3. Examples.

(1.) Three hundred and thirty-three.

(2.) Four hundred and forty-nine.

(3.) Five thousand, three hundred and twenty-five. (4.) Six millions, four thousand and seventy.

(5.) Eighty-five millions, two thousand, four hundred and one.

(6.) One billion, one million, one thousand and one. (7.) Ninety-five billions, eighty-one millions and seventy-five.

(8.) Two hundred and fifty quintillions, six quadrillions, two billions, three hundred and fifty thousand.

(9.) One hundred and twenty-three quintillions, one hundred and twenty-three billions, one hundred and twenty-three.

(10.) Two sextillions, three quintillions, four quadrillions, and four.

(11.) Eleven thousand, eleven hundred and eleven. (12.) Fourteen thousand, fourteen hundred and four

teen.

(13.) One billion, two millions, four hundred and thirtysix.

(14.) Three hundred and nineteen thousand, and five hundred and fifty-five.

(15.) Seventeen millions, one thousand and forty-nine. (16.) Four trillions, two billions, and seventy-four. (17.) Eighteen thousand, three hundred and nine. (18.) One quadrillion, four thousand and two hundred. (19.) One hundred and six trillions, four thousand and two.

(20.) Fifty-nine trillions, fifty-nine billions, fifty-nine millions, fifty-nine thousand, and fifty-nine.

4. The formality of separating the periods, by a point, need not be observed, when the pupil has become familiar with the principles of Notation and acquired a facility in applying them.

§ 9. 1. It will be well for the teacher to illustrate the remark in the note on page 11, that there is no reason in the nature of numbers why the tenfold system of increase in the value of figures, from right to left, should be used rather than any other.

2. It may at once be seen, that other systems might be

formed upon the same principles, and, of course, with the same properties, as to the expression of the greater quantities by the successive rank or place of the figures; and with any other greater or smaller number of significant figures; only that cipher (0) and unit (1) must make part of every such system.

. 3. If no other figures were used but 1 and 0, the value in each would always be successively double of that in the preceding place to the right; and so the whole of the calculation would become a mechanical change of places. The following numbers, 111101, transcribed into our usual decimal system, would be 32, 16, 8, 4, and 1, or 61.

4. If three figures were used, (1, 2, 0,) the increase from right to left would be threefold; the numbers 11101, transcribed into our decimal system, would be 81, 27, 9, and 1, or 118. 20347 would be 208; and so on.

5. So a fourfold, fivefold, sixfold, or any other system might be adopted.

6. The decimal system is, however, most convenient, as is apparent, and is, therefore, in universal use.

§ 10. 1. The Roman method of Notation is by letters. 2. It employs seven letters, I, V, X, L, C, D, M. I, represents one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; M, one thousand.

3. No other number can be expressed by single letters alone.

4. When any other number is to be expressed, a combination is made of the seven letters above, or their value is varied by a sign.

5. A letter placed at the right of one representing an equal or greater value, denotes that the letter at the left is increased by so much; a letter placed at the left of one of a greater value, denotes that so much is taken from that at the right.

§ 11. 1. CASE II. To express numbers by letters.

RULE. Write first the letter which expresses the number nearest that which is to be written. If that be too small, write other letters at the right to increase it, (§ 10. 5); or, if too large, write other letters at the left to educe it to the given number, (§ 10. 5).

3. I is sometimes used instead of D, to represent five hundred, and for every additional O annexed at the right, the number is increased ten times; CIO to represent one ** thousand, and for every Cand put at each end, the number is increased ten times. A line over any number increases its value one thousand times. Thus, IƆƆ, or V,

five thousand; M, one million.

4. By this method of combining letters they may be made to denote all determinate quantities.

5. Examples.

(1.) Nine hundred and forty-three.

(2.) One thousand, eight hundred and fifty-six.

(3.) Two millions, four hundred and twenty-five thousand, four hundred and eleven.

(4.) One billion, two hundred millions, forty thousand, three hundred and fifteen.

(5.) Twenty-five trillions, one hundred and four billions, three hundred and seventy-five millions, and fifty-four.

(6.) Five hundred and thirty-seven millions, two hundred and nineteen thousand, one hundred and three.

(7.) Seven hundred and seventy-seven millions, seven hundred and seven.

(8.) Fourteen thousand, nine hundred and seventy-eight. (9.) Thirty-two millions, twenty-three thousand, two hundred and fifty-six.

II. NUMERATION.

8 12. 1. NUMERATION is the reading of numbers.

2. It is a mere declarative act, consisting, when the numbers are expressed by figures, in calling off each figure by the appropriate name of the place which it occupies; when by letters, in calling off the entire number.

§ 13. 1. CASE I. To read numbers expressed by figures. RULE. Beginning at the left, call each significant figure in order, by the appropriate name of its place and period, except the last, to which the name is not added.

41.080.652.941.600.807.362.546.278.009.650.208; which

is read, forty-one decillions, eighty nonillions, six hundred and fifty-two octillions, nine hundred and forty-one septillions, six hundred sextillions, eight hundred and seven quintillions, three hundred and sixty-two quadrillions, five hundred and forty-six trillions, two hundred and seventyeight billions, nine millions, six hundred and fifty thousands, two hundred and eight.

3. Examples. To be written in words, (equivalent to reading:)

6. The numeration of numbers, expressed by letters, is sufficiently obvious, from the directions given for their notation (§ 10. 5). But to carry out the uniformity of our plan, we subjoin a distinct Case and Rule.

§14. 1. CASE II. To read numbers expressed by letters.

RULE. Begin at the left, and call each letter by name, in order; by the rule for notation by letters, ascertain the number expressed, and declare it.

2. Illustration. MDCCCXLI.

Here we begin at the left, and say, M, D, three Cs, XL, I: one thousand eight hundred and forty-one.

3. Examples. To be written in words (equivalent to reading):

(1.) XL.
(2.) LVI.
(3.) LXXIX.
(4.) CXCIX.

(5.) DCCCLXVII.

(6.) MDCIO.

(7.) MMDC.

(8.) ĪƆ.

(9.) IƆƆDLV.

(10.) VÕIƆLXXXIV.

(11.) MDCLXVI.

(12.) MMCCCCXCXIX.

4. In Latin authors, the Roman method of notation is alone met with.

5. By us, at present, its chief use is to mark divisions and subdivisions in books; and sometimes the number of the year.

6. To express quantity, for Arithmetical calculations, it is not at all employed.

No further reference to it, therefore, will be made in this work; the Arabic method alone will be understood and employed.