Three hundred sixty-five, the days in a year, are expressed thus, 365; 3 being in the place of hundreds, 6 in the place of tens, and 5 in the place of units. After the same manner, the pupil may be required to unite she three orders, and express any number from 99 to 999. T5. We have seen that figures have two values, viz., simple and local. The simple value of a figure is its value when standing. alone; thus, the simple value of 7 is seven. The local value of a figure is its value according to its distance from the place of units; thus, the local value of 7, in the number 75, is 7 tens, or seventy, while its simple value is seven; in the number 756, its local value is seven hundred. Note. From the fact that 10 is 1 more than 9, it follows, as may be found by trial, that the local value of every figure at the left of units, except 9, exceeds a certain number of nines by the simple value of the figure. Take the number 623; 2 (tens) is 2 more than 2 nines, and 6, (hundreds,) 6 more than a certain number of nines. On this principle is founded a method of proof in the subsequent rules, by casting out the nines. T6. Ten hundred make one thousand, which is called a unit of the next higher, or 4th order, consisting of thousands, and is expressed by writing three ciphers at the right hand of the unit 1, giving it a new local value; thus, 1000, one thou sand. To thousands succeed tens and hundreds of thousands, forming units of the 5th and 6th orders. Questions.¶ 4. What are 10 tens called? What do they form? How many places are required to express hundreds? How much does I cipher, placed at the right hand of 1, increase it? 2 ciphers? How do you express two hundred? &c. What are 4 hundreds, 9 tens, and 5 units called? How is one hundred ninety-three expressed? What place does the 3 occupy? the 9? the 1? How do you express the ab sence of an order? How is the number of days in a year expressed? 5. How many values have figures? What are they? What is the simple value? local value? What is the value of 5 in 59? Is it its simple, or a local value? Is the value of 8, in 874, simple or local? of the 7? of the 4? 6. How do you express one thousand? seven thousand? A thousand is a unit of what order? How many thousands are 30 hundreds ? What after thousands, and of what order? The 6th order is what? In writing nine hundrel and two thousand and nine, where do you place eiphers? Why? 7. In this table of the six orders now described, you see the unit 1 moving from right to left, and at each removal forming the unit of a higher order. There are other orders yet undescribed, to form which the unit moves onward still towards the left, its value eing increased ten times by each removal. 1 Note 1. The Ordinal numbers, 1st, 2d, 3d, &c., may be called indices of their respective orders. Note 2. Various Readings. In the number 546873, the left hand figure 5 expresses 5 units of the 6th order, or it may be rendered in the next 'ower order with the 4, and together they may be read 54 units of the 5th order, (ten thousands,) and connecting with the 6, they may be read, 546 units of the 4th order, or 546000. Hence, units of any higher order may be rendered in units of any lower order. To hundreds of thousands succeed units, tens, and hun dreds of millions. T8. To millions succeed billions, trillions, quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decilions, undecillions, duodecillións, tredecillions, &c., to each of which, as to units, to thousands, and to millions, are assigned three places, viz., units, tens, hundreds, as in the following examples: Questions.¶7. How is the unit 1 of the 1st order made a unit of the 2d order? of the 3d order, &c., to the 6th order? What may the ordinal numbers, 1st, 2d, 3d, &c., be called? 7 units of the 6th order tre how many units of the 4th order? The teacher will multiply such questions. What is the least, and what the largest, number which can be expressed by 2 places? 3 places? &c. What after hundreds of thousands? Of what order will millions be? tens of millions? hundreds of millions? 8. What after millions? How many places are allotted to bil lions? to trillions? &c. Give the names of the orders after trillion's In reading large numbers, what is frequently done? Why? The 1st period at the right is the period of what? the 2d? the 3d? the 4th 2 . 