Page. Elementary Principles of Algebraical Calculation Elementary Principles of Geometry Of Proportion as applied to Geometry ... Elementary Principles of Plane Trigonometry Elementary Principles of Spherical Trigonometry Construction and Use of the Mariner's Scale Practical Application of the Principles of Plane Trigonometry Practical Application of the Principles of Spherical Trigonometry Elementary Principles of Navigation...... Middle Latitude and Mercator's Sailing Method of keeping a Sea Journal Principles of the Quadrant, Sextant, &c. ... NAUTICAL ASTRONOMY, Definitions and Principles Practical Methods of finding the Latitude by Meridian and Double Alti- Practical Methods of finding the Variation of the Compass by Observation.... 234 Practical Methods of finding the Time, the Errors and Rates of Chronometers, and the Longitude by Chronometers, and by Lunar Observations.......... 238 Investigations of the preceding practical Rules ERRATA. Note, t signifies from the top, and b, from the bottom. Example 10, for 2 b x read 2 b c. for A B 2 read A B2- for F read E. for cos AC: cos B C read cot A C: cot B C. =, sinC== read sin Ac ✓ sin Cs Answer to Example 17, < B should be 157° 3' 44", or 40 58' 30. Example 1, for 19 44 09 read 19 44 17. Example No. 1, t, for 9m 52s, and 9m 56s, read 8m 27s, and 8m 31s. Ilh 52m 18s P. M. 1h 37m 20s read Ih 35m 32s. It is requested that the above typographical errors may be corrected with a pen. ELEMENTARY PRINCIPLES OF ALGEBRAICAL CALCULATION. When quantities are considered algebraically, they are represented by the letters of the alphabet; the leading letters a, b, c, &c. usually representing such quantities as are conceived to be given or known, and the final letters w, x, y, &c. such as are required or unknown. To abridge analytical expressions, or to indicate the performance of certain operations, several characters are employed, of which the following are the chief: viz. + named plus, signifies addition, minus, subtraction. .. difference, the difference of two quan tities when it is not known which is the greater. Х ... into, multiplication. by, division. equal, denotes the equality of the quantities between which it is placed ; and quantities so connected are said to form an EQUATION. :, : are used to denote proportion. drawn over, or ( ) enclosing several quantities, denotes that altogether they are to be considered as one quantity. ✓ called the root, or radical mark, when by itself, denotes the square root ; w denotes the cube root; and n representing any number whatever, ñ v denotes the nth root. The second power, or the square of any quantity, as 5, or a, is represented by 5 x 5, or 52; a x a, a.d, a a, or a”; the third power, or the cube by 5 x 5 x 5 or 53; a x a x a, a.a.a., a a a, or a', &c. and the number3 2, 3, &c. are called the indices or exponents of the respective powers. The roots of quantities are also often represented by fractional indices; thus, 7t means the square root of ; a's the cube root of a; b* the fourth root of b; cš the fifth root of the cube of c, &c. B Quantities with the sign + prefixed, are called additive, positive, or affirmative quantities, in contradistinction to those which are preceded by the sign –, and which are called subtractive or negative quantities. When no sign is prefixed to a quantity + is understood. Thus 4 means + 4, and c means + c. A number prefixed to a letter is called the numeral coefficient of that letter, and it denotes that the quantity which the letter represents, is to be taken as many times as there are units in the numeral coefficient. Hence, if a represent the greater, and b the less of two quantities, a + b will represent their sum, a b their difference, a x b, a , b by b. a +b.cwe, or (a + b). (ewe) denotes that the sum of a and b is to be multiplied by the difference of c and e. a-b= c + de denotes that the difference of a and b is equal to the square root of c a + b a added to the difference of d and e. atba b, or denotes . b that the sum of a and b is to be divided by their difference. 3 a denotes that the quantity a is to be taken 3 times ; 5 b c denotes that the product of b and c is to be taken 5 times; and 4.a - b denotes that the difference of a and b is to be taken 4 times. OF ADDITION. 5 added to 3 times 5 is 4 times 5; 7 added to twice 7 is 3 times 7; and whatever a may represent, a added to 3 times a is 4 times a. 4 times the product of a and b added to 3 times the product of a and b, is ng times the product of a and b. The difference of b and c added to twice that diffe ence, is three times the same difference. Hence 5a + 3a = 8 a. 7 b c + 11 b c= 18 b c, &c. Again, if 4 and 7 are both to be subtracted from any quantity, the result will be the same if il, or their sum, be subtracted from that quantity; and if 3 times a, and 5 times a, be both to be subtracted from any quantity, it will be equivalent to subtracting their sum, or 8 times a, from the same quantity. Hence 3 a added to 4 a is-70; 2 b c added to -76c is 9 bc, &c. We may therefore conclude, generally, that like quantities with like signs, are added together by prefixing the sum of their numeral coefficients, with the common sign, to the literal part. But when quantities of different kinds are to be joined—as a yard cloth, nor 2 pounds of iron; but a yard of cloth and a pound of iron; and this method of expressing their sum admits of no abbreviation. Also, when quantities to be collected together, are of the same kind but of different denominations, their sum is taken in the same way, by an enumeration of the parts which compose the whole. Thus, if 2 lines of a certain length were required to be added to 3 lines of a different length, the sum would not be 5 times a line of either of the lengths, but 2 times the length of the one, added to 3 times the length of the other. Or the sum of two different quantities a and b, is represented by a + b, not by 2 a or 2 b. 3 b c added to 7 c is 3 bc + pc, &c. Hence the sum of unlike quantities is expressed by the quantities themselves, with their proper signs and coeficients. Again, if 6 is to be added to any quantity, and 2 subtracted, the same result will be produced if 4, or the difference of 6 and 2, be added to that quantity. If 9 is to be subtracted from any quantity, and 3 added to it, this will be equivalent to subtracting 6, or the difference of 9 and 3 from the same quantity. Or the sum of + 6 and 2 is + 4; the sum of 9 and + 3 is the sum of 3 a and + 8 a is + 5a; the sum of + 17 b c a and b 8 13 is 5 bo 응 b 69 Hence, like quantities with unlike signs are added together by prefixing to the literal part, the difference of the coefficients, with the sign belonging to the greater one. And where there are several quantities to be collected together, some of which are alike, and others unlike; the sum will be obtained by adding together the like quantities, and the unlike ones separately. Thus, if a + b were to be added to a + c, the sum would be 2 a + b + c. Finally, as 8 times a and 4 times a are equal to 12 times a, or to 8 + 4 times a ; and 6 b + b is equal to 6 + 1 times b, or to 7 times b; we have generally, a b + cb + db=a++d.b. - 6; . a EXAMPLES IN ADDITION. 1. What is the sum of a, 2 a, 5 a, and 3 a ? Answer, 11 a. 2. What is the sum of 2 ab, 4 a b, and il a b? Answer, 17 a b a 3. What is the sum of 15 and 6 ? 21 o 6 4. What is the sum of 76, 36, and 46 ? Answer, 66. 5. What is the sum of 12 b cd, 3 b cd, and — bcd? Answer, 14 b cd. 6. What is the sum of a + b and a b? Answer, 2 a. 7. What is the sum of a2 + 2 ab + 6% and a’ — ab + 62? Answer, ? a? + 2 62. |