BOOK V. 133 



the length shown by the cord for the side of the major triangle is loi 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 iive 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 incUned toward one side. 



B 



A 



A TRIANGLE HAVING A RIGHT ANGLE AND TWO EQUAL SIDES. 



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 propor- 

 tion 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 Httle 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 



