SECTION I. NOTATION. NOTATION teaches the mode of expressing any number by certain signs, or characters. Numbers can be expressed in words; but this would be entirely impracticable for the purpose of performing arithmetical operations. Two methods have been used for writing numbers. One is by using Roman letters-the other by Arabic figures. Roman Numerals. I. One. V. XX. Twenty. Ten. XXX. Thirty. Seven. XI. Eleven. L. Fifty. IV. Four. VIII. Eight. XII. Twelve. C. Hundred. The Arabic figures are signs of numbers, each having a definite value when written alone, and a relative value when used in connexion with others. Arabic figures. 1, one; 2, two; 3, three; 4, four; 5, five; 6, six; 7, seven; 8, eight; 9, nine; 0, cipher. By the repetition and arrangement* of these figures, any number whatever can be easily expressed, as will be seen in the next section. QUESTIONS.-What are Arabic figures? Can any number whatever be expressed by them? "It is a fundamental law of notation, that a removal of one place towards the left hand increases the value of a figure ten times. With two figures we can express all as far as to nine tens and nine units, or 99. After this comes the hundred, expressed by the figure 1 placed one place farther to the left than it would be, if used to express tens only-as 100. Hence we see that the same figure expresses units ten times greater, in proportion as it is removed from right to left, and by a simple change of place acquires the power of representing successively all the different collections of units, which can enter into the expression of a number. In writing these characters so as to express numbers, it is important to keep in mind the order in which the collections succeed each other, and put ciphers in room of those which are wanting in the numeration, or enunciation of the numbers written. Thus in writing the number three hundred and twenty-four thousand, nine hundred and four, we SECTION II. NUMERATION. NUMERATION is the art of reading figures. There are two modes, which have been adopted in different countries. They may be distinguished into the English and French modes. Both are used to some extent in this country, though the former more generally. Several of the first figures are alike in both. 1. French Mode. Units Tens Tens of thousands Hundreds of thousands Millions. Billions 5 By examining the annexed table, the mode of enumeration is easily understood. The first or top figure, 5, standing alone, has merely its own simple value, i. e. five units. The left hand figure in the second line, 7, being removed one place towards the left, expresses not seven units, but seven tens, or seventy, and with the six at the right hand, seventysix. The left hand figure in the third line, 1, being removed two places to the left, has its value increased to a hundred, and with the figures at the right hand, is 5 3 9 7 6 4 2 1 45 read read one hundred and nine 76 198 5 4 3 2 9087-6 2 1 3 4 65 4 3 2 1987 6 5 4 3 2 109 4 8 1 3 24678 ty-eight. The fourth line extends one place farther to the left, or to the place of thousands; the fifth line to tens of thousands, and so on. The lower line is read thus: Five billions, three hundred and ninety-seven millions, six hundred and forty-two thousands, one hundred and forty-five. QUESTIONS.-What is the first figure called in the French mode? What is the second? Third? Fourth? Fifth? Sixth? Seventh? Eighth? Ninth? put 3 for the hundreds of thousands, 2 for the twenty thousands, or two tens of thousands, 4 for the thousands, 9 for the hundreds; and as the tens come next after the hundreds, and are wanting in this instance, we put a cypher in their place and then write 4 for the units, thus we have 324904."-(Lacroix.) 2. English Mode. It will be seen that this table is in all respects like the preceding till we come to thousands of millions, which are in the French mode called billions ;whereas in this mode the place of billions is considered three figures farther to the left, as is seen in this table. This method is the one used in the following work. Units. Tens Millions . Tens of millions Hundreds of millions Billions 5 76 198 5432 90876 2 1 3 4 6 5 4 3 2 1987 6 5 4 3 4 108 4 8 1 3 2 2 679 5 3 9 7 6 4 2 1 4 5 904 6 7 8 9 4 3 2 1 8 7 6 3 5 7 9 8 2 4 0 7 5 2 1 3 0 7 2 1 7 2 3 4 5 The general rule for the enunciation of any sum is this: To the simple value of each figure join the name of its place, beginning at the left and reading towards the right."