Hypogeiodie

Hypogeiodie is a mathematical art demonstrating how, under the spherical surface of the earth, at any depth, to any assigned perpendicular line (whose distance from the perpendicular of the entrance, and the azimuth likewise with respect to the said entrance, are known), a certain way may be prescribed and traveled; and how any way designed above the surface of the earth may be kept under ground at any limited depth, always going perpendicularly beneath the way designed on the earth; and, conversely, given any way (straight or crooked) under the earth, to lay it out upon the outer face or surface of the earth, so that perpendiculars drawn from the center of the earth to the spherical surface of the earth shall precisely fall in the corresponding points of those two ways. This, with all other cases and circumstances in this, and appurtenances, this art demonstrates. This art is very ample in variety of conclusions, and very profitable in sundry ways to the commonwealth. The occasion of my inventing this art was at the request of two gentlemen who had a certain work (of gain) under ground, and their grounds joined over the work; and by reason of the crookedness, diverse depths, and heights of the way under ground, they were in doubt, and at controversy, over under whose ground the work then was. The name only (before this) was published by me, On the Subterranean Journey; the rest is at God's will. For pioneers, miners, diggers for metals, stone, and coal, and for secret passages under ground between place and place (as this land has diverse), and for other purposes, any man may easily perceive both the great fruit of this art, and also in this art the great aid of geometry.

Hypogeiodie is a mathematical art. It shows how, under the spherical surface of the earth, at any depth, a set course can be laid out and traveled toward any given perpendicular line (whose distance from the perpendicular at the entrance, and its azimuth likewise with respect to that entrance, are known). It shows how any course marked out above the surface of the earth can be kept underground at any set depth, always running directly beneath the course marked on the ground above. And the other way round: given any course under the earth, straight or crooked, how to lay it out on the outer face or surface of the earth, so that perpendiculars drawn from the center of the earth to the spherical surface land exactly on the matching points of those two courses. This art demonstrates all of that, along with every other case and circumstance involved, and everything that belongs to them. The art is very rich in the variety of its conclusions, and it is useful in many ways to the commonwealth. I invented this art at the request of two gentlemen. They had a certain working underground (for gain), and their plots of ground joined above the work; and because of the twists, the different depths, and the heights of the way underground, they were in doubt, and at odds over whose ground the working then lay. Before this, I had published only the name: On the Subterranean Journey. The rest is at God's will. For laborers, miners, and diggers for metals, stone, and coal, and for secret passages underground from place to place (as this land has several), and for other purposes, anyone can easily see both the great fruit of this art and the great help geometry gives to it.