ÆäÀÌÁö À̹ÌÁö
PDF
ePub

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 in expressing decimal parts; for instance, never call 75 seventy-five tenths, but 75-hundredths.

Note.-A Vulgar Fraction may always be turned into a Decimal Fraction, by dividing the figure above the line (or nuinerator) by the figure below the line (or denominator), affixing as many cyphers to the numerator as are required; thus, } = •125.

=

1.000
8

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 miles and 8-tenths of a mile, the figures to the left of the decimal point being integers, and such as are to the right decimals. ADDITION OF DECIMALS.

Arrange all the quantities so that the decimal points may fall in a line directly under each other, and then proceed as in simple addition.

Note. By this arrangement units are added to units, and tens to tens, &c., on the integer (i.c. left) side of the decimal point; and as regards the decimals (to the right of the point), tenths are added to tenths, hundredths to hundredths, &c., and so on. Ex.-Add together 3244 + 56·9 + 4·7

+243 +127.

324.4

56.9

47.

24.3

127.0

Sum 537.3

Ex.-What is the sum of 8′ 5 + 32′ 46 + 987267 +36′ 9 + 362′ 49,

[blocks in formation]

Arrange the two quantities so that the decimal points may fall under each other, and then proceed as in simple subtraction,

[blocks in formation]

Write the quantities under each other, and multiply exactly as in simple multiplication; then, in the product, strike off to the right as many decimal places as there are decimals in both factors; and if there are not so many, the defect must be supplied by prefixing cyphers.

[blocks in formation]
[merged small][ocr errors][merged small][merged small]

For the sake of abbreviation, the second factor (37) has been multiplied by the upper one; and as 37 multiplied by 4 gives 148, and there are four decimal places in the two factors, O is prefixed to 148, and the whole product is a decimal.

DIVISION OF DECIMALS.

Proceed as in simple division, and then 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 many, the defect must be supplied by prefixing cyphers.

[merged small][merged small][merged small][merged small][merged small][ocr errors]

Here 136 into 612 goes 4 times, and leaves remainder 68; to the latter is affixed 0; then 136 into 680 goes 5 times; now the 0 in this case is a decimal, and gives the dividend one decimal place in excess of the divisor; therefore the 5 in the quotient is a decimal.

Ex-Divide 276 by 345
345)276.0(8
276 0

Here the dividend 276 is not divisible by 345, except by affixing 0; after which the divisor goes 8 times into the dividend; now the 0 being a decimal, the 8 in the quotient is also a decimal, according to the rule.

REDUCTION OF DECIMALS.

This is the last rule that we have to consider, and the form it 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.

1. Rule for reducing a lower denomination to the decimal of a higher one. By Table A or B, p. 1, the divisor is 60; therefore write down the given number of minutes (or seconds, as the case may be) and divide them by 60. Ex.-Reduce 42 minutes to the decimal

of an hour.
60)42,0('7
420

Ex-Reduce 33 minutes to the decimal of an hour.

60)33.0(.55

300

300

300

Therefore, 42 min. is seven-tenths of an hour; and if, in this case, the given time had been 10h. 42m,, we should express it, decimally, as 107h,, i.e. 7 hours and 7-tenths of an hour.

Here we see 33 min, is 55-hundredths 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 0, 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

[blocks in formation]

And by a similar process we can make a short table, to which we shall have to refer in the future.

(C) Minutes expressed as the Decimal of an Hour.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

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 circular measure, the hour column becomes decimal of a degree (°); for instance, 18′ = 0·3°, i.e. 18′ 3-tenths of a degree.

NOTE.-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 16m. = 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.”

2. The reverse process of reducing the decimal of a higher denomination into its equivalent lower name is seldom necessary in Navigation; it will therefore be sufficient to say that you must multiply the decimal part of the higher denomination by as many of the lower name as make one of the higher.

For Ex.-In 6 45h., how many hours and minutes?

•45

60

27.00

Therefore 645h. (i.e. 6 hours and 45-hundredths of an hour) = 2h. 27 min.

The beginner is to understand that he must be complete master of the Arithmetic explained in the previous pages before he can make any progress in the study of Navigation.

