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To reduce a Vulgar Fraction to a Decimal of the same value RULE.—Add cyphers at pleasure to the numerator, and divide by the denominator ; the quotient will be the decimal fraction required.

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Every quantity may be considered as a fraction of a larger quantity of the same kind; thus a minute is the go of a degree; an inch, the 12 of a foot, etc: ; and therefore may be reduced to a decimal fraction, as in the following examples :

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RULE.—Multiply the decimal by the number of parts of the next inferior denomination contained in the integer, pointing off in the product as many places for decimals, to the right hand, as the given decimal consists of, and those to the left hand will be an integral number : then multiply the remaining decimals by the number of parts contained in the next inferior denomination, and point off the decimals as before. Proceed thus till it be brought to the lowest denomination. What is the value of '259 of a league? What is the value of .42 of a degree?

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When tenths of a degree or minute are to be reduced into minutes or seconds, it may be expeditiously done by multiplying the tenths by 6, and the product will give the minutes or seconds required : for example, -5 of a degree multiplied by 6 gives 30 minutes, and 9 of a minute, 54 seconds.

On the contrary, to reduce minutes and seconds to tenths of a degree or minute, divide them by 6.

The form this rule most commonly takes for the purposes of navigation is reduction of seconds to the decimal of a minute, or of minutes to the decimal of an hour, generally the latter, as when we have to correct the elements taken from the Nautical Almanac for any other time than Greenwich

noon.

RULE for reducing a lower denomination to the decimal of a higher one. By Table A or B, p. 3, the divisor is 60 ; therefore write down the given number of minutes (or seconds, as the case may be) and divide them by 60.

Examples Reduce 42 minutes to the decimal

Reduce 33 minutes to the decimal of an hour.

of an hour.

60) 42.0 (7

420

60) 33.0 (:55

300

300
300

Therefore, 42 min. is seventenths of an hour; and if, in this case, the given time had been ioh. 42m., we should express it, decimally, as 10.7h., i.e., 10 hours and 7-tenths of an hour.

Here we

see 33 min. is 55hundredths of an hour, and if the given time had been 8h. 33m. we should write 8:55h.

We can adopt a shorter method of getting the same result, as follows: The unit place in the 60 being o, we can reject it if we make the unit place in the minutes a decimal; and the divisor is 6. Thus, using the examples above

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And by a similar process we can make a short table.

(c)

min. hour.
3 = '05
6 = I
9 •15
I2 = 2
15 .25

Minutes expressed as the Decimal of an Hour

min.
hour.

min. hour.
21 = :35

39 .65 24 4

42 7
27 •45

45 -75
30
5

48

.8 33 :55

51 •85 36 .6

54 = 9

18 = 3

But the table may be adapted to various denominations. Thus, using the column of minutes as seconds, the hour column becomes decimal of a minute; or using the column of minutes as minutes (") of angular measure, the hour column becomes decimal of a degree (9); for instance, 18' = 0°:3, i.e., 18' = 3-tenths of a degree.

In practice it is sufficiently accurate to use but one decimal place; thus we should say 14m.

•2 of an hour, since 14 is nearer to 12 than to 18; but 3 of an hour, because 16 is nearer to 18 than to 12; and so for other quantities; always avoid an excess of figures, for otherwise you may

strain at a gnat and swallow a camel."

1бm. .

Proportion or Rule of Three Problems which have the idea of proportion in them are now usually solved by what is called the Unitary Method.

If the given numbers consist of several denominations, they are to be reduced to decimals by the preceding Rules.

Examples If a ship sail 49.5 miles in 8 hours, Suppose a watch or chronometer how many miles will she run in 24 gain 14 seconds in 5 days 6 hours, hours, supposing her to go at the how much will it gain in 17 days 15 same rate ?

hours ? In 8 hours the ship sails 49.5 miles.

6 hours ·25 of a day. 495

.625 of a day. I hour

8

In 5.25 days the watch gains 14 sec.

15 hours

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LOGARITHMS Logarithms are a series of numbers invented, and first published in 1614, by Lord Napier, Baron of Merchiston in Scotland, for the purpose of facilitating troublesome calculations in plane and spherical trigonometry. These numbers are so contrived, and adapted to other numbers, that the sums and differences of the former shall correspond to, and show, the products and quotients of the latter.

Logarithms may be defined to be the numerical exponents of ratios, or a series of numbers in arithmetical progression, answering to another series of numbers in geometrical progression; as,

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0. 1. 2. 3. 4. 5.

6.
7.

8.
1. 10. 100. 1000. 10000. 100000. 1000000. 10000000. 100000000

ind. or log.

geo. pro.

Whence it is evident that the same indices serve equally for any geometrical series; and, consequently, there may be an endless variety of systems of logarithms to the same common number, by only changing the second term 2, 3, or 10, etc., of the geometrical series of whole numbers.

In these series it is obvious that if any two indices be added together, their sum will be the index of that number which is equal to the product of the two terms, in the geometrical progression to which those indices belong : thus, the indices 2 and 6 being added together make 8; and the corresponding terms 4 and 64 to those indices (in the first series), being multiplied together, produce 256, which is the number corresponding to the index 8.

It is also obvious that if any one index be subtracted from another the difference will be the index of that number which is equal to the quotient of the two corresponding terms: thus, the index 8 minus the index 3 = 5; and the terms corresponding to these indices are 256 and 8, the quotient of which, viz., 32, is the number corresponding to the index 5, in the first series.

And, if the logarithm of any number be multiplied by the index of its power, the product will be equal to the logarithm of that power; thus, the index, or logarithm of 16, in the first series, is 4; now, if this be multiplied

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