RULE. As the sum of the two given sides : their difference, :: tangent of half the sum of the opposite angles : tangent of half their difference. The half sum of the sought angles is found by taking half the given angle from 90°; and the half difference being determined by the above proportion, it will merely remain to add this half difference to the half sum to get the greater angle, and to subtract it from the half sum to get the less. The angles being thus found, the unknown side may be computed by Case I. EXAMPLES. 1. In the triangle ABC are given A = 40° 33′ 12′′, b= 153, c = 137, to find the other parts. I. To find B and C. = (B+C)=90° — §A — 90° — 20° 16′ 36′′ = 69° 43′ 24′′; therefore, 2. In the triangle ABC are given C = 32° 18′ 26′′, a= 526, b=378, to find the other angles. A=103° 10′ 21′′, B=44° 22′ 13′′. 3. Given A40° 56′, b=23·859, c=29-271, to find a. a=57.77. 4. Given B = 128° 4′, a = 960, c = 1686, to find the remaining parts. A=18° 21′ 20′′, C=33° 34′ 40′′, b=2400·36. (21.) It now merely remains for us to investigate a rule for the third or last case, where the three sides are given to determine the angles. For this purpose take the longest side AB of the triangle ABC for base: and demit upon it from the vertex the perpendicular CD, which will necessarily fall within the base. With centre C and radius CA, equal to the longer of the two sides AC, CB, describe a circle; and prolong the sides AB, BC to the circumference. Then it is plain that AB(AD — DB) = (AC + CB) (AC — CB) .. AB AC + CB :: AC hence is deduced the following rule. CB: AD - DB; E F Cuse III. When the three sides are given. RULE. As the base or longest side the sum of the other two sides, :: difference of those sides the difference of the segments of the base, made by a perpendicular from the vertical angle. Having thus the sum and difference of these segments, the greater and less segments themselves become known; the former by adding half the difference to half the sum, the latter by subtracting; and thus in each of the two right angled triangles into which the perpendicular divides the proposed triangle the hypothenuse and base will be known, and therefore the angles at the base of the proposed triangle become easily determinable. EXAMPLES. The three sides of a plane triangle are a 25, b = 34, c40. Required the angles. Base =40; sum of sides 59; difference of sides, 9; therefore 2. Given the three sides equal to 4, 5, and 6, to find the angles. 41° 24′ 35′′, 55° 46′ 16′′, and 82° 49′ 9′′. 3. Given a = 1·372, b = 6, c = 5523 to find A by logarithms. A 12° 49' 50". 4. The three sides are a 101·616, b = 153, c = 137, what are angles? A = 43° 33′ 12′′, B=78° 13′ 1′′, C = 61° 13′ 47′′ END OF THE PLANE TRIGONOMETRY. ADDITIONAL NOTES ON THE ELEMENTS OF GEOMETRY. Note (A), page 11. ON THE AXIOMS. IT has been stated upon high authority, and the statement has been recently repeated, that the science of Geometry is independent of those fundamental and indemonstrable principles on which its reasonings have been thought to rest; and an attempt has accordingly been made to establish a system of geometrical truth without axioms; that is, to form a chain that shall have no initial link. Dugald Stewart, whose opinions upon this subject have lately excited some controversy, makes a distinction-no doubt a very proper one-between the truths of Geometry and the reasonings by which they are evolved; maintaining that the former are all implied in the definitions, and that the latter are resolvable into the axioms. The accuracy of this distinction is unquestionable; but it does not justify the assertion that Geometry depends upon the definitions only, and not upon the axioms. Without the reasonings the truths could not be discovered, and, consequently, the science could not exist; and no one will argue that a thing is independent of that which is essential to its existence. That the definitions really involve all the conclusions which the reasoning brings forth is plain. When we define |