perpendicular to the first; or when two ple interest. Thus, any sum at 5 per cent.

straight lines drawn through any point whatever. respectively parallel to the two lines, are perpendicular to each other.

To erect a line perpendicular to a given line AB, at a given point C. Lay off on each side of C convenient and equal distances CA and CB; with A and B as centres, and with a radius greater than CB, describe arcs intersecting each other at E; join E and CE by a straight line, and it will be perpendicular to AB at C. Either

point E, is sufficient

to determine the perpendicular, but by determining both points

C

B

the correctness of the construction may be tested. A third point may be found by using the same centres as before and a different radius, which ought also to fall upon the same line, if the perpendicular is correctly determined.

If it is required to erect a perpendicular to a straight line, AB, at one extremity, B; take any point, C, as a centre, and with CB as

a radius, describe a cir- A cumference of a circle,

cutting the line AB in E. Draw EC, and produce it till it cuts the circumference in D; draw DB; it will be the required perpendicular. Perpendiculars to a straight line through a point, either upon,or without a given straight line, are usually drawn by the aid of a triangular ruler. See Ruler.

PERPENDICULAR IN PERSPECTIVE. A straight line perpendicular to the perspective plane. A perpendicular may be drawn through any point, and every such perpendicular vanishes at the centre of the picture. See Perspective.

PER-PE-TU'I-TY. [L. perpetuitas, everlasting]. In Annuities, the sum of money which will buy an annuity to last for ever. It is equal to the product of the annual value of the annuity, multiplied by the number of years it will take any sum to double at sim

simple interest will double in 20 years, hence the value of a perpetuity of $100 per annum is $2000. In all cases, the perpetuity is equal to the annual payment multiplied by the reciprocal of the rate per cent. at which the perpetuity is computed.

PER-SPECTIVE. [L. per and specio, to see]. The object of perspective is to make such a representation of an object upon a surface as shall present to the eye, situated at a particular point, the same appearance that the object itself would present, were the surface removed.

Perspective consists of two parts: First, the accurate delineation of the principal lines of the picture and Second, the shading and coloring of the picture so as to produce the desired effect of distance, &c. The first part, called linear perspective, is purely mathematical, and this part only will be considered. Perspective drawings may be made upon any surface, but we shall only consider them made upon a plane. The plane upon which the representation is made is called the perspective plane, and is generally supposed to be vertical. The point at which the eye is supposed to be situated is called the point of sight; all that part of space situated on the same side of the perspective plane with the eye, is said to be in front of the perspective plane, all on the other side is said to be behind the perspective plane. A visual ray is any straight line passing through the point of sight. A visual plane is a plane passing through the point of sight. A

visual cone is a cone whose vertex is at the

point of sight.

The perspective of a point, is the point in which a visual ray through the point pierces the perspective plane; if the given point and its perspective are on the same side of the eye, the perspective is said to be real, if on opposite sides it is virtual.

tersection of the perspective plane with the The perspective of a straight line is the invisual plane passing through the line.

The perspective of a curved line is the intersection of the perspective plane with the visual cone passing through the line.

The outline of the perspective of any body, is the intersection of the perspective plane with the enveloping visual cone; the line of

contact of this enveloping cone with the body, | Magnitudes, to be put in perspective, are is called the apparent contour of the body. given by their projections, or by their distanThe term cone is here used in its most en- ces above a horizontal visual plane, and from larged sense. It may sometimes happen that the perspective plane. To find the perspecthe enveloping visual surface may be pyra- tive of any point, draw any two lines through midal, as is the case in finding the perspec- the point, and find their perspectives; their tive of a cube, or other polyhedron, or it may point of intersection is the perspective rebe composed of both conical and pyramidal quired. The most convenient auxiliary lines surfaces; all of these surfaces come under are the perpendicular and a diagonal through the general denomination of conical surfaces. the point. To find the perspective of the The perspective of a body is generally ob- perpendicular, find the point where it pierces tained by finding the perspective of the prin- the perspective plane, and join it by a cipal lines of the body, embracing all those straight line with the centre of the picture: included within the apparent contour. The this will be the perspective. To find the perperspective of any point of a body may be spective of the diagonal, find the point where found by drawing a visual ray through it and the diagonal pierces the perspective plane, determining the point in which it pierces the and join it by a straight line with the proper perspective plane. This operation is tedious, vanishing point of diagonals; this will be the and to shorten the process other methods perspective of the diagonal. To ascertain the have been devised, the best of which is that proper vanishing point of any diagonal, conof diagonals and perpendiculars. The follow-ceive it produced till a part of the diagonal ing definitions of terms are given as necessary comes in front of the perspective plane, then to a complete understanding of this method: if this line inclines to the right, it vanishes A perpendicular is a straight line perpen- at the right hand vanishing point of diagonals, dicular to the perspective plane. A diagonal otherwise it vanishes at the left hand one. is a horizontal line, making an angle of 45° with the perspective plane. Through any point in space one perpendicular and two diagonals can always be drawn.

The centre of the picture is the point in which the perpendicular, through the point of sight, pierces the perspective plane. The horizon is the intersection of the perspective plane with a horizontal visual plane. It passes through the centre of the picture, and is horizontal.

The vanishing point of a line is the point in which a line drawn parallel to it, through the point of sight, pierces the perspective plane. Every system of parallel lines has the same vanishing point, which is a point common to the perspectives of all the lines of the system. The centre of the picture is the vanishing point of all perpendiculars. If a line is parallel to the perspective plane, its vanishing point is at an infinite distance. The vanishing points of diagonals are the points in which the diagonals, through the point of sight, pierce the perspective plane. They are in the horizon of the picture, and at distances from the centre of the picture equal to the distance from the point of sight to the perspective plane.

