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 thousand, ten thousand, twenty thousand, one hundred thou sand, two hundred thousand, &c. The collections of ten hundred 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: 0 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, 96; and the intermediate 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 unit'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=1 110=X 2=11 20=XX |100=C 3=III 30= XXX 300=CCC 1000 M orCI 1100-MC 1200 MCC ¡4=IV 40=XL 400-CCCC 1300-MCCC 15=V 50=L 500 DorI 1400 MCCCC 6=VI 160=LX 600=DC 1500 MD 7=VII 70 LXX 700 DCC 2000=MM 8=VIII 80 LXXX S00 DCCC 5000=100 or v 0=IX 90=XC 900 pcccc 6000=vī 10000 or CC 50000=1000 60000 LX 100000-CCC 1000000=M 2000000=MM 1829 MDCCCXXIX 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. coSextillions. Hund. of Quint. Hund. of Quad. ∞Trillions. Hund. of Mill. Hund. of Thou. Tens of Thou. 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, 673. 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 six figures, and these were subdivided into half periods, as in the following TABLE III. Periods. Sextill. Quintill. Quadrill. Trill. Billions. Millions. Units. Half per. th. un. th. un. th. un. th. un. th. un. th. un. cxt. cxu. Figures. 532,123, 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 read five thousand millions in the latter. The principles of notation are, notwithstanding, the same in both throughout-the difference consisting only in enunciation. EXAMPLES FOR PRACTICE. Write the following in figures: Env.nerate, er vrite the follow Eight. Seventeen. Ninety-three. ing in words: 65 74351234 123 137111055 2040 8900000000 60735 30000010010 123456 222000111002 REVIEW. 1. What is meant by a unit, or one? 2, What is number? 3. How are the numbers formed and named from one to ten? 4. Is the same course pursued with the higher numbers? why not? 5. From what are the names above ten derived? 6. Name the collections of tens. 7. How are the intermediate numbers expressed? 8. Explain the method of expressing number above one hundred. 9. What constitutes the spoken numeration? 10. How is the expression of numbers abridged? 11. What is Notation? How many methods are there? 12. What are the Roman numerals? 13. Are they in general use? 14. Name the Arabic characters. 15. How are numbers above nine expressed by them? 16. What is the name given to the first place, or right hand figure of a number? 17. What to the second place? 18. How would you write two hundred and twenty-two? 19. What is the fundamental law of Notation? 20. How many kinds of value have figures? 21. Upon what does their local values depend? 22. What are the local values called? 23. Repeat the names of the places. 24. What is seen by the first Numeration table? 25. What is the rule for reading numbers? 26. How are large numbers sometimes divided? 27. What is learned from the second table? 28. What names are repeated in every period? 29. What is the difference bétween the French and English methods of numeration? 30. What is Numeration? 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 cents? Here are two collectio 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 he 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, till, by practice, the results arising from the addition of small numbers are committed to memory; and then he will be able to answer the ques ns which involve such additions almost instantaneously. But when the ambers 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 slate, how many cents did he give for both? are 5 tens and 9 book and slate. Here the first number is made up of 3 tens and 6 units, and the second of 2 tens and 3 unita. Now if we add the 3 units of one with the 6 units of the other, their sum is 9 units, and the 2 tens of one added to the 3 tena of the other, their sum is 5 tens. These two results taken together, units, or 59, which is the number of cents given for the The common way of performing the above operation is to write the numbers under one another, so that units shall stand under units, and tens under tens, as at the left hand. Then begin at the bottom of the right hand column, and add together the figures in that column; thus, 3 and 6 are 9, and write the 9 directly under the column. Proceeding to the column of tens, we say, 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. 36 cents. 23 cents. Ans. 59 cents. 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? 216 dolls. Here we write down the numbers as before, and begin with the right hand column-3 and 4 are 7, and 6 are 13; but 13 are 1 ten and 3 units; we therefore write 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, 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. Ans. 363 dolls. 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 ten and so on, and draw line below them. Begin at the bottom, and add up the gures in the right hand column:-if the sum be less than en, write it below the line at the foot of the column; if it e ten, or an exact number of tens, write a cipher, and carry he tens to the next column; or if it be more than ten, and not an exact number of tens, write down the excess of tens, and carry the tens as above. Proceed in the same way with |