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class superior to the former; this again proceeds, (1 ten Thousands, or) Ten Thousands; (2 tens, or) Twenty Thousands; (3 tens, or) Thirty Thousands; and so on up to 10 ten Thousands, or One Hundred Thousands; which is the first of the next superior class; whence proceeding as before we have, 1 Hundred Thousands, 2 Hundred Thousands, 3 Hundred Thousands, &c. up to Ten Hundred Thousands, or I Million, &c. &c. 16. Hence it appears, that 1 ten is ten units; 2 tens, twenty units; 3 tens, thirty units; 4 tens, forty units, &c. : that 1 hundred is ten tens ; 2 hundreds, twenty tens; 3 hundreds, thirty tens, &c.; that 1 thousand is ten hundreds ; 2 thousands, twenty hundreds; 3 thousands, thirty hundreds; and in general that every superior denomination is tenfold the next inferior one; and also that any part of a superior denomination is in like manner tenfold the same part of the next inferior one. 17. It follows, from Art. 12. that there are many intermediate numbers, which, according to the preceding arrangement, must fall under two or more of the foregoing denominations: thus, twenty-five consists of 2 tens and 5 units; six hundred and seventy-eight consists of 6 hundreds, 7 tens, and 8 units; three thousand four hundred and fifty-six consists of 3 thousands, 4 hundreds, 5 tens, and 6 units, &c. &c. Hence a distinct idea of the value of any numbers may be formed from this convenient and beautiful mode of arrangement. 18. Having given a sketch of the general outline, the next thing to be explained is the method of expressing all numbers by the ten digits or figures; in order to which we observe, that each figure, unconnected with any of the other figures, stands merely for its own simple value ; but each has besides a local value, namely, a value which depends on the place it occupies when connected with others ; thus a figure standing in the first or right hand place expresses only its simple value; but if another figure or the cipher be placed to the right of it, then the figure first mentioned expresses ten times its simple value, that is, as many tens as it contains units. If two figures or ciphers be placed to the right of a figure, that figure expresses ten times what it did when it had only one on its right, or one hundred ,times its simple value; and so on continually. 19. Hence appears the use of the cipher, which although it is of no value in itself, yet when placed on the right of any numWOL. I. C

ber, it increases the value of that number tenfold; thus 5 stand-
ing by itself expresses simply five ; but if a cipher be placed on
its right, thus 50, it then becomes fifty, or ten times 5; if two
ciphers be placed, thus 500, it becomes five hundred, or ten times
fifty its former value; let another cipher be placed to the right
of the last, and the number becomes 5000, or five thousand,
which is ten times five hundred, &c.
20. From the two preceding articles, the method of express-
ing any number by figures may be easily inferred: thus, if it be
required to express by figures the number twenty-five, or two
tens and five units, it is evident (art. 18.) that five units must be
expressed by a 5 in the right hand place of the number to be
written, and that the two tens must be expressed by writing a 2
in the second place, or to the left of the 5; thus 25. Six hun-
dred and seventy-eight (or six hundreds, seven tens, and eight
units) is expressed by writing 8 in the (right hand or) first
place, 7 in the second, and 6 in the third; thus 678; in like

manner three thousand four hundred and fifty-six, expressed in T

figures, is 3456, where the 6 represents 6 units, the 5 five tens, or fifty, the 4 four hundreds, and the 3 three thousands.

