Page 151 153 154 155 156 164 165 166 167 168 169 229 176 177 179 20. 187 192 195 197 201 207 207 212 215 220 225 237 Loss and Gain, si 82; 11 88, cx. 1-3. . . MISCELLANEOUS EXAMPLES. | Position, ex. 89—108. of a Triangle, ex. 155–159. Circumference, to find the Diameter, ex. 171–175. of a Globe, ex. 180, 181. of a Cylinder, ex. 185–187. of any Irregular Body, ex. 202, 203. | Mechanical Powers, ex. 192-201. . Forins of Notes, Bonds, Receipts, and Orders, 259 NUMERATION. T 1. A SINGLE or individual thing is called a unit, unity, or one ; one and one more are called two; two and one more are called three; three and one more are called four ; four and one more are called five; five and one more are called six ; six and one more are called seven; seven and one more are called eight; eight and one more are called nine ; nine and one more are called ten, &c. These terms, which are expressions for quantities, are called numbers. There are two methods of expressing numbers shorter than writing them out in words; one called the Roman method by letters,* and the other the Arabic method by figures. The latter is that in general use. In the Arabic method, the nine first numbers have each an appropriate character to represent them. Thus, * In the Roman method by letters, I represents one ; V, five ; X, ten; L, fifty ; C, one hundred ; D, five hundred; and M, one thousand. As often as any letter is repeated, so many times its value is repeated, unless it be a letter representing a less number placed before one representing a greater ; then the less number is taken from the greater ; thus, IV represents four, IX, nine, &c., as will be seen in the following TABLE. LXXXX. or XC. One hundred C. Two hundred CC. Three hundred CCC., Four hundred CCCC. Five hundred D. or 15.* Six Six hundred DC. Seven hundred DCG. VIIII. or IX. Eight hundred DCCC. Nine hundred DCCCC. One thousand M. or CIO. 133. or V.I. Fifty thousand 1500. Hundred thousand CCCI.. or T. One million M. Two million MM. * 15 is used instead of D to represent five hundred, and for every additional an dexed at the right hand, the number is increased ten times. + CIƆ is used to represent one thousand, and for every C and put at each end, the number is increased ten times. A line over any number increases its value one thousand times Forty as, A unit, unity, or one, is represented by this character, 1. 2. Three 3. Four 4. Five 5. Six . Seven 7. Eight 8. Nine 9. Ten has no appropriate character to 'represent it; but is considered as forming a unit of a second or higher in this case, there are no units to be written which of itself signifies nothing; thus, Ten 10. One ten and one unit are called Eleven 11. One ten and two units are called Twelve 12. One ten and three units are called Thirteen 13. One ten and four units are called Fourteen 14. One ten and five units are called Fifteen 15 One ten and six units are called Sixteen 16. One ten and seven'units are called Seventeen 17. One ten and eight units are called Eighteen 18. One ten and nine units are called Nineteen 19. Two tens are called Twenty 20. Three tens are called 30. Four tens are called 40. Five tens are called Fifty 50. Six tens are called Sixty 60. Seven tens are called Seventy Eight tens are called Eighty 80. Nine tens are called Ninety 90. still higher order, consisting of hundreds, represented One hundred 100. One hundred and eleven 111. . Thirty 70. 1 2. There are three hundred sixty-five days in a year. In this number are contained all the orders now described, viz. units, tens, and hundreds. Let it be recollected, units occupy the first place on the right hand; tens, the second place from the right hand; hundreds, the third place. This number may now be decomposed, that is, separated into parts, exhibiting each order by itself, as follows:- The highest order, or hundreds, are three, represented by this character, 3; but, that it may be made to occupy the third place, counting from the right hand, it must be followed by two ciphers, thus, 300, (three hundred.) The next lower order, or tens, are six, (six ters are sixty,) represented by this character, 6 but, that it may occupy the second place, which is the place of tens, it must be followed by one cipher, thus, 60, (sixty.) The lowest order, or units, are five, represented by a single character, thus, 5, (five.) We may now combine all these parts together, first writing down the five units für the right hand figure, thus, 5; then the six tens (60) on the left hand of the units, thus, 65; then the three hundreds (300) on the left hand of the six tens, thus, 365, which number, so written, may be read three hundred, six tens, and five units; or, as is more usual, three hundred and sixty-five. 1ī 3. Hence it appears, that figures have a different value according to the PLACE They occupy, counting from the right hand towards the left. und. ens. Take for example the number 3 3 3, made by the same figure three times repeated. The 3 on the right hand, or in the first place, signifies 3 units; the same figure, in the second place, signifies 3 tens, or thirty ; its value is now increased ten times. Again, the same figure, in the third place, signifies neither 3 units, nor 3 tens, but 3 hundreds, which is ten times the value of the same figure in the place immediately preceding, that is, in the place of tens ; and this is a fundamental law in notation, that a removal of one place towards the left increases the value of a figure TEN TIMES. Ten hundred make a thousand, or a unit of the fourth order. Then follow tens and hundreds of thousands, in the same manner as tens and hundreds of units. To thousands succeed millions, billions, &c., to each of which, as to units and to thousands, are appropriated three places,* as exhibited in the following examples : w Units Hundreds Tens EXAMPLE 1st. EŽAMPLE 2d. 3,1 m 4, 5 9 2, 8 37, 4 6 3, 5 1 2, To facilitate the reading of large numbers, it is frequently practised to point them off into periods of three figures each, as in the 2d example. The names and the order of the periods being known, this division enables us to read numbers consisting of many figures as easily as we can read three figures only. Thus, the above examples are read 3 (three) Quadrillions, 174 (one hundred seventy-four) Trillions, 592 (five hundred ninety-two) Billions, 837 (eight hundred thirty-seven) Millions, 463 (four hundred sixtythree) Thousands, 512 (five hundred and twelve.) After the same manner are read the numbers contained in the following * This is according to the French method of counting. The English, after hundreds of millions, instead of proceeding to billions, reckon thousands, tens and hundreds of thousands of millions, appropriating sić places, instead of three, 10 millions, billions, &c. |