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.

Properties of Numbers.

To render these intelligible to the student, we shall here col lect 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= + +

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.

11. If a number consist of many parts, and each of those parts have a common divisor d, then will the whole number, taken collectively, be divisible by d.

12. Neither the sum nor the difference of two fractions, which are in their lowest terms, and of which the denominator of the one contains a factor not common to the other, can be equal to an integer number.

13. If a square number be either multiplied or divided by a square, the product or quotient is a square; and conversely, if a square number be either multiplied or divided by a number that is not a square, the product or quotient is not a square.

14. The product arising from two different prime numbers cannot be a square number.

15. The product of no two different numbers prime to each other can make a square, unless each of those numbers be a square.

16. The square root of an integer number, that is not a complete square, can neither be expressed by an integer nor by any rational fraction.

17. The cube root of an integer that is not a complete cube cannot be expressed by either an integer or a rational fraction.

18. Every prime number greater than 2, is of one of the forms 4 n + 1, or 4 n 1.

19. Every prime number greater than 3, is of one of the forms 6 n + 1, or 6 n — 1.

20. No algebraical formula can contain prime numbers only.

21. The number of prime numbers is unlimited.

22. The first twenty prime numbers are 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, and 67.

23. A square number cannot terminate with an odd number of ciphers.

24. If a square number terminate with a 4, the last figure but one (towards the right hand) will be an even number.

25. If a square number terminate with 5, it will terminate with 25.

26. If a square number terminate with an odd digit, the last figure but one will be even; and if it terminate with any even digit, except 4, the last figure but one will be odd.

27. No square number can terminate with two equal digits, except two ciphers or two fours.

28. No number whose last, or right hand, digit is 2, 3, 7, or 8, is a square number.