7. If the population of a country, at a given time, be seven millions; and if the yearly rate of increase be th; what will be the population at the end of 35 years ?

8. The population of the United States in 1800 was 5,306,000. What was it in 1780, supposing the yearly rate of increase to be?

9. In what time will the population of a country advance, from four millions to seven millions, if the ratio of increase be To?

00

10. What must be the rate of increase, that the population of a place may change from nine thousand to fifteen thou sand, in 12 years?

If the population of a country is not affected by immigration or emigration, the rate of increase will be equal to the difference between the ratio of the births, and the ratio of the deaths, when compared with the whole population.

Ex. 11. If the population of a country, at any given time, be ten millions; and the ratio of the annual number of births to the whole population be, and the ratio of deaths, what will be the number of inhabitants, at the end of 60 years?

Here the yearly rate of increase-216-870•

5

And the population, at the end of 60 years-31,750,000. The rate of increase or decrease from immigration or emigration, will be equal to the difference between the ratio of immigration and the ratio of emigration; and if this differbe added to, or subtracted from, the difference between the ratio of the births and that of the deaths, the whole rate of increase will be obtained.

Ex. 12. If in a country, the ratio of births be 30,

and if the population this year be 10 millions, what will it

And the population at the end of 20 years, is 12,611,000.

13. If the ratio of the births be

in what time will three millions increase to four and a half millions?

If the period in which the population will double be given; the numbers for several successive periods, will evidently be in a geometrical progression, of which the ratio is 2; and as the number of periods will be one less than the number of terms;

If P-the first term,

A the last term,

n-the number of periods;

Then will A-PX 2", (Alg. 439.)

Or log. A=log. P+nxlog. 2.

Ex 1. If the descendants of a single pair double once in 25 years, what will be their number at the end of one thousand years?

The number of periods here is 40.

And A 2X24°2,199,200,000,000.

2. If the descendants of Noah, beginning with his three sons and their wives, doubled once in 20 years for 300 years, what was their number, at the end of this time?

Ans. 196,608.

3. The population of the United States in 1820 being

9,638,000; what must it be in the year 2020, supposing it to double once in 25 years? Ans. 2,467,333,000.

4. Supposing the descendants of the first human pair to double once in 50 years, for 1650 years, to the time of the deluge, what was the population of the world, at that time?

EXPONENTIAL EQUATIONS.

62. An EXPONENTIAL equation is one in which the letter expressing the unknown quantity is an exponent.

Thus ab, and a-bc, are exponential equations. These are most easily solved by logarithms. As the two members. of an equation are equal, their logarithms must also be equal. If the logarithm of each side be taken, the equation may then be reduced, by the rules given in algebra.

Ex. What is the value of x in the equation 3*-243 ?

Taking the logarithms of both sides, log. 3-log. 243. But the logarithm of a power is equal to the logarithm of the root, multiplied into the index of the power. (Art. 45.)

Therefore (log. 3)X-log. 243; and dividing by log. 3. log. 243 2.38561

-5. So that 35—243.

64. The exponent of a power may be itself a power, as in the equation

where x is the exponent of the power m2, which is the ex

Ex. 4. Find the value of x, in the equation 93-1000.

log. 1000.

3o×(log. 9)=log. 1000. Therefore, 3a—

Then, as 3-3.14. x(log. 3)=log. 3.14

Therefore, x=

log. 3.14

log. 3

477

4999328-1.04.

In cases like this, where the factors, divisors, &c. are logarithms, the calculation may be facilitated, by taking the logarithms of the logarithms. Thus the value of the fraction 4291219 is most easily found, by subtracting the logarithm of the logarithm which constitutes the denominator, from the logarithm of that which forms the numerator.

7 7 13

5. Find the value of x, in the equation

bax +d

m

C

TRIGONOMETRY.

SECTION I.

SINES, TANGENTS, SECANTS, &C.

ART. 71. TRIGONOMETRY treats of the relations of the sides and angles of TRIANGLES. Its first object is to determine the length of the sides, and the quantity of the angles. In addition to this, from its principles are derived many interesting methods of investigation in the higher branches of analysis, particularly in physical astronomy.

72. Trigonometry is either plane or spherical. The former treats of triangles bounded by right lines; the latter, of triangles bounded by arcs of circles.

Divisions of the Circle.

73. In a triangle there are two classes of quantities which are the subjects of inquiry, the sides and the angles. For the purpose of measuring the latter, a circle is introduced.

The periphery of every circle, whether great or small, is supposed to be divided into 360 equal parts called degrees, each degree into 60 minutes, each minute into 60 seconds, each second into 60 thirds, &c., marked with the characters ', "', '"', &c. Thus, 32° 24′ 13" 22" is 32 degrees, 24 minutes, 13 seconds, 22 thirds.

A degree, then, is not a magnitude of a given length ; but