The first ten numbers have each a distinct name. The collection of ten simple units is then considered a unit: it is called a unit of the second order. We speak of the collections of ten, in the same manner that we speak of simple units; thus we say one ten, two tens, three tens, four tens, five tens, six tens, seven tens, eight tens, nine tens. These expressions are usually contracted; and instead of them we say ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, nirety.

The numbers between the tens are expressed by adding the numbers below ten to the tens. One added to ten is called ten and one; two added to ten is called ten and two; three added to ten is called ten and three, &c. These are contracted in common language; instead of saying ten and three, ten and four, &c., we say thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. These names seem to have been formed from three and ten, four and ten, &c. rather than from ten and three, ten and four, &c., the number which is added to ten being expressed first. The sig nification, however, is the same. The names eleven and twelve, seem not to have been derived from one and ten, two and ten; although twelve seems to bear some analogy to two. The names oneteen, twoteen, would have been more expressive; and perhaps all the numbers from ten to twenty would be better expressed by saying ten one, ten two, ten three, &c.

The numbers between twenty and thirty, and between thirty and forty, &c. are expressed by adding the numbers below ten to these numbers; thus one added to twenty is called twenty-one, two added to twenty is called twenty-two, &c.; one added to thirty is called thirty-one, two added to thirty is called thirty-two, &c.; and in the same manner forty-one, forty-two, fifty-one, fifty-two, &c. All the numbers are expressed in this way as far as ninety-nine, that is nine tens and nine units.

If one be added to ninety-nine, we have ten tens. We then put the ten tens together as we did the ten units, and this collection we call a unit of the third order, and give it a name. It is called one hundred.

We say one hundred, two hundreds, &c. to nine hundreds, in the same manner, as we say one, two, three, &c.

The numbers between the hundreds are expressed by adding tens and units. With units, tens, and hundreds we

can express nine hundreds, nine tens, and nine units; which is called nine hundred and ninety-ninc. If one unit be added to this number, we have a collection of ten hundreds; this is also made a unit, which is called a unit of the fourth order; and has a name. The name is thousand.

This principle may be continued to any extent. Every collection of ten units of one order is made a unit of a higher order; and the intermediate numbers are expressed by the units of the inferior orders. Hence it appears that a very few names serve to express all the different numbers which we ever have occasion to use. To express all the numbers from one to nine thousand, nine hundred, and ninety-nine, requires, properly speaking, but twelve different names. will be shown hereafter, that these twelve naines express the numbers a great deal farther.

It

Various methods have been invented for writing numbers, which are more expeditious, than that of writing their names at length, and which, at the same time, facilitate the processes of calculation. Of these the most remarkable is the one in common use, in which the numbers are expressed by characters called figures. This method is so perfect, that no better can be expected or even desired. These figures are supposed to have been invented by the Arabs; hence they are sometimes called Arabic figures. The figures are nine in number. They are exactly accommodated to the manner of naming numbers explained above.*

* Next to the Arabic figures, the Roman method seems to be the most convenient and the most simple. It is very nearly accommodated to the mode of naming numbers explained above. A short description of it may be interesting to some; and it will often be found extremely useful to explain this method to the pupil before the other. The pupil will understand the principles of this, sooner than of the other, and having learned this, he will more easily comprehend the other. He will perfectly comprehend the principle of carrying, in this, both in addition and subtraction, and the similarity of this to the common method is so striking that he will readily understand that also. The pupil may perform some of the cxamples in Sects. I, II, and VIII, Part I, with Roman characters.

THE ROMAN NOTATION.

One was written with a single mark, thus,
Two was written with two marks

Three was written

Four was written

These nine figures are sometimes called the 9 digits. By

But as it

And so on to ten tens, which were written with ten crosses. was found inconvenient to express numbers so large as seven or eight, with marks as represented above, the X was cut in two, thus, and the upper part V was used to express one half of ten, or five, and the numbers from five to ten were expressed by writing marks after the V, to express the number of units added to five.

Six was written
Seven was written
Eight was written

Nine was written

VI

VII

VIII

VIIII

The intermediate numbers between the tens were expressed by writing the excess above even tens after the tens.

XI

Eleven was written

Twelve was written

Twenty-seven was written

XII, &c.
XXVII, &c.

