the first cord, and makes a note of this first side of the minor triangle[1]. Afterward, starting again from that point where the third cord intersects the second cord, he measures the straight space which lies between that point and the opposite point on the first cord, and hi that way forms the minor triangle, and he notes this second side of the minor triangle in the same way as before. Then, if it is necessary, from the angle formed by the first cord and the second side of the minor triangle, he measures upward to the end of the first cord and also makes a note of this third side of the minor triangle. The third side of the minor triangle, if the shaft is vertical or inclined and is sunk on the same vein in which the tunnel is driven, will necessarily be the same length as the third cord above the point where it intersects the second cord; and so, as often as the first side of the minor triangle is contained in the length of the whole cord which descends obliquely, so many times the length of the second side of the minor triangle indicates the distance between the mouth of the tunnel and the point to which the shaft must be sunk; and similarly, so many times the length of the third side of the minor triangle gives the distance between the mouth of the shaft and the bottom of the tunnel.
When there is a level bench on the mountain slope, the surveyor first measures across this with a measuring-rod; then at the edges of this bench he sets up forked posts, and applies the principle of the triangle to the two sloping parts of the mountain; and to the fathoms which are the length of that part of the tunnel determined by the triangles, he adds the number of fathoms which are the width of the bench. But if sometimes the mountain side stands up, so that a cord cannot run down from the shaft to the mouth of the tunnel, or, on the other hand, cannot run up from the mouth of the tunnel to the shaft, and, therefore, one cannot connect them in a straight line, the surveyor, in order to fix an accurate triangle, measures the mountain; and going downward he substitutes for the first part of the cord a pole one fathom long, and for the second part a pole half a fathom long. Going upward, on the contrary, for the first part of the cord he substitutes a pole half a fathom long, and for the next part, one a whole fathom long; then where he requires to fix his triangle he adds a straight line to these angles.
To make this system of measuring clear and more explicit, I will proceed by describing each separate kind of triangle. When a shaft is vertical or inclined, and is sunk in the same vein on which the tunnel is driven, there is created, as I said, a triangle containing a right angle. Now if the minor triangle has the two sides equal, which, in accordance with the numbering used by surveyors, are the second and third sides, then the second and third sides of the major triangle will be equal; and so also the intervening distances will be equal which lie between the mouth of the tunnel and the bottom of the shaft, and which lie between the mouth of the shaft and the bottom of the tunnel. For example, if the first side of the minor triangle is seven feet long and the second and likewise the third sides are five feet, and
- ↑ For greater clarity we have in a few places interpolated the terms “major” and “minor” triangles.
