1the first
cord, and makes a note of this first side of the minor triangle17.
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 in 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.
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 in 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.
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
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
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