Page:De re metallica (1912).djvu/175

Rh the length shown by the cord for the side of the major triangle is 101 times seven feet, that is 117 fathoms and five feet, then the intervening space, of course, whether the whole of it has been already driven through or has yet to be driven, will be one hundred times five feet, which makes eighty-three fathoms and two feet. Anyone with this example of proportions will be able to construct the major and minor triangles in the same way as I have done, if there be the necessary upright posts and cross-beams. When a shaft is vertical the triangle is absolutely upright; when it is inclined and is sunk on the same vein in which the tunnel is driven, it is inclined toward one side.

Therefore, if a tunnel has been driven into the mountain for sixty fathoms, there remains a space of ground to be penetrated twenty-three fathoms and two feet long; for five feet of the second side of the major triangle, which lies above the mouth of the shaft and corresponds with the first side of the minor triangle, must not be added. Therefore, if the shaft has been sunk in the middle of the head meer, a tunnel sixty fathoms long will reach to the boundary of the meer only when the tunnel has been extended a further two fathoms and two feet; but if the shaft is located in the middle of an ordinary meer, then the boundary will be reached when the tunnel has been driven a further length of nine fathoms and two feet. Since a tunnel, for every one hundred fathoms of length, rises in grade one fathom, or at all events, ought to rise as it proceeds toward the shaft, one more fathom must always be taken from the depth allowed to the shaft, and one added to the length allowed to the tunnel. Proportionately, because a tunnel fifty fathoms long is raised half a fathom, this amount must be taken from the depth of the shaft and added to the length of the tunnel. In the same way if a tunnel is one hundred or fifty fathoms shorter or longer, the same proportion also must be taken from the depth of the one and added to the length of the other. For this reason, in the case mentioned above, half a fathom and a little more must be added to the distance to be driven through, so that there remain twenty-three fathoms, five feet, two palms, one and a half digits and a fifth of a digit; that is, if even the minutest proportions are carried out; and surveyors do not neglect these without good cause. Similarly, if the shaft is seventy fathoms deep, in order that it may reach to the bottom of the tunnel, it still must be sunk a further depth of thirteen fathoms and two feet, or rather twelve fathoms and a half, one foot, two digits, and four-fifths of half a digit. And in this instance five feet must be deducted from the reckoning, because these five feet complete the third side of the minor triangle, which is above the mouth of the shaft, and from its