¶ 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 secona 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 tens 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 for 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, thas, 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. ¶ 3. Hence it appears, that figures have a different value according to the PLACE they occupy, counting from the right hand towards the left. Hund. co Tens. Units. 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: ∞ Units → Tens → Units Hundreds Hundreds • Tens Hundreds EXAMPLE 1st. EXAMPLE 20 3 17 4 5 9 2 8 3 7 4 6 3 5 1 2 3, 1 7 4, 5 9 2, 8 3 7, 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 six places, instead of three, ta millions, billions, &c. Note. Should the pupil find any difficulty in reading the following numbers, let him first transcribe them, and point them off into periods. The expressing of numbers, (as now shown,) by figures, is called Ñotation. The reading of any number set down in figures, is called Numeration. After being able to read correctly all the numbers in the foregoing table, the pupil may proceed to express the following numbers by figures: 1. Seventy-six. 2. Eight hundred and seven. 3. Twelve hundred, (that is, one thousand and two hundred.), 4. Eighteen hundred. 5. Twenty-seven hundred and nineteen. 6. Forty-nine hundred and sixty. 7. Ninety-two thousand and forty-five. 8. One hundred thousand. 9. Two millions, eighty thousands, and seven hundreds. 10. One hundred millions, one hundred thousand, one hundred and one. 11. Fifty-two millions, six thousand, and twenty. 12. Six billions, seven millions, eight thousand, and nine hundred. 13. Ninety-four billions, eighteen thousand, one hundred and seventeen. 14. One hundred thirty-two billions, two hundred millions, and nine. 15. Five trillions, sixty billions, twelve millions, and ten thousand. 16. Seven hundred trillions, eighty-six billions, and seven millions. ADDITION OF SIMPLE NUMBERS. T. 1. James had 5 peaches, his mother gave him 3 peaches more; how many peaches had he then? 2. John bought a slate for 25 cents, and a book for eight cents; how many cents did he give for both? 3. Peter bought a waggon for 36 cents, and sold it so as to gain 9 cents; how many cents did he get for it? 4. Frank gave 15 walnuts to one boy, 8 to another, had 7 left; how many walnuts had he at first? and 5. A man bought a chaise for 54 dollars; he expended S dollars in repairs, and then sold it so as to gain 5 dollars; how many dollars did he get for the chaise? 6. A man bought 3 cows; for the first he gave 9 dollars, for the second he gave 12 dollars, and for the other he gave 10 dollars; how many dollars did he give for all the cows? 7. Samuel bought an orange for 8 cents, a book for 17 cents, a knife for 20 cents, and some walnuts for 4 cents; how many cents did he spend ? 8. A man had 3 calves worth 2 dollars each, 4 calves worth 3 dollars each, and 7 calves worth 5 dollars each; how many calves had he? 9. A man sold a cow for 16 dollars, some corn for 20 dollars, wheat for 25 dollars, and butter for 5 dollars; how many dollars must he receive? The putting together two or more numbers, (as in the foregoing examples,) so as to make one whole number, is called Addition, and the whole number is called the sum, or amount. 10. One man owes me 5 dollars, another owes me 5 dollars, another 8 dollars, another 14 dollars, and another 3 dollars; what is the amount due to me? 11. What is the amount of 4, 3, 7, 2, 8, and 9 dollars? 12. In a certain school 9 study grammar, 15 study arithmetic, 20 attend to writing, and 12 study geography; what is the whole number of scholars? SIGNS. A cross, +, one line horizontal and the other perpendicular, is the sign of addition. It shows that numbers, with this sign between them, are to be added together. It is sometimes read plus, which is a Latin word signifying more. Two parallel, horizontal lines,, are the sign of equality. It signifies that the number before it is equal to the number after it. Thus, 5 +3 = 8 is read 5 and 3 are 8; or, 5 plus (that is, more) 3 is equal to 8. In this manner let the pupil be instructed to commit the following ADDITION TABLE. 2+0=2 3+034+0=4 5+0 5 2+1=3 3+1 4 4+1 55+1 6 2+2=4 3+2 = 5|4 + 2 = 65+2= 7 2+35 3+ 64+37 5+3= 8 5+49 9 5+5=10 244 63+4= 74+4=8 2+5=7 3+5= 8 4 2+6= 8 3+6= 94 + 10 5+6=11 2+7=9 3+7=104+ 11. 5+7=12 2+8=10 3+8=11 | 2+9=11 3+9 124913 |