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The importance of the Common Fraction, which is now rendered as simple as an Integral Number, cannot be too highly estimated. It affords the greatest variety of Problems for the general training of the mind. It expresses the Ratio of two numbers, and Ratio is the element of Percentage, and of almost every Arithmetical operation. It greatly abridges the work, especially when Cancellation is applied; it has in a great measure superceded the Decimal and Compound Denominate Number; Avoirdupois Weight is seldom carried below the lb.; and the cwt., qr., and lb. are more simply rendered in lbs. only, and the lbs. are readily reduced to the fraction of a ton. In the measurement of dry-goods, only the yard and the fraction of a yard are used. Troy Weight is seldom reckoned below the oz. and the fraction of the oz. The Apothecary requires the small weights in mixing medicines. Time and Circular Measure cannot be changed; nor is there any attempt made at a change in the great variety of coins and currencies, even by the Metric System.
The Author cannot to highly recommend to the Teacher the use of the Blackboard described on the following page. Great facility in come prehending the combinations and divisions of numbers will be acquired by this method.
This page represents a blackboard with the numbers as high as 72 painted on its margins.
There is also a box containing slips which will cover 4 two, three, four, etc., as high as 12, and numbered 5 accordingly; one of these the student will take in his hand and apply it to the painted numbers to perform addition or subtraction; thus, begin at 1 and take a slip marked 2, then 1 and 2 are 3, 3 and 2 are 5,5 and 2 are 7, 7 and 2 are 9, etc., counting at least the lefthand column; then, to perform subtraction, begin at the bottom of the 1st column; thus, 36 minus 2 equals 34, 34-2=32, 32-2-30, 30-2-28, etc., until the top is reached; then taking a slip marked 3, begin with 1 or 2, or first with 1 and then with 2, and return to the top of the column as before, by subtraction; let this exercise be performed with all the slips, and as the larger numbers are taken, continue the additions to the bottom of the 2d column, and return as before.
For multiplication and division first make a chalk mark after every two figures up to 24, and multiply; thus, once 2 are 2, twice 2 are 4, 3 times 2 are 6, 4 times 2 are 8, etc.; then the number of divisions is 12 and each division has 2 numbers; .. 12 is contained twice in 24, or 2 is contained 12 times, 2 is contained once in 2, in 4 twice, in 6 three times, in 8 four times, in 10 five times, in 12 six times, etc. When the student is familiar with multiplication and division by 2, let the numbers be separated into 3's, then 4's, etc., and let each be continued for 12 divisions; when all the divisions have been performed according to the steps, beginning with 2 and ending with 12, a multiplication and division table will be made.
REM.-In multiplication the product of any two factors is the same by making either the multiplicand and the other the multiplier; so also in division, the divisor and the quotient may be substituted, as the dividend is the product of the divisor and quotient.
REM. The numbers, continued up to 144, should
be painted on the sides of the board.
1. Arithmetic is the science of numbers.
2. A Unit is a single thing; as, a book, one dollar, or simply one.
3. A Number is a unit or a collection of units; as, one, ten, five books, twenty-five dollars.
4. The numbers used in Arithmetic are all formed by combinations of the ten Arabic characters, called Figures; viz., 0, called zero or naught; 1, called one; 2, two; 3, three; 4, four; 5, five; 6, six; 7, seven; 8, eight; 9, nine.
5. Expressing a number either in writing or figures is called Notation, and reading the expression is called Numeration.
6. When numbers are used without reference to any object, they are called Abstract Numbers; as, five, twenty, etc.; but when they are applied to things, they are called Concrete; as, one book, ten men, four dollars, etc.
7. When concrete numbers express values of money, weights, measures, time, etc., they are called Denomi
nate Numbers; as, dollars, pounds, shillings, pounds of weight, ounces, hours, minutes, etc.
8. When different denominations of either kind form but one number, it is called a Compound Number; as, £4 3s. 61., 2 lb. 1 oz. 3 pwt. and 2 gr.
9. Numbers of the same order and the same denomination are termed Like Numbers; other numbers are termed Unlike Numbers.
REM.-Numbers expressing different species of the same genus are unlike, as horses and cows; while the same numbers expressed in the term of the genus are alike, as animals.
MATHEMATICAL TERMS USED IN ARITHMETIC.
1. An affirmative sentence, or anything proposed for consideration, is a Proposition.
2. A self-evident proposition is called an Axiom.
3. A proposition made evident by a demonstration is called a Theorem.
4. When a proposition is used for developing a principle of Arithmetic, it is called a Problem.
5. Propositions given merely for solution, in order to impress the principles on the mind, are called Examples.
6. An obvious consequence of one or more propositions is called a Corollary.
7. An established custom, or an assumption without proof, is called a Postulate.
REM. 1 and 1 are 2, 2 and 1 are 3, 3 and 1 are 4, 5 and 2 are 7, 6 and 3 are 9, etc., is the postulate which forms the basis of Arith.