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If the other angles were required, they can be found by Case 1, or by theo. 1 of this sect.
Add the three sides together, and take half the sum and the differences between the half-sum and each side: then add the complements of the logarithms of the half-sum and of the difference between the half-sum and the side opposite to the angle sought, to the logarithms of the differences of the half-sum and the other sides: half their sum will be the tangent of the angle required.
Example. In the triangle ABC, having the side AB 562, AC 800, and BC 320, to find the angle ABC.
Whose double 128° 4' is the angle ABC. Whence the other angles can be easily found by theo. 1 of this section.
the log. of the square of radius or 20, which is just equivalent to rejecting 20 from the sum of the four logarithms, which should be done, because for every arithmetical complement that is taken 10 must be rejected : but the Ar. Co. of the two sides containing the required angle is taken; consequently 20 should be rejected, which is equal to the log. of the square of radius.
* For the demonstration of this rule the reader is referred to Leslie's Geometry, prop. 12, p. 372.
263.7, and the angle A 46° 30', to find the other angles and the side BC.
Ans. The Z_C 60° 31', the Z B 72° 59', and the side BC 200.
4. In the triangle ABC, having the sides given, viz. AB= 144.8, BC=109, and AC=76, it is required to find the angles by each of the three rules given to Case 4.
Ans. The least angle 29° 49', next greater 54° 07', and the greatest 96° 04'.
Additional eacercises, with their answers.
1. Given the hypothenuse 108, and the angle opposite the perpendicular 25° 36'; required the base and perpendicular. Ans. The base is 97.4, and the perpendicular 46.66. 2. Given the base 96, and its opposite angle 71° 45'; required the perpendicular and the hypothenuse. Ans. The perpendicular is 31.66, and the hypothenuse 101.1. 3. Given the perpendicular 360, and its opposite angle 58° 20'; required the base and the hypothenuse. Ans. The base is 222, and the hypothenuse 423. 4. Given the base 720, and the hypothenuse 980; required the angles and the perpendicular. Ans. The angles are 47° 17' and 42° 43', and the perpendicular 664.8. 5. Given the perpendicular 110.3, and the hypothenuse 176.5; required the angles and the base. Ans. The angles are 38° 41' and 51° 19, and the base 137.8. 6. Given the base 360, and the perpendicular 480; required the angles and the hypothenuse. Ans. The angles are 53° 8' and 36° 52', and the hypothenuse 600. 7. Given one side, 129, an adjacent angle 56° 30', and the opposite angle 81°36'; required the third angle and the remaining sides. Ans. The third angle is 41° 54', and the remaining sides are 108.7 and 87.08. 8. Given one side 96.5, another side 59.7, and the angle opposite the latter side 31° 30'; required the remaining angles and the third side. Ans. This question is ambiguous, the given side opposite the given angle being less than the other given side (see Rule 1);
hence, if the angle opposite the side 96.5 be acute, it will be 57° 38', the remaining angle 90° 52', and the third side 114.2; but if the angle opposite the side 96.5 be obtuse, it will be 122° 22', the remaining angle 26° 8', and the third side 50.32. 9. Given one side 110, another side 102, and the contained angle 113° 36'; required the remaining angles and the third side. Ans. The remaining angles are 34° 37' and 31°47', and the third side is 177.5. 10. Given the three sides respectively 120.6, 125.5, and 146.7; required the angles. Ans. The angles are 51°53', 54° 58', and 73°9'. The student who has advanced thus far in this work with diligence and active curiosity is now prepared to study, with ease and pleasure, the following Part, which comprehends all the necessary directions for the practice of Surveying.
Containing a particular Description of the several Instruments used in Surveying, with their respective Uses.
THE stationary distance, or merings of ground, are measured either by Gunter's chain of four poles or perches, which consists of 100 links (and this is the most natural division), or by one of 50 links, which contains two poles or perches: but because the length of a perch differs in many places, therefore the length of chains and their respective links will differ also.
The English statute-perch is 54 yards, the two-pole chain is 11 yards, and the four-pole one is 22 yards; hence the length of a link in a statute-chain is 7.92 inches.
