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ADDITION OF DEGREES, &c., AND TIME Note.—It will be sufficient here to remark, once for all, that like denominations (as in every other computation of compound quantities) must stand directly under each other ; thus, degrees must be placed under degrees,' under', and " under "; and similarly hours under hours, minutes under minutes, and seconds under seconds.
RULE.-Take the sum of the column of seconds and write it down, if less than 60; if the sum exceeds 60, find how many minutes are contained in it, then write down the remaining seconds, and carry the minutes to the column of minutes. Next, take the sum of the column of minutes, and write it down if less than 60 ; if the sum exceeds 60, find how many degrees are contained in it, and write down the remaining minutes, carrying the degrees to the column of degrees. Finally, take the sum of the column of degrees.
Proceed in the same manner if the quantities are hours, minutes, and seconds.
In (1) the sum of the column of seconds is 146, but we write down 26 and carry 2', because 146” make 2' 26"; the sum of the column of minutes, with 2 to carry is 53', which we write down at once ; the sum of the degrees is 156o.
The operation is the same in (2) and (3).
SUBTRACTION OF DEGREES, &c., AND TIME RULE.-In the lower denominations, when the quantity to be subtracted is less than the other, the process is simple ; if it be greater, we must borrow I of the next higher denomination, which, expressed in terms of the lower, is 6o.
(1) is simple enough.
(2) requires explanation. In the column of seconds we cannot take 46 from 30, but by borrowing I' (i.e., 60") 30" becomes 90", and 46 from 90 leaves 44", which write down. Then, having borrowed under the head of seconds, we say 45 from 19 in the column of minutes, but can get no result unless we borrow 1° (i.e., 60'), and then 45' from 79' leaves 34', which write down. Finally, having borrowed in the minute column, carry I to the 2°, and say 3° from 16° leaves 13o.
The principle is the same in the case of hours, minutes, and seconds.
In Navigation, it is very often required to take the upper line from the lower; this should be practised, as it does not look well to make a transposition of the quantities. (1) Take
76° 0' 52" (2) 127° 17'41" (3) 275o II' 16"
84 48 44 In (1) we say 52" from 60" leaves 8" ; carry 1 to the minutes, then 1' from 60' leaves 59'; finally, carry i to the degrees, and 77o from 90° leaves 13°
(2) and (3) are operations of a similar character, and it will be perceived that bo is borrowed in each of the lower denominations.
MULTIPLICATION OF DEGREES, &C., AND TIME An example will best explain the method ; begin with the column of seconds. (1) Mult.
46' 13" (2) Mult. 106° 4' 53" Ву
424 19 32 Take (1): twice 13 are 26, which write down in column of seconds, since 26 is less than 60. Twice 46 are 92, but 92' = 1° 32'; therefore write down 32 in column of minutes, and carry 1°. Twice 48 are 96, and 1° to carry, make 97o.
97 32 26
DIVISION OF DEGREES, &C., AND TIME An example will best explain the method ; begin with the degrees, then take the minutes, and lastly the seconds.
(1).-Divide 147° 15' 40" by 2.
2) 147° 15' 40"
(2).—Divide 152° 47' 36" by 4.
38 II 54
73 37 50
In (1): 2 into 147 gives 73 and 1 over (i.e., 1° or 60'); then 60 and 15 make 75, and 2 into 75 gives 37' with 1 over (i.e., I' or 60"); finally, 60 and 40 make 100, and 2 into 100 gives 50".
REDUCTION OF DEGREES, &c., AND TIME. An important process in the arithmetic of Angular Measure and Time is Reduction, that is the converting or changing a quantity from one denomination to another without altering its absolute value. Take an example from money : we know that 100 shillings make £5; that is, the shillings here indicated have the same value as the pounds; to get the £5 we divide the 100 by 20 (since 20 shillings make £1); for the reverse process we multiply the 5 by 20, which gives us the 100 shillings, hence the following rule
GENERAL RULE FOR REDUCTION.—Consider how many of the less denomination make one of the greater ; then multiply the higher denomination by this number, if the reduction is to be to a less name; or divide the lower denomination by it, if the reduction be to a higher name.
In Angular Measure and in Time (by Tables, p: 3) 60 of the less denomination make one of the greater ; hence, as the case requires, we must multiply or divide by 6o.
In 5h. 48m. IIS., how many seconds ?
In 505' how many degrees ?
Here, for simplicity, by striking 348 minutes.
off the o from 60, we must also strike 60
off the unit (last) place from 505 ; 20891 seconds.
then say 6 into 50 goes 8 times and 2 Here, 5 multiplied by 60 gives over ; write down 8, and bring down 300, to which add 48, and the result the 5, placing the 2 before it ; the is 348 minutes. Then 348 multiplied result will be 8° 25'. by 60 gives 20880, to which add 11, and the final result is 20891 seconds.
