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to great advantage. The questions composed by the late Martin Clare, F. R. S. have been arranged under their proper rules by Mr. Vyse, in a work entitled, The Tutor's Guide, to which he has added a Key, containing the solutions, the whole forming a very comprehensive system. The ingenious Mr. Keith's Complete Practical Arithmetician is very properly entitled; the work together with the Key certainly form the completest practical treatise extant: the demonstrations added at the end are very clear and satisfactory, and shew that the author has chosen a very modest title for his work. The Rev. Mr. Joyce's System of Practical Arithmetic, published in 1808, is the last work on the subject which we shall notice; this is a very complete and well-written little book, containing a large collection of well-chosen examples, and much information not to be met with in any other work of this nature.
Arithmetic may be considered as a Science, or an Art: as a Science, it treats of the properties of numbers, of their sums, differences, ratios, proportions, progressions, powers, roots, &c. in the most general and abstracted manner; it considers them purely as numbers, and has no reference to any application or use, except that of deducing one property from another, and constituting a necessary link in the chain of universal science. Although this abstracted consideration of numbers is proper for the mathematician, it will be of little use to the learner; he will find, that the quickest and surest way to gain a good and useful knowledge of numbers is to acquire theory from practice, and apply his theory from time to time as he acquires it to practical purposes.
Arithmetic is to be considered as an Art, when it teaches how to perform operations with numbers, and
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 Votation is derived from the Latin nota, a mark, and Vumeration 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
1. 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.
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 pumbers when written.
11. There are ten characters called digits - or figures, by one or more of which every number is expressed : they are written
• 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, sir; 7, seven; 8, eight; 9, nine ; and 0, 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 contimually, 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 (l, l, 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 explaim 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 tem; 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) Sirty; (7 tens, or) Seventy; (8 tens, or) Eighty; (9 tens, or) Ninety; and the next number in this order is (10 tens, or) l 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