INTRODUCTION. NAVIGATION is the art of conducting a ship from one port to another, through the wide and trackless ocean. It may be divided into two parts-namely, Theoretical and Mechanical Navigation. The theoretical is that whereby the navigator discovers in what track the ship must be steered, so as to arrive safe at the intended port, and by which the ship's place on the surface of the earth is determined after she has sailed from a given port, for a given time, at a known rate, and in a known direction; it also determines the ship's place on the surface of the earth by means of astronomical observations. The mechanical part consists in setting the sails in a proper position, turning the rudder, &c., in such a manner as to make the vessel move in the direction required; it, however, can only be learned on ship-board, and in the practice of sailing. The theoretical part alone will be expounded in this treatise, and as it depends entirely on mathematical principles, it is deemed proper to begin by laying down those principles which are of constant recurrence, and without a competent knowledge of which no progress can be made. These necessary branches are Decimal Fractions, nature and use of Algebraic or Analytical Symbols, nature and use of Logarithms and Logarithmic Arithmetic, and Plane and Spherical Trigonometry. The pupil who has mastered these will be in a position to understand all the principles and processes employed in the theoretical part of Navigation. The elementary principles of Geometry will also be necessary; but for a knowledge of that subject, the student is referred to the Explicit Euclid in CHAMBERS'S EDUCATIONAL COURSE. TREATISE ON NAVIGATION. CHAPTER I. DECIMAL ARITHMETIC. 1. As nearly all the calculations in Navigation are performed by means of decimals, it is of the utmost importance that the principles and processes of decimals should be thoroughly mastered. For this purpose the following article should be studied as preparatory to the calculations that follow. 2. A decimal fraction is one that has unity with zeros annexed for its denominator, thus 375 is a decimal fraction, and in the 1000 decimal notation is written 375, without a denominator, but the denominator of every decimal is 1, with as many zeros annexed as there are figures after the decimal point; thus .4825 means 4825 3475 425 •3475 means 10000 1000' 10000' 5,425 means &c. 3. The first figure to the left of the decimal point is the place of units, the first to the right of the point means tenths of a unit, the second to the right of the point means hundredth parts A of a unit, the third to the right means thousandth parts of a unit, and so on; thus 4.3645, the 4 to the left of the point 3 means four units, the 3 in the first decimal place means the IO' 6 100' 6 in the second decimal place means the 4 in the third and the 5 in the fourth decimal place decimal place means 4 1000 Adding both sides of these we obtain the following, 4 + + 5 1000 10000 = 3 6 + ΙΟ ΙΟΟ =3645 in the decimal notation; from which it is evident that every zero between the decimal point and a figure diminishes the value of that figure tenfold. Decimals are commonly divided into the following classes-namely, terminating decimals, repeating decimals, circulating decimals, mixed repeaters, mixed circulates, and interminating decimals. When, in reducing a vulgar fraction, or the lower denominations of a compound quantity to a decimal of the higher, the decimal terminates, it is called a terminating decimal; when, in performing the same operation, the result gives always the same figure, beginning from the decimal point, it is called a pure repeater; but when there are several figures, and then the same figures in the same order, it is called a pure circulate; but when several figures intervene between the decimal point and the repeating or circulating part, it is called a mixed repeater, or a mixed circulate; lastly, when the decimal, though carried ever so far, neither circulates nor repeats, it is called an interminate decimal. 4. Since a figure in the sixth decimal place means, by the above, millionth parts of a unit, and in the seventh place, tenmillionth parts of a unit, if the unit of measure is not very large, these places may generally be neglected without affecting the practical accuracy of the results; and hence in Navigation sufficient accuracy is obtained by carrying repeating and circulating decimals to six or seven places of decimals, and then treating them as terminates. But in this case care should be taken to increase the last figure retained by unity, if the next figure be greater than 5; by attending to this the error is always less than one-half of a unit in the last place retained. 5. CASE I. To change a vulgar fraction to an equivalent decimal. RULE.-Annex zeros to the numerator, and divide by the denominator, the quotient will be the required decimal, and will contain as many figures after the point as there were zeros used in the division; if there are not as many figures, the defect must be supplied by prefixing zeros to the quotient, and placing the point before the zeros. Examples. 5 Change to decimals the following fractions: ; }; Å; ñ1⁄2; and 28. 4)100 = =.25 =.3 } a terminating decimal. a repeating decimal. a circulating decimal. a mixed repeater. = •416 = 28)900000000 = •32142857 Exercises 1. a mixed circulate. 5 (1.) Change into equivalent decimals, ; ; ; ; ; and 11 |