THE VALUE OF A FIGURE, STANDING ALONE, IS CALLED ITS SIMPLE VALUE.

If the figure 1 has a cypher at its right hand, what is its value? Ans. One ten.

If the figure 3 has two cyphers at the right, what is its value?

What is the value of 7 and four cyphers, or 70000? Of 800000? Of 2000? Of 5000000?

Is the value of the figures in these last examples, the same with their simple value, or is it greater, or is it less? Why is it greater? Ans. Because it stands in a higher place. Then,

FIGURES HAVE A VALUE WHICH DEPENDS UPON THEIR PLACE, DIFFERENT FROM THEIR SIMPLE VALUE.

THE VALUE WHICH A FIGure derives FROM ITS PLACE, IS CALLED ITS LOCAL VALUE..

How many units of the first order are equal to one of the second?

How many of the second are equal to one of the third? How many of the third to one of the fourth? Of the 4th to one of the 5th? Of the 6th to one of the 7th? Of the 9th to one of the 10th ? &c. Then

IT TAKES TEN UNITS OF ANY ORDER TO MAKE ONE OF THE NEXT HIGHER; And,

NUMBERS INCREASE FROM RIGHT TO LEFT IN TENFOLD PROPORTION. Here is a picture to illustrate this tenfold increase.

The little square on the right, represents a unit of the first order, and corresponds to the 1 under it. The next diagram represents a unit of the second order, and corresponds likewise to 1, in the 111 second place. The third diagram represents a unit of the third order, and corresponds to 1, in the third place.

You see, then, how rapidly numbers increase. If we were to go on, only two orders higher, that is, to the fifth order, the page would not be large enough to contain the diagram for that order.

You have now been learning to write numbers in figures. This is called NOTATION.

You have, also, been learning to read numbers written in figures. This is called NUMERATION. Then

NOTATION IS THE ART OF WRITING NUMBERS IN FIGURES, And, NUMERATION IS THE ART OF READING NUMBERS, EXPREssed in figures.

OBSERVATIONS ON NUMERATION, FOR ADVANCED PUPILS.

VI. The names thirteen, fourteen, fifteen, &c. seem plainly to be derived from three and ten, four and ten, five and ten, &c., with a slight change in the words. One, two, three, four, and so on, up to ten, are primitive words; that is, they are derived, or formed from no other words in our language. Eleven and twelve, seem likewise to be primitive words, though Mr. Webster thinks that they may have come originally from the words, one left, (after ten,) and two left; twenty, thirty, forty, &c., come from two tens, three tens, &c. All the other numbers up to a hundred, are combinations of those already mentioned, without alteration. Hundred, thousand, and million, are primitive, in our language. Billion, trillion, quadrillion, &c., are formed by combining the Latin numerals with the termination ulion. These names go on much farther than they are given above. After sextillions, we have septillions, octillions, nonillions, decillions, undecillions, duodecillions, tredecillions, &c., but there is seldom occasion to use these terms.

There is a reason why so many derivative names are employed in Numeration. For, otherwise, there would be as many distinct name as numbers, and, it would manifestly be very difficult to remember them all. Indeed, it would be impossible ever to learn them. For, if they were extended, only to 100,000 they would form a considerably greater body of words, than all the rest of the language put together; and, if a man were to employ himself twelve hours in a day in studying them, it would take him nearly twenty years merely to read the names as far as one trillion. Besides, we must discontinue giving names somewhere; and wherever we stop, there is still room to form more. Now, on the plan of numeration in use, the first ten names together with hundred and thousand, (making twelve in all,) are sufficient to express all ordinary numbers. After that, we only need the names of the higher periods. And if it were necessary so far to simplify, we might even dispense with these.

We have ten characters, to be used in writing numbers. The cypher has no value. All the others have a positive value, and are therefore called significant figures. Since characters stop at 9, some means must be contrived to continue the notation of numbers beyond, or the characters are of no use. For this reason, the first number, too great to be expressed by a single character, is considered a unit of a higher order, and is denoted by the figure 1, removed to another place. When we obtain too many of these new units to

be expressed by a single character, we make another order of units still, and so on. Hence, we see, that it was not necessary to stop at 9. Nor was it necessary to continue our characters as far as 9. We might have stopped at 8, or 7, or 6, at pleasure. If we had gone beyond 9, we might have expressed any given number with fewer figures; if we had stopped short of 9, we should have been obliged to use more than are at present necessary.

The origin of the system of counting by tens, seems to have been, that men counted on their fingers, before writing was invented. Thus, instead of thirty, they said, three times all their fingers and thumbs, that is, three tens. So that our scheme of tenfold, or decimal notation, as it is called, owes its origin, probably, to chance. It would be more convenient, on many accounts, to reckon by twelves, instead of by tens; but the decimal notation is now too firmly established to be shaken. The number by which we reckon, that is, which expresses the ratio of increase in any system of Notation, is called the RADIX of that system. The Radix of the decimal Notation is 10.

The word digit comes from the Latin digitus, which means fin. ger, and is applied to figures, because men formerly counted, as before stated, by their fingers. It properly belongs to the cypher, as well as to the other characters, but custom has restricted it to the significant figures.

