532, 123, 410,864,232,012, 345, 862,051,234, 525,411, 213, 673. By this table it will be seen that any number, however large, after dividing it into periods, and knowing the names of the period:s, can be read with the same ease as one consisting of three figures only; for the same names, (hundreds, tens, units,) are repeated in every period, and we have only to join to these, successively, the names of the periods. The first, or right hand period, is read, six hundred seventy-three binits, the second, two hundred forty-three thousands, the the third, four hundred eleven millions, and so on. 76. The foregoing is according to the French numeration, which, on account of its simplicity, is now generally adopted in English books. la the older Arithmetics, and in the two first editions of this work, a period is made to consist of six figures, and these were subdivided into half periis, as in the following TABLC III. iPrsiods. Sextill. Quintill. Quadrill. Trill . Billions. Millions. Units. Half pet. th. un. th. An.. th. II. th, un. th. An. cxt. Cxu. Figures. 1532,123,410,86-1, 232,012, 345,862,051,234,525,411, 243,673 'These two methods agree for the nine first places; but beyond this, the places take different naines. Five billions, for example, in the former method, is read five thousand millions in the latter. The principles of Rotation are, notwithstanding, the same in both throughout-the difference cunsisting only in enunciation:. th. un. EXAMPLES FOR PRACTICE. Write the following in figures: Emerate, or write the followEight. Seventeen. Ninety-three. / ing in words: Three hundred sixty Five thou 7890112 cand four hundred and seven. Thir 63 71351234 ty thousand fifty ninc. Seven 123 137111055 millions. Sixty-four billions. One 2040 8900000000 hundred nine quadrillions, one hun 60735 50000010010 dred nine millions, one hundred nine 123456 222000111002 thousand, one hundred and nine. one? 97,78 ADDITION. REVIEW. ! 1. What is meant by a unit, or 17. What to the second place? 18. How would you write two 2. What is number? hundred and twenty-two? 3. How are the numbers formed 19. What is the fundamental law and named from one to ten? of Notation? 4. Is the same course pursued 20. How many kinds of value with the higher numbers? why not? | have figures? 5. From what are the names a. 21. Upon what does their local bove ten derived? values depend? 6. Name the collections of tens. 22. What are the local values 7. How are the intermediate called? numbers expressed? 23. Repeat the names of the 8. Explain the method of ex. places. pressing number above one hundred. 24. What is seen by the first Nu9. What constitutes the spoken meration table? numeration? 25. What is the rule for reading 10. How is the expression of numbers? mimbers abridged? 26. How are large numbers, some11. What is Notation? How ma- times divided? .ny methods are there? 27. What is learned from the 12. What are the Roman numer- second, table? als? 28. What names are repeated in 13. Are they in general use? every period? 14. Name the Arabic characters. 29. What is the difference boo 15. How are numbers above ninetween the French and English meexpressed by them? thods of numeration? 16. What is the name given to 30. What is Numeration? the first place, or right hand figure 31. What is Arithmetic? of a number? SECTION II, SIMPLE NUMBERS. 77. Numbers are called simple, when their units are all of the same kind, as men, or dollars, &c. 1. ADDITION. ANALYSIS. 78. 1. How many cents are 3 cents and 4 centa? Here are two collections of cents, and it is proposed to find how large a collection both these will make, if put together. The child may not be able to answer the quiestion at once; but having learned how to form numbers by the successive addition of unity, (2, 72,) he will perceive that he can get the answer correctly, either by adding a unit' to 4 three times, or a unit to 3 four times, (7). În this way he must proceed, cill, by practice, the results arising from the addition of small numbers ere committed to memory; and then he will be able to answer the ques tions which involve such additions almost instantaneously. But when the numbers are large, or numerous, it will be found most convenient to write them down before performing the addition. 2. A boy gave 36 cents for a book,' and 23 cents for a skate, how many cents did he give for both? Here the first number is made up of 3 tens and 6 units, and the second of 2 tens and 3 units. Now if we add the 3 units of one with the 6 units of the o-her, their sum is 9 mits, and the 2 tens of one added to the 3 tens of the other, their sum is 5 tens. These two results taken together, are 5 tens and 9 units, or o?, --shich is the number of cents given for the 'book and slate. The common way of performing the above operation is to write the numbers under one another, so that units 36 cents. shall stand under units, and tens under tens, as at the 23 cents. left hand. Then begin at the bottom of the right hand column, and add together the figures in that column; "Aas. 59 cents. thus, 3 and 6 are 9, and write the 9 directly under the column. ' Proceeding to the column of tens, we say, 2 and 3 are 5, and write the 5 directly under the column of tens. Then will the 5 tens and 9 units each stand in its proper place in the answer, making 59. - 3. If a man travel 25 miles the first day, 30 the next, and 33 the next, how far will he travel in the threc days? Ans. 88 miles. 