By the Table of Natural Sines and Cosines... Test of the Accuracy of the Survey...... The Angles and one Side of a Triangle being given, to find the Area..... 225 To determine the Area of a Trapezium, three Sides and the two included To lay out a given Quantity of Land in the form of a Square....... To lay out a given Quantity of Land in the form of a Rectangle, one Side One Side to exceed another by a given Difference To lay out a given quantity of Land in the form of a Triangle or Paral- By a Line through a given Point in the old Line......... By a Line through a given Point in one of the Adjacent Sides....... 283 To divide a Triangle into two Parts having a given Ratio. By a Line through one of the Corners...... By a Line through a Point in one of the Sides. By a Line Parallel to one of the Sides......... To determine the Latitude by a Meridian Altitude of Polaris............... 319 By a Meridian Altitude of the Sun.......... By an Observation on a Star in the Prime Vertical.... 1. Definition. = = EVERY number may be considered as being a power, either integral or fractional, of some other number. Thus, 16 42, 8441-5, and 324 425. When natural numbers are all considered as powers of the same root, the indices of those powers are called the logarithms of the numbers, and the root is called the base of the system. 64 64.1666, 4 = 64a = 64-3333, 8 = = Thus, 2 = 16 = 641 = 64-6666, 326464-8333, 128 256 = 64 641.666; and so on. = 641 — 64.5, = 641.158 Therefore, .1666 is the logarithm of 2, to the base 64; and .3333, .5, .6666, .8333, 1.1666, and 1.666 are the logarithms of 4, 8, 16, 32, 128, and 256 respectively, to the same base. 2. It is well known, that, to multiply two powers of a certain root, we add their indices. If, then, all the natural numbers were expressed as powers of some one base, and the indices of those powers were known, all that would be necessary to determine the product of any two or more of them would be to seek out the corresponding indices, add them, and find the number whose index was equal to their sum: this would be the product required. Thus, in the following table, the numbers are regarded as powers of 2, the indices of the powers being set down opposite the number. To multiply any numbers contained in the column of numbers, headed N, take out the corresponding indices, add these, seek their sum in the column of indices, and opposite thereto, in the column of numbers, is the product required. Suppose, for instance, the product of 32, 1024, and 512 were required: the corresponding indices are 5, 10, and 9. The sum of these is 24; hence, 16777216 is the product required. TABLE OF POWERS OF 2 AND THE CORRESPONDING INDICES. So likewise division may be performed by means of such a table. Ex. Required the quotient of 4194304 by 131072. The indices are 22 and 17. The difference of these is 5. The corresponding number 32 is the quotient required. 3. The table in last article contains only the integral powers of 2. This is sufficient for the purpose of illustration. A complete table contains all the numbers of the natural series, as far as the limits of the table, with the indices, or logarithms. These will in most instances be fractions. Thus, the logarithms corresponding to any of the numbers between 4 and 8 would be 2 and some fraction; |