in the following example. If for 31. I purchase 5 yards of cloth, how much can I have for 127.? If 641. gain 351. in any given time, how much should 3201. gain in the same time? In this example, as the number 320 is not found in the preceding specimen of logarithms, we take the logarithm for 32, the fractional part of which is the same with that of the logarithm for 320, and prefixing the index or characteristic 2, because of the three places of figures in the latter number, we obtain 2,50515 for the logarithm of 320: to this adding the logarithm of 35, and from the sum subtracting the log. of 64, we have the remainder 2,24804, for which the natural number is not within the bounds of the specimen; it must, therefore, be discovered in the way before pointed out we take the nearest number above it in the table, which is, 25527, the log. of 180, and the nearest below it, which is, 23045 the log. of 170; then we state the proportion, as the difference between these two loga rithms, rithms, or 2482, to the difference between their corresponding numbers 170 and 180, or 10, so is the difference between the lowest log. and that above found, or 1259 to a fourth proportional, which will be 5; and this added to the lower number, 170, will give 175, the sum of interest required in the question. Involution of Roots is performed by multiplying the logarithm of the given number by the exponent of the power to which it is to be raised, when the product will be the logarithm of the power required. For example, raise 8 to the. 2nd power, or square. Evolution of Roots is performed by dividing the logarithm of the given number by the exponent of the power, when the quotient will be the logarithm of the required root. For example, extract the square root of 64. The preceding examples will show the manner of using tables of the logarithms of numbers, and those of logarithmic sines, tangents, secants, &c. are employed precisely in the same way, of which examples will be given in the following illustrations of the several cases of plane trigonometry. PROP. I. Fig. 2, Plate 3. The radius of every circle is equal to the chord of 60 degrees, and half the radius is equal to the sine of 30 degrees. Let BDAE be a part of a circle of which C is the centre; at C, with the radius CB, form an angle, BCA, equal to 60 degrees, and join AB, which will then be the chord of 60 degrees. But the sides CA and CB being equal, each being a radius of the circle, the angles at A and B must be equal (Geometry, Prop. 4), and the angle at C being made equal to 60 degrees, the two remaining angles must together be equal to 120 degrees, (page 383, and Geom. Prop. 7), and each of them equal to 60 degrees: the three angles, therefore, of the triangle BAC being all equal, the three sides must also be equal (Geom. Prop. 4), and AB must be equal to CA or CB; but CA and CB are each of them a radius of the circle, consequently AB, the chord of 60 degrees, is equal to the radius of the circle. Again, if from the centre C, the line CD be drawn bisecting the angle ACB, and consequently bisecting the chord AB in the point F, CD will be a radius, and AF will be the sine of 30 degrees (page 382). But the triangles ACF and FCD having the side AC equal to CB, and the side CF common, as also the angle BCF equal by construction to the angle FCB, the remaining side AF will be equal to the remaining side FB, (Geom. Prop. 3), consequently the sine of 30 degrees will be equal to one half of AB, that is, of the radius of the circle. PROP. |