2°. If this product has a decimal part, multiply this decimal by the number of parts which the unit of the present denomination contains of the next inferior denomination to that just before employed; this product is the quantity which the given decimal contains of the next denomination.

3°. Proceed (if there still be decimals), in like manner, to the lowest denomination in which the decimal is required to be expressed.

Ex. 3. Find the number of inches and eighths in 0.48 of a foot.

of }

The next inferior denomination to that of feet is inches, of \
which the number in a foot is...

The next proposed inferior denomination to inches is eighths,
of which the number in an inch is

Ex. 4. What is the value of 625 of a cwt.?

The next inferior denomination to that of a cwt. is qrs., of
of }

which the number in a cwt, is

699'430 feet.

7יס

X 60

Ans. 608 eighths.

.625

X 4

2500 qrs.

X 28

Find the value of the following expressions:

1.25 cwt.; 375 qrs.; 625 lbs.; 975 ton; 7768 ton; 24956 ton; 0675 cwt.

2. 491 day; 343 week; 534 year; 53058875 day..

3. 2957795 degree; 7.85425 degrees; 64 3825 degrees; 10.8725 degrees.

4.

487956324 ton of sea water in cubic feet and inches (35 ft. = 1 ton).

5. Express the same in gallons of 277 274 cubic inches.

I.

2.

MISCELLANEOUS EXAMPLES.

The height of the highest mountain is about 28000 feet: what decimal is that of the earth's diameter, which is 8000 miles?

The parallax of the star a Centauri is given as o'9187, or if of a second: show by how much the vulgar fraction differs from the decimal fraction.

3. Reduce 29d 12h 44m 21.82 to decimals of a day.

4. Add together 2'095 hours, '07 days, '05 weeks, and express the same as the decimal of 365.25 days.

5. A nautical mile is 6082.66 feet, and an imperial mile 5280 feet; express each of these miles as decimals of the other. Also find how near the results are to the decimal values of and 3.

6. A sidereal day is 23h 56m 45.09; express this as a decimal of a common day-that is, of 24h-and give the result to nine decimal places.

7. If 90 degrees correspond to 100 French grades, how many degrees are there in the sum of 41.45 degrees and 41°45 grades.

8. A mètre is 39'37079 English inches, a kilomètre is 1000 mètres; express as decimals of each other a kilomètre and an English mile.

9. If the length of a degree of latitude is 365030 feet, and a mètre one ten-millionth of 90 degrees: find its length in feet.

10. Express in figures: Thirty-four and two thousandths, and by it divide 28255662. What alteration must be made in the quotient if the decimal part in the dividend be moved eight places to the left?

11. The sidereal year being 365d 6h 9m 98.6, and the tropical year 365d 5h 48m 497: reduce their difference to the decimal of a tropical year.

12. Supposing the velocity of electricity be 288000 miles per second, and the earth's circumference to be 25000 miles: calculate to seven places of decimals the time of transmission of an electric telegraph to the antipodes.

13. A French mètre is 39'37 inches nearly: show that a foot is equal to 304 mètre, ncarly.

14. Find the sum, difference, and product of 86.25 and 39'625, and divide the sum of the three results by 6·25.

15. The circumference of a circle is 3.14159 times its diameter. Find the circumference of circles whose diameters measure 137 feet, 196 yards, and 28-342 miles, respectively.

16. What is the difference between the fifteenth and sixteenth parts of 297'9832 ?

17. A penny is '08975 inches thick: find the height of a pile of 1000 pennies.

18. A cubic inch of water weighs 252.458 grains, and the weight of an imperial gallon or water is 10 lbs. avoirdupois: find (to three places of decimals) the number of cubic inches in an imperial gallon, there being 7000 grains in 1 lb. avoirdupois.

19. A cubic foot weighs 445 lbs.: what does a pound measure in inches?

20. A gallon of water weighs 10 lbs., and measures 277 274 cubic inches: what is the weight of a cubic foot and the measure of a ton?

