PLANE

, or Plain, in Geometry, denotes a Plane figure, or a surface lying evenly between its bounding lines. Euclid.

Some desine a Plane, a surface, from every point of whose perimeter a right line may be drawn to every other point in the same, and always coinciding with it.

As the right line is the shortest extent from one point to another, so is a Plane the shortest extension between one line and another.

Planes are much used in Astronomy, conic sections, spherics, &c, for imaginary surfaces, supposed to cut and pass through solid bodies.

When a Plane cuts a cone parallel to one side, it makes a parabola; when it cuts the cone obliquely, an cllipse or hyperbola; and when parallel to its base, a circle. Every section of a sphere is a circle.

The sphere is wholly explained by Planes, conceived to cut the celestial bodies, and to fill the areas or circumferences of the orbits. They are differently inclined to each other; and by us the inhabitants of the earth, the Plane of whose orbit is the Plane of the ecliptic, their inclination is estimated with regard to this Plane.

Plane of a Dial, is the surface on which a dial is supposed to be described.

, in Mechanics. A Horizontal Plane, is a Plane that is level, or parallel to the horizon.

Inclined Plane, is one that makes an oblique angle with a horizontal Plane.

The doctrine of the motion of bodies on Inclined Planes, makes a very considerable article in mechanics, and has been fully explained under the articles, MECHANICAL Powers, and Inclined Plane.

Plane of Gravity, or Gravitation, is a Plane supposed to pass through the centre of gravity of the body, and in the direction of its tendency; that is, perpendicular to the horizon.

Plane of Reflection, in Catoptrics, is a Plane which passes through the point of reslection; and is perpendicular to the Plane of the glass, or reflecting body.

Plane of Refraction, is a Plane passing through the incident and refracted ray.

Perspective Plane, is a Plane transparent surface, usually perpendicular to the horizon, and placed between the spectator's eye and the object he views; through which the optic rays, emitted from the several points of the object, are supposed to pass to the eye, and in their passage to leave marks that represent them on the said Plane.—Some call this the Table, or Picture, because the draught or Perspective of the object is supposed to be upon it. Others call it the Section, from its cutting the visual rays; and others again the Glass, from its supposed transparency.

Geometrical Plane, in Perspective, is a Plane parallel to the horizon, upon which the object is supposed to be placed that is to be drawn.

Horizontal Plane, in Perspective, is a Plane passing through the spectator's eye, parallel to the horizon.

Vertical Plane, in Perspective, is a Plane passing through the spectator's eye, perpendicular to the geometrical Plane, and usually at right angles to the perspective Plane.

Objective Plane, in Perspective, is any Plane situate in the horizontal Plane, of which the representation in perspective is required.

Plane of the H<*>ropter, in Optics, is a Plane passing through the horopter AB, and perpendicular to a Plane passing through the two optic axes CH and CI. See the fig. to the article Horopter.

Plane of the Projection, is the Plane upon which the sphere is projected.

Plane Angle, is an angle contained under two lines or surfaces.—It is so called in contradistinction to a solid angle, which is formed by three or more Planes.

Plane Triangle, is a triangle formed by three right lines; in opposition to a spherical and a mixt triangle.

Plane Trigonometry is the doctrine of Plane triangles, their measures, proportions, &c. See TRIGONOMETRY.

Plane Glass, or Mirror, in Optics, is a glass or mirror having a slat or even surface.

Plane Chart, in Navigation, is a sea-chart, having the meridians and parallels represented by parallel straight lines; and consequently having the degrees of longitude the same in every part. See Chart.

Plane Number, is that which may be produced by the multiplication of two numbers the one by the other. Thus, 6 is a plane number, being produced by the multiplication of the two numbers 2 and 3; also 15 is a Plane number, being produced by the multiplication of the numbers 3 and 5. See Number.

Plane Place, Locus Planus, or Locus ad Planum, is a term used by the ancient geometricians, for a geometrical locus, when it was a right line or a <*>ircle, in opposition to a solid place, which was one of the conic sections.

These Plane Loci are distinguished by the Moderns into Loci ad Rectum, and Loci ad Circulum. See LOCUS.

Plane Problem, is such a one as cannot be resolved geometrically, but by the intersection either of a right line and a circle, or of the circumferences of two circles. Such as this problem following: viz, Given the hypothenuse, and the sum of the other two sides, of a rightangled triangle; to find the triangle. Or this <*> Of four given lines to form a trapezium of a given area.

Plane Sailing, in Navigation, is the art of working the several cases and varieties in a ship's motion on a Plane chart <*> or of navigating a ship upon principles| 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.

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| Table, illustrated with various examples, in my Treatise on Mensuration, 2d edit. pa. 509 &c.