the foundation of the application of arithmetic to geometry. This connexion will be understood from the following illustration.

which are likewise equal, into four equal parts by the points 8, 9, 10. Join the points 1, 1; 2, 2; 3, 3; 10, 10; then the figure ABCD is evidently divided into 8 times 4, or 32 smaller rectangles, any one of which, as A 1 E8, is called a square, because it is a rectangle whose sides are all equal, A 1 being equal to A 8. Now each of these 32 squares has its sides one inch in length, and is termed a square inch; consequently the figure A B C D contains 8 times 4, or 32 square inches. In a similar manner, if A B = 8 feet, and A C = 4 feet, then the figure will contain 32 square feet, and so on.

91. Suppose now that A B is 67 inches, or 5 feet 7 inches in length, and AC is 38 inches, or 3 feet 2 inches in length; then, as we have seen, the rectangle A B C D will contain 67 times 38, or 2546 square inches. But as 1 square foot contains 144 square inches, if we divide 2546 by 144, we get 17 square feet and 98 square inches for the content of the rectangle. If, now, 5 feet 7 inches be multiplied by 3 feet 2 inches, the result ought to be the same as we have here obtained. To effect this multiplication, place the factors so that feet may be under feet, inches under inches, and proceed as in the multiplication of decimals, recollecting to carry 1 for every 12, because it is this number which connects the different denominations in the duodecimal system. In order to observe the similarity of the processes in the multiplication of decimals and duodecimals, we shall give the mode of multiplying 5'7 by 3.2, in connexion with the other.

Thus the processes are precisely similar, and the fractional ones serve to illustrate the other, because if the denominators 10 and 12 were omitted, the corresponding multiplications would have precisely the same

form. In the duodecimal multiplication it is important to notice that while the product of feet by feet is square feet, and the product of inches ty inches is square inches, the product of feet by inches is neither square feet nor square inches, but rectangles, each one foot in length and one inch in breadth, that is, twelfths of a square foot, or 12 square inches. 92. As another example, let it be required to multiply 23 feet 7 inches 9 parts by 19 feet 3 inches 6 parts.

Let 10 be expressed by and 11 by ; then 23 transformed to the duodecimal scale will be expressed by one-twelve + 11 or 1 ɛ, and 19 by one-twelve +7 or 17; consequently the multiplication may be made entirely in the duodecimal scale, or partly in that scale and partly in the decimal scale in the following manner :

Ft. In. Pts.
23 7 9

In this way the multiplication of 23 by 19 is made in the decimal scale, while all the others are effected by the duodecimal scale, and the result is obtained in square feet, twelfths of square feet, hundred and fortyfourths of square feet or square inches, and so on. But in the other process, the result 320 2016 is expressed entirely in the duodecimal scale; therefore 320 must be transformed from the duodecimal to the decimal scale, and then the results are precisely the same by both methods. If the multiplication be effected by taking aliquot parts of the multiplicand, the process would be as in the margin, where the multiplicand is multiplied by 19, as in Compound Multiplication, and then the aliquot parts are taken.

2x9+1

47 3

6

9

7

6

23

7

9

In. Ft. 425

2
1

Pt. In.

6

3 11 3 6

1 11 7 9

RATIO, PROPORTION, AND PROGRESSIONS.

93. The ratio of one number to another is the relation which the former has to the latter, the comparison being made by considering what multiple, part, or parts, or, which is the same thing, what fraction the first is of the second. Thus the ratio of 8 to 12 is the same as that of 2 to 3, 2

2

or of to 1; because in each pair of numbers the first is of the second.

3

Hence if the ratio of any two numbers be represented by a fraction, there must be a tacit reference to unity. In this manner the ratio of 20 to 4

The ratio of one number to another is expressed by two points placed between them, as 3: 4. The numbers are called the terms of the ratio, the former being called the antecedent and the latter the consequent.

94. If the first of four numbers has to the second the same ratio which the third has to the fourth, the four numbers are said to form a propor

hence, these ratios being equal, the four numbers 2, 3, 10, 15, are proportional. This proportion is expressed in the following manner; 2:3:10: 15, and is read as 2 is to 3, so is 10 to 15. In a proportion thus expressed, the numbers 2 and 15 are called the extreme terms, or simply the extremes, and the numbers 3 and 10 the means.

hence, if four numbers are in proportion, the product of the extremes is equal to the product of the means.

95. The order of the terms of a proportion may be changed, provided that in the new arrangement the product of the extremes is equal to that of the means. Thus in the proportion 2: 3 :: 10: 15, the following arrangements may be made :

2: 3:10: 15
2:10: 3:15

15: 10 :: 3:2

15: 3:10: 2

:

3: 2::15: 10

3:15:: 2:10 10: 2:15: 3 10: 15: 2:3

96. Also if the corresponding terms of two proportions be multiplied together, the products thence arising will give four numbers, which are proportionals. Thus if 30: 15 :: 6 : 3 and 2 : 3 :: 4: 6 be two proportions, then we have

hence 30 × 2 : 15 × 3 :: 6 × 4 : 3 × 6, or 60 : 45 :: 24 : 18.

