3. What is the solid content of a grindstone 32 inches in diameter, and three inches thick?

1. In 9 hhd. 15 gal. 3 qts. of wine, how many quarts?

2. In twelve pipes of wine, how many pints?

Ans. 12096.

REMARK. It is thought unnecessary to give examples of reduction under all the tables of Weights, Measures, &c.; as the attentive scholar will readily understand the correct process, from the exercises he has already had. The learner mnay, however, find it useful to turn to those tables under which no example is here given-state questions of his own accord-and, having found their true answers, write them down with the other examples in his CIPHERING-BOOK.

FRACTIONS.*

FRACTIONS are parts of an unit, or whole number.

When & whole is expressed by figures, the number is called an integer :but when a part, or some parts, of a thing are denoted by figures, as, one fourth, two thirds, three tenths, &c. of a thing, these figures are called fractions.

Fractions are divided into two kinds, VULGAR and DECIMAL.

VULGAR FRACTIONS.

A VULGAR FRACTION is that which can have any denominator; and is expressed by two numbers written one above the other, thus-, with a line between.

The figure above the line is called the numerator,

The figure below the line is called the denominator,

5

8

The denominator (which is the divisor in division) shows how many parts the integer is divided into ;-and the numerator (which is the remainder after division) shows how many of those parts are meant by the fraction.

Fractions are either proper, improper, single, compound, or mixed. Any whole number may be made an improper fraction, by drawing a line under it, and putting unity, or 1, for a denominator; as, 9 may be expressed, fractionwise, thus-, and 12 thus,, &c.

1. A single, or simple fraction, is a fraction expressed in a simple form; as,,, 70, &c.

2. A compound fraction is a fraction expressed in a compound form, being a fraction of a fraction, or, two or more fractions connected together; as, of 2, 4 of 5 of 1; which are read thus-one half of three fourths, two sevenths of five elevenths of nineteen twentieths, &c.

*The term, FRACTION, signifies a broken part or parts of any thing or number; and these parts can be represented by figures, as well as whole things or numbers. It was shewn on page 29, that fractions arise from the operations of division; and hence we may see the necessity of understanding something of the arithmetick of Vulgar Fractions, even though it be in some respects" a tedious and intricate rule.”

134. What are fractions?- -135. What is a Vulgar Fraction ?—136. How is it written?—137. What is the figure above the line called?- -138, What is the one below the line called?.. -139. What is the meaning, or the use, of these terms?140. Are there different kinds of Vulgar Fractions?141. How can a whole number be made an improper fraction ?-142. What is a simple fraction ?-143. What is a compound fraction?

3. A proper fraction is a fraction, whose numerator is less, than its denominator; as, 1, 2, 3, &c.

4.

An improper fraction is a fraction, whose numerator is larger than its denominator; as, 5, 3, 3,

2. &c.

5. A mixed number is composed of a whole number and a fraction; as, 73, 35,4, &c., that is, seven and three fifths, &c.

6. The common measure of two or more numbers, is that number, which will divide each of them without a remainder: Thus, 5 is the common measure (or divisor) of 10, 20, and 30; and the greatest number which will do this is called the greatest common measure.

7. A number, which can be measured by two, or more, numbers, is called their common multiple: And, if it be the least number which can be so measured, it is called the least common multiple; thus, 40, 60, 80, 100, are multiples of 4 and 5; but their least common multiple is 20.

8. A prime number is that, which can only be measured (that is, divided) by itself, or an unit; as, 5,,&c.

9. A perfect number is equal to the sum of all its aliquot parts.*

PROBLEM I.

To find the greatest common measure of two or more numbers.

RULE 1. If there be two numbers only, divide the greater by the less, and this divisor by the remainder, and so on, always dividing the last divisor by the last remainder, till nothing remain ; then will the last divisor be the greatest common measure required.

2. When there are more than two numbers, find the greatest common measure of two of them, as before; then of that com

*The aliquot parts are the parts of a number, that will measure it without any remainder; as, 3 is an aliquot part of 12. And 2, 3, and 6, are aliquot parts of 6. And 2, 4, 7, 14, and 28, are aliquot parts of 28. Now, if 6 be divided by 2, 3, and 6, the respective quotients will be 3, 2, and 1, whose sum is equal to 6, the original number. And if 23 be divided by 2, 4, 7, 14, and 28, the several quotients will be 14, 7, 4, 2, and 1, whose sum is equal to 28, the given number. 6 and 28 are therefore perfect numbers, according to the definition. The sum of all the aliquot parts of any number, which is not a perfect one. will be either less or greater than the given number, and never precisely equal to it; as may be proved by a similar operation.

144 What is a proper fraction ?—145. What is an improper fraction ?—— -146. What is a mixed number ?- 147. What is a common measure of two or more numbers?148. What is a common multiple ?—149. What is the difference between a prime and a perfect number ?- -150. How do you find the greatest common measure of two or more numbers?

mon measure and one of the other numbers, and so on, through all the numbers, to the last; then will the greatest common measure, last found, be the answer.

3. If 1 happens to be the common measure, the given numbers are prime to each other, and found to be incommensurable, or in their lowest terms.*

EXAMPLES.

1. What is the greatest common measure of 36 and 96?

2. What is the greatest common measure of 1224 and 1080 ?

Ans. 72,

PROBLEM II.

To find the least common multiple of two or more numbers.

RULE 1. Divide by any number that will divide two or more of the given numbers without a remainder, and set the quotients, together with the undivided numbers, in a line beneath.

2. Divide the second line as before, and so on, till there are no two numbers that can be divided; then the continued product of the divisors and quotients will give the multiple required.t

*The truth of this rule may be shown from the first example; for since 12 divides 24, it also divides 24+12, or 36. Again, since 12 divides 24 and 36, it also divides 36×2+24, or 96.

The reason of this rule may be shown from the first example, thus: It is evident, that 3X5X8X10-1200 may be divided by 3, 5, 8, and 10, without a remainder; but 10 is a multiple of 5; therefore, 3×5×8×2=240 is also divisible by 3, 5, 8, and 10.— Also, 8 is a multiple of 2; therefore, 3×5×4×2=120 is also divisible by 3, 5, 8, and 10; and is evidently the least number that can be so divided.

151. What is the rule for finding the greatest common multiple of two or more numbers?

EXAMPLES.

1. What is the least common multiple of 6, 10, 16, and 20?

*5)6 10 16 20

*2)6 2 16

4

*2)3 1 8

*3 1 *4

2

1

We survey the given numbers, and find that 5 will divide two of them, viz. 10 and 20, which we divide by 5, bringing into a line with the quotients, the numbers, which 5 will not measure : again, we view the numbers in the second line, and find 2 will measure them all, and we get 3, 1, 8, 2, in the third line, and find that 2 will measure 8 and 2, and in the fourth line get 3, 1, 4, 1 all 5×2×2×3×4=240 Ans. prime; we then multiply the prime numbers and the divisors continually into, each other, for the number sought, and find it to be 240.

* * * * *

2. What is the least number, which can be divided by the 9 digits separately, without a remainder?

Ans. 2520.

REDUCTION OF VULGAR FRACTIONS,

Is the bringing them out of one form into another, in order to prepare them for the operations of Addition, Subtraction, &c.

CASE I.

To abbreviate or reduce fractions to their lowest terms.*

* A fraction is in its lowest terms, when represented by the least numbers possible: when reduced to its lowest terms is.

thus

152. What is reduction of vulgar fractions?-153. When is a fraction in its lowest terms?

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