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It may always be known whether the angle sought is greater or less than a right angle; for if the square of the side opposite to it be greater than the sum of the squares of the other two sides, it is an obtuse angle (12. 2); but if less, it is an acute angle (13. 2). Hence this case is not ambiguous.
75. Remark. The angle C, in case 2, may have two values, one greater, the other less than a right angle. For an angle and its supplement having the same sine (34), the angle belonging to the sine of the angle C found by the analogy, may be either that which is found in the tables, or the supplement of it.
This ambiguity does not arise from any defect in the solution of the problem, but from a circumstance essential to the problem; for when AC, the side opposite to the given angle B, is less than the other given side AB, there are two triangles, each of which has the sides AB, AC, and the angle B of the same magnitude; but yet these two triangles are not equal, the angle opposite to AB in one triangle being the supplement of the angle opposite to AB in the other. The truth of this appears as follows.
From the centre A, with the radius AC, describe an arc cutting BC in c, and join A, c. Then ACc is an isosceles triangle, and therefore the angles ACc and AcC are equal. Therefore the angle AcB, which is the supplement of AcC, is also the supplement of ACB. Now ACB, AcB are the angles found by the analogy in the solution. But the sine found by the analogy may be the sine of either of the angles ACB, AcB (34); and either of these angles, one of which will be acute, and the other obtuse, may be the angle opposite to AB in the proposed triangle.
Again, the triangles ABC, ABc have the side AB and the angle B common, and the sides AC, Ac equal; but they have not the remaining sides equal, and the remaining angles equal, namely, BC=Bc, the angle B c A=BCA, and BAc=BAC. Consequently the triangles ABC, ABc are not equal.
Thus, when the given side AB is greater than AC, and consequently the angle C greater than B, there are two triangles which satisfy the conditions of the problem. But when AC is greater than AB, the intersections C and c fall on opposite sides of the angle B, and therefore the two triangles have
not the angle B common to both. In this case the angle required being necessarily less than B is an acute angle, and therefore the solution ceases to be ambiguous.
From the two angles ACB, AcB the two angles BAC, BAC may be found. The angle BAC is the supplement of the sum of the angles B and C (32. 1), therefore the sine of BAC is the same as the sine of B+C. The angle B Ac AcC—B (32. 1)=ACc-B. Hence, the angle C being found, sine C: s. (C+B): AB: BC, and s. C: s. (C-B) :: AB: BC.
76. If the three angles of an oblique triangle be denoted by A, B, C, and their opposite sides by a, b, c, then propositions IV, V, VI, VII, may be expressed by general equations, by means of which all the cases of plane triangles may be resolved.
or log. tan. (A ~ B)=log. (a ~ b)—log. (a+b)+log. cot. C.
3. BD DC=
or log. (BD ~ DC)=log. (a+b)+log. (a ~ b)—log. c.
In this equation, when the angle A is acute, b2+c2 will be greater than a2; and when A is obtuse, b2+c2 will be less than a2. If any three of these four parts (beside radius) be given, the fourth part may be found from these equations.
Observations preparatory to the Practical Solution of the Cases of Plane Triangles.
77. Logarithms are a set of artificial numbers adapted to the common or natural numbers, 1, 2, 3, 4, 5, &c. for the purpose of facilitating arithmetical calculations. The addition and subtraction of logarithms correspond to the multiplication and division of common numbers. Therefore logarithms are generally used in trigonometrical and astronomical calculations, to avoid the tedious operations of multiplication and division in finding a fourth proportional to three given numbers.
78. Before the learner attempts the solution of the following cases of triangles he ought to know decimal fractions, practical geometry, and the use of the tables of the logarithms of
numbers, and of sines and tangents. The construction and use of these tables are generally prefixed to them in the collections of mathematical tables, of which the best are those of Taylor, 4to, Callet, 8vo, Hutton, 8vo, Mackay, 8vo, and the tables commonly called the Requisite Tables, 8vo.*
79. The sines, tangents, &c. are also called the natural sines, tangents, &c. of the arcs or angles to which they belong; and the logarithms of the numbers by which they are represented, are called the logarithmic sines, tangents, &c. As one or other of these lines makes a part of every trigonometrical operation, they have been calculated to a given radius, for every degree, minute, and (sometimes) second of the quadrant, and ranged in tables for practical use. Hence, by means of such tables, the sine, tangent, &c. of any arc or angle may be found by inspection; and, on the contrary, the arc or angle to which any sine, tangent, &c. belongs, may also be found by inspection.