And similarly for any other quantity. Thus, 763 seconds (of
time) reduced to minutes give And similarly for any other
12m. 43s. tity. Thus, 18' 42" reduced to "give 1122". (1) In 98° 0' 56" how many' (minutes) and " (seconds of arc) ?
Answer-5880'; and 352856". (2) In 3h. 59m. 145. how many minutes and seconds ?
Answer—239min.; and 14354 sec. (3) In 43062 seconds how many hours ?
Answer-Ilh. 57m. 425. Enough has now been said on the arithmetic of the circle and of time, and we next proceed to explain the nature of Decimals, without a knowledge of which we should be unable to use either the Nautical Almanac or the usual Nautical Tables that aid us in finding a ship's place from day to day.
Numbers such as 1, 52, 148, and so on to millions or more, consisting of any whole number of units, are called Integers, but when we speak of a number which is a portion of 1, or unity, as of an eighth (t), a quarter (1), a third (!), a half (t), two-thirds (3), or three-fourths (#) of anything, say of a mile, we mean a fractional part of the mile, and the arithmetic of these values is termed Vulgar Fractions. In computations connected with Navigation, the same values are expressed decimally-i.e., as Decimal Fractions—because by their aid we get better results with fewer figures than if we used vulgar fractions.
Decimal fractions are distinguished by a dot placed before the figure, and are read as tenths, hundredths, thousandths, etc., only ; thus ·I stands for Io (one-tenth), 3 for 3 (three-tenths), •25 for 15 (twenty-five hundredths), •125 for 10 (125-thousandths), and so on.
The relation of Decimal to Vulgar Fractions may be illustrated as follows : The vulgar fraction }, when written decimally, becomes -5, i.e.,
or 5 divided by 10, because 5 parts of anything that is divided into 10 parts is the same as one-half of the whole (unit); similarly, 4 becomes a decimal in the form of .25, i.e., 25 divided by 100, because 25 parts of 100 parts is the same thing as one-fourth.
Cyphers after (or affixed to) decimal parts do not alter their value ; thus 5:50, or -500, each express an equal value—viz., Po Po, or 0%, i.e., half a unit. But cyphers before (or prefixed to) decimal parts decrease the value tenfold for each cypher ; thus, while -5 is B or }, .05 is only 1ő or zo; and similarly .005 becomes to co or zdo:
This explanation is not to be taken as trivial, or out of the course of our main subject, because when we come to speak of the Nautical Almanac, we shall find most of the corrections extend to hundredths and thousandths.
Caution.--After what has been written here, avoid a very common error expressing decimal parts; for instance, never call -75 seventy-five tenths, but 75-hundredths; and never call it decimal seventy-five, but decimal seven five.
Note.—A Vulgar Fraction may always be turned into a Decimal Fraction, by dividing the figure above the line (or numerator) by the figure below the line (or denominator), affixing as many cyphers to the numerator as are required ; thus, $
1.000 = •125. Having explained the nature of decimals, it may now be stated that in the arithmetic of Navigation the numbers are not entirely decimals, but in the majority of instances consist of whole numbers (or integers) and decimals, as when we write 28.8 miles for 28 and 8-tenths of a mile, the figures to the left of the decimal point are integers, and such as are to the right are decimals.
Addition and Subtraction of Decimals RULE.—Addition and subtraction of decimals are performed exactly the same as in whole numbers, observing always to place the decimal points so that they may stand directly under one another; and thus figures of the same denomination will range properly.
2.304 127.414 262-394
Examples in Addition (2) 65:
(3) 720·1464 2463
Examples in Subtraction (5) 246.25
(7) 176•014 (8) 174:
171.439 Nore.--In example (8) as there are no figures in the upper line above 561 (in the lower one), 000 is considered to be expressed in the upper line, and then the subtraction is made in the
Multiplication of Decimals RULE.—Multiply the given numbers together as if they were whole numbers, and point off as many decimals in the product, counting from the right hand, as there are decimals in the multiplicand and multiplier together.
When it happens that there are not so many figures in the product as there should be decimals, supply the defect by prefixing cyphers on the left hand (see Example 3).
Division of Decimals Proceed as in simple division, but introducing cyphers into the dividend if required; and then, the division having been made, strike off in the quotient as many decimal places as the dividend has decimal places in excess of the divisor ; if there are not so mariy, the defect must be supplied by prefixing cyphers.
When, after the division, there is a remainder, cyphers may be added to the dividend, and the operation continued as before until either there be no remainder, or a sufficient degree of exactness has been obtained in the quotient.
Divide 276 by 345 Divisor. Dividend. Quotient.
Divisor. Dividend. Quotient. 136) 612.0 (4:5
345) 276.0 (8 544