For the more ready reckoning the links of a four-pole chain, there is a large ring, or sometimes a round piece of brass, fixed at every 10 links; and at 50 links, or in the middle, there are
two large rings. In such chains as have a brass piece at every 10 links, there is the figure 1 on the first piece, 2 on the second, 3 on the third, &c. to 9. By leading therefore that end of the chain forward which has the least number next to it, he who carries the hinder end may easily determine any number of links: thus, if he has the brass piece number 8 next to him, and six links more in a distance, that distance is 86 links. After the same manner 10 may be counted for every large ring of a chain which has not brass pieces on it; and the number of links is thus readily determined. The two-pole chain has a large ring at every 10 links, and in its middle, or at 25 links, there are two large rings; so that any number of links may be the more readily counted off, as before. The surveyor should be careful to have his chain measured before he proceeds on business; for the rings are apt to open by frequently using it, and its length is thereby increased, so that no one can be too circumspect in this point. In measuring a stationary distance, there is an object fixed in the extreme point of the line to be measured; this is a direction for the hinder chainman to govern the foremost one by, in order that the distance may be measured in a right line; for if the hinder chainman causes the other to cover the object, it is plain the foremost is then in a right line towards it. For this reason it is necessary to have a person that can be relied on at the hinder end of the chain, in order to keep the foremost man in a right line; and a surveyor who has no such person should chain himself. The inaccuracies of most surveys arise from bad chaining, that is, from straying out of the right line, as well as from other omissions of the hinder chainman : no person, therefore, should be admitted at the hinder end of the chain of whose abilities, in this respect, the surveyor is not previously convinced; since the success of the survey, in a great measure, depends on his care and skill. In setting out to measure any stationary distance, the fore man of the chain carries with him ten iron pegs pointed, each about ten inches long; and when he has stretched the chain to its full length, he at the extremity thereof sticks one of those pegs perpendicularly in the ground; and leaving it there, he draws on the chain till the hinder man checks him when he arrives at that peg: the chain being again stretched, the fore man sticks down another peg, and the hind man takes up the former; and thus they proceed at every chain's length contained in the line to be measured, counting the surplus links contained between the last peg and the object at the termination of the line, as before : so that the number of pegs taken
up by the hinder chainman expresses the number of chains: to which, if the odd links be annexed, the distance line required in chains and links is obtained, which must be registered in the field-book, as will hereafter be shown. If the distance exceeds 10, 20, 30, &c. chains, when the leader's pegs are all exhausted, the hinder chainman, at the extremity of the 10 chains, delivers him all the pegs; from whence they proceed to measure as before, till the leader's pegs are again exhausted, and the hinder chainman at the extremity of these 10 chains again delivers him the pegs, from whence. they proceed to measure the whole distance line in the like manner; then it is plain, that the number of pegs the hinder chainman has being added to 10, if he had delivered all the pegs once to the leader, or to 20 if twice, or to 30 if thrice, &c., will give the number of chains in that distance; to which if the surplus links be added, the length of the stationary distance is known in chains and links. It is customary, and indeed necessary, to have red, or other coloured cloth fixed to the top of each peg, that the hinder man at the chain may the more readily find them; otherwise, in chaining through corn, high grass, briers, rushes, &c. it would be extremely difficult to find the pegs which the leader puts down: by this means no time is lost, which otherwise must be, if no cloths are fixed to the pegs, as before. It will be necessary here to observe, that all slant, or inclined surfaces, as sides of hills, are measured horizontally, and not on the plane or surface of the hill, and is thus effected.
PL. 8. fig. 4.
Let ABC be a hill; the hindmost chainman is to hold the end of the chain perpendicularly over the point A (which he can the better effect with a plummet and line, than by letting a stone drop, which is most usual), as d is over A, while the leader puts down his peg at e : the eye can direct the horizontal position near enough; but if greater accuracy were required, a quadrant applied to the chain would settle that. In the same manner the rest may be chained up and down; but in going down, it is plain the leader of the chain must hold up the end thereof, and the plummet thence suspended will mark the point where he is to stick his peg. The figure is sufficient to render the whole evident, and to show that the sum of the chains will be the horizontal measure of the base of the hill: for de=Ao, fg= op, hi-pg, &c.; therefore de-Hfg+hi, &c.–Ao-Hop-Hpg, &c.