BOOK V. 



135 



generally speaking, it is not deep ; but there are usually several, all 

 inclined, and one always following the other. Therefore, if a tunnel is seventy- 

 seven fathoms long, it will reach to the middle of the bottom of a shaft when 

 six fathoms and two feet further have been sunk. But if all such inclined 

 shafts are seventy-six fathoms deep, in order that the last one may reach 

 the bottom of the tunnel, a depth of seven fathoms and two feet remains to 

 be sunk. 



TRIANGLE HAVING AN OBTUSE ANGLE AND TWO EQUAL SIDES. 



If a minor triangle is made which has an obtuse angle and three unequal 

 sides, then again the sides of the large triangle cannot be equal. For 

 example, if the first side of the minor triangle is six feet long, the second 

 three feet, and the third four feet, and the cord along the side of the greater 

 triangle one hundred and one times six feet, that is, one hundred and one 

 fathoms, the distance between the mouth of the tunnel and the bottom of 

 the last shaft will be a length one hundred times three feet, or fifty fathoms ; 

 but the depth that lies between the mouth of the first shaft and the bottom of 

 the tunnel is one hundred times four feet, or sixty-six fathoms and four feet. 

 Therefore, if a tunnel is forty-four fathoms long, the remaining distance to 

 be driven is six fathoms. If the shafts are fifty-eight fathoms deep, the 

 newest will touch the bottom of the tunnel when eight fathoms and four 

 feet have been sunk. 



TRIANGLE HAVING AN OBTUSE ANGLE AND THREE UNEQUAL SIDES. 



If a minor triangle is produced which has all its angles acute and its 

 three sides equal, then necessarily the second and third sides of the minor 

 triangle will be equal, and likewise the sides of the major triangle frequently 

 referred to will be equal. Thus if each side of the minor triangle is six feet 

 long, and the cord measurement for the side of the major triangle is one 

 hundred and one times six feet, that is, one hundred and one fathoms, then 

 both the distances to be dug will be one hundred fathoms. And thus if the 

 tunnel is ninety fathoms long, it will reach the middle of the bottom of the 

 last shaft when ten fathoms further have been driven. If the shafts are 



