

August, 1872.

ARITHMETIC.

DEFINITIONS.

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one

1. Any one thing, regarded as to its singleness, is termed a UNIT.

2. Two or more units of the same kind, considered collectively, are called a NUMBER. We say

number of birds,” “ a number of trees," "a number of stars,” &c. : the unit (whose repetition constitutes the number) being—in the first case, a bird; in the second, a tree; in the third, a star; &c.

Almost as soon as it can observe anything, a child is able to distinguish between object and a “number” of such objects-between, for example, one apple and a number of apples, between one marble and a number of marbles, &c. Further observation enables the child in a very short time to distinguish between a larger and a smaller number of objects, all of the same kind—between, say, five apples and three apples, between ten marbles and seven marbles, &c. Comparing a number of objects of one kind with the same number of objects of another kind (a number of apples, for instance, with the same number of marbles), the child soon begins to see that the mere number is quite distinct from the nature of the objects in other words, that the perception of what the objects are is quite distinct from that of how many there are; and so the mind gradually acquires the power of contemplating numbers without reference to any objects in particular–or, as we say, of contemplating numbers in the abstract. Thus, a boy who has three apples in one pocket, and three marbles in another, cannot fail to see that, although an apple and a marble are two very different things, the number of apples is the same as the number of marbles ; and the contemplation of this sameness gives the boy the abstruct notion, as it is called, of the number « three.”

3. A number is said to be CONCRETE or ABSTRACT according as it is considered with or without reference to particular things.

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B

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“Five books,” “eight shillings,“ten days,&c. are examples of concrete numbers-mention being, in each case, made of what the units are, as well as of hour many there are; but “five,” "eight,” “ten,” &c.--considered without reference to any units in particular--are abstract numbers.

Although the word “number” conveys the idea of plurality, it is not unusual to speak of “the number one”—just as we say - the number three, “the number five,” &c. : “ one” being popularly regarded as the smallest number. On the other hand, the term “unit ” is sometimes applied (but only in a secondary sense) to a collection of individuals. Thus, when mention is made of "five regiments” of soldiers, the mind is disposed to regard five as the “number," and a regiment as the “unit;" when we speak of “ eight mease ” of herrings, eight may be regarded as the “number,” and a mease (five hundred) as the

unit;" and when we say 66 ten dozen" of wine, ten may be taken as the “number,” and a dozen (bottles) as the “unit.”

4. ARITHMETIC is the Science of Numbers.

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NOTATION AND NUMERATION.

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5

7

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DECIMAL OR ARABIC SYSTEM.* 5. Every number can be expressed by means of one or more of the following ten figures, the first nine of which are called digits :3

6 one two three four five six seven eight nine cipher,

or nought. 6. The art of writing down numbers in figures is called NOTATION ; the art of expressing numbers in words (spoken or written), NUMERATION.

7. The largest number that can be expressed by means of one figure is nine. Every larger number is represented by a combination of two or more figures.

* In this system we reckon by tens: hence the name DECIMALfrom decem, Latin for “ten.” The system was introduced into Europe by the Arabs, about 800 years ago : hence the name ARABIC. The Hindoos, however, are the people with whom the system is supposed to bave originated.

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