tive; this, with the perspective of the diameter already found, are conjugate diameters of the ellipse of perspective. These methods, with suitable modifications, serve to find the perspectives of circles, however situated. The principles already explained, serve to find the perspective of any body, whatever may be its form, and also the perspective of its shadow. The number of balls in a complete triangular pile is equal to the sum of the series, 1, 1+2, 1+2+3, &c... to 1+2+3+. +n, or, 1+3+6+..+ n(n+1) 2 The formula for sum- is The principles of mathematical perspective are intimately connected with the arts of design, and a knowledge of their application is indispensable to the architect, the engraver, S = na + and the skillful mechanic. The practice of perspective is particularly necessary to the painter and the sculptor. Perspective alone enables us to represent fore-shortenings with accuracy, and its aid is required in the accurate delineation of even the simplest of natural objects. OBLIQUE PERSPECTIVE. The perspective is said to be oblique when the perspective plane is taken obliquely to the principal face of the object delineated. PARALLEL PERSPECTIVE. The perspective is said to be parallel when the perspective plane is taken parallel to the principal face of the object represented. + Series, 2d 66 n(n-1)(-2) da 1.2.3 -d2+&c... (1). PILING SHOT AND SHELLS. Shot and shells are generally piled at arsenals, navy yards, &c., in regular piles of a pyramidal or wedge-shaped form. The piles are named from the form of their bases, triangular, square and rectangular. The number of balls in the top layer is 12, in the next layer 23, in the next, 32, and so on. To find the number of balls in a pile of n layers, we have the series, 1 4 9 16 25 36, &c. The triangular pile is made up of a succession of triangular layers, equilateral, and diminishing from bottom to top, so that the 1st order of diff., 3 5 7 9 11 &c. Substituting these in formula (1) and reduc- | annexed figure. The top layer contains ing, we have, (m+1) balls, the second layer contains 2 (n+n+1)... (3), ·((n+m)+(n+m) +(m+1)).. (4). Where space is an object, the rectangular pile is preferable to either of the others, and balls that can be piled upon a given area, the longer the pile, the greater the number of having a given breadth. One long pile is more economical of space than two or more short ones. The square pile occupies most space for the number of balls contained in it. PINT. A unit of measure of capacity, equivalent to one-eighth of a gallon, or about 39 cubic inches. See Measures. PLAN. In Descriptive Geometry and Surveying, a representation of the horizontal projection of a body. The plan of an object is the same as its horizontal projection. The term is particularly applied to architectural drawings. PLANE. [L. planus, even, flat]. A suris the number of balls in the triangular face face such that, if any two points be taken at of each pile, and the next factor in each case pleasure and joined by a straight line, that denotes the number of balls in the longest line will lie wholly in the surface. A plane side of the base, plus the number in the sides is supposed to extend indefinitely in all of the base opposite, plus the number in the directions. A plane may be generated by a top parallel row, we have the following practical rule for finding the number of balls in any pile. straight line moving in such a manner as to also be generated by revolving one straight nate one variable; the resulting equation will line about another straight line, perpendicular be the equation of the projection of their to it, as an axis of revolution. The rectilineal equation of a plane may be reduced to the form, z = cx + dy + g. intersection on the plane of the other two. If the equation of a plane is in which x, y and z, are the co-ordinates of every point of the surface, and c, d and g, constants. The plane is given when c, d and g are known, and may be constructed by points, as follows: Assume any two values for x and y, and substitute them in the equation; there and the equations of a straight line are will result a corresponding value for z, which x = az + a, and y = b + ß, with the assumed value of x and y will be the line and plane will be parallel when the co-ordinates of a point that may be constructed by known principles. In like man 1 -acbd 0; a = C and b -d; ner, any number of points may be found and they will be at right angles when Planes are generally constructed by finding the points in which they cut the co-ordinate axes, and then passing a plane through these three points. Planes are given in Descriptive Geometry by their traces, that is, by their intersections with the planes of projections; they may, in like manner, be determined analytically. To find the equation of the trace of a plane upon the planes XY XZ and YZ respectively: make in the equation of the plane, z, y and x, respectively equal to 0 in the equation; the resulting equations will be the equations of the required traces. Two traces will be sufficient to fix the position of a plane. are and in general, the angle which they make sin A 1 V1+ a2 + b2 √ 1 + c2 + d2 To find the point in which a line pierces a plane, combine their equations and find the corresponding values of the variables; these will be the co-ordinates of the required point. PLANE ANGLE. A portion of a plane lying between two straight lines, meeting at a point. The lines are called sides of the anSee Angle. gle, and their common point is the vertex. PLANE CHART. A chart constructed so that the parallels of latitude and longitude are represented by straight lines parallel to each other, and at the same distance from each If we take two planes, whose equations other, in every latitude. PLANE CURVE. A curve all of whose points lie in the same plane. PLANE DIRECTOR. A plane parallel to every element of a warped surface of the first class. See Warped Surface. PLANE FIGURE. A portion of a plane limited by lines either straight or curved. When the bounding lines are straight, the figure is rectilinear, and is called a polygon. When they are curved, the figure is curvilinear. PLANE GEOMETRY. That part of Geometry which treats of the relations and properties of plane figures. HORIZONTAL PLANE A plane parallel to the surface of still water, or, parallel to a tangent plane to the earth's surface at the OBJECTIVE PLANE. In Surveying, the horizontal plane upon which the object to be delineated is supposed to stand. It is usually taken as the horizontal plane of projection. PLANE OF A DIAL. The plane upon which the hour lines of the dial are constructed. See Dial. larly employed in the filling in of a trigono metrical survey; it is also of some use in land surveying. Where exactness is required, the plane table is of little value, but in making approximate sketches, it commends itself or account of the rapidity with which operations can be carried on. The plane table consists of a square board or limb, mounted upon a tripod. Two leveling plates are attached, one to the tripod and the other to the limb, and are connected by a ball and socket joint. Four leveling screws, working through one leveling plate and against