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The History of Arithmetic is involved in so much darkness and uncertainty, that it is impossible now to ascertain at what period, or by what ration it was first cultivated as a science. It was known to the Egyptians, Phenicians and Chaldeans several centuries before the Christian era, but whether it was invented, or very much improved by these natious is, at least, doubtful The system of Arithmetic now in general use was introduced into Europe by the Arabians about the year 960

Nature having furnished a convenient and universal standard of computation in the fingers of the hands, we find that ten, their number, is adopted as the basis of the numeration of nearly all nations. People in the early ages of the world, doubtless had recourse to their fingers to aid them in their computations in the same manner that cbildren have now. The finger of both hands being all counted over it would become necessary to repeat the operation, and setting apart à finger as often as the operation was repeated, when they were all once set apart there would be a collection of units, equal to len collections, of ten units each ;-denoting this larger collection by a digit, or finger, they would go on to form other collections of the same nature. Thus, undoubtedly, originated our decimal scale of Arithmetic.

Although the basis and scale of numeration have been the same, the methods of votation, or of expressing numbers by characters, bas been different in different nations The most simple, and probably the earliest method of d-noting numbers, was by straight marks. One thing that favours this opinion is the fact that a straight mark, or I, is used to denote a single unit in several systems, particularly the Roman and Arabic.

The Romans originally denoted the numbers from one to four by I's, probably intended to represent the four fingers of one hand, and five they denoted by the letier, V, perhaps suggested by the appearance of the hand with the thumb extended The numbers from five to ten were denoted by adding the l's, or digits of the other hand to the V, and ten, or two fives, was denoted by two V's one inverted under the other forming an X, &c.

It has been supposed, and the supposition is not without some foun. dation, that the Arabic characters, which are now in general use, were originally formed by a combination of straight lines One was denot. ed, as at present, by a perpendicular straight line, two was denoted by two horizontal and equal straight lines, thus =, which, being formed hastily with a pen wonld naturally assume the form z, and by degrees became rounded into 2 ; three, being denoted by three equal borizontal lines, thus =, became, in like manger changed to 3 or 3, four being denoted by four equal lines in the form of a square, thu. o. became, in lime, changed to 4, fire being denoted by five equal

lines, thus , becaine 5, six being denoted by six equal lines -becamse, 6, and so on.

The principles of our aritmetical notation appear lo be incapable or farther simplification; a scale different from the decimal, wouid how. ever on some accounts be preferable, as the subdivisions of the units of different orders, would contain a greater number of aliquot parts, or exact divisors, of those units. Supposing 8 to have been the basis of the scale then 4 and 2 would be exact measures of it, and } and would each be expressed by a single digit; or if the basis had been 12, Iben 6,4, 3 and 2 would have been its divisors, and }, }, {and 1-6th would each be expressed by a single figure. But since the decimal scale is universally established, the nearer we can bring tbe different denominations of weights, measures, &c. to conform to this scale, the more simple and easy will be all our computations. The experiment was tried by the establishment of our Federal Currency and has succeed. ed completely, and were the same seriously attempted in our weights and ineasures we have no doubt of its oltimate success and of its benoficial results.




SECTION 1. Notation and Numeration. 70. An individual thing taken as a standard of comparison, is called unity, a unit, or one.

71. Number is a collection of units, or ones.

72. Numbers are formed in the following manner; one and one more are called two, two and one, three, three and one, four, four and one, five, five and one, sit, six and one, seven, seven and one eight, eight and one, nine, nine and one, ten; and in this way we might go on to any extent, forming collections of units by the continual addition of one, and giving to each collection a different name. But it is evident, that, if this course were pursued, the names would soon become so numerous that it would be utterly impossible to remember them. Hence lias arisen a method of combining a very few names, so as to give an almost infinite variety of distinct expressions. These names, with a few exceptions, are derived from the names of the nine first numbers, and from the names given to the collections of ten, a hundred, and a thousand units. The nine first numbers, whose names are given above, are called units, to distinguish them from the collections of tens, hundreds, &c. The collections of tens are named ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety.(6) The intermediate numbers are expressed by joining the names of the units with the names of tbe tens. To express one ten and four units, we say fourteen, to express two tens and five units, we say twenty-five, and others in like man

What is meant by a unit, or one? I higher numbers? why not?
What is number?

