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130 by carelessness into a slight error, this at the end will produce great errors. Now these triangles are of many shapes, since shafts differ among themselves and are not all sunk by one and the same method into the depths of the earth, nor do the slopes of all mountains come down to the valley or plain in the same manner. For if a shaft is vertical, there is a triangle with a right angle, which the Greeks call ὀρθογώνιον and this, according to the inequalities of the mountain slope, has either two equal sides or three unequal sides. The Greeks call the former τρίγωνον ἰσοσκελές the latter σκαληνόν for a right angle triangle cannot have three equal sides. If a shaft is inclined and sunk in the same vein in which the tunnel is driven, a triangle is likewise made with a right angle, and this again, according to the various inequalities of the mountain slope, has either two equal or three unequal sides. But if a shaft is inclined and is sunk in one vein, and a tunnel is driven in another vein, then a triangle comes into existence which has either an obtuse angle or all acute angles. The former the Greeks call ἀμβλυγώνιον, the latter. That triangle which has an obtuse angle cannot have three equal sides, but in accordance with the different mountain slopes has either two equal sides or three unequal sides. That triangle which has all acute angles in accordance with the different mountain slopes has either three equal sides, which the Greeks call τρίγωνον ισόλευρον or two equal sides or three unequal sides.

The surveyor, as I said, employs his art when the owners of the mines desire to know how many fathoms of the intervening ground require to be dug; when a tunnel is being driven toward a shaft and does not yet reach it; or when the shaft has not yet been sunk to the depth of the bottom of the tunnel which is under it; or when neither the tunnel reaches to that point, nor has the shaft been sunk to it. It is of importance that miners should know how many fathoms remain from the tunnel to the shaft, or from the shaft to the tunnel, in order to calculate the expenditure; and in order that the owners of a metal-bearing mine may hasten the sinking of a shaft and the excavation of the metal, before the tunnel reaches that point and the tunnel owners excavate part of the metal by any right of their own; and on the other hand, it is important that the owners of a tunnel may similarly hasten their driving before a shaft can be sunk to the depth of a tunnel, so that they may excavate the metal to which they will have a right.

The surveyor, first of all, if the beams of the shaft-house do not give him the opportunity, sets a pair of forked posts by the sides of the shaft in such a manner that a pole may be laid across them. Next, from the pole he lets down into the shaft a cord with a weight attached to it. Then he stretches a second cord, attached to the upper end of the first cord, right down along the slope of the mountain to the bottom of the mouth of the tunnel, and fixes it to the ground. Next, from the same pole not far from the first cord, he lets down a third cord, similarly weighted, so that it may intersect the second cord, which descends obliquely. Then, starting from that point where the third cord cuts the second cord which descends obliquely to the mouth of the tunnel, he measures the second cord upward to where it reaches the end of