differential of the logarithm will be the differential of (a+x) divided by (a + x,) [SEE PAGE 280;] and, since the two members of the preceding equation are equivalent, their differential coefficients are equivalent, which will form another equation. The differential coefficients of the members of this last equation will also be equivalent, &c

Hence,

(1) log. (a + x) =A+Bx+Cx2+Dx2+ Ex1+ F x3, &c.

B+2Cx+3Dx2+4 Ex2+5 F x1, &c.

Since the members of each of these equations are equiva lent, the equations must be true, whatever x may be. Make x = 0 in each of the equations, and we have

Substitute these values of A, B, C, &c., in equation (1,)

Subtract 2.7182818 from 3, and there remains .2817182. Substitute 2.7182818 for a, in equation (A,) and .281782 for x, and we have

Log. of (2.7182818+.2817182)=log. of 3=1+
X.2817182X(371)2 X(-2817182)2, &c.

This series is very converging, a few terms will give the logarithm of 3, to the base e, or 2.7182818, very nearly.

1

с

Multiply it by, which is .434294484, and it will give the logarithm of 3 to the base 10.

Assume a = 2.7182818, and x = .7182818

and substitute in equation B, and we will obtain the logarithm of 2.

We can, then, easily find the logarithms of 4, 5 and 6. In order to find the logarithm of 7, substitute 6 for a, and 1 for x, in equation (A,) and we shall find the logarithm of 7. 2. Find the logarithms of 11, 13, 17, 19, 23.

SECTION CXX.

Miscellaneous Examples.

Solve the following equations:

2.

3.

4.

{

= 60

=

= 75

x(x + y + z)
y (x + y + z)
z(x + y + z) = 90

x2+ y2+z2 = 1400
x2+y+z2 = 1020
(x + y + z) y = 1200

́ x2 + y2 — x — y = 98
xy+
+x+y=71

x2 + y2 + x2 + y2+ 2x2 y2 = 238632 5. { y1 + z1 + y2 + z2 + 2 y2 z2 =

6.

7.

8.

1640

x2 + y2 + z2 = (275100 — x2 — y2 — z2)}

9. A farmer has two flocks of sheep, each containing 147. From the first flock he sells a certain number, and from the second, a number expressed by the same digits inverted. The sum of the digits is 12. What are the numbers sold from each flock, supposing twice as many remain in the one as in the other?

10. There are three numbers in geometrical progression, the difference of whose difference is 6; and their sum is 42 What are the numbers?

11. There are two numbers whose difference is 4; and, their product multiplied by the sum of their squares is 480. What are the numbers?

12. Suppose 100 dollars are put at interest, at 6 per cent. simple interest, and 50 dollars are placed at compound interest at the same rate; in what time will the two amounts be equal?

13. A person has 3 horses and a saddle, which of itself is worth 220 dollars. Now, if the saddle be put on the back of the first horse, it will make his value equal to that of the sec

ond and third; but, if it be put on the back of the second, it will make his value double that of the first and third: and, if it be put on the back of the third, it will make his value triple that of the first and second. What is the value of each horse?

14. Given, the sum of the squares of two numbers = 195, the sum of their cubes = 1799, and their continued product = 385, to find the numbers.

=

15. It is required to find three numbers, such that the product of the first and second subtracted from the sum of their squares shall be equal to 13, the product of the first and third subtracted from the sum of their squares shall be equal to 19, and the product of the second and third subtracted from the sum of their squares shall be 21.

16. Divide 140 into two such parts, that the greater being divided by the less, and the less by the greater; and the greater quotient being multiplied by 8, and the less by 41, the two products shall be equal.

17. A teacher gave a student two numbers to multiply to gether. When he had finished the multiplication he proved it by dividing the product by the multiplier, which was the smaller factor: the quotient was 227, and there remained 113; hence it was multiplied wrong. When the student had found out the error, he said that he had calculated only 1000 too little in the multiplication. One factor was 75 more than the other. What numbers were given to be multiplied?

18. The population of a city contains 20000 persons, we know that the population has increased regularly

yearly. What was the population 10 years ago?

and

19. In how many years will the population of a place be ten times as great as it is at present; supposing the yearly increase to be 3 persons in every hundred?

20. Let x and y be two quantities which depend upon each other in such a way, that when the n determinate values a a' a", &c., are assigned to y, the x has the corresponding determinate values b b' b", &c.

Ans. Assume (1) x=A+By+Cy2+Dy3, &c., to n terms. Substitute, first, a for y and b for x, and we have one equation, then b' for x, and a' for y, and we have another equation, &c. It is evident we shall have as many equations as there are corresponding determinate values of x and y. From akese equations A, B, C, D, &c., may be found.

ANNOTATIONS.

Introductory Lessons.

In every question there is one or more numbers to be found from certain relations which are given. These relations are called the data of the question or problem. In algebra we use certain signs or symbols to represent numbers; commonly the letters of the alphabet: the last letters, according to custom, are used to represent unknown numbers, and the first to represent numbers which are known. The relations of numbers are shown partly by the positions of these letters, and partly by the algebraic signs used in connexion with them.

In sections 1 and 2 the relations of the numbers are such, that when we assume x to represent one of them, we can find a certain number of x's equal to a given number; as 3 x equal to 12, from which x is easily found. The number which shows how often a letter is taken, as 3 in the example above, is called the coefficient of that letter.

When we have a number of x's equal to a given number, divide the given number by the coefficient of x, and it will give the value of x.

In the subsequent sections, the equations formed at first are more complicated; but they are reduced to the same form with those in Section I. by applying the principles contained in the following axioms.

AXIOM 1.—If the same number be added to each of two equal numbers, the two sums will be equal.

AXIOM 2.-If the same number be subtracted from two equal numbers, the remainders will be equal.

AXIOM 3.-If equal numbers be multiplied by the same number, the products will be equal.

AXIOM 4.-If equal numbers be divided by the same number, the quotients will be equal.

AXIOM 5.-The same powers or the same roots of equal numbers are equal.

See illustration of axiom 2 in question 16, Section IV.

These axioms are constantly used, and should be always referred to, as no reduction of an equation can be understood without them.

SEC. I. TO VI. ALGEBRA.

The questions do not differ from those in the introductory