THEOREMS.
A right-angled triangle (the only kind generally necessary to be treated of for mining purposes) is that which has one right angle in it. The longest side, or that opposite to the right angle, is called the hypothenuse; the other two are called the legs or sides, dr the base and perpendicular: or, by Euclid's definition, "In a right-angled triangle, the side opposite to the right angle is called the nYPoTIIENUSE; and of the other sides, that upon which the figure is supposed to stand is called the BASE, and the remaining side the PERPENDICULAR." The three angles of every triangle are together equal to two right angles, or 180 degrees.
The greater side of every triangle has the greater angle opposite to it.
The squares of two sides of a triangle are together double the square of half the base, and of the square of a straight line drawn from the vertex to bisect the base.
The sum of the three angles of every plane triangle being equal to half a circle, or 180 degrees, it therefore follows that if either acute angle, in such triangle, be taken from 90°, the remainder will be the other acute angle, or the complement.
The supplement of any angle is what that angle wants of 180°: hence the supple ment of any one angle is always equal to the sum of the other two.
A few other properties of right-angled triangles may be worthy of notice, viz.: when the angle opposite the base is 30°, the hypothenuse is exactly double the length of the base.
When the angles are 45°, the base and perpendicular are equal.
When the angle opposite the base is 60°, the hypothenuse is double the length of the perpendicular.
To show how a knowledge of the foregoing theorems may be rendered useful in mining practices, suppose in the triangle A B C, figure 153, the base B A represented a drift or cross-cut, and the side A C a seam, making an angle with the base of 66° 30': consequently, the angle A must be 23° 30', because it requires that number of degrees to constitute a right angle, the complement of the angle A, or 180°, the supplement of the triangle A B C.
Again, suppose the angle C of the slope C A, figure 154, were found to be 39° 30': then the opposite angle A must contain 50° 30'.
We now approach towards the actual use of the tables, and have succeeded, we hope, in clearing all impediments out of the learner's way, so that he will find no difficulty in readily applying the numbers to dialling operations. We have previously set a few examples of the mere act of taking out the primes, and have studiously endeavored to render every thing as perspicuous and comprehensible as the nature of the work would possibly admit. But should any one have gone thus far and still find an obscurity hang over him, so that he cannot penetrate into the nature of the subject as he would wish or as he may have expected, yet let him not be discouraged: this will always be the case with every one who calculates on fully comprehending any thing connected with the mathematics by definition or description only. Let him steadily, attentively, and per severingly proceed with the examples, and, if he is properly interested in the matter, he will soon find the subject open with perspicuity and demonstration on his mind, and convey to him the incontrovertible assurance of the truth of the calculations, as well as the correctness of his own views, ideas, or conceptions of the subject.