the unknown quantity must be equal to 3 x 15 or 45, that quantity itself is equal to 455 or 9. Generally one of the extremes is equal to the product of the means divided by the other extreme; and one of the means is equal to the product of the extremes divided by the other mean.

4. We may, without affecting the correctness of a proportion, subject the several terms which compose it to all the changes which can be made, while the product of the extremes remains equal to that of the means. Thus, for 5: 3 :: 15:9, which

gives 5 x 93 x 15, we may

I. Change the places of the means without changing those of the extremes, or change the places of the extremes without changing those of the means: this is denoted by the term alternando.

II. Put the extremes in the places of the means; this is called invertendo.

3:59:15

III. Multiply or divide the two antecedents or the two consequents by the same number.

It also appears, with regard to proportions, that the sum or the difference of the antecedents is to that of the consequents, as either antecedent is to its consequent.

And, that the sum of the antecedents is to their difference, as the sum of the consequents is to their difference.

If there be a series of equal ratios represented by §

Therefore, in a series of equal ratios, the sum of the antecedents is to the sum of the consequents, as any one antecedent is to its consequent.

If there be two proportions, as 30: 15 :: 6 : 3, and 2:3::4:6, then multiplying them term by term we shall have 30 x 2: 15 x 36 x 4:3 × 6, which is evidently a proportion, because 30 × 2 × 3 × 6 15 x 3 x 6 x 4 = 1080. Thus, also, any powers of quantities in proportion are in proportion; and conversely of the roots.

=

2:36:9)

2:36:9
2:36:9

22: 32 :: 62 : 92

23:33:: 63: 93

Rule of Three.

When the elements of a problem will form a proportion of which the unknown quantity is the last term, a simple calculation will determine it, and the problem is said to belong to the Golden Rule, or Rule of Three. The operation is regulated by the foregoing principles of proportion.

Of the three given numbers, two are called the terms of supposition, and the other the term of demand. Now if the term of demand be greater or less than the other term of the same kind, and the question require the term sought to be respectively greater or less than the other, the question belongs to the Rule of Three direct: otherwise it belongs to the Rule of Three inverse.

For the Rule of Three direct we have this

RULE. Write the three given terms in the following order, viz. let that which implies or asks the demand be put in the third place, and the other of the same kind in the first: then will the remaining term, which is similar to the fourth or required one, occupy the second place. Having thus disposed the numbers, called stating the question, reduce the first and third terms to one and the same denomination; and if the second term be a compound one, reduce it to the lowest name mentioned. Multiply the second and third terms together, and divide the product by the first, and the quotient will be the answer, in the same denomination to which you reduced the

second term.

When the second term is a compound one, and the third a composite number, it is generally better to multiply the second term, without any previous reduction, by the component parts of the third, as in compound multiplication, after which divide the compound product by the first term, or, by its factors. (Here the first and third terms are homogeneous, in a given ratio, the second and fourth in the same.)

For the Inverse Rule.

State the question and reduce the terms in the direct rule: then multiply the first and second terms together and divide the product by the third, and the quotient will be the

answer.

A distinct rule is usually given for the working of problems in Compound Proportion; but they may generally be solved with greater mental facility by means of separate statings.

Note. The Rule of Three receives its application in ques

tions of Interest, Discount, Fellowship, Barter, &c.

F

Properties of Numbers.

To render these intelligible to the student, we shall here collect a few definitions.

1. An unit, or unity, is the representation of any thing considered individually, without regard to the parts of which it is composed.

2. An integer is either a unit or an assemblage of units; and a fraction is any part or parts of a unit.

3. A multiple of any number is that which contains it some exact number of times.

4. One number is said to measure another, when it divides it without leaving any remainder.

5. And if a number exactly divides two, or more numbers, it is then called their common measure.

6. An even number is that which can be halved, or divided into two equal parts.

7. An odd number is that which cannot be halved, or which differs from an even number by unity.

8. A prime number is that which can only be measured by 1, or unity.

9. One number is said to be prime to another when unity is the only number by which they can both be measured.

10. A composite number is that which can be measured by some number greater than unity.

11. A perfect number, is that which is equal to the sum of all its aliquot parts: thus 6=+&+8•

Prop. 1.-The sum or difference of any two even numbers is an even number.

2. The sum or difference of any two odd numbers is even ; but the sum of three odd numbers is odd.

3. The sum of any even number of odd numbers is even; but the sum of any odd number of odd numbers is odd.

4. The sum or difference of an even and an odd number is odd.

5. The product of an even or an odd number, or of two even numbers, is even.

6. An odd number cannot be divided by an even number, without a remainder.

7. Any power of an even number is even.

8. The product of any two odd numbers is an odd number. 9. The product of any number of odd numbers is odd; and every power of an odd number is odd.

10. If an odd number divides an even number, it will also divide the half of it.