ber, it increases the value of that number tenfold; thus 5 standing by itself expresses simply fire; 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 fire 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 fire thousand, which is ten times five hundred, &c. > 20. From the two preceding articles, the method of expressing 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 hundred 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 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 fol lowing table are to be read. 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 hun~ dred 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. QUADRILLIONS. TRILLIONS. BILLIONS. MILLIÓNS. UNITS. Units of 6 17,8 3 4. 1 3 0,9 2 7. 6 3 1,8 2 9. 4 0 3,1 7 2. S 9 5,2 6 3. 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 hun. dred 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; nine 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 1 23; then, supplying the vacant places by ciphers, the number will become 100002030, which is what was required. 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 nine. 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 ... 1002 ... 23456 30405 ... 987654 ... 100200 9080070 .. 81726354 ... 701820734 ... ... 234567 ... 10200300040000. ... 24. The following characters are employed to mark the connection 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 × (named into) denotes multiplication. The mark (named by) denotes division. The mark ✅ 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. ADDITION. 25. Simple Addition teaches to collect two or more whole numbers into one, which is called their sum. The mark for Addition is + plus, (more,) and shews that the number which follows the sign, is to be added to the number standing before it. 26. To add single figures together. RULE I. Begin at the bottom, and find what number will The word Addition is derived from the Latin addo, to put to; sum from summa; and proof from probo, to prove, or make out. b This rule depends on the method of notation; (Art. 12.) thus, in the first example, if I want to know the sum of 7 and 4, I must evidently resolve each into the units of which it is composed, and then count all the units in both, one by one, to find the amount; and this practice I must follow until my mind acquires from habit, a sufficient dexterity in numbering to do without it. The only method that a person totally ignorant of Addition could employ, would be this; he would write down all the ones in each of the numbers to be added, and then count the whole. Thus ex. 1. would stand according to such a method, These ones being counted, are found to amount to 21, which is the sum of the given numbers. Simple as this explanation may appear, it is plain that the reason of the rule can be shewn on no other principle. By this method this following Table was first calculated. and 5 in the top line, and at the point where the two rows of figures (one vertical, the other horizontal) meet stands 9, which is the sum of 4 and 5. To find the sum of 7 and 6, look for 7 on the left, and 6 at top, and at the point In like manner where the lines containing 7 and 6 meet stands 13, their sum. 5 and 9 are found to be 14; 8 and 3 are 11; 9 and 9 are 18, &c. &c. Those who |