note one parcel of twelve: and the figures 203 would denote the number two hundred and ninety-one; for the 2, standing in the third column, would denote two parcels of twelve times twelve each, that is, two hundred and eightyeight. And certainly if this duodecimal notation had been originally adopted, and the language accommodated to it by affording distinct names for the several combinations of twelve, it would have possessed a considerable advantage over the decimal notation, which proceeds by combinations of tens. For the number twelve admitting four divisors, (namely, 2, 3, 4, 6,) while the number ten can be evenly divided only by 2 and 5, we would be much less frequently involved in fractional remainders than at present. And if all the divisions of measures, weights, coins, &c. ran in the same duodecimal progression, the practical advantages would be great. But it appears from the structure of all known languages, that numeration by tens has been adopted by all nations in all ages, rather than the numeration by twelves, or any other number. And this is obviously to be accounted for from the natural circumstance of the number of our fingers; the fingers being, in the origin of society, the readiest instrument to assist numeration; and still, indeed, frequently employed for that purpose by the illiterate peasantry. So that we may conclude, that if nature had furnished man with twelve fingers instead of ten, the duodecimal numeration would have been as general as the decimal now is; and languages would have abounded as much with names for the combinations of twelve, as they now do with names for the combinations of ten. To express Numbers by Figures, according to the Arabic Notation. 13. RULE.-Make a sufficient number of ciphers or dots, and divide them into periods; then, commencing at the left, place in their proper positions, beneath the ciphers or dots, the significant figures necessary for expressing the proposed number. If any places remain unoccupied, let them be filled with ciphers. Thus, the method of expressing the number two hundred and five millions, twenty thousand, seven hundred and nine, will be found in the following manner : and thence, by filling the unoccupied places, 205,020,709, By practice, the learner will soon be enabled, in most cases, to dispense with the ciphers or dots. Exercises in Notation. Express the following numbers in figures: Ex. 1. Fifty-two. 2. Three hundred and fifteen. 3. Four hundred and five. 4. One thousand three hundred and four. 5. Seven thousand and eighty-four. 6. Nine thousand and nine. 7. Six thousand and seventy. 8. Twenty thousand and seventy-three. 9. Four hundred and six thousand and nine. 10. Six hundred and fifty thousand and ninety. 11. Seven millions seven thousand and ten. 12. Twenty-five millions three hundred thousand. 13. Eleven millions two hundred and ten. 14. One hundred and ten millions and twenty thousand. 15. One billion ten millions two thousand. 16. One trillion one hundred billions two millions. 17. One quadrillion and nineteen millions. 18. Nine hundred billions six millions and five. 19. The world was created two thousand three hundred and forty-eight years before the Deluge; three thousand two hundred and fifty-one years before the building of Rome; four thousand and four years before the birth of Christ; five thousand four hundred and ninety-six years before the discovery of America; and five thousand eight hundred and thirty-one years before the present time (1827.) Let each of these numbers be expressed in figures. 20. The following numbers express the distances of the primary planets from the sun, in American miles; express them in figures.-Mercury, thirty-seven millions; Venus, sixty-nine millions; the Earth, ninety-five millions; Mars, one hundred and forty-five millions; Vesta, two hundred and twenty-five millions; Juno, two hundred and fifty-three millions; Ceres, two hundred and sixty-two millions seven hundred and fifty thousand; Pallas, two hundred and sixtythree millions; Jupiter, four hundred and ninety-four millions; Saturn, nine hundred and six millions; Uranus, or Herschel, one billion eight hundred and twenty-two millions. To express in Words the Numbers denoted by Lines of Figures. 14. RULE.-Commencing at the right hand, divide the given figures into periods of three figures each, till not more than three remain. Then the first period towards the right hand contains units or ones, the second thousands, the third millions, &c. as in the Numeration Table: and, therefore, commencing at the left side, annex to the value expressed by the figures of each period, except that of the units, the name of the period. Thus, the expression 37053907 becomes, by division into periods, 37,053,907, and is read thirty-seven millions fiftythree thousand nine hundred and seven, the term units or ones, at the last, being omitted. By practice, the pupil will soon find it unnecessary to divide into periods any lines of figures except those of considerable magnitude. Exercises in Numeration. Write down in words, or name, the numbers signified by the following expressions : What is magnitude or quantity? What is the term of comparison called, by which we measure any quantity ? What is a number? And what gives rise to whole numbers? What is numeration? What is arithmetic ? How are whole numbers formed? How is the number one hundred formed? How many hundreds compose a thousand ? How many figures are usually necessary to express numbers? What are their names? How do you write a number composed of units, tens, hundreds, &c. by means of figures? Can all numbers be written by means of nine figures? What other figure, besides those nine, is necessary to express all numbers? What is the use of the character cipher or zero ? What is the rule for expressing numbers by figures? How do you read in words a number denoted by a line of figures? ADDITION OF WHOLE NUMBERS. 15. The operation by which we express the total value or amount of several given numbers in one sum, is called addition of whole numbers. For example, six dollars and nine dollars, expressed in one sum, are fifteen dollars. To perform the operation of addition, it is necessary that the learner should be able to assign the sum of any two given numbers not exceeding nine; and for this purpose he shoul be exercised in the following table. ADDITION TABLE. 2 and 2 and 3 and 4 and 5 and 6 and 7 and 4 6 3 and 9=128=128=139=15 8=117=117=128=14 8 and! 8=16 9=17 6-8 4 7 4 and 7 and 7 9 5 8 4 8 5 and 6 and 7=149 and To enable the learner to acquire accuracy and despatch in addition, it is proper to train him to add in the following manner, till he can do it with facility. Since 6 and 6 are 12, 26 and 6 are 32; (here it should be pointed out to him that 12 and 32 end in the same figure :) since 9 and 7 are 16, 39 and 7 are 46; since 8 and 6 are 14, 88 and 6 are 94; since 9 and 6 are 15, 9 and 16 are 25; since 8 and 9 are 17, 8 and 99 are 107, &c. 16. In addition, we successively take the sum of the digits standing in each column, and combine those sums into one total. The reason of commencing from the right hand column, or place of units, and proceeding from right to left, is, that we may carry on the combination of the sums of the several columns as we proceed. Thus, in adding together 509 and 293, the sums of the numbers standing in the several columns are 12 units, 9 tens, or 90, and 7 hundreds, or 700. Now, adding the 1 ten contained in the 12 units to the 9 tens, (the sum of the second column,) we have 10 tens, or 1 hundred; which added to the 7 hundreds, (the sum of the third column,) gives 8 hundreds; and these combined with the 2 units in the sum of the first column give 802 as the total. By proceeding from right to left, we are saved the trouble of writing the several sums of the several columns separately, and afterwards combining them by a second addition. We write down under each column the right hand figure of its sum, and carry the other figures to the next column. But the same result will be obtained by reDeated additions, proceeding from left to right, or taking the ums of the columns in any order. And in this way, the ung scholar may be made to prove his work. |