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Arithmetic of Navigation ... ...
Protractor and Diagonal Scale ...
Variation of the Compass ...
Deviation of the Compass ...
.. Ship’s Reckoning ... ... ... Mercator's Chart ... ...
Examination in Chart (Board of Trade)
Signs and Abbreviations used on Charts
Adjustments of the Sextant (Examination Paper)
To Read the Sextant ... ... ... ... ... ...
Deviation of the Compass (Examination Paper)
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... (Appendix) International (Commercial) Code of Signals
(Appendix) Seamanship ... ... ... ... ... ... ... ... ... (Appendix)
1,16 1, 12
YOUNG SEAMEN AND YACHTS MEN.
Navigation has been defined as the art of conducting-in the sense of piloting or guiding—a ship from one port to another ; and this piloting on the open ocean, far out of sight of beacons, landmarks and lighthouses, rests on the determination of DIRECTION, DISTANCE, and RELATIVE PositiON,—combining observation with calculations that are the practical application of Geometry and Astronomy. It is the purpose of this little work to explain the use of the instruments, and the kind of calculations, that are required in Navigation, as well as the uşual methods that are adopted from day to day during a voyage: to find what is called the Ship's PLACE AT ŞEA, which is the basis on which rests the direction of her course towards the port of destination.
ARITHMETIC OF NAVIGATION. THE first thing to be considered is-Do you understand the Arithmetic of Navigation ?-if you do not, it can be explained in a few pages, and these the beginner would do well to carefully read; he who is proficient in this knowledge can pass on to the next subject Circles and Angles.”
It is of course taken for granted that you are well up in the four rules of simple arithmetic-addition, subtraction, multiplication, and division-which are as constantly required in Navigation as in daily business transactions. .
Let us begin with the arithmetic of the Circle and of Time, the parts of which in both are divided sexagesimally, or, in other words, sixty of a less denomination make one of a greater. Two short TABLES furnish us with the basis of computation.
(A) Divisions of the CIROLE, OR ANGULAR MEASURE.
60 minutes . . . . . . . . . . l degree o. and these terms are respectively marked !", so that 5° 51' 28" is to be read 5 degrees, 51 minutes, 28 seconds.
(B) MEASUREMENT OF TIME.
60 minutes . . . . . . . . . . 1 hour (h.) but in this instance the terms are respectively marked h. m. B., so that 6b. 31m. 248. is to be read 6 hours, 31 minutes, 24 seconds.
It will here be perceived that though the lower denominations in hoth Tables are known by similar names-seconds and minutes—yet have they different signs to distinguish them; nor are these signs (which represent values ) interchangable, for, as will be presently shown, a second of time has 15 times the value of a second of angular measure, and so also as regards the minutes.
This appears to be the proper place for drawing attention to the ARITHMETICAL SIGNS which are most frequently used in computation; they are as follows: o equal to, is the sign of Equality; as 60 minutes = 1 hour; that is, 60 minutes are
equal to 1 hour. plus (more), is the sign of Addition; as 8 + 7 = 15; that is, 8 added to 7 is equal
to 15. - minus (less), is the sign of Subtraction; as 9 – 3 = 6; that is, 9 lessened by 3 is
equal to 6. X multiplied by, is the sign of Multiplication; as 9 x 12 = 108; that is, 9 multiplied
by 12 is equal to 108. divided by, is the sign of Division; as 84 - 12 = 7; that is, 84 divided by 12 is equal to 7.
REDUCTION. · One of the earliest processes in our arithmetic will be Reduction, which 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 denomina. tion by it, if the reduction be to a higher name.
In Angular Measure and in Time (by Tables p. 1), 60 of the less denomination make one of the greater; hence, as the case requires, we must multiply or divide by 6n, E..—In 5b. 48m. 118., how many seconds ? | Ex.-In 506' how many degrees ? 5h. 48m, 118.
Here, for simplicity, by striking off the
O from 60, we must also strike off the unit 20,891 seconds.
(last) place from 505; then say 6 into 50
goes 8 times and 2 over; write down 8, Here, 5 multiplied by 60*gives 300, to
and bring down the 5, placing the 2 before which add 48, and the result is 348
it; the result will be 8° 25'. minutes. Then 348 multiplied by 60 gives
And similarly for any other quantity, 20,880, to which add 11, and the final
Thus, 7638. reduced to minutes give result is 20,891 seconds.
12m. 438. And similarly for any other quantity. Thus, 18' 42' reduced to give 1,122".
Note. Before proceeding to the next rules, it will be sufficient 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 so with hours, minutes, and seconds.
ADDITION. 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.
h. m. 8. . (1) 28 17 49
) 3 46 16
(3) 108 49
. 3 49 20
Sum 112 41
11 23 36
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. 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 1 of the next higher denomination, which, expressed in ternis of the lower, is 60.
h. m. 8. . . From 18 49 30 (2) 16 19 30
(3) 16 8 42 Take 7 2016
2 44 46
6 3 48 Rem. 11 2915
13 34 44
10 4 54
(1) is simple enough.
(2) requires explanation. In the column of seconds we cannot take 46 from 30, but by borrowing 1' (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 1 to the 2°, and say 3° frum 16° leaves 130,
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) 275 11 16
360 Rem. 13 698
52 42 19
84 48 44 In (1) we say 52" from 60% leaves 8"; carry 1 to the minutes, then l' from 60ʻ leaves 59'; finally, carry 1 to the degrees, and 77° from 90° leaves 13o.
(2) and (3) are operations of a similar character, and it will be perceived that 60 is borrowed in each of the lower denominations.
(1) Mult. 48 46 13 - (2) Mult. 106 4 63
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 97°.
Division. An example will best explain the method ; begin with the degrees, then take the minutes, and lastly the seconds. Ex. 1.- Divide 147° 15' 40" by 2. Ex. 1.—Divide 162° 47' 36' by 4. 2)147 15 40
4)152 47 36 .73 37 50
38 11 54 In Ex. 1, 2 into 147 gives 73 and 1 over (i.e. 1° or 60'); then 60° and 15 make 75, and 2 in 75 gives 37 with 1 over (i.e. l' or 60%); finally, 60 and 40 make 100, and 2 into 100. gives 50.
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.
i. DECIMALS. When we speak of an eighth (f), a quarter (4), a third (1), a half (1), twothirds ($), 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. But, 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, &c., only; thus, :1 stands for to (onetenth), •3 for (three-tenths), .25 for to (twenty-five hundredths), 125 for
(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. Y 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, 7 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-quarter. .
Cyphers after (affixed to) decimal parts do not alter their value; thus •5, 50, or •500, each express an equal value-*, , or $, i.e. half a unit. But cyphers before (prefixed to) decimal parts decrease the value tenfold for each cypher; thus, while •5 is % or , 05 is only too or zb, and similarly •005 becomes odo.
This explanation is not to be taken as trivial, or out of the course of our main