In a similar manner it may be shown that if the terms of a proportion be squared or cubed, or if their square roots or cube roots be extracted, the results will still constitute a proportion.

97. Hence, since the product of the extremes is equal to that of the means, if the product of the means be divided by one of the extremes, the quotient will be the other extreme; or if the product of the extremes be divided by one of the means, the quotient will be the other mean. The operation by which the fourth term of a proportion is found, any three of them being known, is called the Rule of Three.

98. Numbers may be also compared by observing how much the one differs from the other. Thus the difference between 8 and 19 is 11;

and if we take two other numbers connected in the same manner, that is, two numbers whose difference is 11, as 51 and 62, then the four numbers 8, 19, 51, 62, form what is usually termed an arithmetical proportion. It will be obvious that 62 +8 5119; that is, the sum of the extremes is equal to the sum of the means.

=

99. A set or series of numbers is said to be in continued arithmetical proportion, or in arithmetical progression, when the difference between every two succeeding terms of the series is the same. Thus, the two series of numbers,

2, 5, 8, 11, 14, etc., and 100, 90, 80, 70, 60, etc.,

are in arithmetical progression, the common difference of the former being 3, and that of the latter 10. It is obvious that if we take any three terms of the first of these series, as 5, 8, 11, the sum of the extremes will be equal to twice the mean, that is, 5+11 2 x 8; since 8 is just as much above 5 as it is below 11. Hence half the sum of any two numbers is the arithmetical mean between them.

100. A set or series of numbers is said to be in continued proportion, or in geometrical progression, when the ratio of any term to the preceding term of the series is the same. Thus the two series of numbers,

are in geometrical progression, the common ratio of the former being 2, and that of the latter If we take any three terms of the first of these

2 3

=

series, as 4, 8, 16, the product of the extremes will be equal to the square of the mean, since 4 × 16 82; hence the square root of the product of any two numbers is the geometrical mean between them.

The properties of a series of quantities, either in arithmetical or geometrical progression, are investigated and applied in the Algebra, and require no further notice here.

RULE OF THREE.

101. This is the most extensive and useful rule in Arithmetic, and it is termed the The Rule of Three, because in it three quantities are given and a fourth is to be found, to which that one of the three quantities which is of the same kind with it shall have the same ratio as one of the remaining two has to the other of these.

Suppose it required to find what 17 yards will cost, if 52 yards cost £10. 88. Reducing £10. 8s. to shillings, we get 208 shillings for the

cost of 52 yards; therefore, the cost of 1 yard must be

208
52'

or 4 shil

lings, and since the cost of 17 yards will be 17 times the cost of 1 yard; therefore 17 yards will cost 17 times 4 shillings, or 68 shillings, which is £3. 88. Hence £10. 8s. has the same ratio to £3. 8s which 52 yards has to 17 yards, and the proportion is thus written :

52 yards: 17 yards :: £10. 8s. : £3. 8s.

Again, were it required to find in how many days 27 men would finish a piece of work which 15 men, working at the same rate, could accomplish in 18 days; it is evident that the more men there are employed

the less time will they require to finish it, and vice versa. In this case there exists a proportion, but the order of it is inverted; because, if 15 men require 18 days, 27 men will require, not more, but less than 18 days; consequently 18 days must have the same ratio to the number of days required which 27 men has to 15 men; and therefore the proportion will be

27 men: 15 men :: 18 days: x days,

where x denotes the number of days which 27 men would require; consequently (97) the extreme term of the proportion is

102. When it is proposed to resolve a question by means of a Rule of Three, first ascertain whether the solution can depend on proportion, and, if it does, then assign to each term the place which it ought to occupy. But as the placing of the numbers constitutes the chief difficulty, the following remarks will be found useful in assisting the student to write down correctly the first three terms of the proportion.

Place that quantity which is of the same kind with the fourth or required quantity as the third term of the proportion; then consider, from the nature of the question, whether the fourth quantity is to be less ⚫ or greater than that in the third term. If the fourth quantity is to be less than the third, place the less of the two other quantities in the second term; but if the fourth is to be greater than the third, place the greater of the two others in the second term.

If the quantities consist of several denominations, reduce the first and second to the same denomination, the lowest in either, and reduce the third term to the lowest denomination in it; then (97) the fourth term is obtained by multiplying the second and third terms and dividing their product by the first term. The quotient is the fourth quantity or number required, and it is in the same denomination to which the third term was reduced.

If the first and second terms be divided or multiplied by the same number, their ratio will not be altered; and if all the numbers be regarded as abstract numbers, we may also divide the first and third terms by the same number without destroying the proportion.

Ex. 1. Find the value of 36 cwt. 1 qr., if 2 cwt. 2 qrs. 10 lb. cost £4. 7s. 94d.

Here the third term must be money, and as 36 cwt. 1 qr. will cost more than 2 cwt. 2 qrs. 10 lb., the greater of these quantities must be placed in the second term, and the first three terms of the proportion must be arranged as follows::