Ans. 19.105. Ans. .9911. 5. Required the square root of 365. Of the Arithmetical Complements of Logarithms. When it is required to subtract several logarithms from others, it will be more convenient to convert the subtraction into an addition, by writing down, instead of the logarithms to be subtracted, what each of them wants of 4. Required the cube of 7.503. Ans. 422.37. 5. Required the 7th power of .32513. Ans. .0003841. PROBLEM VI. To extract any root of a number logarithmically. RULE. Divide the logarithm of the given number by the index of the root, that is by 2 for the square root, by 3 for the cube root, &c. and the quotient will be the logarithm of the required root. Note. When the index of the logarithm is negative, and does not exactly contain the divisor, increase the index by a number just sufficient to make it exactly divisible by it, and carry the units borrowed, as so many tens, to the left hand figure of the decimal part; then Ans. 19.105.
what the first figure, on the right hand, wants of 10, and what every other figure wants of 9; this remainder is called the Arithmetical Complement. Thus, if the logarithm be 2.53061, its arithmetical complement will be 7.46939. If one or more figures to the right hand be ciphers, write ciphers in their place, and take the first significant figure from 10, and the remaining figures from 9. Thus, if the logarithm be 4.61300, its arithmetical complement will be 5.38700. In any operation, where the arithmetical complements of logarithms are added to other logarithms, there must be as many 10's subtracted from the sum, as there are arithmetical complements used. As an example, let it be required to divide the product of 76.4 and 35.84, by the product of, 473.9 and 4.76. ! GEOMETRY. DEFINITIONS. 1. GEOMETRY is that science wherein the properties of magnitude are considered. 2. A point is that which has position, but not magnitude. 3. A line has length but not breadth. 4. A straight, or right line, is the shortest line that can be drawn between any two points. 5. A superficies or surface has length and breadth, but not thickness. 6. A plane superficies is that in which any two points being taken, the straight line which joins them lies wholly in that superficies. 7. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line, as A, Fig. 1. Note. When several angles are formed about the same point, as at B, Fig. 2, each particular angle is expressed by three letters, whereof the middle letter shows the angular point, and the other two, the lines that form the angle; thus, CBD or DBC signifies the angle formed by the lines CB and DB. D |