21. It having been calculated that the air-pump of a marine engine can lift 8616·96 tons of salt water in 6 hours, by considering that a cubic foot of salt water weighs 64 lbs.: what is the error and actual weight of the salt water, when the true weight of a cubic foot is 64-16875 lbs. ?

22. If a sovereign weighs 123′274 grains: how many sovereigns will weigh 10 lbs. 8 oz. 8 dwts. 5 grs. ?

23. Divide 2021 by 1000, 20°21 by '001, 23'0142 by 121, 23014200 by '0121, and 2301'430° by 0.0012100.

24. If the mean diameter of the earth be 504979200 inches in length, express its length in feet, yards, poles, furlongs, and miles.

25. A cubic foot of gold is extended by hammering, so as to cover an area of 6 acres: find the thickness of the gold in decimals of an inch, correct to the first two significant figures. Į

ON LOGARITHMS.

70. Logarithms are numbers arranged in Tables for the purpose of facilitating arithmetical computations. They are adapted to the natural numbers, 1, 2, 3, . . . . in such a manner that by means of them

the operation of Multiplication is changed into that of Addition;

71. Take any whole numbers, as 18, 813, 6489; the first consists of two, the second of three, and the third of four figures or digits. Again, in the mixed number 739.815, the whole number or integral part (739) consists of three digits.

72. By multiplying a number by itself, one, two, three, &c., times successively, we obtain the second, third, fourth, &c., powers of that number; hence, a power of a number is the number arising from successive multiplication by iwi. Thus, 3 × 3 = 9 is the square or second power of 3; and 5 × 5 × 5 = 125, the cube or third power of 5; and so on.

These operations are denoted by means of Indices, or sraall figures plane on the right of the numbers, a little above the line; thus, 2a = 2 × 2 = 49 33 = 3 × 3 × 3 = 27, and 2 = 2 × 2 × 2 × 2 × 2 = 32, where the Index or exponent denotes the number of factors employed.

73. When there are a series of numbers, such that each is found from the previous one by the addition or subtraction of the same number, they are said to be in arithmetical progression. 1, 3, 5, 7, 9, 11, &c., are in arithmetical or equi-different progression, since each number is found by adding 2 to the immediately preceding.

*No proof can here be offered that numbers must exist possesing the properties under which we call them logarithms; neither can any account be here given of the methods of computing such logarithms. The reader will accept the statements that if such numbers exist, bearing the properties aforesaid, they are called logarithms. He must also accept the tables which are published, recording logarithms for the several numbers to which they profess to belong, though he cannot at present verify the computations of these several logarithms; and he will be informed how he may use these tables to effect with comparative ease many calculations which would otherwise be most laborious.

The truth is, though it requires for its demonstration higher algebra than this work presupposes the reader to be acquainted with, that not only has every number a logarithm, but it has an infinite variety of logarithms, constructed, as the term is, on different scales or bases. The base of any system of logarithms is defined by the fact that in that system unity is its logarithm. Any number might be used as a base; but in fact there are only two numbers which are ever really used.

e.

The one is an unterminating decimal, 27182818

denoted generally by the letter This is the base of what is called the natural or Naperian system; and the advantage of it consists in the ease with which logs. are computed, to this base; but which we cannot here explain.

The other is 10, which is the base in ordinary use, and with this base log. 101. Logarithms to this base are the only ones which will now be considered in their practical use,

74. Again, the numbers 3, 6, 12, 24, &c., are in geometrical progression, for each number is formed from the one immediately preceding by multiplying by 2. If we take the following series of powers, 31, 32, 3, 3, 3, &c., we find that the exponents proceed in arithmetical progression, and the quantities themselves in geometrical progression.

75. Def.-Logarithms are a series of numbers in arithmetical progression answering to another series in geometrical progression, so taken that o in the former corresponds with 1 in the latter.

I

Thus, o, 1, 2, 3, 4, 5, 6, &c., are the logarithms or arithmetical series, and 1, 2, 4, 8, 16, 32, 64, &c., are the numbers or geometrical series, answering thereto-the latter being called the natural number. Or, o, 1, 2, 3, 4, 5, the logarithms,

and 1, 5, 25, 125, 425, 5125, the corresponding numbers.

