Let him also separate them into periods of six figures, according to the English method, and then numerate and read them. It will be seen, by reference to the foregoing tables, that the French and English methods of numeration agree as far as nine places of figures, which is as far as we generally wish to extend numbers in the ordinary business operations of life. Numbers could be chosen which should be widely different, and still would be read precisely the same by the two methods. For instance, the French method of reading 103900000000000 is the same as the English method of reading 103000900000000000000, each reading being one hundred and three trillions, nine hundred billions. The same is the case with infinite other numbers which might be selected. Hence the importance of knowing which system of numeration is employed. Twenty billions in the English system is a thousand times twenty billions in the French system. ROMAN NOTATION. 8. The Romans, as well as many other nations, expressed numbers by certain letters of the alphabet. The Romans made use of only seven capital letters, viz. : I for one ; V for five ; X for ten; L for fifty; C for one hundred ; D for five hundred ; M for one thousand. The other numbers they expressed by various repetitions and combinations of these letters, as in the following TABLE. 1 expressed by I. 66 II. As often as any character is repeated, so many 66 V. 66 VI. 66 VII. 3 expressed by III. times is its value re4 IV, or IIII. peated. 5 A less character be6 fore a greater, diminishes 7 its value. A less char8 66 VIII. acter after a greater, in9 creases its value. 10 50 66 IX. 6 X. CG 66 L. 100 66 C. 500 A bar (-) over any 1000 number, increases it 1000 2000 fold. 5000 << 6 D. By what means did the Romans express numbers? In this notation, how did repeating a letter affect the value which it represented ? How was the value of a character affected when one of less value was placed before it? How when a character of less value was placed after it? How was the value affected by a bar drawn over it? ADDITION OF SIMPLE NUMBERS. 9. SIMPLE ADDITION is putting together several numbers of the same kind or denomination. The sum total which is obtained by adding several numbers together, is called the amount. Before explaining the method of adding numbers, we will show the use of the two symbols =, +. The symbol =, is called the sign of equality, and when placed between two quantities, it indicates that they are equal. Thus $1=100 cents, implies that one dollar is equal to one hundred cents. The symbol +, is called the sign of addition, and when placed between two quantities, indicates that those quantities are to be added. Thus 3+4=7, denotes that the sum of 3 and 4 is equal to 7. The symbol + is generally read plus; a Latin word, meaning more. What is simple addition? What is the result obtained by adding several numbers together, called ? Describe the symbol of equality. Describe that of addition. By the assistance of these two symbols we may form the following 1. Where the sums of the several columns are less than ten ; Add together 2432, 3343 and 4122. Set the numbers under each other: | units under units; tens under tens ; OPERATION hundreds under hundreds ; thousands under thousands. Draw a line below the whole. Add first, the column of units. Set the sum 7 under the column of units; next add the tens; set the sum 9 under the column of tens-next add the hundreds ; set the sum 8 under 4 1 2 2 the column of hundreds. Lastly, add 9 8 9 7 the thousands, and set the sum 9 under the column of thousands. The whole amount is, then, nine thousand eight hundred and ninety-seven. Add 6264, 2532 and 1203. Ans. 9999. Add 4132, 1001 and 1423. Ans. 6556. 2. Where the sums of the several columns equal or exceed ten; wo Thousands. Hundreds Ten Thousands. oo Hundreds. o Co o Units. What is the sum total of the following numbers : 3758, 4903, 7006, 3713, 3721. Place the numbers as directed in the preceding example. The sum OPERATION of the numbers in the units' column is 21—that is, 2 tens and 1 unit. Set the 1 under the units' column, and carry the 2 to the next or tens' column. The sum of the tens' column thus increased is 10 4 9 0 3 tens; that is, 1 hundred and no 7 0 6 tens. Place a zero under the tens' 3 7 1 3 1 7 2 column, and carry the 1 to the hundreds' column. The sum of the 2 3 1 0 1 hundreds column, so increased, is 31 hundreds; that is, 3 thousands and 1 hundred. Set the 1 under the hundreds' column, and carry the 3 to the thousands' column. The sum of this column, so increased, is 23 thousands, or 2 tens of thousands and 3 thousands. Set the 3 under the thousands' column, and carry the 2 to the tens of thousands' place; or, what is the same thing, set down the whole of the sum of the last column. 10. From what has now been explained, we know that ten units are equal to one ten, ten tens are equal to one hundred, ten hundreds are equal to one thousand, and so on; ten of any order are equal to one of the next superior order. Hence, for adding numbers of the same denomination, we deduce this RULE. I. Place the numbers to be added under each other, so |