$ 1.20, of which 1lb. will be worth 24 cents, the price of the required mixture. Therefore, we must take 3lb. at 26 cents, and 2lb. at 21 cents, to make a mixture worth 24 cents per pound, which is the same result as was obtained in the operation. Hence the

RULE.-1. Place the prices of the ingredients under each other, in the order of their value, and connect the price of each ingredient which is less in value than the price of the mixture, with one that is greater.

2. Then place the difference between the price of the mixture, and that of each of the ingredients, opposite to the price with which it is connected, and the number set opposite to each price is the quantity of the ingredient to be taken at that price.

NOTE. There will be as many different answers as there are different ways of connecting the prices, and by multiplying and dividing these answers they may be varied indefinitely.

EXAMPLES FOR PRACTICE.

2. A farmer wishes to mix corn at 75 cents a bushel, with rye at 60 cents a bushel, and oats at 40 cents a bushel, and wheat at 95 cents a bushel; what quantity of each must he take to make a mixture worth 70 cents a bushel ?

3. I have 4 kinds of salt worth 25, 30, 40, and 50 cents per bushel; how much of each kind must be taken, that a mixture might be sold at 42 cents per bushel?

4. My swamp hay is worth $12 per ton, my salt hay $ 15, and my English hay $20; how much of each kind must be taken, that a ton may be sold at $ 18?

ART. 307. When the quantity of one ingredient is given to find the quantity of each of the others.

QUESTIONS.-What is the rule for alligation alternate? How can you obtain different answers? Are they all true?

Ex. 1. How much sugar, that is worth 6, 10, and 13 cents a pound, must be mixed with 20lb. worth 15 cents a pound, so that the mixture will be worth 11 cents a pound?

By the conditions of the question we are to take 20lb. at 15 cents a pound; but by the operation we find the difference at 15 cents a pound to be only 5lb., which is but of the given quantity. Therefore, if we increase the 5lb. to 20, the other differences must be increased in the same proportion. Hence the propriety of the following

RULE. Take the difference between each price and the mean price, as before; then say, as the difference of that ingredient whose quantity is given is to each of the differences separately, so is the quantity given to the several quantities required.

EXAMPLES FOR PRACTICE.

2. A farmer has oats at 50 cents per bushel, peas at 60 cents, and beans at $1.50. These he wishes to mix with 30 bushels of corn at $1.70 per bushel, that he may sell the whole at $1.25 per bushel; how much of each kind must he take?

3. A merchant has two kinds of sugar, one of which cost him 10 cents per lb., and the other 12 cents per lb.; he has also 100lb. of an excellent quality which cost him 15 cents per lb. Now, as he ought to make 25 per cent. on his cost. how much of each quantity must be taken that he may sell the mixture at 14 cents per lb.

ART. 308. When the sum of the ingredients and their mean price are given, to find what quantity of each must be taken.

Ex. 1. I have teas at 25 cents, 35 cents, 50 cents, and 70

QUESTION. Art. 307. What is the rule for finding the quantity of each of the other ingredients when one is given?

cents a pound, with which I wish to make a mixture of 180lb., that will be worth 45 cents a pound. How much of each kind must I take?

By the conditions of the question, the weight of the mixture is 180lb.; but by the operation we find the sum of the differences to be only 60lb., which is but of the quantity required. Therefore, if we increase 60lb. to 180, each of the differences must be increased in the same proportion, in order to make a mixture of 180lb., the quantity required. Hence the

RULE. - Find the differences as before; then say, as the sum of the differences is to each of the differences separately, so is the given quantity to the required quantity of each ingredient.

EXAMPLES FOR PRACTICE.

He

2. John Smith's "great box" will hold 100 bushels. has wheat worth $2.50 per bushel, and rye worth $2.00 per bushel. How much chaff of no value must he mix with the wheat and rye, that if he fill the box, a bushel of the mixture may be sold at $ 1.80 ?

3. I have two kinds of molasses, which cost me 20 and 30 cents per gallon; I wish to fill a hogshead, that will hold 80 gallons, with these two kinds. How much of each kind must be taken, that I may sell a gallon of the mixture at 25 cents per gallon and make 10 per cent. on my purchase?

4. A lumber merchant has several qualities of boards; and it is required to ascertain how many, at $10 and $ 15 per thousand feet, each, shall be sold on an order for 60 thousand feet, that the price for both qualities shall be $12 per thousand feet.

QUESTION. Art. 308. How do you find what quantity of each ingredient must be taken when the sum and mean price are given?

§ XLIII. PERMUTATION.

ART. 309. PERMUTATION is the method of finding how many different changes or arrangements may be made of any given number of things.

ART. 310. To find the number of different arrangements that can be made of any given number of things.

Ex. 1. How many different numbers may be formed from the figures of the following number, 432, making use of three figures in each number?

Ans. 6.

In the 1st operation, we have made all the different arrangements that can be made of the given figures, and find the number to be 6. In

the second operation, the same result is obtained by simply multiplying together the first three of the digits, a number equal to the number of figures to be arranged. Hence the

RULE.-Multiply all the terms of the natural series of numbers, from 1 up to the given number, continually together, and the last product will be the answer required.

EXAMPLES FOR PRACTICE.

2. My family consists of nine persons, and each person has his particular seat around my table. Now, if their situations were to be changed once each day, for how many days could they be seated in a different position?

3. On a certain shelf in my library there are 12 books. If a person should remove them without noticing their order, what would be the probability of his replacing them in the same position they were at first?

4. How many words can be made from the letters in the word "Embargo," provided that any arrangement of them may be used, and that all the letters shall be taken each time?

QUESTIONS. Art. 309. What is permutation? Art. 310. What is the rule for finding the number of arrangements that can be made of any given number of things?

§ XLIV. MENSURATION OF SURFACES.

ART. 311. A SURFACE is a magnitude, which has length and breadth without thickness.

The surface or superficial contents of a figure, are called its

area.

ART. 312. An ANGLE is the inclination or opening of two ines, which meet in a point.

A right angle is an angle formed by one line falling perpendicularly on another, and it contains 90 degrees.

An acute angle is an angle less than a right angle, or less than 90 degrees.

An obtuse angle is an angle greater than a right angle, or more than 90 degrees.

THE TRIANGLE.

ART. 313. A TRIANGLE is a figure having three sides and three angles. It receives the particular names of an equilateral triangle, isosceles triangle, and scalene triangle.

It is also called a right angled triangle when it has one right angle; an acute angled triangle, when it has all its angles acute; and an obtuse angled triangle, when it has one obtuse angle.

The base of a triangle, or other plane figure, is the lowest side, or that which is parallel to the horizon; as, C D. The altitude of a triangle is a line drawn from one of its angles perpendicular to its opposite side or base; as, A B.

An equilateral triangle is a figure which has its three sides equal.

QUESTIONS. — Art. 311. What is a surface? What are the superficial contents of a figure called?-Art. 312. What is an angle? What is a right angle? An acute angle? An obtuse angle? -Art. 313. What is a triangle? What particular names does it receive? When is it called a right angled triangle? When an acute angled triangle? When an obtuse angled triangle? What is the base of a triangle? What the altitude? What is an equilateral triangle?