[Illustration 135b (Surveying Triangle): Triangle having an obtuse angle and three unequal 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.
[Illustration 136a (Surveying Triangle): A triangle having all its angles acute and its three sides equal.]
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 ninety-five fathoms deep, the last will reach the bottom of the tunnel when it is sunk a further depth of five fathoms.
[Illustration 136b (Surveying Triangle): Triangle having all its angles acute and two sides equal, A, B, unequal side C.]
If a triangle is made which has all its angles acute, but only two sides equal, namely, the first and third, then the second and third sides are not equal; therefore the distances to be dug cannot be equal. For example, if the first side of the minor triangle is six feet long, and the second is four feet, and the third is six feet, 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 the distance between the mouth of the tunnel and the bottom of the last shaft will be sixty-six fathoms and four feet. But the distance from the mouth of the first shaft to the bottom of the tunnel is one hundred fathoms. So if the tunnel is sixty fathoms long, the remaining distance to be driven into the mountain is six fathoms and four feet. If the shaft is ninety-seven fathoms deep, the last one will reach the bottom of the tunnel when a further depth of three fathoms has been sunk.