21. Numeration, or the reading of numbers, is effected in the following manner; point to the first (or right hand) figure of any number, and call it units; point to the second, and call it tens; to the third, and call it hundreds; to the fourth, and call it thousands; to the fifth, and call it tens of thousands; to the sixth, and call it hundreds of thousands; to the seventh, and Call it millions; to the eighth, and call it tens of millions; to the ninth, and call it hundreds of millions; to the tenth, and call it thousands of millions; to the eleventh, and call it tens of thousands of millions; to the twelfth, and call it hundreds of thousands of millions, &c. &c. Then (beginning at the left) read the figures back again from left to right, adding to the name of each figure the denomination you gave it when reading from right to left: in this manner the numbers in the following table are to be read,

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Here the denominations are placed over the figures, those in the first column being units, those in the second tens, those in the third hundreds, &c. wherefore the first line of the table will be nine (units), the second ninety-eight, the third nine hundred and eighty-seven, the fourth nine thousand eight hundred and seventysir, &c. and the last one hundred and twenty-three thousands four hundred and fifty-six millions, seven hundred and eighty-nine thousands, five hundred and sixty-seven. When a number contains one or more ciphers, the denominations which the ciphers occupy are to be omitted in reading; thus, 405 is read four hundred and five; here are no tens : 30 is read thirty; here are no units : 70003 is read seventy thousands and three ; here the denominations of tens, hundreds, and thousands are wanting. 22. The method of classing numbers as above explained may be extended to any length: but the most convenient method of assisting the mind to form an idea of large numbers is to divide them into periods of six figures each, beginning at the right, calling the first period units, the second millions, the third billions, &c. according to the following table: where it must be remarked, that each period contains units, tens, hundreds, thousands, tens of thousands, and hundreds of thousands, of the denomination marked over that period.

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The right hand place of each denomination is units of that denomination: but we do not pronounce the word units in reading, except at the right hand place of all; instead of it we say, millions, billions, &c. naming the right hand figure of each period simply by the denomination marked over that period. 23. When any number expressed in words is required to be expressed in figures, if the learner is at a loss how to do it, he may make as many dots (placing them in a line from right to left) as there are places in the number to be written, calling the right hand dot the place of units, the second the place of tens, and so on; then under the said place of units put the units figure of the number to be written; under the place of tens put the tens figure of the number; under the place of hundreds put the hundreds figure of the number, &c. and if at last there be any dot without a figure under it, the place must be supplied by a cipher. Thus to write the number four thousand three hundred and fifty-six in figures—here are units, tens, hundreds, and thousands ; four dots ' ' ' ' must therefore be made; the left hand dot representing the place of thousands, 4 must be placed under it; under the next dot, or place of hundreds, 3 must be placed; under the next, which represents the place of tens, 5 must be placed; and 6 under the right hand dot, which represents the place of units; thus 4356. To write in figures one million two thousand and thirty; here we want the place of units, tens, &c. up to millions; mine dots will therefore be necessary, thus, . . . . . . . . . ; put 1 in the millions place, 2 in the thousands place, and 3 in the tens place, and you will have

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i 2 3 ; then, supplying the vacant places by ciphers, the

ExAMPLEs. Write in figures the following numbers. 1. Twenty-four. 2. Three hundred and sixty-two. 3. Seven thousand two hundred and forty. 4. Ninety thousand. 5. Eight hundred and ten. 6. One million and mine. 7. Sixty-seven thousand two hundred and one. 8. Two hundred million three hundred thousand four hundred. 9. One million and sixty-four. 10. Thirty thousand three hundred and thirty-three. 11. Five hundred billions. Write or express in words the following numbers. 14. . . 23 . . . 70 . . . 123 . . . 590 . . . 509 . . . 4321 . . . 5040 . . . ] 002 . . . 23456 . . . 30405 . . . 987654 . . . 100200 . . . 234567 . . . 908007O . . . 81796354 . . . 701820734 . . . 10200300040000. 24. The following characters are employed to mark the conmection of numbers, or to denote certain operations. The mark+ (named plus, or more) denotes addition. The mark — (named minus, or less) denotes subtraction. The mark x (named into) denotes multiplication. The mark -- (named by) denotes division. The mark w/ is called a radical sign; and a line drawn over two or more numbers, (serving to connect them,) thus 3×4, is called a vinculum. The mark = is the sign of equality. The further use of these characters will be explained in the proper places.

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