The Roman Method is by letters; thus, I expresses one, V five, X ten, L fifty, C one hundred, D five hundred, M one thousand. By the different arrangement and repetition of these seven letters all numbers may be expressed. These letters express only simple value. As often as the same letter is repeated, its value is repeated; thus, X represents ten, and XX twenty, the left hand letter expressing no more than that at the right. If a letter expressing less value, be placed after one expressing greater value, the effect is addition; thus, XI represents eleven. But if the letter expressing less value be placed at the left, the effect is subtraction, thus, IX represents nine. * I is sometimes used instead of D to represent five hundred, and every additional placed at the right hand, increases the number ten times. † CID is sometimes used to express one thousand, and every C and placed at each end, increases the number ten times, thus C expresses one thousand, and CC100 ten thousand. NOTE.-A line drawn over a number increases it a thousand times, thus, X expresses ten thousand, and XX twenty thousand. It may now be seen from the two methods of notation, that every number, above unity, is formed from the continual addition of unity to itself; and it has justly been said that NATURE, in forming the human hand, furnished us with the first elements of calculation. For the simplest method of expressing one, is a single line (1) which ac cording to the Roman notation, is written I, which was for one finger, and two, by two lines, (1) three by three lines, ( | | ) and four by four lines, (14D). As this exhausts all the fingers of one hand, the next number, five, would naturally be represented by a new character as V, formed by the opening between the thumb and forefinger, then six would be by VI, and so on to ten, which might be expressed by two V's, but to express it in the least room possible, they must be written together, thus, X, ten. Thus they proceeded from the different arrangement and repetition of these characters to fifty, which they found necessary to represent by some new combination of lines. Proceeding on the supposition of a quintuple scale, it would require three lines, or a new combination of two; they naturally concluded that it would be sufficiently distinct, if the two lines formed a right angle, thus L represents fifty; accordingly one hundred was written which, for the ease of writing, was afterwards rounded off and became C. From the reasoning advanced, five hundred would require a new character, and as the next combination most simple is three lines forming a triangle, it was written thus, from the Greek and one angle afterwards being rounded off, it is now represented by the Roman D. And one thousand, which was represented by two D's has since by means of contraction, been represented by M. To facilitate the progress of the young student in writing numbers according to the Arabick notation, where ciphers come between or after the significant figures, the following table is annexed. Tens. Thousands. Hundreds. Units. Tens of Thousands. Hundreds of Thousands. Millions. Tens of Millions. Hundreds of Millions. 10 1 0 4 1001 10050 10000 1 1600500 1 0 8 0 0 1 0 1 10 100 9000 606 30 3 3 0 3 Ten. One hundred and four. Let the student be required to express in words the following sums: 79, 65, 105, 1005, 1010, 100300, 9846, 3804, 64890. Write in figures the following sums: Seventy-nine, ninety-seven, one hundred and one, one thousand two hundred, one hundred thousand three hundred, ninety-nine thousand nine hundred and ninety-nine. Let the student be required to express in words the following sums: IV, IX, XX, XC, CCX, XXV, MDC, LVI, DLX, DCC, CVII. Questions to be answered by the student. NOTE. It is the interest of every teacher to advance his pupils as fast as possible, and to do that, he must teach thoroughly. He should class his students and question them, at least as often as once a day, on the sums, the rules and the principles on which they are founded; and the student should make this his maxim: never to pass on from rule to rule, while there remains any thing back of him unconquered. QUESTIONS ON NUMERATION AND NOTATION. What is Arithmetick? A. The art or science which treats of the nature and properties of numbers. What is unity or unit? A. That by which every thing is called one. What is an integer or whole number? A. Some entire quantity as 8, 16, 20, &c. Why is an integer so called? A. In opposition to fractions, as one-half, two-thirds. How many methods have we of expressing numbers? A. Two, the Arabick and Roman. What is the Arabick method? A. By ten characters or figures. How many of them are significant of value?