ART. 3. The Roman notation, so called from its originating with the ancient Romans, employs in expressing numbers 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.

All the other numbers are expressed by the use of these letters, either in repetitions or combinations; as, II expresses two; IV, four; VI, six, &c.

By a repetition of a letter, the value denoted by the letter is represented as repeated; as, XX represents twenty; CCC, three hundred.

By writing a letter denoting a less value before a letter denoting a greater, their difference of value is represented; as, IV represents four; XL, forty. By writing a letter denoting a less value after a letter denoting a greater, their sum is represented; as, VI represents six; XV, fifteen.

A dash (-) placed over a letter increases the value denoted by the letter a thousand times; as, V represents five thousand; IV, four thousand.

QUESTIONS. Art. 3. Why is the Roman notation so called? By what are numbers expressed in the Roman notation? What effect has the repetition of a letter? What is the effect of writing a letter expressing a less value before a letter denoting a greater? What of writing the letter after another denoting a greater value? How many times is the value denoted by a letter increased by placing a dash over it? Repeat the table.

The Roman notation is now but little used, except in numbering sections, chapters, and other divisions of books.

EXERCISES IN ROMAN NOTATION.

The learner may write the following numbers in letters:

1. Ninety-six.

2. Eighty-seven.

3. One hundred and ten.

4. One hundred and sixty-nine.
5. Two hundred and seventy-five.

6. Five hundred and forty-two.

7. One thousand three hundred and nineteen. 8. One thousand eight hundred and fifty-eight.

Ans. XCVI.

ART. 4. The Arabic notation, so called from its having been made known through the Arabs, employs in expressing numbers ten characters or figures, viz. :

1, 2, 3, 4, 5, 6, 7. 8, 9, 0. one, two, three, four, five, six, seven, eight, nine, cipher. The first nine are sometimes called digits, from digitus, the Latin signifying a finger, because of the use formerly made of the fingers in reckoning. The cipher, also, has sometimes been called naught, or zero, from its expressing the absence of a number, or nothing, when standing alone.

ART. 5. The particular position a figure occupies with regard to other figures is called its PLACE; as in 32, counting from the right, the 2 occupies the first place, and the 3 the second place, and so on for any other like arrangement of figures.

The digits have been denominated significant figures, because each has of itself a positive value, always representing so many units, or ones, as its name indicates. But the size or value of the units represented by a figure differs with the place occupied by the figure.

For example, there are written together to represent a number three figures, thus, 366 (three hundred and sixty-six). Each of the figures, without regard to its place, expresses units, or ones; but these units, or ones, differ in value. The 6 occupying the first place represents 6 single units; the 6 occupying the second place repreQUESTIONS. - What use is now made of Roman notation? Art. 4. How many characters are employed in the Arabic notation? What are the first nine called, and why? What is the cipher sometimes called? What does it represent when standing alone?- Art. 5. What is meant by the place of a figure? What have the digits been denominated? Why? How does the size or value of units represented by figures differ?

sents 6 tens, or 6 units each ten times the size or value of a unit of the first place; and the 3 occupying the third place represents 3 hundreds, or 3 units each one hundred times the size or value of a unit of the first place.

ART. 6. The cipher becomes significant when connected with other figures, by filling a place that otherwise would be vacant; as in 10 (ten), where it gives a ten-fold value to the 1; and 120 (one hundred and twenty), where it gives a ten-fold value to the 12; and 304, where it has the same effect, by filling an intervening place, causing the 3 to represent three hundreds, instead of three tens.

ART. 7. The simple value of a figure is the value its unit has when the figure stands alone; or, in a collection, when standing in the right-hand place. Thus 6 alone, or in 26, expresses a simple value of six single units, or ones.

The local value of a figure is the value its unit has when the figure is removed from the right-hand place, and depends upon the place the figure occupies.

The local value of figures will be made plain by the following table and its explanation.

{

The figures in this table are read thus:

Nine.

Ninety-eight.

Nine hundred eighty-seven.

Nine thousand eight hundred seventy-six.
Ninety-eight thousand seven hundred sixty-five.
Nine hundred eighty-seven thousand six hundred
fifty-four.

Nine millions eight hundred seventy-six thousand
five hundred forty-three.

QUESTIONS. Art. 6. When does a cipher become significant?- Art. 7. What is the simple value of a figure? What is the local value of a figure? What is the design of this table?

It will be noticed in the preceding table, that each figure in the right-hand or units' place expresses only its simple value, or so many units; but, when standing in the second place, it denotes so many tens, or ten times its simple value; and when in the third place, so many hundreds, or one hundred times its simple value; when in the fourth place, so many thousands, or a thousand times its simple value, and so on; the value of any figure being always increased ten-fold by each removal of it one place to the left hand.

NUMERATION.

ART. 8. NUMERATION is the art of reading numbers when expressed by figures.

ART. 9.

There are two methods of numeration in common use the French and the English.

ART. 10. The French method is that in general use on the continent of Europe and in the United States. It separates figures into divisions called periods, of three places each, and gives a distinct name to each period.

FRENCH NUMERATION TABLE.

Hundreds of Sextillions.

Tens of Sextillions.
Sextillions.

Hundreds of Quadrillions.
Tens of Quadrillions.
Quadrillions.

∞ Hundreds of Quintillions.
Tens of Quintillions.
→ Quintillions.

∞ Hundreds of Trillions

127, 894, 237, 8 67, 123, 67 8, 478,

Period of Sextillions.

6 38.

QUESTIONS. Art. 7. What value has a figure standing in the right-hand or units' place? What in the second place? What in the third? How do figures increase from the right toward the left? - Art. 8. What is numeration? What are the two methods of numeration in common use? Where is the French method more generally used? Repeat the French Numeration Table. What are the names of the different periods in the table? What is the value of the numbers in the table expressed in words?

The value of the numbers represented in this table, expressed in words, is, One hundred twenty-seven sextillions, eight hundred ninety-four quintillions, two hundred thirty-seven quadrillions, eight hundred sixty-seven trillions, one hundred twenty-three billions, six hundred seventy-eight millions, four hundred seventyeight thousand, six hundred thirty-eight.

The names of the periods above Sextillions, in their order, are, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecillions, Sexdecillious, Septendecillions, Octodecillions, Novemdecillions, Vigintillions, &c.

ART. 11. The successive places occupied by figures are often called orders. Hence, a figure in the right-hand or units' place is called a figure of the first order, or of the order of units; a figure in the second place is a figure of the second order, or of the order of tens; in the third place, of the order of hundreds, and so on. Thus, in the number 1847, the 7 is of the order of units, 4 of the order of tens, 8 of the order of hundreds, and 1 of the order of thousands, each figure expressing as many units as its name indicates of that order to which it belongs; so that we read the whole number, one thousand eight hundred and forty-seven.

ART. 12. From the preceding table and explanation, we deduce the following rule for numerating and reading numbers expressed by figures according to the French method.

RULE. Begin at the right hand, and point off the figures into periods of THREE places each.

Then, commencing at the left hand, read the figures of each period, adding the name of each period excepting that of units.

EXERCISES IN FRENCH NUMERATION.

The learner may read orally, or write in words, the numbers represented by the following figures :

QUESTIONS. Art. 10. What are the names of the periods above sextillions? Art. 11. What are the successive places of the figures in the table called? Of what order is the first or right-hand figure? The second? The third? &c. Art. 12. What is the rule for numerating and reading numbers according to the French method?