The number corresponding to the tabular logarithm is 34.09 And the tabular difference is 128: The 63 being annexed to the tabular number 34.09 gives 34.0963 for the number answering to the logarithm 1.532708. 2. Required the number answering to the logarithm 3.233568. The given logarithm is Their difference is Tabular difference 253)64.00(25 64 Hence the number sought, is 1712.25, marking four places for integers since the characteristic is 3. MULTIPLICATION BY LOGARITHMS. 14. When it is required to multiply numbers by means of their logarithms, we first find from the table the logarithms of the numbers to be multiplied; we next add these logarithms together, and their sum is the logarithm of the product of the numbers (Art. 2). The term sum is to be understood in its algebraic sense; therefore, if any of the logarithms have negative characteristics, the difference between their sum and that of the positive characteristics, is to be taken, and the sign of the greater prefixed. Here the 1.865314 cancels the +2, and the 1 carried from the 3. Multiply 3.586, 2.1046, 0.8372, and 0.0294, together. In this example the 2, carried from the decimal part, cancels 2, and there remains I to be set down. DIVISION OF NUMBERS BY LOGARITHMS. 15. 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. 3). 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. Here, 1 carried from the decimal part to the 3 changes it to , which being taken from 2, leaves ✪ for the characteristic. 3. To divide 37.149 by 523.76 log 37.149 1.569947 log 523.76 2.719133 4. To divide 0.7438 by 12.9476 log 0,7438 1.871456 log 12.9476=1.112189 Here, the 1 taken from I, gives 2 for a result, as set down. ARITHMETICAL COMPLEMENT. 16. The Arithmetical complement of a logarithm is the number which remains after subtracting this logarithm from 10. 10-9.274687 0.725313. Thus of 9.274687. 0.725313 is the arithmetical complement 17 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. 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. GEOMETRY is the science which has for its object the measurement of extension. Extension has three dimensions, length, breadth, height, or thickness. 2 A line is lengh without breadth, or thickness. The extremities of a line are called points: a point, therefore, has neither length, breadth, nor thickness, but position 3. A straight line is the shortest distance from one point to another. 4. Every line which is not straight, or composed of straight lines, is a curved line. Thus, AB is a straight line; ACDB is a broken line, or one composed of straight A lines; and AEB is a curved line. E B The word line, when used alone, will designate a straight line; and the word curve, a curved line. 5. A surface is that which has length and breadth, without height or thickness. 6. A plane is a surface, in which, if two points be assumed at pleasure, and connected by a straight line, that line will lie wholly in the surface. 7. Every surface, which is not a plane surface, or composed of plane surfaces, is a curved surface. 8. A solid or body is that which has length, breadth, and thickness; and therefore combines the three dimensions of extension. 9. When two straight lines, AB,AC, meet each other, their inclination or opening is called an angle, which is greater or less as the lines are more or less inclined or opened. The point of intersection A is the vertex of the angle, and the lines AB, A AC, are its sides. B The angle is sometimes designed simply by the letter at the vertex A; sometimes by the three letters BAC, or CAB, the letter at the vertex being always placed in the middle. Angles, like all other quantities, are susceptible of addition, subtraction, multiplication, and division, Thus the angle DCE is the sum of the two angles DCB, BCE; and the angle DCB is the difference of the A |