3, 0 8 2, 7 1 5, 2 0 3, 1 7 4, 5 9 2, 8 3 7, 4 6 3, 5 1 2 To facilitate the reading of large numbers, we may 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 those of only three figures. Thus, in looking at the above examples, we find the first period at the left hand to contain one figure only, viz., 3. looking under it, we see that it stands in the 9th period from units, which is the period of septillions; therefore we read it 3 septillions, and so on, 82 sextillions, 715 quintillions, 203 quadrillions, 174 trillions, 592 billions, 837 millions, 463 thousands, 512. By T9. From the foregoing we deduce the following principles : Numbers increase from right to left, and decrease from left to right, in a ten-fold ratio; and it is A FUNDAMENTAL LAW OF THE ARABIC NOTATION; that, Questions. 9. How do numbers increase? how decrease, and in what proportion? To what is 1 ten equal? 1 hundred? 1 thousand? &c. To what are 10 units equal? 10 hundreds? &c. What is a fundamental law of the Arabic notation? What is notation? numera tion? How do you write numbers? read numbers? If you were to write a number containing units, tens, hur reds, and millions, but no thousands, how would you express_it? of Millions. Units Hundreds - Tens co Units of Thousands. Hundreds - Tens Units of Units. I. Removing any figure one place towards the left, increases its value ten times, and II. Removing any figure one place towards the right, decreases its value ten times. The expressing of numbers as now shown is called Notation. The reading of any number set down in figures is called Numeration. To write numbers. - Begin at the left hand, and write in their respective places the units of each order mentioned in the number. If any of the intermediate orders of units be omitted in the number mentioned, supply their respective places with ciphers. To read numbers.-Point them off into periods of three figures cach, beginning at the right hand; then, beginning at the left hand, read each period separately. Let the pupil write down and read the following numbers: # Two million, eighty thousand, seven hundred and five. Eighty-four quintillion, seven quadrillion, one hundred million grains of sand. T10. Numbers are employed to express quantity. Quantity is anything which can be measured. Thus, Time is quantity, as we can measure a portion of it by days, hours, &c. Distance is quantity, as it can be measured by miles, rods, &c. By the aid of numbers quantities may either be added together, or one quantity may be taken from another. Arithmetic is the art of making calculations upon quantities by means of numbers. Questions.¶ 10. Numbers are employed to express what? What is quantity? By what is a quantity of grain measured? a quantity of cloth? What is arithmetic? What is an abstract number? a denominate number? What is the unit of a number? What is the unit value of 8 bushels? of 16 yards? of 20 pounds of sugar? of 3 quarts of milk? of 9 dozen of buttons? of 18 tons of hay? of 16 hogsheads of molas es? A number applied to no kind of thing, as 5, 10, 18, 36, is alled an abstract number. A number applied to some kind of thing, as 7 horses, 25 lollars, 250 men, is called a denominate number. The unit, or unit value of a number, is one of the kind which the number expresses; thus, the unit of 99 days is 1 day; the unit of 7 dollars is 1 dollar; the unit of 15 acres is 1 acre. In like manner the unit of 9 tens may be said to be 1 ten; the unit of 8 hundred to be 1 hundred; the unit of 6 thousand to be 1 thousand, &c. ADDITION OF SIMPLE NUMBERS. T11. 1. James had 5 peaches, his mother gave him 3 more; how many had he then ? Ans. 8. Why? Ans. Because 5 and 3 are 8. 2. Henry, in one week, got 17 merit marks for perfect lessons, and 6 for good behavior; how many merit marks did he get? Ans. Why? 3. Peter bought a wagon for 36 cents, and sold it so as to gain 9 cen's; how many cents did he get for it? 4. Frank gave 15 walnuts to one boy, 8 to another, and had 7 left; how many walnuts had he at first? 5. A man bought a chaise for 54 dollars; he expended 8 dollars in repairs, and then sold it so as to gain 5 dollars; how many dollars did he get for the chaise? The putting together of 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. 6. One man owes me 5 dollars, another owes me 6 dollars, another 8 dollars, another 14 dollars, and another 3 dollars what is the sum due to me? 7 What is the amount of 4, 3, 7, 2, 8, and 9 dollars? 8. 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? Questions.¶11. What is addition? What is the answer, of number sought, called? What is the sign of addition? What does it show? How is it sometimes read? Whence the word plus, and what 's its signification? What is the sign of equality, and what does it |