[Illustration 137 (Surveying Triangle): A triangle having all its angles acute and its three sides unequal.]
If a minor triangle is produced which has all its angles acute, but its three sides unequal, then again the distances to be dug cannot be equal. For example, if the first side of the minor triangle is seven feet long, the second side is four feet, and the third side is six feet, and the cord measurement for the side of the major triangle is one hundred and one times seven feet or one hundred and seventeen fathoms and four feet, the distance between the mouth of the tunnel and the bottom of the last shaft will be four hundred feet or sixty-six fathoms, and the depth between the mouth of the first shaft and the bottom of the tunnel will be one hundred fathoms. Therefore, if a tunnel is fifty fathoms long, it will reach the middle of the bottom of the newest shaft when it has been driven sixteen fathoms and four feet further. But if the shafts are then ninety-two fathoms deep, the last shaft will reach the bottom of the tunnel when it has been sunk a further eight fathoms.
This is the method of the surveyor in measuring the mountain, if the principal vein descends inclined into the depths of the earth or the transverse vein is vertical. But if they are both inclined, the surveyor uses the same method, or he measures the slope of the mountain separately from the slope of the shaft. Next, if a transverse vein in which a tunnel is driven does not cut the principal vein in that spot where the shaft is sunk, then it is necessary for the starting point of the survey to be in the other shaft in which the transverse vein cuts the principal vein. But if there be no shaft on that spot where the outcrop of the transverse vein cuts the outcrop of the principal vein, then the surface of the ground which lies between the shafts must be measured, or that between the shaft and the place where the outcrop of the one vein intersects the outcrop of the other.
[Illustration 138 (Hemicycle): A--Waxed semicircle of the hemicycle. B--Semicircular lines. C--Straight lines. D--Line measuring the half. E--Line measuring the whole. F--Tongue.]
[Illustration 138A (Surveying Rods): A--Lines of the rod which separate minor spaces. B--Lines of the rod which separate major spaces.]