Or, 0, 1,
o,

and I, 10, 100, 1000, 10000, 100000, the corresponding numbers. In which it will be seen, that by altering the common ratio of the geometrical series, the same arithmetical series may be made to serve as logarithms of any series of numbers. As above, when the common ratio of

the geometrical series are 2, 5, and 10 respectively.

I

76. The common ratio in the geometrical series corresponding to the common difference of 1 in the arithmetical series is called the base of the system. Thus, the base of the first specimen exhibited is 2, the base of the second is 5, and the base of the third is 10.

In the specimens just exhibited we have, in each, taken two ascending progressions, but they might equally well have been two descending progressions, or the one descending and the other ascending. Logarithms, however, as now used in practice, are limited to the case of two progressions, either both ascending or both descending the former giving the logarithms of integers, the latter of fractional numbers.

But a better way of considering logarithms is as follows:

77. Def.-The logarithm of a number to a given base is the index of the power to which the base must be raised to give the number.

For instance, if the base of a system of logarithms be 2, 3 is the logarithm of 8, because 8 = 23 = 2 X 2 X 2.

And if the base be 5, then 3 is the logarithm of the number 125, because 125=53 = 5 × 5 × 5.

There may be thus as many different systems of logs. as we please; but, for practical use, it is necessary to select and adhere to one. That usually

employed now is called Briggs' system.

78. We now proceed to describe what is called the common system of logarithms. In the common system of logarithms unity is assumed to be the logarithm of 10; that is, 10 is the constant base. All the logarithms registered

in the Tables commonly used, are indices of the radix or base 10; a Table of logarithms of numbers is in fact nothing more than a Table of the exponents of 10 placed against the several numbers themselves. Accordingly

I

I

Now, if the above Tables were amplified by the insertion of the logarithms of all the numbers between 1 and 10, between 10 and 100, &c., we should have a Table of logarithms of all numbers from 1 to 10000; and whatever may be the difficulty of determining the intermediate logarithms, it is at once easily seen that the logarithms of all numbers between 1 and 10, i.e., between 10° and 101 must lie between 0 and 1, and will be o + a fraction, that is, a decimal less than 1; of all numbers between 10 and 100, i.e., between 101 and 10 must lie between 1 and 2, and will be 1 + a fraction, or a decimal between 1 and 2; of all between 100 and 1000 will be 2 + a fraction, and so forth; or the integral part of each intermediate logarithm will be one less than the number of integral figures in the quantity of which it is the logarithm. Thus, the logarithms of 2, 3, 4, &c., to 9, have o as the integral part; those of 10, 11, 12, &c., to 99, have I as the integer; those of 100, 101, 102, &c., to 999, have 2 as the integer; and so forth. Hence Tables of logarithms usually supply only the fractional or decimal part; the integral part is always known from the number of integers in the value whose logarithm is wanted. Very few logs. can be expressed in terminating decimals, but this causes little inconvenience, since a log. carried to six or seven decimal places is sufficiently exact for all common purposes.

79. The integers 1, 2, 3, 4, &c., which are the logarithms of 10 and its powers (see 78), are chief indices, and the logarithms intermediate to these, as for instance 1778151 (which is the logarithm of 60) cosisting of an integer and a decimal fraction, though they are also indices, are usually referred to as consisting of an index* and mantissa†, the integral part being specially termed the index or characteristic, because it indicates, by being one less, how many integral places are in the corresponding natural number, and the annexed decimal being called the mantissa.

EXAMPLE. In the log. 4616339, the figure (4) standing to the left of the decimal point is the characteristic or index, and the remaining portion ('616339) is the mantissa or decimal part.

In order to avoid confusion from the use of the word "index" to signify two things, we shall throughout this work employ the term characteristic when speaking of logarithms, and index when speaking of roots or powers.

Mantissa, a Latin word signifying an additional handful; something over and above an exact quantity.