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Arithmetic to protect himself from the frauds of money-changers and stewards, who took advantage of the ignorance of their employers. In his preparation for this work, he professes to have read all the books which had been published on this subject, adding, also, that there was hardly any nation which did not possess such books.

About the year 1540, Robert Record, Doctor in Physic, printed the first edition of his famous Arithmetic, which was afterward augmented by John Dee, and subsequently by John Mellis, and which did much to advance the science and practice of Arithmetic in England in its early stages. This work, which is now quite a curiosity, effectually destroys the claim to originality in some things of which authors much more modern have obtained the credit. In it we find the celebrated case of a will, which we have in the Miscellaneous Questions of Webber and Kinne, and which, altered in language and the time of making the testament, is the 2nd Miscellaneous Question in the present work. This question is, by his own confession, older than Record, and is said to have been famous since the days of Lucas de Burgo. In Record it occurs under the "Rule of Fellowship." Record was the author of the first treatise on Algebra in the English language.

In 1566, a complete work on Practical Arithmetic was published by Nicolas Tartaglia, an Italian, and one of the most eminent mathematicians of his time.

From the time of Record and Tartaglia, works on Arithmetic have been too numerous to mention in an ordinary history of the science. De Morgan, in his recent work (Arithmetical Books'), has given the names of a large number, with brief observations upon them, and to this the inquisitive student is referred for further information in regard both to writers and books on this subject since the invention of Printing. It is remarkable that De Morgan knew next to nothing of any American works on Arithmetic. He mentions the " American Accountant" by William Milns, New York, 1797, and gives the name of Pike (probably Nicholas Pike) among the names of which he had heard in connection with the subject. Of the compilation of Webber and the original work of Walsh, he seems to have been entirely ignorant.

The various signs or symbols, which are now so generally used to abridge arithmetical as well as algebraical operations, were introduced gradually, as necessity or convenience taught their importance. The earliest writer on Algebra after the invention of printing was Lucas de Burgo, above mentioned, and he uses p for plus and m for minus, and indicates the powers by the first two letters, in which he is followed by several of his successors. After this, Steifel, a German, who in 1544 published a work entitled Arkhmetica Integra, added considerably to the use of signs, and, according to Dr. Hutton, was the first who employed -f and — to denote addition and subtraction. To denote the root of a quantity he also used our present sign J, originally r, the initial of the word radix, root. The sign to denote equality, was introduced by Record, the above-named English mathematician, and for this reason, as he says, that " noe 2 thynges can be moar equalle," namely, than two parallel lines. It is a curious circumstance that this same symbol was first used to denote subtraction. It was also employed in this sense by A1bert Girarde, who lived a little later than Record. Girarde dispensed with the vinculum employed by Steifel, as in 3~+4,and substituted the parenthesis (3 + 4), now so generally adopted. The first use of the St. Andrew's cross, X, to signify multiplication, is attributed to William Oughtred, an Englishman, who in 1631 published a work entitled Clavis Mathematics, or Key of Mathematics.

It was intended to notice several other works, ancient and modern, but the length to which this sketch has already extended forbids it.

We had thought of alluding to the ancient philosophic Arithmetic, and the elevated ideas which many of the early philosophers had of the science and properties of numbers. But a word must here suffice. Arithmetic, according to the followers of Plato, was not to be studied "with gross and vulgar views, but in such a manner as might enable men to attain to the contemplation of numbers; not for the purpose of dealing with merchants and tavern-keepers, but for the improvement of the mind, considering it as the path which leads to the knowledge of truth and reality." These transcendentalists considered perfect numbers, compared with those which are deficient or superabundant, as the images of the virtues, which, they allege, are equally remote from excess and defect, constituting a mean between them; as in the case of true courage, which, they say, lies midway between audacity and cowardice, and of liberality, which is a mean between profusion and avarice. In other respects, also, they regard this analogy as remarkable; perfect numbers, like the virtues, are "few in number and generated in a constant order; while superabundant and deficient. numbers are, like vices, infinite in number, disposable in no regular series, and generated according to no certain and invariable law."

NOTE TO TEACHERS.

For the convenience of those who require a less extended course, eeveral entire Articles, and some examples, have been marked (°), to be omitted at the option of the teacher.

ARITHMETIC.

