as that for numbers greater than unity. In order to retain this relation, the subtraction indicated is not performed, and the logarithm is written in the form 0.36961, or 9.3696 10, or with the new symbol 1.3696, thus retaining the positive mantissa 0.3696 and its connection with the sequence of digits 2, 3, 4, 2.

It is to the simplicity of the relations between the mantissa of the logarithm and the sequence of digits in the number, the characteristic of the logarithm and the position of the decimal point in the number, that the system of common logarithms owes its superiority to all other systems for purposes of computations. (See, for example, the rule for shifting the decimal point in the system of logarithms to the base e, given on page 31 of the Tables.)

83. Computation by means of Common Logarithms. The value of a product, quotient, power or root may be found by logarithms by means of the Theorems 7, 8, 9, 10 in Section 81. The following examples will illustrate the methods.

To find a product by means of logarithms.

EXAMPLE 1. Find the distance around the earth at the equator, if the equator is regarded as a circle of radius 3963.

We have C 2π 3963, whence by Theorem 7, log C = log 2 + log π + log 3963. Before turning to the table of logarithms, write out a blank form as indicated on the left. When this is filled in it gives the computation on the right.

To find a quotient by means of logarithms.

EXAMPLE 2. Find x = (0.003468)/(0.4783).

X

x = 2.489

We have, by means of Theorem 8, Section 81, log x =

log 0.003468

The mantissa of the numerator being less than the mantissa of the denominator, we add 1 to and subtract 1 from the characteristic of the numerator in order to avoid a negative mantissa in the difference.

If two measurements are made to four figures their product or quotient should not contain more than four significant figures (Section 26). The product or quotient may be obtained by a four-place table of logarithms to the correct number of significant figures. In general, the use of a table of logarithms in computing products and quotients conforms to the rule for the number of significant figures, if the number of significant figures in the measurements agrees with the number of decimal places in the logarithms.

To find a power by means of logarithms.

EXAMPLE 3. Find x = (0.008964)*.

By Theorem 9, Section 81, log x = 4 log (0.008964)

EXAMPLE 4.

Find x = (0.6785)*. By Theorem 10, log x = ₫ log 0.6785.

1. Compute the values of the following expressions by means of logarithms. In each case, write a blank form for the computation before turning to the tables.

(a) (3.462) (23.14). (b) 4795/2439.

(c) 32.8611.

11

(d) 543.2

(h) 56.34/973.4. Suggestion: Write the characteristic of the numerator in the form (3 – 2).

(i) v.03764. Suggestion: The characteristic of the given number may be written in the form (1 − 3).

2. Find the distance to a tower 59.47 feet high from a point from which the angle of elevation of the top of the tower is 21°.87.

3. Find the area of an isosceles triangle whose base is 21.5 inches and whose vertex angle is 27°.16. Note that it is sufficient to find the logarithm of the altitude and not the altitude itself.

4. Apportion a claim for $35.55 damages on freight which traveled on the M.C. 428 miles, N.Y.C. 175 miles, I.C. 235 miles, if each road pays in proportion to the number of miles traveled on it.

5. Find the cost of 1000 rivets weighing 4 ounces each, the output be ing at the rate of 78 per hour, if the cost of raw material is $4.50 a ton, if labor is $0.375 an hour, and the overhead expense is $0.123 an hour.

6. What is the cost of 6744 pounds of chromic acid 95% pure, if acid 90% pure costs 7 cents a lb.?

7. A cellar 230 feet by 330 feet by 15.5 feet is excavated by means of a scoop with a capacity of cubic yards which makes 15 trips an hour. Estimate the cost of drawing away the dirt if a driver and team cost $3.75 a day of 10 hours and draw two loads of 331 cubic feet each time.

8. The average daily circulation of a newspaper for six semi-annual periods from 1912 to 1915, expressed in hundred thousands, was 210, 229, 230, 246, 260 and 298. Find the percentage of increase in circulation for each period over the preceding.

9. Solve the following quadratic equations using logarithms wherever allowable.

(a) 2.13x2 + 4.76x – 3.82 = 0.

(b) 32.6x2 - 87.5x + 43.7 = 0.

10. Find the radius of a parallel of latitude through a point whose latitude is 45° N., assuming the radius of the earth to be 3963.