— A. Nine, the tenth is of no value. Can all numbers be expressed by these ten characters? A. Yes. By whom was the Arabick notation introduced into Europe? A. By the Arabs. How long since? A. About one thousand years. Where did they obtain their knowledge? A. From India. What are the words at the head of the numeration table in numerating nine figures? A. Units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions. In what direction are whole numbers numerated? A. From the right to the left. How are decimals numerated? A. From the left to the right. What is a decimal? A. Something less than a unit. What are the words made use of, in numerating three figures in decimals? A. Tenth parts, hundredth parts, thousandth parts. In what proportion do figures increase from the right to the left? A. Ten-fold when they express the same significant value. If the left hand figure expresses greater simple value than that at the right, what is the proportion? A. Greater than ten-fold. If the left hand figure expresses less simple value than that at the right, what is the proportion? A. Less than ten-fold. What is numeration? A. Expressing figures in words. What is notation? A. Writing figures for words. What value do the Arabick characters express? A. Simple and local. What value does a figure express standing alone? A. Simple. When a number is expressed by more than one figure, what value does that at the right express? A. Simple. What do those at the left express? A. Local. What is the Roman method of notation? A. By seven letters. What are they? A. I, V, X, L, C, D, M.What value do they express? A. Simple. SIMPLE ADDITION, Ja collecting two or more numbers of the same denomination in The number arising from the operation of the work, is called the sum or amount. one sum. RULE. I. Write the numbers to be added, so that units may stand under units, tens under tens, hundreds under hundreds, &c. and draw a line under the whole. II. Then add the right hand column, and if the sum be less than ten, write it directly under the column added, if it exceed ten, write down the right hand figure and add the left hand figure or figures, to the first figure of the next column. Observe the same rule in all the left hand columns; setting down the whole amount of the last column. PROOF. Add the columns downwards, as you added upwards, and if the same amount is produced, the work is presumed to be right. Two parallel lines denote equality; thus, 6 and 2-8, that is 6 and 2 are equal to 8. For the sake of brevity, addition is often denoted by the following charcter, thus, 2+6 signifies the amount of 2 and 6, that is, 2+ 6==8. ADDITION TABLE. 2 and 1 are 3/3 and 9 are 12/5 and 5 are 107 and 1 are & 4310 135 + 6 =117+ 2 = 9 123456 3 53 11 145 7 127 3 10 8 14 9 4 8 10 4 234 63 12 155 4 1 2 وو 10 157 8 دو دو 12/6 4 10 " 45. 54 NOTE. 1st. To acquire a facility in adding, the student should carefully examine this table previous to his working sums. He will thereby render his task comparatively easy. NOTE. 2d. To test the student's knowledge of the table, the teacher should question him on the table, occasionally varying the order, and sometimes mention the last of the two numbers to be added first. FAMILIAR QUESTIONS. NOTE. The student should be required mentally to answer each of these sums, directly after he has read the question. how 1. John had 5 apples, and James gave him two more ; many had he after receiving the two from James? 2. Peter gave 4 shillings for a grammar, and 2 shillings for an English Reader; what did both cost him? 3. William has 5 peaches and Charles 4; how many have they both? How many then do five and four make? 4. George Washington served as President 8 years, and John Adams 4 years; how long did they both serve? 5. James Monroe served as President of the United States 8 years, James Madison 8 years, and J. Q. Adams 4 years; how long did the three serve? How many then do 8 and 8 and 4 make? 6. Joseph paid 3 shillings for a slate, 5 shillings for an arithmetick, and 2 shillings for paper; how much did he pay out? 7. Henry was two days in learning addition, he learned subtraction in one, multiplication in two, and division in three; how long was he going over the four rules? How many are two, one, two and three? 8. Charles has 3 notes, one of 6 dollars, one of 8 dollars and 1 of 4 dollars; what is the amount of the three? 9. A man bought 8 bushels of wheat for 12 dollars, and eight bushels of corn for 6 dollars; how much did he pay for both? How many are 12 and 6 ? 10. A farmer paid 10 dollars for a plough, and five for a harrow; what did he for both? pay |