Archimedes Project

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deduced from the notion of the earth's being an extended Plane.

This principle, though notoriously false, yet places being laid down accordingly, and a long voyage broken into many short ones, the voyage may be performed tolerably well by it, especially near the same meridian.

In P<*>ain Sailing it is supposed that these three, the rhumb line, the meridian, and parallel of latitude, will always form a right-angled triangle; and so posited, as that the perpendicular side will represent part of the meridian, or north and south line, containing the difference of latitude; the base of the triangle, the departure, or east-and west line; and the hypothenuse the distance sailed. The angle at the vertex is the course; and the angle at the base, the complement of the course; any two of which, besides the right angle, being given, the triangle may be protracted, and the other three parts found.

For the doctrine of Plane Sailing, see Sailing.

Plane Scale, is a thin ruler, upon which are graduated the lines of chords, sines, tangents, secants, leagues, rhumbs, &c; being of great use in most parts of the mathematics, but especially in navigation. See its description and use under Scale.

Plane Table, an instrument much used in landsurveying; by which the draught, or plan, is taken upon the spot, as the survey or measurement goes on, without any future protraction, or plotting.

This instrument consists of a Plane rectangular board, of any convenient size, the centre of which, when used, is fixed by means of screws to a three-legged stand, having a ball and socket, or universal joint, at the top, by means of which, when the legs are fixed on the ground, the table is inclined in any direction. To the table belongs,

1. A frame of wood, made to fit round its edges, for the purpose of fixing a sheet of paper upon the table. The one side of this frame is usually divided into equal parts, by which to draw lines across the table, parallel or perpendicular to the sides; and the other side of the frame is divided into 360 degrees, from a centre which is in the middle of the table; by means of which the table is to be used as a theodolite, &c.

2. A magnetic needle and compass screwed into the side of the table, to point out directions and be a check upon the sights.

3. An index, which is a brass two<*>foot scale, either with a small telescope, or open sights erected perpendicularly upon the ends. These sights and the fiducial edge of the index are parallel, or in the same Plane. General Use of the Plane Table.

To use this instrument properly, take a sheet of writing or drawing paper, and wet it to make it expand; then spread it flat upon the table, pressing down the frame upon the edges, to stretch it, and keep it fixed there; and when the paper is become dry, it will, by shrinking again, stretch itself smooth and flat from any cramps or unevenness. Upon this paper is to be drawn the plan or form of the thing measured.

The general use of this instrument, in land-surveying, is to begin by setting up the table at any part of the ground you think the most proper, and make a point upon a convenient part of the paper or table, to repre- sent that point of the ground; then fix in that point of the paper one leg of the compasses, or a fine steel pin, and apply to it the fiducial edge of the index, moving it round the table, close by the pin, till through the sights you perceive some point desired, or remarkable object, as the corner of a field, or a picket set up, &c; and from the station point draw a dry or obscure line along the fiducial edge of the index. Then turn the index to another object, and draw a line on the paper towards it. Do the same by another; and so on till as many objects are set as may be thought necessary. Then measure from your station towards as many of the objects as may be necessary, and no more, taking the requisite offsets to corners or crooks in the hedges, &c; laying the measured distances, from a proper scale, down upon the respective lines on the paper. Then move the table to any of the proper places measured to, for a second station, fixing it there in the original position, turning it about its centre for that purpose, both till the magnetic needle point to the same degree of the compass as at first, and also by laying the fiducial edge of the index along the line between the two stations, and turning the table till through the index the former station can be seen; and then fix the table there: from this new station repeat the same operations as at the former; setting several objects, that is, drawing lines towards them, on the paper, by the edge of the index, measuring and laying osf the distances. And thus proceed from station to station; measuring only such lines as are necessary, and determining as many as you can by intersecting lines of direction drawn from different stations.

Of Shifting the Paper on the Plane Table. When one paper is full of the lines &c measured, and the survey is not yet completed; draw a line in any manner through the farthest point of the last station line to which the work can be conveniently laid down; then take the sheet off the table, and fix another fair sheet in its place, drawing a line upon it, in a part of it the most convenient for the rest of the work, to represent the line drawn at the end of the work on the former paper. Then fold or cut the old sheet by the line drawn upon it; apply it so to the line on the new sheet, and, as they lie together in that position, continue or produce the last station line of the old sheet upon the new one; and place upon it the remainder of the measurement of that line, beginning at where the work left off on the old sheet. And so on, from one sheet to another, till the whole work is completed.

But it is to be noted, that if the said joining lines, upon the old and new sheet, have not the same inclination to the side of the table, the needle will not respect or point to the original degree of the compass, when the table is rectified. But if the needle be required to respect still the same degree of the compass, the easiest way then of drawing the lines in the same position, is to draw them both parallel to the same sides of the table, by means of the equal parallel divisions marked on the other two sides of the frame.

When the work of surveying is done, and you would fasten all the sheets together into one piece, or rough plan, the aforesaid lines are to be accurately joined together, in the same manner as when the lines were transferred from the old sheets to the new ones.

See more full directions for the use of the Plane