11. The area of a sphere in terms of the radius r is given by the equation S 4πr2. Find the area of the surface of the earth considered as a sphere of radius 3963 miles.

12. The volume of a sphere in terms of the radius is V = πr3. Find the volume of the earth (see 11). If the average density of the earth is 5.2, and the weight of a cubic foot of water is 62.4, find the weight of the earth in tons.

13. Find the area of the orbit of the earth assuming it to be a circle of radius 92,800,000 miles. How many miles does the earth travel in a day if we assume a year to be equal to 365 days?

14. Kepler's third law of planetary motion states that for different planets the squares of the times of describing their orbits are proportional t2 d3 to the cubes of the mean distances from the sun. That is T2 ᎠᏰ

Given the earth's mean distance, D = 92,800,000, its periodic time T = 365, and the periodic time of Mars t = 686, find the mean distance of Mars from the sun.

15. If the distance of the earth from the sun changed from 93,000,000 to 90,000,000, assuming Kepler's third law, show that the year would be shortened by about 17.6 days. (Let T = 365.3.) Since no appreciable diminution in the year has been noted from ancient times to the present, what inference can be drawn?

16. In railroad surveying a simple curve is defined as a circular arc joining two tangents.

The degree of curve, D, is the angle at the center of the circle subtended by a chord of 100 feet.

A

B

T

FIG. 132.

(a) Find the degree of curve if the radius is 800 feet.

(b) Find the radius if the degree of curve is 5o.

(c) Find the length of the arc AB of a 4° curve if the central angle = 28°.

and

(d) The exterior angle at V is 52°, the tangent distance, T, is to be 800 feet. Find the radius and the degree of curve. How far will the curve pass from the vertex, V?

(e) Find T if it is required that the degree of curve shall not exceed 4° for two tangents with an external angle of 40°.

17. If a body is constrained to move in a curved path, a force directed outward arises which is called the centrifugal force. For example, as a train rounds a curve the pressure of the flanges of the wheels against the outer rail of the curve is a centrifugal force. The magnitude of this force Wv2 in pounds is C where W = weight in pounds, v = velocity in feet gr per second, r = radius of the circle in feet and g is the acceleration due to gravity= 32.2.

(a) An automobile weighing 2 tons rounds a curve whose radius is 600 feet at a velocity of 25 miles an hour. What is the magnitude of the force tending to make it skid?

(b) The weight of a mass situated at the equator of the earth is decreased by the centrifugal force due to the rotation of the earth on its axis. Find this decrease for a weight of 500 pounds. What would be the decrease in latitude 60° N.?

(c) At what angle should a circular automobile speedway one mile in circumference be banked for a speed of 100 miles an hour in order that there shall be no tendency to skid?

18. During the war with Spain in 1898, a chain letter was started for the benefit of the Red Cross. The person starting it wrote to 10 friends

numbering each letter 1; each of these was to write to 10 friends numbering their letters 2, etc. The chain was to be completed by letters numbered 100. If the chain had not been broken how many letters would have been numbered 100? Approximately how many letters in all would there have been? How does this compare with the population of the world?

84. Solution of Triangles. The formulas (1), page 181, and the law of sines involve multiplication and division only, and hence the computations in the solution of right triangles and in Cases I and II of oblique triangles may be effected expeditiously by means of logarithms.

The law of cosines involves addition and subtraction and hence is not well adapted to logarithms. In Chapter VIII formulas will be derived by means of which the computations in the solution of Cases III and IV of oblique triangles can be effected by means of logarithms.

EXAMPLE. To find the distance from a point A on the shore of a bay to a point B off shore, a point C is taken on the shore 350 feet from A. The angles BAC and BCA are found to be 84°.13 and

The logarithms of sin B and sin C may be obtained directly from page 20 of the Tables.

A

Checking the solution of a triangle. In any numerical computation it is of the utmost importance that the accuracy of the results be checked in some way.

If ABC is a right triangle, an excellent check is given by the Pythagorean theorem, c2 = a2+ b2, which can be adapted to a2 logarithmic computation by writing it in the form

a = √ c2 − b2 = √(c − b) (c + b).

(1)

Notice that the possibility of expressing c2 - b2 as a product is the basis of the adaptation to the use of logarithms, and that