To express ten Xs, or ten tens, that is, one unit, of the third order, or one hundred, three marks were used, thus, C. And to avoid the inconvenience of writing seven or eight Xs, the C was divided, thus E, and the lower part L used to express five Xs, or fifty.

To express ten hundreds, four dashes were used, thus, M. This last was afterwards written in this form CƆ and sometimes CIƆ, and was then divided, and I was used to express five hundreds.

These dashes resemble some of the letters of the alphabet, and those letters were afterwards substituted for them.

The resembles the I; the V resembles the V; the X resembles the X., the L resembles the L; the C was substituted for the ; the resembles the D and the M resembles the M.

these nine characters all numbers whatever may be expressed.

To express ten, we make use of the first character 1. But to distinguish it from one unit, it is written in a new place, thus 10; the 0, which is called zero or a cipher, being placed on the right. The zero 0 has no value, it is used only to occupy a place, when there is nothing else to be put in that place.

Twenty-four

One thousand, eight hundred, and twenty-six MDCCCXXVI

A man has a carriage worth seven hundred and sixty-eight dollars, and two horses, one worth two hundred and seventy-three dollars, and the other worth two hundred and forty-seven dollars; how many dollars are the whole worth?

These numbers may be written as follows:

Operation.

DCCLXVIII dolls.

CCLXXIII dolls. CCXXXXVII dolls.

MCCLXXXVIII dolls.

To add these numbers together it is easy to see that it will be the most convenient to commence on the right, and count the Is first. We find eight of them, which we should write thus VIII, but observing that

*It is usual to write four IV, instead of IIII, and nine IX, instead of VIIII, and forty XL, instead of XXXX, and ninety XC, instead of LXXXX, &c. in which a small character before a large, takes out its value from the large. This is more convenient when no calculation is to be made. But when they are to be used in calculation, the method given in the text is best.

Eleven is written thus, 11, with two 1s. The 1 on the left expresses one ten; and the one on the right expresses one unit, or one added to ten. Twelve is written 12; the I on the left signifies one ten, and the 2 on the right signifies two units, and the whole is properly read ten and two.

there are more Vs we set down only III, reserving the V and counting it with the other Vs. Counting the Vs we find two, and the ons which we reserved makes three. Three Vs are equivalent to one X and one V. We write the V and reserve the X. Counting the Xs, we find seven of them, and the one which was reserved makes eight. Eight Xs are equivalent to LXXX. We write the three Xs and reserve the L. Counting the Ls, we find two of them, and the one which was reserved makes three. Three Ls are equivalent to CL. We write the L and reserve the C. Counting the Cs, we find six of them, and the one which was reserved makes seven. Seven Cs are equivalent to DCC. We write the CC and reserve the D. Counting the Ds we find one, and the one which was reserved makes two. Two Ds are equivalent to M. The whole sum therefore is MCCLXXXVIII dollars.

The general rule for addition, therefore is, to begin with the characters which express the lowest numbers and count all of each kind together without regard to their value, only observing that five Is make one V, and that two Vs make one X, and that five Xs make one L, &c., and setting them down accordingly.

A man having one hundred and seventy-eight dollars, paid away seventy-nine dollars for a horse; how many had he left?

Operation.
CLXXVIII dolls.
LXXVIIII dolls.

To perform this operation we begin at the right hand, and take the Is from the Is, the Vs from the Vs, &c. But a difficulty imme LXXXXVIIII dolls. diately occurs, for we cannot take IIII from III; it is necessary therefore to take the IIII from VIII, that is, from IIIIIIII, which leaves III; these we set down. Since we have used the V in the upper line, it will be necessary to take the V in the lower line from one of the Xs, that is from VV. V from VV, leaves V, which we set down. Having used one of the Xs, there is but one left. We cannot take XX from X, we must therefore use the L, which is equivalent to five Xs, which, added to the one X, make XXXXXX; from these we take XX and there remain XXXX, which we set down. Since the L in the upper line is already used, it is necessary to take the L in the lower line from the C which is equivalent to LL; one I taken from these, leaves L, which we set down. The whole remainder therefore is LXXXXVIIII dolls.

Hence the general rule for taking one number from another, expressed by the Roman characters, is, to begin with the characters expressing the lowest numbers, and take those of the same kind from each other, when practicable, but if any of the numbers to be subtracted exceed those from which they are to be taken, a character of the next highest order must be taken, and reduced to the order required, and joined with the others from which the subtraction is to be made. This process is called subtraction