ten-fold; thus in the number 91, the proportion is ninety-fold, that is, the left hand figure expresses ninety times as much as the one at the right; but when the left hand figure expresses less significant value than that at the right, the proportion is less than ten-fold, thus in the number 19, the left hand figure expresses only one more than that at the right.

We numerate whole numbers from the right hand to the left, as may be seen from the tables already given; but decimals which are parts of integers, must be numerated from the left to the right, as may be seen from the following

NOTE. As integers increase in a ten-fold proportion counted from units to the left, so decimals decrease in the same proportion counted from the left to the right.

The comma (,) placed between whole numbers and decimals, is called SEPARATRIX OF DECIMAL POINT-The figures at the left of the separatrix are whole numbers, those at the right are decimals.

Numeration is expressing in words what is written in figures; thus, 66, when expressed in words, reads, sixty-six.

Notation is writing in figures what is proposed in words, thus, sixtysix, in words, is written in figures, 66.

The Arabick method has an advantage over the Roman, on account of the figures expressing simple and local value. A figure standing alone expresses simple value; thus 3, simply expresses three, but when we write down more than one figure, all except the right hand figure, have a local value; thus, in the number 433, the left hand figure stands in the place of hundreds and expresses four hundred, according to the laws of notation, and the next figure at the right, stands in the place of tens, and from its local placing expresses 30, and the next at the right, standing in the place of units, expresses its simple value three.

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 addi tion; thus, XI represents eleven. But if the letter expressing less value be placed at the left, the effect is subtraction, thus, IX represents nine.

10 is sometimes used instead of D to represent five hun dred, and every additional placed at the right hand, increases the number ten times.

† CIO is sometimes used to express one thousand, and every C and placed at each end, increases the number ten times; thus CIO expresses one thousand, and CC 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, ( three by three lines, ( | | ) and four by four lines, () 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 represents fifty; accordingly one hundred was written C 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, A, from the Greek A; and one an gle 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. S

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."

.2 Millions.

Hundreds of Millions.
Tens of Millions.

10

104

1 0 0 1 10050 10000 1 1 6 0 0 5 0 0 1 0 8 0 0 1ΟΥ 1 0 1 0 0 9 0 0 0 606 30 3 3 03

Let the student be required to express in words the follow ing 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, téns, hundreds, thou sands, 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,

Is collecting two or more numbers of the same denomination in one Sum. The number arising from the operation of the work, is called the sum or amount.

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 are equal to 8.

2

For the sake of brevity, addition is often denoted by the following haracter+, thus, 2+6 signifies the amount of 2 and 6, that is 2+ 6-8.

2 and

ADDITION TABLE.

1 are 313 and 9 are 1215 and 5 are 1017 and 1 are 8 13 5 + 6 = 117 + 2 = 9

2 + 2 =

9

10

11

12

12345678

43

10

11

145

7 127

3 10

"

12

11

"

"

74

55

9

147

12

13