79. 4. A man bought a pair of horses for 216 dollars, a sleigh for 84 dollars, and a harness for 63 dollars; what did they all cost him? Here we write down the numbers as before, and be. 216 dolls. gin with the right hand colunm-3 and 4 are 7, and 6 84 dolls. are 13; but 13 are l'ten and 3 units; we therefore write 63 dolls. the 3 under the column of units, and carry the 1 ten te the column of tens, saying, 1 to 6 are 7, and 8 are 15, Ans. 363 dolls. and 1 are 16. But 16 tens are 1 hundred and 6 tens; We therefore write the 6 under the column nf tens, and carry the 1 into the column of hundreds, saying, 1 to 2 are 3, which we write down in the place of hundreds, and the work is done. From what precedes, the scholar will be able to understand the following definition and rule. SIMPLE ADDITION. 80. Simple Addition is the uniting together of several simple numbers into one whole or total number, called the sum, tor amount. RULE. -81. Write the numbers to be added 'under one another, 'with units under units, tens under tens, and so on, and draw a line below' them. Begin at the bottom, and add up the 'figures in the right hand column:-if the sum be less than ten, write it below the 'line at the foot of the column ; if it be ten, or an exact number of tens, write a cipher, and carry the tens to the next column; or if it be more than ten, and aot an exact number of tens, write down the excess of tens, mond-carry the tens as above. Proceed in the same way with 82. SIMPLE ADDITION. the columns of tens, hundreds, &c. always remembering, that ten units of any one order, are just .equal to one unit of the next higher order. PROOF. 82. Begin at the top, and reckon each column downwards, ard if their amounts agree with the former, the operation is supposed to have been rightly performed. NOTE.—No method of proving an arithmetical operation will demonstrate the work to be correct; but as we should not be likely to commit errors in both operations, which should exactly balance each other, the proof renders the correctness of the operation highly probable. QUESTIONS FOR PRACTICE. 5. According to the census 9. In a certain town there of 1820, Windsor contained are 8 schools, the number of -2956 inhabitants, Middlebury scholars in the first is 24, in 2535, Montpelier 2008, and the second 32, in the third 28, Burlington 2111; how many in the fourth 36, in the fifth inhabitants were there in 26, in the sixth 27, in the sethose four towns? venth 40, and in the eighth Operation. 58; how many scholars in all 2956 Windsor. the schools? Ans. 251. 2535 Middlebury. .10. Sir Isaac Newton wa's 2308 Montpelier. 2111 Burlington. born in the year 1642, and was 85 years old when he di. 9910 Total. ed; in what year did he die? Ans. 1727. 9910 Proof. 11. I have 100 bushels of wheat, worth 125 dollars, 150 6. A man has three fields, bushels of rye, worth 90 dolone contains 12 acres, ano- lars, and 90 bushels of corn, ther 23 acres, and the other worth 45 dollars; how many 47 acres; how many acres are bushels have I, and what is it there in the whole? worth? Ans. 340 bush. Ans. 82. worth 260 dolls. 7. A person killed an ox, 12. A man killed 4 hogs, the meat of which weighed one'weighed 371 pounds, one :842 pounds, the ‘hide 105 510 pounds, one 472 pounds, "pounds, and the tallow 92 and the other 396 pounds; pounds; what did they all | what did they all weigh? weigh? Ans. 839. Ans. 1749 pounds. 8. How many dollars are '13. The difference hetween 2565 dollars, 7009 dollars, and two numbers is 5, and the 1796 dollars, when added toge- least number is 7:; ther? Ans, 10870 dolls. lithe greater? .Ang. 12 e bags? 14. The difference between 1 and 14 from Woodstock to itwo numbers is 1448, and the Windsor; how far is it from least number is 2575 ; what Burlington to Windsor? is the greater? Ans, 4023. Ans. 99 miles. 15. There are three bags 18. How many days in a of money, one contains 6462 common year, there being in dollars, one 8294 dollars, and January 31 days, in February the other 5749 dollars; how 28, in March 31, in April 30, many dollars in the three in May 31, in June 30, in JuAns. 20485 dolls. ly 31, in August 31, in Sep 16. According to the cen tember 30, in October 31, in sus of the United States in November 30, and in Decem 1820, there were 3995053 free her 81 days? Ans. 365. white males, 3866657 free, white females, and 1776289 bis age, said that he was 9 19. A person being asked persons of every other descrip-years old when his youngest tion; what was the whole brother was born, that his bronamber of inhabitants at that ther was 27 years old when time? Ans.9637999. his eldest son was born, and 17. It is 38 miles from Bur- that his son was 16 years old; Hington to Montpelier, 47 from what was the person's age? Montpelier 0 Woodstock, Ans. 52 years. 21. 22. 23. 23213 2424612 8192735 9876987 89862184 16423 1234567 214268 7986698 409613ES 21230 7654321 1541320 4343434 695646 90418 2112710 40212 2121212 94965 20. 24. .25. 2746+390+100179976+432176639=25067, Ans. ..6.99548216+.4826852+19181716=63551264, Ans. · 2. MULTIPLICATION. ANALYSIS. 83. We have seen that Addition is an operation by which several numbers are united into one sum. Now it frequently happens that the numbers to be added are all equal, in wbich case the operation may be abridged by a process called Multiplication. 3. If a book cost 5 cents, what rv:ll 4 such books cost? |