by 8 times the whole value of all the smaller figures; or would express 8 times as many units, or ones, in value, as the smaller figures do, although the places occupied by each of the given sums are equal. It is, therefore, obvious, that the larger the significant figures used in numeration, in any given number €. places, the greater will be the value expressed; and the smaller the significant figures, expressing a value, the greater will be the units, or ones, between their respective values. Thus, every significant figure up to 9, is composed of so many units or ones, as the figure denotes; and, whether it retains a simple or local value, the same truth is clearly visible; for 9, 99, or 999, are each composed of so many ones.

These examples are deemed sufficient to ensure accuracy in the learner, relative to the Notation and Numeration of whole numbers, and also how to write and read any proposed sum correctly..

Intimately connected with the Notation and Numeration of whole numbers, is that of Decimals; the latter being predicated on the principle of tens, and derive their name from the Latin numeral "decem," which signifies ten. Decimals are parts of a whole number, or a broken number ingeniously framed on the principle of tens, or a tenfold proportion; but with this difference from whole numbers, viz. whole numbers increase, as has already been shown, in a tenfold ratio, from right to left; whereas Decimals, although constructed on the principle of tens, or tenth parts, yet decrease in value from the left to the right. Decimals, therefore, are not units, or whole numbers, but expressions of a part of a unit or whole number. They are distinguished from whole numbers, by placing a comma, usually called a separatrix, before them: or, when a whole number and a Decimal are united, the separatrix is placed between the whole number and the Decimal. Decimals always suppose the unit or whole number, of which the Decimal is the expression of a part, to be divided into such a number of parts, if they were to be expressed, as would invariably be composed of unity or one, with one or more ciphers annexed to it. Thus the parts of the unit, or whole number, which are usually distinguished by the term denominator, would be either 10, 100,

1000, &c. if they were required to be expressed; but this is never necessary to be done; for the decimal value is fully known by the numeration of the decimal numbers, which are always expressed.

As decimals are counted from the left hand to the right, each Decimal derives its value according to its distance from the place of units, or the separatrix. If the decimal figure stand in the first place at the right hand of the separatrix, it occupies the place of tenths; if in the second place, it becomes hundredths; if in the third place, thousandths; decreasing in value in each place to the right, in a tenfold ratio, as may be seen by the following table.

The whole numbers, numerated from right to left, give 7 millions, 654 thousands and 321.

The Decimals are numerated from left to right; the first figure after the separatrix, is 2 tenths; the second is 3 hundredths; the third is 4 thousandths; the fourth is 5 ten-thousandths; the fifth is 6 hundred-thousandths; the sixth is 7 millionths. All are read together thus: 234 thousand 567 millionths of one unit.

The left-hand figure of the Decimal always expresses the principal value of the given Decimal, as it stands next to unit's place; or it signifies so many parts out of unity or one, which m this case would be divided into ten parts. The second decimal figure is of far less value, as it denotes only so many hundredth parts of an unit, or an unit divided into a hundred parts; the 3 in the second place would require only 3 parts to be

taken out of a hundred; and thus the ratio is decreased tenfold by the farther removal of this figure from the separatrix to the right. If, therefore, the left hand figure of the Decimal were ,4, viz. 4 tenths, by annexing ever so many nines to the right hand of it, the ,4 would never become ,5, or 5 tenths, which would be equal to the half of an unit, or one. Although the annexing of one 9 to the 4 tenths, would make the value 49 hundredths, leaving one hundredth part short of 5 tenths, which is equal to one half of unity; then add a second 9 to the Decimal, which will make 499 thousandths, leaving the deficiency only one thousandth part short of making the,4 to possess the value of 5 tenths, still it is not 5 tenths, and never could be made such, by the addition, or annexing of ever so many nines, although the addition of each 9, made the deficiency 9 parts out of ten nearer than before the last 9 was annexed. The de ficiency would, indeed, be exquisitely small, yet the left-hand figure of the Decimal could never be thus increased to the value of one tenth; or the 4 tenths to the value of 5 tenths. The fact then is clearly discoverable; that numbers may approach nearer to each other for ever, and yet never meet. It is also obvious, that as Decimals decrease in value from left to right, they consequently increase, in the same ratio as whole numbers, trom right to left.