80. Upon a table of sines and tangents, and the doctrine of similar triangles, depends the practical solution of the several cases of plane trigonometry, which may be performed either by the natural or the logarithmic sines, tangents, &c., as occasion requires. The logarithmic sines and tangents are commonly used, because the calculations by them are performed by addition and subtraction; but the natural sines and tangents require the more tedious operations of multiplication and division.
81. Each of the cases of rectilinear triangles admits three different methods of solution. 1. By geometrical construction. 2. By arithmetical calculation. 3. Instrumentally.
In the first method, the triangle is constructed by laying down the sides from a scale of equal parts, and the angles from a scale of chords, or a protractor. The unknown parts of the triangle thus constructed are found by measuring them on the same scale or instrument from which the known parts were taken.
In the second method let the analogy be formed according to the proper rule above delivered; then, if the natural numbers be used, multiply the second and third terms together, and divide the product by the first; the quotient will be the fourth term required. If logarithms be used, add the logarithms of the second and third terms, and from the sum sub
* F. Nichols proposes to reprint De La Lande's Stereotype Tables of the Logarithms of Numbers, Sines and Tangents, with the addition of other useful tables, in one volume 18mo. Designed chiefly for the use of students in the Universities.
tract the logarithm of the first; the remainder will be the logarithm of the fourth term; and the number answering to that logarithm in the tables will be the number sought.
In the third method, where the rule commonly called Gunter's Scale is used, form the analogy, as in the last method; then extend the compasses, on the logarithmic lines described on one side of the scale, from the first term to the second or third, as they happen to be of the same name; that extent will reach from the other term to the fourth term required, both extents being directed toward the same end of the scale.
The second method, in which the operation is performed by logarithms, is generally practised. The other two methods are chiefly of use as checks on the arithmetical calculations, or in certain simple cases, where approximate values of the quantities sought are deemed sufficient; but must not be applied when accuracy is required.
82. In any operation, when one or more logarithms are to be subtracted from the others, it will be sometimes more convenient to take their complements (or what each logarithm wants of 10.0000000) instead of the logarithms, and then to add all the logarithmic terms together, and to subtract from the index of their sum as many times 10 as there were logarithms to be subtracted. Thus, if the log. to be subtracted be 3.4932758, its complement 6.5067242 may be added; and if the log. to be subtracted be 9.07432600, its complement 0.92567400 may be
If the index of the logarithm, whose complement is to be taken, be greater than 10, subtract the logarithm from 20, and the remainder will be its complement. Thus, the complement of the log. 12.4907327 is 7.5092673. If this complement be added to another logarithm, 20 must be subtracted from the index of the sum.
If the logarithm of a decimal is to be subtracted, add 10 to the index (which is negative), and take the complement of the rest of the figures, as before. Thus, the complement of the log. 8.5972648 is 2.4027352.
83. In trigonometrical calculations the sine, cosine, tangent, &c. of the same angle often occur. Therefore let all the analogies be formed (if there be more than one), before you begin the numerical operations, and leave the terms blank, that they may be afterward filled up with their proper logarithms. You will thus perceive what angles are repeated in the analogies, and therefore can, at one opening of the tables, take out all the logarithms belonging to the same angle, and write them in
their proper places. By this means you will prevent the loss of time and labour, which would happen by taking out the logarithms of the angles separately.
84. In the subsequent calculations the angles are mostly found to the nearest minute, as they appear in the common tables of logarithms; which is sufficient for almost all purposes. The seconds, if required, may be found by the rules prefixed to tables of logarithms. In Taylor's Tables the sines, &c. extend to seconds through the quadrant; and degrees, minutes, and seconds can be found at once by inspection. See Appendix.
85. Practical Solution of the Cases of Right-angled Triangles. Fig. page 21.
Case 1. In any right-angled triangle ABC, given the hypothenuse AC 324, and the angle at the base BAC 48° 17', to find the other parts of the triangle.
Draw an indefinite line AC. From the point A, with the chord of 60° as radius, describe an arc, upon which lay off the angle A=48° 17', taken from the same line of chords. From some convenient scale of equal parts lay down AC=324. From A, through the extremity of the arc which measures the given angle, draw an indefinite line AB; and from C let fall a perpendicular CB upon AB. Measure the sides AB, BC on the same. scale from which AC was taken, and they will be found to be 215 and 242 nearly.
The angle C, being the complement of A, is 90°— 48° 17' 41° 43'.
* The learner must construct the following problems on paper, according to the directions given for the construction of each. To save expense, the actual construction of the figures is omitted in this work; but the rules for constructing them will be sufficiently illustrated by inspecting the triangles in page 18 and 21. The construction of most of the examples is so plain and simple, that it is not necessary to give particular rules for all the cases of rectilinear triangles.