the other, serve to regulate the PLANE OF PROJECTION. One of the planes lateral motions of the table with respect to to which points are referred in descriptive the axis of the instrument. The limb may geometry for the purpose of determining be moved in azimuth around the axis of the their relative position in space. See Descrip- instrument, which motion may be checked tive Geometry. by a clamp screw; small motions in azimuth PLANE OF RAYs. In Shades and Shadows, are then communicated by a tangent screw. a plane parallel to a ray of light. PERSPECTIVE PLANE. The plane upon which the perspective of an object is drawn. See Perspective. PRINCIPAL PLANE. In Spherical Projections, the plane upon which the projection of the different circles of the sphere are projected. It is generally taken through the centre of the sphere to be projected. PLANE PROBLEM. A problem which can be solved geometrically, by the aid of the right line and circle only. PLANE SAILING. The method of computing the position of a ship and her path, under the supposition that the surface of the earth is a plane. See Navigation. PLANE SCALE. A scale upon which are graduated chords, sines, tangents, secants, rhumbs, geographical miles, &c. The scale is principally used by navigators in their computations, in plotting their courses, &c. The limb of the instrument is made horizontal by the aid of a small detached spirit level, by laying it over two of the leveling screws, and bringing the bubble to the centre, then placing it over the other two, and again bringing the bubble to the centre. с L F B The upper face of the limb is bordered by a brass plate about an inch in width, and its centre is marked by a steel pin, F. The perimeter of the limb is graduated to degrees and fractions of a degree, as follows; suppose a circle to be described with F as a centre, and tangent to the sides of the brass plate. Let the circle be graduated to the required unit, and then suppose straight lines to be drawn from the centre, F, through these points of division; the intersection of these lines with the brass PLANE SURVEYING. That branch of Sur- plate are marked and numbered from the veying in which the curvature of the earth's point I through 180° around to L, and from surface is not taken into consideration. In L through 180° around to I again. In some this branch the surface of the earth is re- plane tables the numbering is from 0° to 360°. garded as a plane. Such is the ordinary field There are generally diagonal scales of equal and topographical surveying, where only very parts, DC and AB cut in the plates. Used in limited portions of the earth's surface are plotting. Near the outer edges of the limb, considered. See Surveying. two small grooves are made to receive two PLANE TABLE. An instrument used in plates of brass, DC and AB, which are drawn surveying for plotting in the field without the to their places by means of milled-headed necessity of taking field notes. It is particu- screws, which pass through the table from the under side and screw firmly into the plates. base. Clamp the limb and make it truly horThe object of these plates is to secure the izontal. Mark the point corresponding to the paper on which the drawing is to be made. end of the base by a needle, and pressing the By loosening the screws, and pushing up the ruler against the needle, direct the sights to plates, the paper may be introduced; then, the other extremity of the base. With a fineby turning the screws back again, the plates pointed pencil, draw a straight line along the are drawn down, and the paper is held tightly. beveled edge of the ruler, and lay off on it, The paper might be slightly moistened, which from a scale of equal parts, the length of the would secure a smoother surface when it is base, and mark the second point; then direct dried. A ruler accompanies the table, with the sights, in succession, to all the principal two sights like compass sights, or sometimes objects that are visible from the first station, with a telescope, in their stead. One edge of and draw pencil lines along the edge of the the ruler is beveled, and this edge is so placed ruler. Next plant the plane table so that the that it is in the plane of the openings through the sights. The sights are constructed so as to fold down for convenience in carriage. A compass is sometimes attached for determining the bearings of lines. The plane table, as described, is used for two distinct purposes; 1st. To measure horizontal angles; and 2d, For determining the shorter lines of a survey, both in extent and position. To measure a horizontal angle. Place, by means of a plumb line, the centre of the table exactly over the angular point; then level the table and clamp the limb; after which place the ruler with the sights raised so that the beveled edge shall rest against the steel pin at the centre; direct the sights to the left hand object, and read the reading at each end of the ruler, and take a mean of the results for the first reading. Then direct the sights to the right hand object, and take the second reading in like manner. If the ruler has not passed over the 0° point of the limb, the excess of the second reading over the first is the value of the required angle. If the ruler has passed the 0° point, the first reading must be subtracted from 180° and the difference added to the second reading; the sum will be the value of the angle required. second end of the plotted base line shall be over the second end of the base line in the field; level the instrument, and having placed the needle at the second end of the base line, bring the beveled edge of the ruler to coincide with the plotted base, and then turn the limb of the instrument till the sights are directed to the first end of the base; clamp the limb and direct the sights, in succession, to every object sighted from the first station, marking the points of intersection of these lines with the corresponding ones from the first station. To illustrate let it be required to determine the relative position of several houses. From station A, and on one of the line of the triangulation, as AB measure the base line AB, which we will suppose equal to 300 yards. Place the plane table over A, and sight to the corners of the houses, and mark the lines 1, 2, 3, 4, &c. Then move the table to B, place the plotted line AB in the direction from B to A, and sight to the same corners as before, and draw the lines as in the figure; the points at which they intersect the corresponding lines before drawn, determine the plot of the corners of the houses; To determine lines in extent and position. Having fastened a sheet of paper on the table, examine the lines and objects which are to be determined in position and select for a base a convenient line which is connected with some point of the triangulation, taking care that as many prominent objects as possible may be seen from its two extremities. Then place the plane table over one extremity of the base, the front lines of the houses may then so that the point on the paper corresponding be drawn on the paper, and upon these the may be exactly over the extreme point of the plots of the houses themselves may be con |