I am about to forestall a little here, and I do so in order to group together such calculations as are of a similar character; they will not be wanted until you have acquired a good knowledge of the subject, but they are nevertheless such as are in constant requisition in Navigation, and it will be better to refer back than to introduce them hereafter. If you prefer to leave these subordinate computations for the present, go on to "Circles and Angles," p. 9; if you chose to look them through now, I will endeavour to explain them as simply as possible.

RELATION OF CIRCULAR MEASURE TO TIME.

The circumference of every circle is divided into 360 equal parts, called degrees; now the earth, whose circumference as a sphere is 360°, turns on its axis in the direction of this circumference once in 24 hours; therefore the 360° of circular measure are the equivalent of 24 hours; dividing 360 by 24 we get 15, and hence 15° of circular measure are the equivalent of 1 hour. We say, astronomically, that as a complete rotation of the earth on its axis is performed in 24 hours, meridians 15° asunder are thus brought to the sun at regular intervals of one hour. This gives us the following Table:—

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small][merged small]

=

TO CONVERT TIME INTO CIRCULAr Measure (Arc), multiply the time by 15 for by 3 and 5 successively, since 3 × 5 15); then the seconds of time become seconds (") of arc, the minutes of time become minutes (') of arc, and the hours become degrees.

Or, if we turn the hours into minutes, and divide by 4, then the minutes of time give degrees, the seconds of time give minutes (') of arc, and the remainder (if any) multiplied by 60 and divided by 4 gives seconds (") of arc.

[blocks in formation]

TO CONVERT CIRCULAR MEASURE (OR ARC) INTO TIME, the shortest method is to multiply the degrees and minutes (and) of arc by 4, when the minutes (') of arc become seconds of time, and the degrees become minutes of time which reduce to hours by dividing by 60.

Ex.-Convert 137° 26′ into its equivalent time.

[blocks in formation]

Both these computations relate to the conversion of Longitude (which is circular measure) into Time, and Time into Longitude.

CIRCLES AND ANGLES.

We will draw a circle. Open a pair of dividers (compasses) so that the distance between the two points is equal to the distance from c to A (Fig. 1); with one point fixed at c, let the other point, starting from A, revolve once round the fixed point, and the geometrical plane figure called a CIRCLE has been described; the continuous curved line A B D E A bounding the circle is called the CIRCUMFERENCE, and c is the centre of the circle, from which the circumference is everywhere equally distant.

[blocks in formation]

In a similar manner a circle of any size can be drawn; as, for example (Fig. 1), the circle with the dotted line.

NOTE. The terms circle and circumference are very generally used the one for the other, but this is not strictly correct; the circumference bounds the circle, and the circle is the entire area, space or surface included within the circumference.

Any straight line, as в E, drawn from one part of the circumference to another, but passing through the centre of the circle, divides the figure into two equal parts, each of which is called a SEMICIRCLE, as B A E and

[graphic]

BDE.

The line BE is called the DIAMETER of the circle.

Any straight line, as c E or c B, drawn from the centre of the circle to the circumference is a RADIUS; and it is evident that the line c в is equal to the line CE, which together make up the diameter B E, therefore the diameter is twice the radius, and the radius the half of the diameter.

If two diameters cross each other so that they divide each semicircle into two equal parts, and consequently the circumference into four equal parts, each part -as B A, A E, E D, and D B-is called a QUADRANT.

An ARC is any portion (long or short) of the circumference, as a F, or B F. The circumference of every circle is divided into 360 equal parts, called DEGREES; each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds; which are respectively marked '". (See p. 1, Table A.)

Since the circumference of a circle contains 360°, the half of it (a semicircle) contains 180°, and the quarter of it (a quadrant) contains 90°.

All that has been said in respect to the outer circle in the fig. applies equally to the inner one.

An ANGLE may be described as the inclination of two straight lines towards each other, or, as the opening between two straight lines which meet in a point; thus F C A, or simply c, is the angle (Fig. 1) contained by the two lines c F and OA;* these lines are called legs.

When an angle is spoken of without reference to the lines by which it is formed, it is named by the single letter at the point, as c; when the lines are referred to, the three letters are introduced, the letter at the point being the central one, as F C A.

« ÀÌÀü°è¼Ó »