The vanishing point of rays is the point in which a ray of light through the point of sight pierces the perspective plane; the vanishing point of horizontal projections is the point in which the projection of the same ray on the horizontal plane through the point of sight intersects the horizon of the picture. These two points are in the same straight line, perpendicular to the horizon. When the former is assumed or given, the latter can be found by drawing through it a straight line perpendicular to the horizon and finding the point in which it intersects the horizon.

The shadow which any point casts upon any surface, lies upon the ray of light, and upon the projection of that ray upon the surface. Hence, to find the perspective of the shadow cast by any point upon a horizontal plane, find the perspective of the projection of the point upon the plane, and join it by a straight line with the vanishing point of horizontal projections of rays. Join the perspective of the point with the vanishing point of rays: the point in which these two lines intersect, is the perspective required. These principles are enough to find the perspective of all bodies, and the perspectives of their shadows; but, certain constructions, in par

ticular cases, serve to facilitate the operations tersecting in Q; then is Q the perspective of of finding the perspectives of bodies and of the shadow cast by the point whose perspectheir shadows. tive is N. Draw through Q a line parallel to AB, and limited by OR. The figure HRQ TSH is the perspective of the outline of the shadow cast on the horizontal plane.

To illustrate the rules above given, let it be required to find the perspective of a cube and its shadow in the horizontal plane.

For convenience, take the perspective plane through the front face of the cube, and let the horizontal plane of the base be taken as the plane on which the shadow is cast, and suppose AB to be the line of intersection of

The following rules for finding the perspectives of circles, are of much use in practical operations:

First. To find the perspective of a horizontal circle.

to AB, and find its perspective HG; then are DE and HG conjugate diameters of the ellipse of perspective, which may, therefore, be constructed by known rules.

these planes. Assume DD' parallel to AB, as the horizon of the picture. Assume C as the centre of the picture, D and D' equally Draw a diameter AB of the circle perpendistant from C, as the vanishing points of dicular to the perspective plane, find its perdiagonals, R as the vanishing point of rays spective DE, and bisect it in F; draw the of light, and H' as the vanishing point of perspective of a diagonal through F, and find projections of rays. Construct the square the diagonal KC, of which it is the perspecHL, to represent the front face of the cube, tive; draw the chord LM, through C, parallel and it will be its own perspective. The four edges that pierce the perspective plane at H, K, L and M, are perpendiculars, and their indefinite perspectives are found by drawing through these points straight lines to C. The diagonal through the back left hand upper vertex pierces the perspective plane at M, and MD' is its perspective. The point O, in which this line intersects LC, is the perspective of the back left hand upper vertex. Through O draw ON parallel to AB, and it will be the perspective of the back upper edge of the cube; draw NS perpendicular to AB, and this will be the perspective of one of the back vertical edges of the cube: the figure HONSLK is the perspective of the cube.

In the figure, we have supposed the horizontal plane of projection to have been revolved so that the part behind the ground-line falls below the ground-line. The perspective will always be an ellipse, or some of its particular cases when the perspective of all its points are real; that is, when a plane, parallel to the perspective plane, through the point of sight passes entirely in front of the circle.

Second. To find the perspective of any circle whatever. Draw two tangents to it parallel to the perspective plane; then draw To find the perspective of its shadow on the diameter through their points of contact, the horizontal plane. Draw MR it is the and find its perspective: draw the perspecperspective of the ray through M ; draw HH';| tive of a diagonal through its middle point, it is the perspective of the horizontal projec- and find the diagonal corresponding to it; tion of the same ray, and R, their point of through the point in which it intersects the intersection, is the perspective of the shadow diameter taken, draw a chord of the circle of M on the plane. Draw NR and RC, in- parallel to the tangent, and find its perspec

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,

The formula for sum-
method of differences,
ming a series by the

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 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.

1 3 6 10 15 21, &c. 1st order of diff., 2 3 4 5 6 &c.

2d

3d

66

Hence,

&c.

&c.

a=1, d=2, d. 1, d.=0, d=0, &c... Substituting these in formula (1), and reducing, we have

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 22, in the next, 32, and so To find the number of balls in a pile of

on.

The triangular pile is made up of a suc-n layers, we have the series, cession 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

n (n+1) (2n+1)

S=

(3).

2 (m2), the third, 3 (m + 3), and so on. To find a formula for the number of balls in a complete rectangular pile, we have the series

The rectangular pile is formed as in the

1.(m+1), 2(m+2), 3(m+3), 4(m+4)&c. | Add to the number of balls in the longest 1st ord. diff., m+3, m+5, m+7,

&c. side of the base the number in the parallel side &c. opposite, and also the number in the parallel &c. top row; multiply this sum by one-third of the number of balls in the triangular face of the pile, and the result will be the number of balls in the pile.

a=m+1, d1 =(m+3), d=2, d1=0, d=0, &c. Substituting in formula (1) and reducing, we have,

This rule is easy to remember, and is equally applicable to each of the three forms; (4). it is often called the workman's rule.

In order to find the number of balls in an

Where space is an object, the rectangular

incomplete pile, compute the number that the pile is preferable to either of the others, and pile would contain if complete, and the num-balls that can be piled upon a given area, the longer the pile, the greater the number of ber required to complete it; the difference of these two numbers is the number of balls in the pile. Formulas (2), (3) and (4), may be written

2

(n+n+1)... (3),

((n+m)+(n+m)
+(m+1)).. (4).

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 393 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.

PLĀNE. [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 touch a given straight line, and continue parallel to its first position. A plane may