From what are the names above ten How are the numbers formed and I derived ? named from one to ten?

Name the collections of tens. Is the same course pursued with the I

ner. The collections of ten tens, or hundreds, are expressed by placing before them the names of the units; as, one hundred, two hundred, and so on to nine hundred. The intermediate numbers are formed by joining to the hundreds the collections of tens and units. To express two hundred, four tens, and six units, we should say two hundred forty-six. The collections of ten hundreds are called thousands, which take their names from the collections of units, tens and hundreds, as, one thousand, two thousand, - ten thousand, twenty thousand, - one hundred thousand, two hundred thousand, &c. The collections of ten hundred thousands are called millions, the collections of ten hun. dred millions are called billions, and so on to trillions, quadrillions, &c. and these are severally distinguished like the collections of thousands. The foregoing names, combined according to the method above stated, constitute the spoken numeration.

73. To save the trouble of writing large numbers in words, and to render computations more easy, characters, or symbols, have been invented, by which the written expression of num. bers is very much abridged. The method of writing numbers in characters is called Notution. The two methods of notation, which have been most extensively used, are the Roman and the Arabic.* The Roman numerals are the seven following letters of the alphabet, I, V, X, L, C, D, M, which are now seldom used except in numbering chapters, sections, and the like. The Arabic characters are those in common use. They are the ten following: O cipher, or zero, 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 uipe. The above characters, taken one at a time, denote all the numbers from zero to

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How are the intermediate numbers | abridged ? expressed?

What is Notation? How many Explain the method of expressing methods are there?

number above one hundred. What are the Roman numerals? What constitutes the spoken nu Are they in general use? meration ?

Name the Arabic characters, How is the expression of nuinbers |

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pine inclusive, and are called simple units. To denote num. bers larger than pine, two or more of these characters must be used. Ten is written, 10, twenty, 20, thirty, 30, and so on to ninety, 90; and the intermediate numbers are expressed by writing the excesses of simple units in place of the cipher; thus for fourteen we write, 14, for twenty-two, 22, &c. (13) Hence it will be seen that a figure in the second place denotes a number ten times greater than it does when standing alone, or in the first place. The first place at the right hand is therefore distinguished by the name of unit's place, and the second place, which contains units of a higher order, is called the ten's place. Ten teps, or one hundred, is written, 100, two hundred, 200, and so on to nine hundred, 900, and the intermediate numbers are expressed by writing the excesses of tens and units in the teps' and units' places, instead of the ciphers. Two hundred and twenty-two is written, 222. Here we have the figure 2 repeated three times, and each time with a different value. The 2 in the second place denotes a number ten times greater than the 2 in the first; and the 2 in the third, or hundredz' place, denotes a number ten times greater than the 2 in the second, or ten's place; and this is a fundamental law of Notation, that each removal of a figure one place to the left hand increases its value ten times.

74. We have seen that all numbers may be expressed by repeating and varying the position of ten figures. In doing this, we have to consider these figures as having local values, which depend upon their removal from the place of units. These local values are called the names of the places : which may be learned from the following



Hund, of Quint.
Tens of Quint.
Hunds, of Quad.
Tens of Quad.

cHund, of Trill.

Tens of Trill.
Hund. of Bill.
Tens of Bill.
A Billions.
Hund, of Mill.
Tens of Mill.

cHund. of Thou.

Tens of Thou.


líow are numbers above nine ex., What is the fundamental law of no. pressed by them?

tation ? What is the name given to the first How many kinds of value have fig.

place, or right hand figure of a ! ures? number?

Upon what does their local values What to the second place ?

depend ? How would you write two hundred What are the local values called ? and twenty two?

| Repeat the names of the places.

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