will be in proportion compounded on the quantities of matter they contain, and of the velocities wherewith they

move.

112. There are two bodies the one contains 25 times the matter of the other (or twenty-five times heavier) but the lesser moves with 1000 times the swiftness of the greater: in what proportion are the forces by which they are moved? Answ. lesser 40 to 1.

113. There are two bodies, one of which weighs 100lb. the other 60lb. but the lesser body is impelled by a force 8 times greater than the other, the proportion of the velocities wherewith these bodies move is required?

Answ. as 13 to 1

1. In comparing the motions of bodies, if their volocities be equal, the spaces described by them are in direct proportion of the times in which they are described.

2. If the times be equal, then the spaces described will be as their velocities,

3. If the times and velocities be unequal, the spaces will be in a proportion compounded of the times and velocities.

114. There are two bodies one of which moves forty times swifter than the other, but the swifter body has moved but one minute, whereas the other has been in motion two hours: the ratio of the spaces described by these two bodies is required?

Answ. the swifter to the slower as 1 to 3

115. Suppose one body to move thirty times swifter than another; as also the swifter to nove 12 minutes, the other only 1, what difference will there be between the spaces by them described, supposing the last has moved 60 inches? Answ. 1795 feet.

110. There are two bodies, one whereof has described 50 miles, the other only 5, but the first hath moved with 5 times the velocity of the second: what is the ratio then of the times they have been describing those spaces?

Answ. the first body hath been.in motion double the time of the latter.

QUESTIONS

QUESTIONS.

2. What is the Rule of Three?

A. That which teaches from three numbers given to find a fourth proportional.

2. How are the given numbers to be managed?

4. The first and third must be reduced to the same name. viz. the lowest mentioned in either, and the second likewise to its lowest name.

2. How must the numbers be stated?

A. So that the first and third may be of one name or kind.

2. How is the operation performed?

A. Multiply the second and third together, and divide their product by the first, the quotient is the fourth num ber sought, in the same name with the second.

HTT

CHAP. VIII.

THE RULE OF THREE INVERSE.

TITHERTO the questions proposed were such that the first number of the stating was always to the third as the second was to the fourth required, which is called direct proportion, or the Rule of Three Direct: and this is the most useful and general property of proportionals, as will appear by the sequel.

But (the question being stated according to the Rule before laid down) the nature and conditions of several questions are such, that, as the first is to the third, so reciprocally must the fourth be to the second: that is, the greater the third is in proportion to the first, the less must the fourth be in respect of the second: or the less the third is in proportion to the first, the greater the fourth must be in proportion to the second. This is called reciprocal or inverted, or indirect proportion, or the Rule of Three Inverse.

The principal difficulty that will embarrass the learner, will be, to distinguish when the proportion is direct, and when indirect. This is done from an attentive consideration of the sense and tenor of the question proposed, for if thereby it appears that when the third term of the stating

I

ting is less than the first, the answer must be less than the second, or when the third is greater than the first, the answer must be greater than the second; then the proportion is direct.

But if the third is less than the first, and yet the sense of the question requires the fourth to be greater than the second, or if the third being greater than the first, the answer must be less than the second, the proportion is indirect.

Application.

First, this question being proposed. If 12 men make 4 perch of ditching in one day, how many perch will 24 men make in the same time? The stating will stand thus:

where it is very manifest that 24 men will do more in the same time, than 12 at the same proportion of working, viz. in proportion as 24 to 12 i. e. twice as much, viz. 8 perches: so that 24 men being more than 12, and requiring the answer more than 4 perches (the second number) this question is direct.

But if the question proposed were this, viz. if 12 men make 16 perch of ditching in 4 days, in what time will 24 men perform the same at the same rate of working?

Here 16 is a superfluous term, having no corresponding term, which being rejected, state the other terms of the question.

In which stating it is very evident that 24 men will perform 16 perch in less time than 12 men, and that therefore the fourth required must be less than the second, in the same proportion as the third is greater than the first, therefore the proportion is inverse or indirect.

Rule for the Operation.

The question being stated as already directed in the Rule

of

of Three Direct, multiply the first and second numbers together, and divide the product by the third, the quotient is the answer required, in the same name with the second.

2. What is the Rule of Three Inverse ?

A. When three numbers are given to find a fourth, which shall have such proportion to the second as the first to the third.

2. How is a question distinguished whether it belongs to the Rule of Three Inverse or Direct?

A. If more do more or less do less respect,
It is a question in the Rule Direct;
But less requiring more, and greater less,
A question of the Inverse Rule express.

Examples.

2. 1. There was a certain building raised in 8 months by 120 workmen; but the same being demolished, it is required to be rebuilt in 2 months: how many men must be employed about it? Answ. 480 men.

2. If 28s. will pay for the carriage of an hundred weight 150 miles; how far may 6Cwt, be carried for the same money? Answ. 25 miles.

3. If for 51 5s. I have 14Cwt. carried 136 miles; how many miles may I have 24 Cwt. carried for the same money? Answ. 79 miles.

4. If a footman perform a journey in 3 days, when the days are 16 hours long, how many days will he require of 12 hours long to go the same journey in?

Answ. 4 days.

5. How many yards of plush are sufficient to make a cloak of equal magnitude with one which had in it 4 yards of 7 quarters wide, when the plush is but 3 quarters wide? Answ. 9 yards of plush.

6. How many yards of canvas that is ell wide, will be sufficient to line 20 yards of sey, that is 3 quarters wide? Answ. 12 yards.

7. If a man performs a journey in 6 days, when the dayis 8 hours long; in what time will he do it, when the day is 12 hours long. Answ. 4 days.

8. IfI lend my friend 100l. for 6 months, (allowing the month to be 30 days) how long ought he to lend me 1000l. to requite my kindness? Answ. 18 days.

9. If 6 mowers.can mow a field in 12 days, in what time will 24 mowers do it? Answ. 3 days.

10. Suppose 800 Soldiers were placed in a garrison, and their provisions computed sufficient for 2 months; how many soldiers must depart that the provisions may serve Answ. 480 men.

them 5 months?

11. Admit that I lent to a friend on his occasion 100l. for 6 months, and he promised me the like kindness when I desired it; but when I came to request it, he could lend me only 751. The question is, how long I may keep his money to recompense my courtesy to him?

Ansu. 3 months.

A LEVER OF THE FIRST ORDER.

A Lever of the first order hath the power at one of its ends, the weight to be raised is put at the other, and the fulcrum or prop somewhere between them.

In this order, the power applied at one end will be reciprocally proportional to the distances of those ends from the fulcrum or point supported: or in the steel yards as the distance of the weight from the point of suspension.

Examples.

12. What weight will a man be able to raise, who presses with the force of an hundred weight and an half on

the