10lbs. to 15lbs. being the same as that of 2s. to 3s., we have the proportion

10lbs. 15lbs. :: 2s.: 3s.;

but we cannot have the proportion

10lbs. 2s. :: 15lbs.: 3s.

as no ratio subsists between 10lbs. and 2s. or between 15lbs. and 3s.

Nor indeed can we even in the first of these forms multiply together the concrete quantities so that the product of the extremes equals the product of the means; but what we do in finding any term in such cases, is to consider merely their numerical values, because the ratios being abstract magnitudes will remain the same whatever be the nature of the quantities they are used to compare. See the Appendix.

135. Ratio and Proportion as here used are generally called Geometrical Ratio and Geometrical Proportion, because they are employed in Geometry in the same sense: also, Arithmetical Ratio and Arithmetical Proportion are sometimes used to express the Differences of two or more numbers and their relations to each other, exactly in the same manner as we have throughout applied Ratio and Proportion to denote their Quotients and the relations subsisting among two or more such.

7

5

2

1

5'

Thus, of 7 and 5, the geometrical ratio is 7 : 5=whereas their arithmetical ratio is 7-5 = 2: also, the numbers 3, 4, 15, 20 form a geometrical proportion,

but 4, 3, 2, 1 constitute an arithmetical

When necessary, the relations of numbers considered in the latter point of view may be determined by means of the equality 432-1, in a manner similar to what has been done above.

136. If three numbers as 18, 13 and 8 be in what is called continued Arithmetical proportion, then 18-13 = 13-8; and if 13 +8 be added to both members of this equality, we shall have

18+8=13+13;

that is, the Sum of the Extremes is equal to twice the Arithmetical Mean between them; and therefore the arithmetical mean is equal to half their sum.

In the same manner, 16, 8 and 4 are said to be in continued geometrical proportion, because

and multiplying both sides of this equality by 8 x 4, we obtain

16 × 48 × 8;

or, the Product of the Extremes is equal to the Square of the Geometrical Mean between them and consequently the geometrical mean between two numbers is equal to the Square Root of their product.

These terms and the corresponding operations form the chief substance of the next Chapter, and they have been noticed in this only because they appear to arise immediately out of what has been considered in it.

Applications of Ratio and Proportion.

137. Ratio and Proportion will now be applied and exemplified under the following heads.

(1) The Rule of Proportion.

(2) Simple and Compound Interest.
(3) The Natures and Transfers of Stocks.
(4) Discount or Rebate.

(5) The Equation of Payments.

(6) The Rule of Fellowship.

(7) The Rule of Alligation.
(8) The Doctrine of Exchanges.

I. THE RULE OF PROPORTION.

138. DEF. As has been observed in The Rule of Three of which this is only another name, we have here three quantities either simple or compound given to find a fourth which shall complete the proportion; and this is called a fourth proportional to the three quantities proposed taken in order.

159. Assuming as an Axiom, that Effects have the same relation or ratio to each other as the Causes which produce them under the same circumstances, it is evident that in any two cases of the same kind we shall have the following proportion :

First Cause: Second Cause :: First Effect: Second Effect;

and then, what was said in Articles (130) and (131) will enable us to find any one term if the three others be supposed to be given.

To avoid the trouble of writing the name of the required term or quantity at length, we shall always denote it by the simple symbol x which must be treated in the same way as any other number: and it may occupy any place in the proportion either by itself or as a factor either integral or fractional with given numbers, as in the following Examples.

Ex. 1. If 5 men can mow 12 acres of grass in a certain time; how many acres will 16 men be able to mow in the same or an equal time?

and therefore by the Articles just referred to, we find

5 × x = 16 × 12 = 192 :

whence, x =

192
5

ac. = 38 ac. 1 ro. 24 po.

Ex. 2. If 8oz. of bread be sold for 6d. when wheat is at £15 a load; what should be the price of wheat when 12oz. are sold for 4d.?

If the price of a load of wheat be regulated by, so as to be proportional to, the price of an ounce of bread, since, in the former case the price of 1oz. = §d. = 3d., and in the latter the price of 1oz. = 1d. = d.,

These examples, the causes in which are simple terms being dependent upon only one magnitude, are instances of what is called Direct Proportion, because the effect is greater or less in the same proportion as the cause is greater or less.

Ex. 3. If 10 men can perform a piece of work in 12 days; how many days will it take 8 men to do the same?

Here, the causes will evidently be to each other as 10 × 12 to 8 × r; and the effects are the same, and may therefore be represented by 1, or any other symbol:

whence, 10 x 12 : 8 × x :: 1:1;

therefore 8 × x =

10 x 12 = 120,

Ex. 4. How much in length, that is 3ft. 9in. broad, will be equal to what is 37ft. 9in. long, and 7ft. 6in. broad? Here, the first cause = 45in. × æ in.;

the second cause = 90in. × 453in.;

and the effects are to be equal:

therefore 45 × x: 90 × 453 :: 1:1;
whence, 45 x x = 90 × 453,

In these two examples, the entire causes are compound quantities, depending upon two subordinate causes; and because the effect is the same, each subordinate cause is less or greater according as the other is greater or less, constituting what is called Inverse Proportion.

Ex. 5. If a person can perform a journey of 100 miles in 12 days of 8 hours each; how far will he be able to travel in 15 days of 9 hours each?

Here, 12 × 8 and 15 × 9 are the causes, and the distances travelled 100 and x are the effects: whence,

Ex. 6. If 60 bushels of corn feed 6 horses for 50 days; in how many days will 15 horses consume 75

bushels?

The causes are 6 × 50 and 15 × x, and the effects are 60 and 75 bushels: therefore

In the former of these examples, the distances travelled are in the compound ratio of the numbers of days and their lengths: and in the latter, the numbers of bushels have the same ratio as that which is compounded of the numbers of horses and days.

Ex. 7. If 25 labourers can dig a trench 220 yards long, 3ft. 4in. wide and 2ft. 6in. deep, in 32 days of 9 hours each: how many would it require to dig a trench half a mile long, 2ft. 4in. deep and 3ft. 6in. wide, in 36 days of 8 hours each?

These examples, the causes and effects being simple and compound quantities consisting of their respective