The mode of dividing numbers into periods of three figures, is called the FRENCH mode of diding because the French arithmeticians first used it. The E. to divide into periods of nalf periods of three figures.

six figures, and these, sometin

As far as the order of Heus of Millions, both systems employ the same names for the several orders; but, after that, the English goes on with thousands of millions, tens of thousands of millions, and hundreds of thousands of millions; instead of Billions, tens of Billions, and hundreds of Billions, and, thus, brings the Billion order, where the French has the Trillion. The French is much most convenient. In the English, there is no thousand period, the period of units extending as far as millions, and, thereby embracing all the thousands. No other period is lost, but, as more figures are put into each, the higher periods are carried farther to the left. Of course, the names Billion, Trillion, &c, stand for much larger numbers in the English system, than in the French. An English Billion, for example, is a thousand French Billions, and an English Trillion is a million of French Trillions.

A perfect knowledge of the difference between the two may be gained by comparing the following Tables.

Though the invention of our numerical characters is commonly ascribed to the Arabs, there can be little doubt that they owe their origin to the philosophers of India. Arabian writers attribute the honor to the Indians, and though the opinion has been controverted by very learned men, no conclusive, nor even very weighty argument, has been brought forward against it. An attempt has been made to prove that the Arabs derived their characters from the Greeks, and transmitted them to the nations farther east; and, in support of this opinion, an alleged similarity between the figures themselves, and the letters used by the Greeks, to denote numbers, has been insisted on; but this resemblance is altogether imaginary, and would, probably, never have been discovered except by one, anxious to establish a favorite theory. There is a tradition, among the Indians, that their numbers, as well as their knowledge of the science of Geometry,

Tartary. We find nothing

m a nation in the north of to corroborate this opinion, since no traces of an enlightens are to be found in that part of the world. But whatever may been the origin of numbers, there is no obscurity as to the channel, through which we have received them. About the middle of the 7th century, the Arabs overran Persia and Egypt, and very soon extended their conquests over all northern Africa. . In the early part of the 8th century, they advanced into Spain, bringing with them the arts and learning, which they had acquired from conquered nations, and likewise, their own system of numeration. About the middle of the 11th century, the Arabic characters were introduced into England. On the continent, their use had already become very extensive; and long before the discovery of America, it had become general throughout the civilized world.

We have mentioned that the notation by tens, or the decimal notation, had its origin, most probably, in accident; and that numbers might have been made to increase in any other ratio. Near the beginning of the second century, and of course, long before the introduction of the Arabic characters into Europe, a different scheme was actually introduced. It is called the sexagesimal notation, from the fact, that its Radix is 60. It was introduced by Ptolemy, an Egyptian philosopher, and was probably derived, like the decimal notation, from the East, though the invention is commonly ascribed to him. The Indians and Chinese employ it at the present day

in reckoning time, using periods of 60 years, instead of centuries. Their smaller divisions are similar. We have remnants of it left in the division of time into minutes and seconds, and likewise, in the division of the circle, into degrees, minutes, and seconds, for which latter purpose, Ptolemy seems to have intended it. This division, in both cases, is often carried to thirds, fourths, &c. It is not, however, to be supposed, that 60 different characters were employed. The common mode of writing, that is, the Roman, or Greek, was used as far as sixty, and then the same notation was used over again for the next higher order, with an accent () drawn down at the right. For the third order, two accents were used ("). For the fourth, three, ("") and so on. These accents answered the same purpose as the Arabic cypher; except that when different orders were written together, the accents were retained, where the cypher would be dropped. This was necessary, because several characters were employed within the same order. Without retaining the accents, we should not be able to determine the dividing point between a higher and a lower order. To illustrate by Arabic characters, 31'23 signifies 31 sixties and 23. For some purposes, this notation would be convenient, and for many, inconvenient. These cannot be explained here.

Of the modes of notation, employed before the introduction of the Arabic, all wer more or less objectionable. The Roman combined more advantages than any other, and, as it has not yet entirely fallen into disuse, some knowledge of it is absolutely necessary. The following is a brief account of

THE ROMAN NOTATION.

We have mentioned that the decimal scale was probably suggested by the number of the human fingers. But some nations instead of counting by the fingers of both hands, only employed those of one, and, therefore, fell into the habit of reckoning by fives. In other words, they employed five as the Radix of their system of Numeration. This is still the case with some uncultivated tribes in Africa, and on this continent. The Roman Notation, though adapted both to the quinary and decimal scales, is most simply explained by the former, from which, indeed, it seems to have sprung.

Most nations have coincided in expressing unity or one by a single mark, thus, I. This indeed seems the most natural mode that could have been invented, and this was the mode employed in the Roman scale. This character resembles the letter I. That letter was, consequently, afterwards used in its stead. Two was expressed by two marks, II; three by three marks, III; and four by four marks, IIII. Five employs all the fingers on one hand to express it, and seems therefore to have been made the Radix of the system. A peculiar charac ter was therefore given it, consisting of two lines, joined at one end, thus, V. For this, the letter V was afterwards employed. Six was expressed, of course, by this character and one mark, thus VI; seven, thus VII; eight, VIII; nine, VIIII. As ten is two fives, the character V was used twice to express it, and for convenience, these seem to have been joined, thus X. For this, the letter X was substituted. Eleven would then be XI; twelve, XII; thirteen, XIII; fourteen, XIIII; fifteen, XV; sixteen, XVI; seventeen, XVII; eighteen, XVIII; nineteen, XVIIII; twenty, being two tens, was of course, XX; twenty-one, XXI; and so on. Thirty was xxx, and forty xxxx. Then, as a particular character was written for five, fifty naturally received a particular character likewise. 1. (afterwards, the letter L,) was, therefore, put for fifty, LX for sixty, LXX for