say, two hundred forty-six. The collections of ten hundreds are called thousands, which take their names from the collections of units, tens and hundreds, as, one thousand, two thoro sand, ten thousand, twenty thousand, one hundred thous sand, two hundred thousand, &c. The collections of ten hun dred thousands are called millions, the collections of ten hundred millions are called billions, and so on to trillions, quadrillions, &c. and these are severally distinguished like the collections of thousands. The foregoing names, combined according to the method above stated, constitute the spoken numeration. 73. To save the trouble of writing large numbers in words, and to render computations more easy, characters, or symbols, have been invented, by which the written expression of numbers is very much abridged. The method of writing numbers in characters is called Notation. The two methods of notation, which have been most extensively used, are the Roman and the Arabic.* The Roman numerals are the seven following letters of the alphabet, I, V, X, L, C, D, M, which are now seldom used, except in numbering chapters, sections, and the like. The Arabic characters are those in common use. They are the ten following: O cipher, or zero, 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine. The above characters, taken one at a time, denote all the numbers from zero to nine inclusive, and are called simple units. To denote numbers larger than nine, two or more of these characters must be used. Ten is written 10, twenty 20, thirty 30, and so on to ninety, 90; and the intermédiate numbers are expressed by writing the excesses of simple units in place of the cipher; thus for fourteen we write 14, for twentytwo, 22, &c.(13) Hence it will be seen that a figure in the second place denotes a number ten times greater than it does when standing alone, or in the first place. The first place at the right hand is therefore distinguished by the name of unil's place, and the second place, which contains units of a *A comparison of the two methods of notation is exhibited in the following TABLE. 1=I 110=X 1100=C 11000=M orCIO 10000= orCCI 2=II 20=XX 200=CC 1100=MC 3=III 30=XXX 300=CCC 1200=MCC 4=IV 40=XL 400=CCCC 1300=MCCC 100000=CCC1053 15=V 50=L 500=D orld 1400=MCCCC 1000000=M 6=VÌ 160=LX 600=DC 1500MD 2000000= 7=VII 70=LXX 700 - DCC 2000=MM 1829==MDCCCXXIX 18=VIII 80=LXXX SON=DCCC 5000=1ɔɔ or v A=IX 90=XC 900=ncccc16000=yt 150000=1025 =MM higher order, is called the ten's place. Ten tens, or one hundred, is written, 100, two hundred, 200, and so on to nine hundred, 900, and the intermediate numbers are expressed by writing the excesses of tens and units in the tens' and units' places, instead of the ciphers. Two hundred and twenty-two is written, 222. Here we have the figure 2 repeated three times, and each time with a different value. The 2 in the second place denotes a number ten times greater than the 2 in the first; and the 2 in the third, or hundreds’ place, denotes a number ten times greater than the 2 in the second, or ten's place; and this is a fundamental law of Notation, that each removal of a figure one place to the left hand increases its value ten times. 74. We have seen that all numbers may be expressed by repeating and varying the position of ten figures.' In doing this, we have to consider these figures as having local values, which depend upon their removal from the place of units. These local values are called the names of the places: which may be learned from the following TABLE I. Sextillions. Tens of Bill. Tens of Mill. Tens. By this table it will be seen that 2 in the first place denotes simply 2 units, that 3 in the second place denotes as many tens as there are simple units in the figure, or 3 tens; that 2 in the third place denotes as many hundreds as there are units in the figure, or 2 hundreds; and so on. Hence to read any number, we have only to observe the following Rule.- To the simple value of each figure join the nume of its place, beginning at the left hand, and reading the figures in their order towards the right. The figures in the above table would read, three sextillions, four hundred fifty-six quintillions, seven hundred fifty-four quadrillions, three hundred seventy-eight trillions, four hundred sixty-four billions, nine hundred seventy-four millions, three hundred one thousand, two hundred thirty-two. 75. In reading very large numbers it is often convenient to divide them into periods of three figures each, as in the following 532, 123,410,864,232,012, 345, 862,051,234, 525,411, 243, 678. By this table it will be seen that any number, however large, after dividing it into periods, and knowing the names of the periods, 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 -units, 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. In the older Arithmetics, and in the two first editions of this work, a period is made to consist of șix figures, and these were subdivided into half periods, as the following TABLE III. Periods. Sextill. Quințill. Quadrill. Trill. Billions. Millions. Units. Half per. th. un. th. un. th. An, th, un. the un. txt. xu. Figures. 1532,123, 410,864, 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 names. Five billions, for example, in the former method, is rcad five thousand millions in the latter. The principles of notation are, notwithstanding, the same in both throughout--the differenca consisting only in enunciation. th. An. EXAMPLES FOR PRACTICE. Write the following in figures: Env:nerate, or write the follow. Eight. Seventeen. Ninety-three. ing in words: 'Thrce hundred sixty. Five thou 7890112 sand four hundred and seven. Thir. 65 74351234 ty thousand fifty nine. Seven 123 137111055 millions. Sixty-four billions. One 2040 8900000000 hundred nine quadrillions, one hun 60735 30000010010 dred nine millions, one hundred nine 123456 222000111002 thousand, one hundred and nine. one? 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. Wbat 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? numbers 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 be. 15. How are numbers above nine tween the French and English me expressed 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. 71. 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 cents? Here are two collection 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 question at once; but having learned how to form numbers by the successive addition of unity, (2, 72,) he will perceive that be can get the answer correctly, either by adding a unit to 4 three times, or a unit to 3 four times, (7). In this way he must proceed, cill, by practice, the results arising from the addition of sınall numbers are committed to memory; and then be will be able to answer the ques ciens which involve such additions almost instantaneously. But when the Rumbers are large, or numerous, it will be found most convenient to writo them down before performing the addition. 2. A boy gave 36 cents for a book, and 23 cents for a slate, how many centa did be 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 oʻher, their sum is 9 waits, 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 59, which 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 caluma, and add together the figures in that column; Ans. 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, Daking 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 three 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 be216 dolls. gin with the right hand column3 and 4 are 7, and 6 84 dolls. are 13; but 13 are 1 ten and 3 units; we therefore write 63 dolls. the 3 under the column of units, and carry the 1 ten to 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 of 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, or amount. RULE. 81. Write the numbers to be added under one another, with units under units, tens under ter 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 Rot an exact number of tens, write down the excess of tens, and carry the tens as above. Proceed in the same way with |