i 3 2 BOOK V. 



the first cord, and makes a note of this first side of the minor triangle 17 . 

 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 sub- 

 stitutes 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 



17 For greater clarity we have in a few places interpolated the terms " major " and 

 " minor " triangles. 