DEFINITIONS.

Article 1, Quantity is anything that can be increased, diminished, or measured; as time, weight, lines, surfaces, and solids.

2. A unit is a single thing or quantity regarded as a whole.

3. An abstract unit is one that has no reference to any particular thing or quantity.

4. A concrete unit is one that has reference to some particular thing or quantity.

5. A number is an expression of quantity, representing either a unit or a collection of units.

6. An abstract number is a number whose unit is abstract; as, one, six, nine.

7. A concrete or denominate number is a number whose unit is concrete; as, one dollar, six pounds, nine men.

8i A simple number is a unit, or a collection of units, either abstract, or concrete of a single kind or denomination; as, 1, 15, 1 book, 13 dollars.

9. The unit of measure of any quantity is one of the same kind with that by which the quantity is measured or compared; as, in the abstract number, six, the abstract unit is that of measure or comparison; and in six pounds, the concrete unit, one pound, is that of measure or comparison.

10. Arithmetic is the science of numbers and the art of computing by them. It treats of the properties and relations of numbers, and teaches the methods of applying the principles of the science to practical purposes.

11. An axiom is a self-evident truth.

12. A problem is a question proposed for solution, or something to be done.

13. An operation is the process of finding, from given quantities, others that are required.

14. A sign is a symbol employed to indicate the relations of quantities, or operations to be performed upon them.

15. A rule is a direction for performing an operation.

16. An example is a particular application of a general principle or rule.

17. The principal or fundamental processes of arithmetic are Notation and Numeration, Addition, Subtraction, Multiplication, and Division.

SIGNS.

18. The sign of equality, two short horizontal lines, =, is read equal, or equal to, and denotes that the quantities between which it is placed are equal to each other. Thus, 12 inches = 1 foot, signifies that 12 inches are equal to 1 foot.

19. The sign of addition, an erect cross, -f-, is read plus, and, or added to, and denotes that the quantities between which it is placed are to be added together. Thus, 8 -(- 6 signifies that 6 is to be added to 8.

20. The sign of subtraction, a short horizontal line, —, is read minus, or less, and denotes that the quantity on the right of it is to be subtracted from the quantity on the left. Thus, 8 — 6 signifies that 6 is to be subtracted from 8.

21. The sign of multiplication, an inclined cross, X, is read times, or multiplied by, and denotes that the quantities between which it is placed are to be multiplied together. Thus, 7x6 signifies that 7 is to be multiplied by 6.

22. The sign of division, a horizontal line between two dots, -4-, is read divided by, and denotes that the quantity on the left of it is to be divided by that on the right. Thus, 42-5-6 signifies that 42 is to be divided by 6.

23. The sign of aggregation, a parenthesis, ( ), including several numbers, or a vinculum, , drawn over them,

indicates that the value of the expression is to be used as a single number. Thus, (17 -)- 3) X 5, indicates that the sum of 17 and 3, or 20, is to be multiplied by 5; and 12 -f- (9 — 3) 2, indicates that the difference between 9 and 3 divided by 2, or 3, is to be added to 12.

AXIOMS.

24. Arithmetic, in common with other branches of the mathematics, is based upon axioms, few in number, and universally admitted to be so clearly true, that no process of reasoning can make them plainer; as,

1. If the same quantity, or equal quantities, be added to equal quantities, the sums will be equal.

2. If the same quantity, or equal quantities, be subtracted from equal quantities, the remainders will be equal.

3. If the same quantity, or equal quantities, be added to unequal quantities, the sums will be unequal.

4. If the same quantity, or equal quantities, be subtracted from unequal quantities, the remainders will be unequal.

5. If equal quantities be multiplied by the same quantity, or equal quantities, the products will be equal.

6. If equal quantities be divided by the same quantity, or equal quantities, the quotients will be equal.

7. If the same quantity be both added to and subtracted from another, the value of the latter will not be changed.

8. If a quantity be both multiplied and divided by the same quantity, its value will not be changed.

9. If two quantities be equally increased or diminished, their difference will not be changed.

10. Quantities which are equal to the same quantity are equal to each other.

11. Quantities which are like parts of equal quantities are equal to each other.

12. The whole of a quantity is greater than any of its parts.

13. The whole of a quantity is equal to the sum of all its parts.

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