DIVISION OF NUMBERS BY LOGARITHMS. 12. When it is required to divide numbers by means of their logarithms, we have only to recollect, that the subtraction of logarithms corresponds to the division of their numbers (Art. 4). Hence, if we find the logarithm of the dividend, and from it subtract the logarithm of the divisor, the remainder will be the logarithm of the quotient. This additional caution may be added. The difference of the logarithms, as here used, means the algebraic difference; so that, if the logarithm of the divisor have a negative characteristic, its sign must be changed to positive, after diminishing it by the unit, if any, carried in the subtraction from the decimal part of the logarithm. Or, if the characteristic of the logarithm of the dividend is negative, it must be treated as a negative number. 2. To divide 0.06314 by .007241. log 0.06314= 2.800305 Here, 1 carried from the decimal part to the 3, changes it to 2, which being taken from 2, leaves 0 for the characteristic. 3. To divide 37.149 by 523.76. log 37.149=1.569947 log 523.76 2.719133 Here, the 1 taken from 1, gives 2 for a result, as set down. ARITHMETICAL COMPLEMENT. 13. The Arithmetical complement of a logarithm is the number which remains after subtracting the logarithm from 10. Thus, of 9.274687. 10 9.274687 0.725313. 0.725313 is the arithmetical complement 14. We will now show that, the difference between two logarithms is truly found, by adding to the first logarithm the arithmetical complement of the logarithm to be subtracted, and then diminishing the sum by 10. and Let a= the first logarithm, b= the logarithm to be subtracted, c=10-b= the arithmetical complement of b. Now the difference between the two logarithms will be expressed by a—b. But, from the equation c=10-b, we have c-10=-b, hence, if we place for b its value, we shall have a-b=a+c-10, which agrees with the enunciation. When we wish the arithmetical complement of a logarithm, we may write it directly from the table, by subtracting the left hand figure from 9, then proceeding to the right, subtract each figure from 9 till we reach the last significant figure, which must be taken from 10: this will be the same as taking the logarithm from 10. 2 Hence, to perform division by means of the arithmetical complement, we have the following RULE. To the logarithm of the dividend add the arithmetical complement of the logarithm of the divisor; the sum, after subtracting 10, will be the logarithm of the quotient. In this example, the sum of the characteristics is 8, from which, taking 10, the remainder is 2. 1. EXTENSION has three dimensions, length, breadth, and thickness. 2. GEOMETRY is the science which has for its object: 1st. The measurement of extension; and 2dly. To discover, by means of such measurement, the properties and relations of geometrical figures. 3. A POINT is that which has place, or position, but not magnitude. 4. A LINE is length, without breadth or thickness. 5. A STRAIGHT LINE is one which lies in the same direction between any two of its points. 6. A BROKEN LINE is one made up of straight lines, not lying in the same direction. 7. A CURVE LINE is one which changes its direction at every point. The word line when used alone, will designate a straight line; and the word curve, a curve line. 8. A SURFACE is that which has length and breadth 9. A PLANE is a surface, such, that if any two of its points be joined by a straight line, such line will be wholly in the surface. 10. Every surface, which is not a plane surface, or composed of plane surfaces, is a curved surface. 11. A SOLID, or BODY is that which has length, breadth, and thickness: it therefore combines the three dimensions of extension. 12. An ANGLE is the portion of a plane included between two straight lines which meet at a common point. The two straight lines are called the sides of the angle, and the common point of intersection, the vertex. Thus, the part of the plane includ ed between AB and AC is called an angle: AB and AC are its sides, and A its vertex. An angle is sometimes designated 4 simply by a letter placed at the vertex, B as, the angle A; but generally, by three letters, as, the angle BAC or CAB,—the letter at the vertex being always placed in the middle. 13. When a straight line meets another straight line, so as to make the adjacent angles equal to each other, each angle is called a right angle; and the first line is said to be perpendicu lar to the second. 14. An ACUTE ANGLE is an angle less than a right angle. نے 15. An OBTUSE ANGLE is an angle |