(Thompson.) APPLICATION. Write in figures the following numbers. 1. Twenty-seven. 2. Two hundred and forty-three. 3. Five thousand, four hundred and twenty-nine. 4. Seventeen thousand, two hundred and five. 5. One hundred and twenty-five thousand, three hundred and ninety-two. 6. Three millions, seven hundred thousands, four hundred and eleven. Write in words the following numbers. 39,402.-1600.79,411.-927,500.-2840701. Numbers may be considered in two ways. One is when no particular denomination is mentioned, to which their units belong; in this they are called abstract numbers. The other is, when the denominations of their units are specified, as when we say, two men-five miles-six hours; these are called concrete numbers. "The formation of numbers by the successive union of units is evidently independent of the nature of their units, and this is also the case with the properties resulting from this formation, by which properties we are able to compound and decompound numbers, which is called calculation."-(Lacroix.) There are no limits to the numbers which may be enumerated. As far as names can be applied, numbers can be read; and as names may be applied indefinitely, figures may be read indefinitely. The learner may read the following numbers. Units. 3567 5 4 3 2 1 0 1 879 1 4 5 6 7 3 3 2 Write 23 sextillions in figures-15 heptillions-178 octillions-25 nonillions-1687 decillions. A RULE in Arithmetic, is the statement of a principle in its practical form, ‚—so as to furnish the necessary information to the learner, with regard to the course he is to pursue in solving a given example. A DEMONSTRATION shows the reasons on which a rule is founded. The rule gives the directions, and the demonstration shows the reasons for such directions. When the rule, and the reasons on which it is founded, are understood, the practical application is easy. The simple or fundamental rules, are four. All those which follow, are merely combinations and changes of these. The learner will be repaid for a very particular and faithful attention to them, before he proceeds to those which follow. QUESTIONS.-What is the first method in the English mode called? What the second? Third? Fourth? Fifth? Sixth? &c. &c. How do the modes agree? How differ? In how many ways may numbers be considered? What is said of the formation of numbers by successive units? How far can you enumerate? What do you mean by a rule? What is a demonstration? Are these of any use? How many fundamental rules? For what will you be repaid? SECTION III. ADDITION. By SIMPLE ADDITION, is meant the putting together of units, or collections of units, so that the whole may be included in one number. Three dollars, five dollars, and eight dollars, when added make sixteen, and of course may be written or spoken of as one number. 1. A man sold at one time 14 bushels of wheat, and at another time 27; how many bushels did he sell? Ans. 41. 2. What is the amount of 20, 44 and 173, when added together? Ans. 237. 3. A Bookseller receives four boxes of books; the first box contains 115, the second 146, the third 98, and the fourth 225; how many books does he receive in all? Ans. 584. 4. Pennsylvania has 53 counties, New York 56, and Virginia 115; how many counties in these three states? Ans. 224. 5. A man killed an ox, the meat of which weighed 642 pounds, the hide 105 pounds, and the tallow 92 pounds; what did they all weigh? Ans. 839. 6. A farmer gathered 12 bushels of apples from one tree, 16 from another, 19 from another, 11 from another, 22 from another, and 5 from another; how many bushels did he gather from all the trees? Ans. 85. 7. What is the sum of 412, 27, 1632, and 4? Ans, 2075. 8. According to the last census, New Hampshire has 269533 inhabitants, Vermont has 230679, and Massachusetts has 610014; what then is the whole number of inhabitants in these three states? Ans. 1160226. 9. A man has four flocks of sheep; in the first flock are 425 sheep and 217 lambs, in the second 300 sheep and 199 lambs, in the third 911 sheep and 881 lambs, in the fourth 218 sheep and 90 lambs; how many sheep and lambs has he? Ans. 3241. 10. A tree was broken off by the wind 12 feet from the ground, the part broken off was 37 feet long; length of the tree before it was broken? QUESTIONS.-What is Simple Addition? rule for Addition? what was the Ans. 49 feet. Can you give a |