Ciphers prefixed, that is, placed at the left hand of a Decimal, diminish the value of the Decimal in a tenfold ratio, as the significant Decimal is thereby removed one place farther from the separatrix. Thus five, expressed decimally, would be written, 5 tenths, which, being the half of ten, and ten would constitute the whole, or unity, so five tenths would be equivalent to the half of unity, or one. But if a cipher be placed before the five, it would be expressed thus, ,05 hundredths, or only five parts taken out of a hundred parts; or, if two ciphers were prefixed, as, ,005 thousandths, the value would be only five parts out of a thousand parts; because the significant figure in the Decimal, being thus removed farther from the separatrix, by the prefixing of each cipher, is continually decreased tenfold by each removal. But ciphers annexed, or joined to Decimals, do not alter their decimal value. Thus 5, tenths,

,50 hundredths, or ,500 thousandths, are each respectively equal to one half; for 5 is the half of ten, 50 the half of a hundred, and 500 is the half of a thousand. Hence the effect of ciphers with Decimals, is directly the reverse with that of whole numbers.

From this brief survey of the nature of Decimals, it is evident they can be connected with whole numbers, in the operations of Arithmetic, with the greatest facility, by carefully preserving the separatrix in its proper place, so as clearly to distinguish the Decimals from the whole numbers.

The prolixity which has been indulged on this highly important subject of Numeration, justly considered the basis of every arithmetical operation, has arisen from a consciousness that its nature and principles were generally very imperfectly understood by the learner, if understood at all. Students are desirous of acquiring a knowledge of the whys and wherefores of any branch of science. If the nature and principles of a science are clearly explained, and the rules rendered perspicuous and intelligible by the author, the learner might be enabled to acquire a knowledge of it, with but very little aid from a teacher. His task then becomes pleasant, and his researches are accompanied with profit and delight. But if the explanations and illustrations of the nature and principles are not rendered explicit and luminous by the author, and the task devolves upon the living teacher, all knowledge or assistance might be withheld from the learner. It might happen, in such a case, that the deficiency of the former, and the neglect, or total incompetency of the latter, would leave the student in entire darkness relative to an important branch which he was anxious to investigate and acquire. Hence it is, that a deep-rooted prejudice is frequently excited against the pursuits of a branch of science, or of the arts and sciences generally, arising solely from the sources already suggested, viz. deficiency in defining, or neglect, or incompetency, in instructing. In no branch is it more requisite, that the learner should be well versed, than in a general idea of the nature and principles of Notation. Without his, he is wholly disqualified from entering upon any subsequent rule in Arithmetic, as he could not, in such a situation,

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acquire any proper knowledge of the nature, powers, and operations of numbers. With this view of the subject, the learner cannot be too diligent in acquiring an adequate knowledge of this fundamental rule, by repeated and thorough revisions of its nature and principles, and by exercise in many and varied examples, which he can readily supply, before he shall proceed. to subsequent rules.

A few examples only will here be given to exercise the learner. Let him write and read the following:

100 millions 55 thousand one hundred and three.

1000 millions ten hundred thousand and twenty.

202 millions 2 thousand 300 hundred, and sixty-seven hundredths decimal.

1 billion 20 millions 20 thousand and twenty, with the Decimals of three hundred and seventy-nine millionths.

One hundred thousand and one hundred, with five tenths, decimal.

Twenty-five thousand and three, with four thousandths de

cimal.

Seventy-five millions and twenty, with eight thousand six hundred and seventy-seven ten-thousandths, decimal.

Six millions sixty thousand and sixteen, with one millionth, decimal.

Questions relative to Notation and Numeration.

1. How is Aritnmetic defined?

2. Whence is the term derived; and what does it signify? 3. How many and what are its radical rules?

4. What is the first defined to be, or what is it denominated? 5. What is the difference between Notation and Numeration?

6. Why are both words necessary as a technical term for this rule ?

How is Numeration, in its general acceptation, defined? 8. What does it teach?

9. How many characters or figures are used in computation": 10. What are the first nine figures called; and why?