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ARITHMETICK.

SECTION I.

DEFINITIONS.

.# Definition is the import or meaning of a word, expressed by other words. Quantity is any thing which can be increased, decreased or measured. [Number, lines, time, motion, space and matter, are quantities.] JMathematicks is the science which treats of the numbering and measuring of quantities. •An Unit is one, or a single thing of any kind. JNumber is an unit, or several units collected together... [One, two, three and ten, are numbers expressed by words.] Arithmetick is that part of Mathematicks which treats of numbers; or, it is the science of numbering. 4 Representative is a word, or character, standing

for some other thing. [1 is a representative of a single thing of any kind.]

EXPLANATION OF ARITHMETICAL CHARACTERS OR

FIGURES.

Numbers are generally expressed in Arithmetick by the ten following characters or figures —1, 2, 3.

* *

4, 5, 6, 7, 8, 9, 0°. Each of these figures has its own particular name, by which it is always called when standing alone. These names are called the common names of the figures. The word written before each figure below, is its common name;—one 1, two 2, three 3, four 4, five 5, six 6, seven 7, eight 8, nine 9, cipher 0. Each figure, when standing alone, always denotes the same number, which is called its simple value. The unit, or 1, is the least figure used, and all the others, except the cipher, are made up of as many units as are expressed by their common name; thus, 1 represents one single thing, 2 represents, or stands for two ones, 3—three ones, 4—four ones, 5–five ones, 6—six ones, 7—seven ones, 8—eight ones, 9—nine ones. The cipher never represents any thing of itself. Its use will be explained hereafter. When figures are written alone, they do not express quantity; for this they are dependent upon words or other characters; for if we write the figure 6, and the question be asked, six what? there is no definite answer to the question. When figures stand alone, they express a certain number of units which do not represent quantity, and they can only be said, in such situations, to be indefinite, or abstract numbers. But if we write a word or character expressing quantity in connection with the figure, the word or character denotes what particular kind of quantity is meant, and the figure expresses how much of that quantity is taken into calculation ;—thus, in the expressions, 5 dollars, 5 hours, the words dollars and hours denote particular kind of quantity, and the figure 5 expresses the number of parts of each quan: tity considered; in this respect, figures represent real values. From what has been said, it appears that parts of as many quantities can be expressed by the same

* These figures are called Arabic, because they were brought into Europe by the inhabitants of Arabia: Historians suppose they were brought into Arabia from India. They were first used in Eu‘rope about the year 1150.

figure, as we can find words to express different quantities, and that the real value of each unit in the figure, will vary in the same manner as the quantities

: H themselves; for in the phrases, 2 dollars, o 2 years, 2 acres, 2 men, the figure expresses the same number of parts of each quantity, but an unit in each place has a different value.

COMBINATION of Figures, or The ME-
THOD OF WRITING TWO OR MeRE Fi-
GURES IN CONNECTION.

When several figures are written in connection, the right hand figure expresses the same number of units, and is called by the same name, as though it stood alone; but the figure in the second place expresses ten times as many units as it would if it stood alone, or at the right hand, and is called as many tens as it would express units if it stood by itself. The figure in the third place denotes as many hundreds as it would units if it stood alone, or as many tens as it would units if it stood in the second place. The figure in the fourth place signifies as many thousands as it would hundreds in the third place. If we write four 1’s in connection, the 1 at the right hand is an unit; the 1 in the second place, is a collection of ten units represented by one figure; the 1 in the third place, a collection of a hundred units, and the 1 in the fourth place, a collection of a thousand units:

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Thus, 1 1

Ur In the adjoining to. let us suppose To the character under Úto be a tube or hollow ease, containing one cent resting upon its edge. Now, if we wish to represent this cent by a figure, the figure 1 standing directly under the character U, will express it. Next, let us suppose the tube under Y to be ten times as long as that under U, and filled with cents standing the same as the other, it would of course contain ten, and the figure 1, standing directly under it in the second place, represents the same number. Lastly, let us suppose the tube under H, to be ten times as long as that under Y, or made by placing ten other tubes, one above another, each equal to that under Y; it would hold ten times as many cents as the tube Y, or a hundred times as many as the tube U. But as a figure in the third place expresses one hundred units, the figure 1 standing under the tube H, represents as many cents as that tube would contain. Each unit in this-number represents one cent; and whenever any other quantity is represented by figures, each unit will express one of the equal parts into which such a quantity is divided. To express the numbers between one and ten, we make use of the other figures, as 2, 3, 4, 5, &c. The names of all the figures standing in the second place, except the 1, are contractions; thus, 2 tens is called twenty-3 tens, thirty,+4 tens, forty, —5 tens, fifty,+6 tens, sixty-7 tens, seventy,+8 tens, eighty-9 tens, ninety. The numbers between ten and twenty, twenty and thirty, &c. are expressed by placing a figure to the right hand of the tens. % express ten and one, ten and two, &c. by writing the figures 1, 2, 3, &c. to the right hand of the ten; thus, 11, 12, 13; and they may be read ten and one, ten and two, ten and three, and so on; but to express them in a shorter manner, we make use of the contractions, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, and nineteen. . To express the numbers between twenty and thirty, thirty and forty, &c. instead of saying two tens and one, two tens and two, three tens and one, three tens and two, &c. we repeat the contractions, twenty-one, twenty-two, thirty-one, thirty-two, &c. Any figure,

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when standing in any place to the left of the right hand place, represents ten times as many units, as it would do if it stood in the next place to the right. From this principle, by the use of the nine figures and the cipher, we can express an indefinite number of quantities. In the expression, 2 dollars, we express the number of dollars by the figure 2 ; but if we write 1 to the right hand of the 2, the 2 is removed into the second place, (21 dollars,) and denotes twenty dollars; and by writing another 1, (211 dollars,) to the right hand, the 2 is removed into the third place, and represents two hundred dollars. We might continue this operation at pleasure, and every time we place a figure to the right hand, those to the left would express ten times as many as they did before; so that it is easy to perceive, that we might express as many different quantities by the same figure, as we can place figures to the right hand. All the other figures increase in the same manner. The cipher, which is sometimes called zero, never has any value of itself, whether written alone or in connection with other figures; but when it is written with other figures, it alters their value. Thus, 1 denotes one, but if we write a cipher to the right hand of it, (10,) the 1 is removed into the second place, and signifies ten. But if the cipher be placed at the left hand of a figure, (01,) it does not alter the value of the figure, because it does not remove the figure from the right hand place. The value of a figure, when it does not stand at the right hand, is called its local walue.

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A Simple JName of numbers, is a word by who one place of figures in the same half period is dis

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