Cube .0006724 4.82765 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 EXAMPLES. 1. Required the cube root of 391.27. 3) Logarithm of 391.27 is 2.59248 Cube root 7.314 0.86416 2. Required the square root of .08593. 2) Logarithm of .08593 is -2.93414 Square root .29314 -146707 3. Required the cube root of .7596. 3) Logarithm of .7596 is -1.88058 4. Required the cube root of .0000613. 3) Logarithm of .0000613 is -5.78746 5. Required the square root of 365. Ans. 19.105. 6. Required the 5th root of .9563. Ans. .9911. 7. Required the 4th root of .00079. Ans. .16765 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 10.00000, which may readily be done, by writ ing down 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 tens 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. 473.9 Ar. Co. 7.32431 4.76 Ar. Co. 9.32239 35.84 log. 1.55437 76.4 log. 1.88309 Quotient 1.214 0.08416 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 is that which 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. 8. The magnitude of an angle depends on the inclination which the lines that form it have to each other, and not on the length of those lines. Thus the angle DBE is greater than the angle ABC, Fig. 3. 9. When a straight line stands on another straight line so as to incline to neither side, but makes the angles on each side equal, then each of those angles is called a right angle, and the line which stands on the other is said to be perpendicular to it. Thus ADC and BDC are right angles, and the line CD is perpendicular to AB, Fig. 4. 10. An acute angle is that which is less than a right angle, as BDE, Fig. 4. 11. An obtuse angle is that which is greater than a right angle, as ADE, Fig. 4. 12. Parallel straight lines are those which are in the same plane, and which, being produced ever so far both ways, do not meet, as AB, CD, Fig. 5. 13. A figure is a space bounded by one or more lines. 14. A plane triangle is a figure bounded by three straight lines, as ABC, Fig. 6. 15. An equilateral triangle has its three sides equal to each other, as A, Fig. 7. 16. An isosceles triangle has only two of its sides equal, as B, Fig. 8. 17. A scalene triangle has three unequal sides, as ABC, Fig. 6. 18. A right angled triangle has one right angle, as ABC, Fig. 9; in which the side AC, opposite to the right |