398. In bodies of the same shape,

Two corresponding lines are in the

two.

same ratio as

any

other

The ratio of two corresponding surfaces is the square of the ratio of two corresponding lines.

The ratio of two corresponding volumes is the cube of the ratio of two corresponding lines.

Conversely,

The ratio of two corresponding lines is the square root of the ratio of two corresponding surfaces, and the cube root of the ratio of two corresponding volumes.

EXERCISE LXXIX.

1. If the diameter of the moon be reckoned at 2000 mi., and that of the earth at 8000 mi., find the ratio of the surface of the moon to that of the earth. Also, find the ratio of the volume of the moon to that of the earth.

2. If the diameter of the earth be reckoned at 8000 mi., that of Jupiter at 84,000 mi., and that of the sun at 880,000 mi., find the ratios of their volumes.

3. If the diameters of two circles be 20 in. and 40 in. respectively, find the ratio of their circumferences and the ratio of their surfaces.

4. If the areas of two circles be 8000 sq. in. and 36,000 sq. in. respectively, find the ratio of their diameters to the nearest thousandth of an inch.

5. If the volumes of two spheres be 100 cu. in. and 1000 cu. in. respectively, find the ratio of their diameters to the nearest thousandth of an inch.

6. If two stacks of hay of the same shape contain 4 t. 6 cwt. and 1 t. 8 cwt. respectively, find the ratio of their heights.

7. If an ox 7 ft. in girth weigh 1500 lbs., what will be the girth of a similar ox weighing 2500 lbs.?

8. The surface of a pyramid is 560 sq. in. What is the surface of a similar pyramid whose volume is 27 times as great?

9. The volume of a pyramid is 1331 cu. in. What is the volume of a similar pyramid whose surface is 4 times as great?

10. If a well-proportioned man 5 ft. 10 in. high weigh 160 lbs., what should a man 6 ft. high weigh, to the nearest tenth of a pound? What should be the height, to the nearest tenth of an inch, of a man weighing 210 lbs. ?

11. A three-gallon jug and a one-gallon jug are of the same shape. What, to the nearest thousandth, is the ratio of their diameters?

12. Two hills have exactly the same shape; one is 900 ft. high, the other 1200 ft. Find the ratio of their surfaces, and also the ratio of their volumes.

13. A ball 3 in. in diameter weighs 4 lbs. ; another ball of the same metal weighs 9 lbs. Find the diameter

of the second ball to the nearest thousandth of an inch.

14. If Apollo's altar were a perfect cube 10 ft. on a side, what, to the nearest hundredth of an inch, would be the dimensions of a new cubical altar containing twice as much stone?

15. A man standing 40 ft. from a building 24 ft. wide observed that, when he closed one eye, the width of the building hid from view 90 rods of fence which was parallel to the width of the building. Find the distance from the eye of the observer to the fence.

16. A bushel measure and a peck measure are of the same shape. Find the ratio of their heights.

CHAPTER XXII.

LOGARITHMS.

399. In the common system of notation the expression of numbers is founded on their relation to ten.

Thus, 3854 indicates that this number contains 103 three times, 102 eight times, 10 five times, and four units.

400. In this system a number is represented by a series of different powers of 10, the exponent of each power being integral. But, by employing fractional exponents, any number may be represented (approximately) as a single power of 10.

401. When numbers are referred in this way to 10, the exponents of the powers corresponding to them are called their logarithms to the base 10.

For brevity the word "logarithm" is written log.

It is evident that the logarithms of all numbers between

402. The fractional part of a logarithm cannot be expressed exactly either by common or by decimal fractions; but decimals may be obtained for these fractional parts, true to as many places as may be desired.

If, for instance, the logarithm of 2 be required; log 2 may be posed to be

sup

Then 10 = 2; or, by raising both sides to the third power, 10 = 8, a result which shows that is too large.,

Suppose, then, log 2.

Then 1000 = 2, or by raising both sides to the tenth power, 103 210. That is, 1000 1024, a result which shows that is too small.

=

=

Since is too large and too small, log 2 lies between and ; that is, between .33333 and .30000.

=

f=.2.

1000

In supposing log 2 to be, the error of the result is 108 In supposing log 2 to be, the error of the result is 1000-10242.024; log 2, therefore, is nearer to than to . The difference between the errors is .2-(-.024) = .224, and the difference between the supposed logarithms is .33333 — .3 = .03333. The last error, therefore, in the supposed logarithm may be consid ered to be approximately of .03333.0035 nearly, and this added to .3000 gives .3035, a result a little too large.

By shorter methods of higher mathematics, the logarithm of 2 is known to be 0.3010300, true to the seventh place.

403. The logarithm of a number consists of two parts, an integral part and a fractional part.

Thus, log 2 = 0.30103, in which the integral part is 0, and the fractional part is .30103; log 20 = 1.30103, in which the integral part is 1, and the fractional part is .30103.

404. The integral part of a logarithm is called the characteristic; and the fractional part is called the mantissa.

405. The mantissa is always made plus. Hence, in the case of numbers less than 1 whose logarithms are minus, the logarithm is made to consist of a minus characteristic and a plus mantissa.

406. When a logarithm consists of a minus characteristic and a plus mantissa, it is usual to write the minus sign over the characteristic, or else to add 10 to the characteristic and to indicate the subtraction of 10 from the resulting logarithm.

Thus, log .21.30103, and this may be written 9.30103 – 10.

407. The characteristic of a logarithm of an integral number, or of a mixed number, is one less than the number of integral digits.

Thus, from 401, log 10, log 101, log 1002. Hence, the logarithms of all numbers from 1 to 10 (that is, of all numbers consisting of one integral digit), will have 0 for characteristic; and the logarithms of all numbers from 10 to 100 (that is, of all numbers consisting of two integral digits), will have 1 for characteristic; and so on, the characteristic increasing by 1 for each increase in the number of digits, and therefore always being 1 less than that number.

408. The characteristic of a logarithm of a decimal fraction is minus, and is equal to the number of the place occupied by the first significant figure of the decimal.

=

Thus, from 401, log .1 = -1, log .01 -2, log .001-3. Hence, the logarithms of all numbers from .1 to 1 will have - 1 for a characteristic (the mantissa being plus); the logarithms of all numbers from .01 to .1 will have - 2 for a characteristic; the logarithms of all numbers from .01 to .001 will have 3 for a characteristic; and so on, the characteristic always being minus and equal to the number of the place occupied by the first significant figure of the decimal.