134 BOOK V. 



depth there must be deducted half a fathom, two palms, one and a half digits 

 and the fifth part of half a digit. But if the tunnel has been driven to a 

 point where it is under the shaft, then to reach the roof of the tunnel the 

 shaft must still be sunk a depth of eleven fathoms, two and a half feet, one 

 palm, two digits, and four-fifths of half a digit. 



If a minor triangle is produced of the kind having three unequal sides, 

 then the sides of the greater triangle cannot be equal ; that is, if the first 

 side of the minor triangle is eight feet long, the second six feet long, and the 

 third five feet long, and the cord along the side of the greater triangle, not 

 to go too far from the example just given, is one hundred and one times 

 eight feet, that is, one hundred and thirty-four fathoms and four feet, the 

 distance which lies between the mouth of the tunnel and the bottom of the 

 shaft will occupy one hundred times six feet in length, that is, one hundred 

 fathoms. The distance between the mouth of the shaft and the bottom of the 

 tunnel is one hundred times five feet, that is, eighty-three fathoms and two feet. 

 And so, if the tunnel is eighty-five fathoms long, the remainder to be driven 

 into the mountain is fifteen fathoms long, and here, too, a correction in 

 measurement must be taken from the depth of the shaft and added to the 

 length of the tunnel ; what this is precisely, I will pursue no further, since 

 everyone having a small knowledge of arithmetic can work it out. If the 

 shaft is sixty-seven fathoms deep, in order that it may reach the bottom of 

 the tunnel, the further distance required to be sunk amounts to sixteen 

 fathoms and two feet. 



A TRIANGLE HAVING A RIGHT ANGLE AND THREE UNEQUAL SIDES. 



The surveyor employs this same method in measuring the mountain, 

 whether the shaft and tunnel are on one and the same vein, whether the vein 

 is vertical or inclined, or whether the shaft is on the principal vein and the tunnel 

 on a transverse vein descending vertically to the depths of the earth ; in the 

 latter case the excavation is to be made where the transverse vein cuts the 

 vertical vein. If the principal vein descends on an incline and the cross- vein 

 descends vertically, then a minor triangle is created having one obtuse angle or 

 all three angles acute. If the minor triangle has one angle obtuse and the two 

 sides which are the second and third are equal, then the second and third 

 sides of the major triangle will be equal, so that if the first side of the minor 

 triangle is nine feet, the second, and likewise the third, will be five feet. Then 

 the first side of the major triangle will be one hundred and one times nine 

 feet, or one hundred and fifty-one and one-half fathoms, and each of the 

 other sides of the major triangle will be one hundred times five feet, that is, 

 eighty-three fathoms and two feet. But when the first shaft is inclined, 



