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to be passed over by these points, then the angles will be in proportion to the parts of the arc passed over; and if the whole circle be divided into any number of equal parts, as, for example, 360, the num ber of these comprehended between the points of the compasses will denote the magnitude of the angle. This is sufficiently clear; but there is another circumstance which beginners are not always sufficiently aware of, and which, therefore, requires to be well attended to: it is, that the angle will be neither enlarged nor diminished by any change in the length of the legs, provided their position remains unaltered; because it is the inclination of the legs, (and not their length) or the space between them, which constitutes the angle. So that if a pair of compasses, with very long legs, were opened to the same angle as another smaller pair, the intervals between their respective points would be very different; but the number of degrees on the circles, supposed to be applied to each, would be equal, because the degrees themselves on the smaller circle would be exactly proportioned to the shortness of the legs. This property renders the admeasurement of angles very easy, because the diameter of the measuring circle may be varied at pleasure, as convenience requires.

In practice, however, the magnitude of instruments is limited on each side. If they are made very large, they are difficult to manage; and their weight, bearing a high proportion to their strength, renders them liable to change their figure, by bending when their position is altered: but, on the contrary, if

they are very small, the errors of construction and graduation amount to more considerable parts of the divisions on the limbs of the instrument.

GENERAL PRINCIPLES OF CALCULATION.

Before we proceed any farther, I shall slightly notice the general principles of the calculations we are going to use.

Plane Trigonometry is the art of measuring and computing the sides of plane triangles, or of such whose sides are right lines.

In most cases of practice, it is required to find lines or angles whose actual admeasurement is difficult or impracticable. These mathematicians teach us to discover by the relation they bear to other given lines or angles, and proper methods of calculation.

Finding the comparison of one right-line with another right-line more easy than the comparison of a right-line with a curve, they measure the quantities of the angles not by the arc itself, which is described on the angular point, but by certain lines described about that point.

If any three parts of a triangle are known, the remaining unknown parts may be found either by construction or by calculation.

If two angles of a triangle are known in degrees and minutes, the third is found by subtracting their sum from 180 degrees; but if the triangle be right

angled, either angle in degrees, taken from 90 deg. gives the other angle.

Before the required side of a triangle can be found by calculation, its opposite angle must be given or found.

The required part of a triangle must be the last of four proportionals, written in order one under the other, whereof the three first terms are given or known.

Against the three first terms of the proportion, are to be written the corresponding numbers taken from tables, which have been constructed to facilitate calculation.

These tables are called logarithms; and are so contrived, that multiplication is performed by addition, and division by subtraction.

If the value, then, of the first term of your proportion be taken from the sum of the second and third, you obtain the value of the fourth, or quantity required; because the addition and subtraction of logarithms corresponds with the multiplication and division of natural numbers.

To avoid even the subtraction of the first term, when radius is not one of the proportionals, some chuse to add the arithmetical complement.

To find the arithmetical complement of a logarithm, begin at the left hand, and write down what each figure wants of 9, and what the last figure wants of 10. The number thus found is to be added to the second and third values; the sum, rejecting the borrowed index, is the tabular number expressing the

440 GENERAL PRINCIPLES OF CALCULATION.

quantity required: thus the arithmetical complement of 2.6963564 is 7.3036436.

To find the logarithm of a given number. Here you must remember that the integral part of a loga rithm is called its index, because it denotes the number of figures in the natural number answering to the logarithm. The decimal part of every logarithm belongs equally to a whole number, a mixed number, or a decimal number; that is, they are expressed by the same figures, in the same order, but the index varies according to the value of the expression. The index of a logarithm is always an unit less than the number of figures in the integer number, of which it is the logarithm.

Hence the following general rule for finding the index of a logarithm. To the left of the logarithm, write that figure or figures which expresses the dis-tance from unity, of the highest place digit in the given number, reckoning the unit's place 0, the next place 1, the next to that 2, the next to that 3, &c.

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By attending to the following example, it will easy for you to find the logarithm of a given number, and the number corresponding to a given. logarithm.

Thus, let the number be 7854. One column gives the decimal part; the next the logarithm completed with the indexes.

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Tables of logarithms are also constructed for sines, tangents, &c. of an arc: these are to be taken out from the tables, according to their respective value.

Spherical trigonometry is the science of calculating the triangles formed on the surface of a globe, by three arches of great circles: the smaller circles of a sphere are not noticed in the calculations of spherical trigonometry. This science is too intricate to be any way explained in this Essay; we must therefore content ourselves with only giving the proportions necessary to answer our purpose.*

OF THE QUADRANT, AND ITS USES.

Every circle being supposed to be divided into 360 equal parts or degrees, it is evident that 90 degrees or one-fourth part of a circle, will be sufficient to measure all angles formed between a line perpendicular to the horizon, and other lines which are not

*For one of the most familiar and best treatises on this subject, I refer the reader to "An Introduction to the Theory aud Practice of Plane and Spherical Trigonometry," Svo. by Mr. Thomas Keith, 2d. edition, 1810. EDIT.

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