to apply them to use in trade and business, and in the common affairs of life. Surely arguments cannot be necessary to prove that no art is more generally useful than this. Whatever our occupations or engagements in life may be, in every trade, business, and employment, to every individual, rich and poor, the knowledge of numbers is necessary. But we need not enlarge on this subject; a small degree of experience and observation will be sufficient to convince the candid enquirer of the great usefulness of Practical Arithmetic.

In commencing his mathematical studies, the learner will begin with Notation; this and Numeration he must endeavour to understand well, as what are usually called the four fundamental rules depend immediately on the structure of our excellent system of numbers. Addition and Subtraction follow in order; and next the Multiplication-table, which must be learned sufficiently perfect, that it may be repeated through from one end to the other, either backwards or forwards, without mistake or hesitation. Having acquired a perfect knowledge of the table, Multiplication and Division, which follow next in order, will not be found difficult. To pass through these rules in a blundering and aukward manner, although it may satisfy a lazy dunce, will not be sufficient for him who aspires to knowledge: if any operation is not perfectly understood, so as to be performed with tolerable ease, the previous examples ought to be worked over again, and repeated until it is. Having passed through the rules in the order they stand in this book, and occasionally consulted the notes, so as to understand the reasons on which the rules are founded, their connection with each

The word Notation is derived from the Latin nota, a mark, and Numeration from numerus, a number.

other, and dependence on self-evident principles, the learner may proceed to Algebra; he will find very little difficulty in that if he understands the arithmetical part well.

1.

DEFINITIONS.

AN unit is that which is known by the name of one. 2. Number is either an unit; a collection of two or more units; or one or more parts of an unit.

3. A whole number is that which consists of one or more units.

4. A broken number or fraction is that which consists of one or more parts of an unit.

5. An even number is that which can be divided into two equal whole numbers.

6. An odd (or uneven) number is that which cannot be divided into two equal whole numbers.

7. An integer is any whole quantity or thing, considered as a whole; the word is used in opposition to a part.

WHOLE NUMBERS.

8. Arithmetic of whole numbers teaches how to calculate or compute by whole numbers.

9. The fundamental rules of Arithmetic are Notation and Numeration, whence are derived Addition, Subtraction, Multiplication, and Division: in the proper application of these rules the whole art of Arithmetic consists.

NOTATION AND NUMERATION.

10. Notation teaches how to write or express numbers by appropriate characters, either singly, or by a proper combination of two or more characters; and Numeration shews how to read numbers when written.

a

11. There are ten characters called digits or figures, by one or more of which every number is expressed: they are written

a From the Latin digitus, a finger. The want of figures to express numbers probably gave rise to digital or manual Arithmetic, in which numbers were expressed, and calculations performed, by the different positions of the hands and fingers. This appears to us a childish play, but it was formerly a serious study, as appears from the elaborate account of it, given by venerable Bede, in his Opera, p. 227, &c. Some of the eastern nations still employ this method, and they are said to surpass us in the expedition and accuracy of their calculations.

and named as follows; 1, one, or unity; 2, two; 3, three; 4, four; 5, five; 6, six; 7, seven; 8, eight; 9, nine; and O, nought, (or nothing.)

12. Unity, or one, is the least of all whole numbers, and may be considered as the root or origin of all the rest; for if unity be increased by itself, and if the result be increased by unity, and again, if the last result be increased by unity, and so on continually, the several results will constitute the entire system of whole numbers. For example, unity or 1 increased by itself becomes (1, 1, or) 2; again, 2 increased by unity becomes (1, 1, 1, or) 3; in like manner 3 increased by unity becomes (1, 1, 1, 1, or) 4, and so on indefinitely.

13. The nine first numbers are all that can be expressed by single figures; to denote all higher numbers it is necessary to combine two, three, or more figures, and sometimes to employ one or more ciphers.

14. It has been shewn in the preceding article, that all numbers originate in unity, and successively arise, by the continual increase of the preceding number by unity, and that the nine figures represent the nine first numbers; also that higher numbers require a combination of two or more figures. Before we explain the method of combination, it will be necessary to shew the manner of classing numbers, which has been universally adopted for the convenience of computation, and is indispensable where high numbers are concerned.

15. Numbers are classed and ranged under the following denominations, viz. Units, Tens, Hundreds, Thousands, Tens of Thousands, Hundreds of Thousands, Millions, &c. The first nine numbers constitute the class of units: the number which next follows the last of this class (or 9) is ten; this is the first number of the class of Tens; this class proceeds thus, (1 ten, or) Ten; (2 tens, or) Twenty; (3 tens, or) Thirty; (4 tens, or) Forty; (5 tens, or) Fifty; (6 tens, or) Sixty; (7 tens, or) Seventy; (8 tens, or) Eighty; (9 tens, or) Ninety; and the next number in this order is (10 tens, or) 1 Hundred, which is the first number of the next superior class; this class proceeds thus, 1 Hundred, 2 Hundreds, 3 Hundreds, and so on up to 10 Hundreds, which is 1 Thousand, or the first number of the next superior class; which in like manner proceeds thus, 1 Thousand, 2 Thousands, 3 Thousands, &c. up to 10 Thousands, which is the first number of the next

class superior to the former; this again proceeds, (1 ten Thousands, or) Ten Thousands; (2 tens, or) Twenty Thousands; (3 tens, or) Thirty Thousands; and so on up to 10 ten Thousands, or One Hundred Thousands; which is the first of the next superior class; whence proceeding as before we have, 1 Hundred Thousands, 2 Hundred Thousands, 3 Hundred Thousands, &c. up to Ten Hundred Thousands, or 1 Million, &c. &c.

16. Hence it appears, that 1 ten is ten units; 2 tens, twenty units; 3 tens, thirty units; 4 tens, forty units, &c. : that 1 hundred is ten tens; 2 hundreds, twenty tens; 3 hundreds, thirty tens, &c.: that 1 thousand is ten hundreds; 2 thousands, twenty hundreds; 3 thousands, thirty hundreds; and in general that every superior denomination is tenfold the next inferior one; and also that any part of a superior denomination is in like manner tenfold the same part of the next inferior one.

17. It follows, from Art. 12. that there are many intermediate numbers, which, according to the preceding arrangement, must fall under two or more of the foregoing denominations: thus, twenty-five consists of 2 tens and 5 units; six hundred and seventy-eight consists of 6 hundreds, 7 tens, and S units; three thousand four hundred and fifty-six consists of 3 thousands, 4 hundreds, 5 tens, and 6 units, &c. &c. Hence a distinct idea of the value of any numbers may be formed from this convenient and beautiful mode of arrangement.

18. Having given a sketch of the general outline, the next thing to be explained is the method of expressing all numbers by the ten digits or figures; in order to which we observe, that each figure, unconnected with any of the other figures, stands merely for its own simple value; but each has besides a local value, namely, a value which depends on the place it occupies when connected with others: thus a figure standing in the first or right hand place expresses only its simple value; but if another figure or the cipher be placed to the right of it, then the figure first mentioned expresses ten times its simple value, that is, as many tens as it contains units. If two figures or ciphers be placed to the right of a figure, that figure expresses ten times what it did when it had only one on its right, or one hundred times its simple value; and so on continually.

19. Hence appears the use of the cipher, which although it is of no value in itself, yet when placed on the right of any num