it has been found advantageous to measure the quantities of angles, not by the arc itself, which is de-, scribed on the angular point, but by certain lines described about that arc. If any three parts of a plain triangle be given, any required part may be found both by construction and calculation.* If two angles of a plain triangle are known in degrees, minutes, &c. the third angle is found by subtracting their sum from 180 degrees. In a right-angled plain triangle, if either acute angle (in degrees) be taken from 90 degrees, the remainder will express the other acute angle. When the sine of an obtuse angle is required, subtract such obtuse angle from 180 degrees, and take the sine of the remainder, or supplement. If two sides of a triangle are equal, a line bisecting the contained angle, will be perpendicular to the remaining side, and divide it equally. Before the required side of a triangle can be found by calculation, its opposite angle must first be given, or found. The required part of a triangle must be the last term of four proportionals, written in order under one another, whereof the three first are given or known. In four proportional quantities, either of them may be made the last term; thus, let A, B, C, D, be proportional quantities. As first to second, so is third to fourth, A:B::C:D. As second to first, so is fourth to third, B:A:: D:C. As third to fourth, so is first to second, C:D::A:B. As fourth to third, so is second to first, D:C:: B:A. Against the three first terms of every proportion, or stating, must be written their respective values taken from the proper tables. *This is imperfectly stated by several writers. One of the given parts must be a side. A triangle consists of six parts, viz. three sides and three angles. EpIT. If the value of the first term be taken from the sum of the second and third, the remainder will be the value of the fourth term or thing required; because the addition and subtraction of logarithms corresponds with the multiplication and division of natural numbers. If to the complement of the first value, be added the second and third values, the sum rejecting the borrowed index, will be the tabular number expressing the thing required; this method is generally used when radius is not one of the proportionals. The complement of any logarithm, sine, or tangent, in the common table, is its difference from the radius 10.000.000, or its double 20.000.000. CANONS FOR TRIGONOMETRICAL CALCULATION. 1. The following proportion is to be used when two angles of a triangle, and a side opposite to one of them, is given to find the other side. As the sine of the angle opposite the given side,. is to the sine of the angle opposite the required side so is the given side to the required side. 2. When two sides and an angle opposite to one of them is given, to find another angle; use the following rule: As the side opposite the given angle, is to the side opposite the required angle; so is the sine of the given angle, to the sine of the required angle. The memory will be assisted in the foregoing cases, by observing that when a side is wanted, the proportion must begin with an angle; and when an angle is wanted, it must begin with a side. 3. When two sides of a triangle and the included angle are given, to find the other angles and side. As the sum of the two given sides is to their difference; so is the tangent of half the sum of the two unknown angles, to the tangent of half their dif ference. Half the difference thus found, added to half their sum, gives the greater of the two angles, which is the angle opposite the greatest side. If the third side is wanted, it may be found by solution 1. 4. The following steps and proportions are to be used when the three sides of a triangle are given, and the angles required. Let A B C, plate 9, fig. 30, be the triangle; make the longest side AB the base; from C the angle opposite to the base, let fall the perpendicular CD on AB, this will divide the base into two segments AD, BD. The difference between the two segments is found by the following proportion: As the base AB, or sum of the two segments, is to the sum of the other sides (A C+ BC); so is the difference of the sides (A C B C), to the difference of the segments of the base (A D D B). Half the difference of the segments thus found, added to the half of AB, gives the greater segment AD, or subtracted, leaves the less D B. In the triangle ADC, find the angle AC D, by solution 2; for the two sides AD, and AC, are known, and the right angle at D is opposite to one of them. The complement of A CD, gives the angle A. Then in the triangle AB C, you have two sides AB, BC, and the angle at A, opposite to one of them, to find the angles C, and B. OF THE LOGARITHMIC SCALES ON THE SECTOR. There are three of these lines usually put on the sector, they are often termed the Gunter's lines, and are made use of for readily working proportions; when used, the sector is to be quite opened like a straight rule. -If the 1 at the beginning of the scale, or at the left hand of the first interval, be taken for unity, then the 1 in the middle, or that which is at the end of the first interval and beginning of the second will express the number 10; and the ten at the end of the righthand of the second interval or end of the scale, will represent the number 100. If the first is 10, the middle is 100, and the last 1000; the primary and intermediate divisions in each interval, are to be estimated according to the value set on their extremities. In working proportions with these lines, attention must be paid to the terms, whether arithmetical or trigonometrical, that the first and third term may be of the same name, and the second and fourth of the same name. To work a proportion, take the extent on its proper line, from the first term to the third in your compasses, and applying one point of the compasses to the second, the other applied to the right or left, according as the fourth term is to be more or less than the second, will reach to the fourth. Example 1. If 4 yards of cloth cost 18 shillings, what will 32 yards cost? This is solved by the line of numbers; take in your compasses the distance between 4 and 32, then apply one foot thereof on the same line at 18, and the other will reach 144, the shillings required. Example 2. As radius to the hypothenuse 120, so is the sine of the angle opposite the base 30° 17' to the base. In this example, radius, or the sine of 90, and the sine of 30° 17′ taken from the lines of sines, and one foot being then applied to 120 on the line of numbers, and the other foot on the left will reach to 604 the length of the required base. The foot was applied to the left, because the legs of a right-angled triangle are less than the hypotheunse. Example 3. As the cosine of the latitude 51° 30', (equal the sine of 38° 30') is to radius, so is the sine of the sun's declination 20° 14', to the sine of the sun's amplitude. Take the distance between the sines of 38° 30′ and 20° 14′ in your compasses; set one foot on the radius, or sine of 90°, and the other will reach to 330°, the sun's amplitude required. CURIOUS AND USEFUL TRIGONOMETRICAL The following problems, though of the greatest use, and sometimes of absolute necessity to the surveyor, are not to be found in any of the common treatises on surveying. The maritime surveyor can scarce proceed without the knowledge of them; nor can a kingdom, province, or county be accurately surveyed, unless the surveyor is well acquainted with the use and application of them. Indeed no man should attempt to survey a county, or a sea coast, who is not master of these problems. The second problem, which is peculiarly useful for determining the exact situation of sands, or rocks, within sight of three places upon land, whose distances are well known, was first proposed by Mr. Townly, and solved by Mr. Collins, Philosophical Transactions, No. 69. There is "no problem more useful in surveying, than that by which we find a station, by observed angles of three or more objects, whose reci procal distances are known: but distance and bearing from the place of observation are unknown. "Previous to the resolution of these problems, another problem for the easy finding the segment of a circle capable of containing a given angle, is necessary, as will be clear from the following observation. "Two objects can only be scen under the same angle, from some part of a circle passing throngh those objects, and the place of observation. "If the angle under which those objects appear, be less than 90°, the place of observation will be somewhere in the greater segment, and those objects will |