and consequently the difference of the square of the longest, and either of the others, is the square of the remaining one.

15. If the height of a fort be 15 feet, and surrounded by a ditch 24 feet wide, what must be the length of a ladder to reach from the outside of the ditch to the top of the fort? Ans. 28.3+feet.

16. What is the height of a castle, when a line 212 feet long will just reach from the top of the castle to the opposite bank of a river, known to be 20 yards broad? Ans. 203.332+feet.

CUBE ROOT.

When the cube root of any number is required. 1. Prepare the given number, by separating it into periods of three figures each from the units' place.

Note.-When whole numbers and decimals, or decimals only, are given, the same observation is to be made, as to the manner of separating the figures into periods, as in the square root.

2. Find the greatest root contained in the left-hand period; place it to the right of the given number, and its cube under the first left-hand period, and take their difference; bring down the next period, and place it to the right of said difference for a dividend.

3. Square the root, and multiply the square by 3, for a defective divisor.

4. Try how often the defective divisor is contained in the dividend, omitting the two right-hand figures, and place the number of times it is contained to the right of the ascertained root, and its square to the right of the defective divisor, supplying the place of tens with a cipher, if the square be less than 10.

5. Multiply the last figure by all the figures of the root previously ascertained, and multiply that product by 30; then add the product to the divisor to complete it.

Multiply and subtract as in long division, and bring

lown the next period for a new dividend continually, until all the periods have been brought down.

Note. When the cube root of a vulgar fraction is required, reduce it to its lowest terms, and extract the cube root of the numerator for a new numerator, and the cube root of the denominator for a new denominator; when the numerator and denominator, or both, have remainders, reduce the fraction to a decimal, and extract the cube root.

2. When a mixed number is given, reduce the fraction to a decimal, and extract the cube root.

Questions.

How do you prepare a given sum for the extraction of the cube root?

What is to be noted, when a whole number and decimal, or decimal only, is given?

What is next to be done, after dividing the given number into periods of three figures?

When you have found the greatest root contained in the left-hand period, and placed its cube under the first period, what is to be done next?

How do you proceed, after having squared the ascertained root, and multiplied it by 3?

When you have tried how often the defective divisor is contained in the dividend, omitting the left-hand period, &c., what is to be done next?·

When you have multiplied the last figure by all the figures of the ascertained root, and by 30, what is next to be done?

When the cube root of a vulgar fraction is required, how do you find it?

When the cube root of a mixed number is required, how do you find it?

Examples.

1. What is the cube root of 3796416?

3796416(156 Ans.

1

Defective div. and square of 5, 325)2796. +150 complete divisor, 475 3275

Defective div. and square of 6, 67536)421416 +2700 complete divisor, 70236 421416

2. What is the cube root of 7532641? Ans. 196.02.+ 3. What is the cube root of 12.1138475?

Ans. 2.296.+ 4. What is the cube root of 5382674? Ans. 175.2.+. 5. What is the cube root of .37862135? 6. What is the cube root of 46.295363543?

Ans. .723.+

ALLIGATION is a rule which enables us to resolve questions concerning the mixture of several simples into one compound quantity.

Case 1.

When the quantity and rates of the simples are given, to find the rate of a mixture compounded of these simples.

Rule.

1. Find the value of each quantity according to their respective costs.

2. As the whole of the quantities

Is to one of its parts,

So is the total amount of their value

To the value of one of its parts.

Case 2.

When the prices of several simples are given to find how much of each, at their respective rates, will be required to make a mixture at any proposed price.

Rule.

1. Place all the rates of the simples under each other, and link each rate which is less than the mean rate, with one or more that is greater.

2. Take the difference between each rate, and the mean price placed opposite the respective rate with which it is linked, which will be the quantity required.

Note.-If all the given prices are greater or less than the mean price, they must be linked to a cipher. Different modes of linking produce different answers.

Case 3.

When the prices of all the simples, the quantity of one of them, and the mean price of the whole mixture, are given to find the quantities of the rest.

Rule.

1. Place the mean rate and the several prices; link them, and take their differences, as in the preceding case. 2. As the difference of the same name with the quantity given,

Is to the differences respectively,

So is the given quantity

To the several required quantities.

Case 4.

When the prices of the several simples, the quantity to be coinpounded, and the mean price, are given to find the quantity of each simple.

Rule.

1. Link the several prices, and take their differences, as before.

2. As the sum of the differences

Is to the difference opposite each price,
So is the quantity to be compounded
To the quantity required.

What is Alligation ?

Questions.

When the quantities and the rates of the simples are given to find the rate of a inixture compounded of these sunples, how do you work?

By what rule do you work, when the price of several simples is given to find how much of each, at their respective rates, will be required to make a mixture at any proposed price?

How do you proceed, when the price of all the simples, the quantity of one of them, and the mean price of the whole, are given to find the quantities of the rest?

How do you proceed, when the prices of the several simples, the quantity to be compounded, and the mean price, are given to find the quantity of each simple ?

Case 1.

1. If a person have 4 lb. of tea, at 90 cents per lb., 8 lb. at 75 cents, and 6 lb. at 110 cents, to be mixed together, what will a pound of the mixture be worth?

2. A grocer has 2 cwt. of coffee, at 25 dollars per cwt., 4 cwt. at 20 dollars 50 cents per cwt., and 7 cwt. at 18 dollars 62 cents per cwt., which he will mix together: what will 1 cwt. of this mixture be worth? Ans. $20 18 cts.