terminating planes, is said to be regular; an irregular | is, as the base AE to the base CF, so is the solid AB prism is one in which the ends are irregular polygons. to the solid CD.

The line joining the centre of the terminating planes of a regular prism is called the prismatic axis, and the altitude or height of the prism is equal to the length of this line.

The parallelograms which form the sides of prisms are always equal in number to the sides of the base, and prisms are classed accordingly. The prisms which have a triangular base are called triangular prisms; those which have a quadrangular base are called quadrilateral prisms. Hexagonal prisms are those having a hexagon for their base; and polygonal prisms are those of which the base is a polygon. The parallelopiped and the cube are two quadrangular prisms, which, being of great importance in the arts, require special attention. The parallelopiped is bounded by six four-sided figures, of which every opposite two are parallel. It is called rectangular when its base is a rectangle; and when these six planes are all rectangles placed perpendicularly to each other, it is said to be right, and oblique when the planes are inclined to each other. We can scarcely look around us without seeing rectangular parallelopipedons. Beams of timber, hewn stones, the box of a gardener's wheelbarrow, the bodies of carts, are almost invariably rectangular parallelopipedons; and this form is generally given to houses and to rooms. It is one of the propositions of the 11th book of Euclid, that if a solid be contained by six planes, two and two of which are parallel, the opposite planes are similar and equal parallelograms; thus let the solid CDGH (which is a parallelopiped) be contained by the parallel planes AC, GF; BG, CE; FB, AE; its opposite planes are similar and equal parallelograms.

B

D

A

H

G

F

E

Parallelopipeds, when cut by a plane passing through the diagonals of two of the opposite planes, are formed into two equal triangular prisms.

C

F

B

Let AB be a solid parallelopiped, and DE, CF the diagonals of the opposite parallelograms AH, GBnamely, those which are drawn betwixt the equal angles in each; and because CD, FE are each of them parallel to GA, though not G in the same plane with it, CD, FE are parallel; wherefore the diagonals CF, DE are in the plane in which the parallels are, and are themselves parallels; and the plane CDEF shall cut the solid AB into two equal parts.

A

D

E

H

A

Whatever can be proved respecting the properties
of parallelopipeds, is equally true with respect to those
The dice used by gamblers are
of cubes, because a cube is merely a parallelopiped
with square sides.
cubes; and tables, footstools, and rooms often receive
a cubic form. Just as the unit of measure for plane
surfaces is a square, so the unit of measure for solids
is a cube, and the length of each side of the cube is an
inch, a foot, a yard, or any other lineal unit which may
be fixed. To measure the solidity of a solid is to as-
certain how many cubic inches, or cubic feet, &c. are
contained therein. The solidity of a parallelopiped is
found by multiplying the length by the breadth, and
the product by the height; the result gives the number
of cubes contained within the figure.

Because the triangle CGF is equal to the triangle CBF, and the triangle DAE to DHE; and that the parallelogram CA is equal and similar to the opposite one BE; and the parallelogram GE to CH; therefore the planes which contain the prisms CAE, CBE are equal and similar, each to each; and they are also equally inclined to one another, because the planes AC, EB are parallel, as also AF and BD, and they are cut by the plane CE; therefore the prism CAE is equal to the prism CBE, and the solid AB is cut into two equal prisms by the plane CDEF.

Def. The insisting straight lines of a parallelopiped, mentioned in the following propositions, are the sides of the parallelograms betwixt the base and the plane parallel to it.

E

F

C

Let 1, 1, and h, be the length, breadth, or thickness, v=lth. and the height, and v the volume or solid content; then Let AF be a right rectangular parallelopiped. Let lineal units, as 4 inches, its thickits length AB be ness BC 2 inches, and its height AD 3 inches. The solid can evidently be di- D vided into three equal portions by planes through G G and H, parallel to the base AC; and into four equal H portions by means of planes through K, L, M, parallel to the side BF; and into two equal portions by a plane through I, parallel to BD. Each of the small cubes into which the solid is now divided, is a cubic inch; the number of cubic inches in the lowest portion HC is 4 x 2, or 8, and in the second and uppermost portion there are as many; and in them all, therefore, there are 4 × 2 × 3, or 24 -that is, to find the cubic contents of the solid, find the continued product of the length, breadth, and height.

The solidity of all prisms, or of any parallelopiped, may be found by multiplying the area of the Let b denote the base, and the base by the height. bh. height; then A pyramid is an angular solid, M` which has a polygon for its base, and triangles for its sides or faces; Parallelopipeds are equal when they are upon equal these sides all meet in one point, and form a solid angle, bases and of the same altitude. This fact is very evi- which is called the vertex of the pyramid. Pyramids, dent, and is constantly acted upon when the solids like prisms, may be either regular or irregular. When are right; for instance, in making book-shelves, each the base of a pyramid is regular, the line joining its shelf is made equal to the others that is, it is made vertex or summit is called its aris, and when the axis is equally long and equally broad, and has the same alti-perpendicular to the base, it is then a regular pyramid. tude or thickness. Solid parallelopipeds which have They are also said to be triangular, quadrilateral, polythe same altitude, are to one another as their bases: gonal, &c. according as the base is a triangle, a quadrithus let AB, CD be solid parallelopipeds of the same lateral, a polygon, &c. The Pyramids of Egypt are quadaltitude; they are to one another as their bases-thatrilateral, having square bases and four similar and

621

equal triangular sides. An obelisk is also a pyramid, of a cone. The area of the base is multiplied by the and has a square base and triangular sides; but the altitude, which, as before shown, is the rule for finding height is very great in proportion to the extent of the the solidity of a cylinder; one-third of the result, therebase. The solidity of a pyramid is found by multiplying fore, gives the solidity of the cone. the area of the base by the perpendicular height, and one-third of the product is the answer: 12= = bh. Example.-Find the solidity

of a rectangular pyramid, the length and breadth of its base being 6 and 4 feet, and its altitude 20 feet.

2= bh = × 6 × 4 × 20 = 160 cubit feet.

S

Q

V

N

SPHERICAL GEOMETRY.

A sphere or globe is a solid having one continued curved surface, and which is conceived to be generated by the revolution of a semicircle about its diameter: balloons and cricket-balls are spheres.

Spherical geometry consists in the investigation of the properties of spheres.

Every point on the surface of a sphere is equally distant from a point in the middle of the sphere called its centre; any line drawn from the centre to the circumference is called a radius, and any line drawn P through the centre, and terminated at both extremities by the circumference, is termed a diameter. When the diameter is perpendicular to the plane of a circle of the sphere, it is termed an axis, and the extremities of the axis are called the poles. Circles of the sphere, whose planes pass through the centre, dividing the sphere into two equal parts, are called great circles, and all others are small circles. By the distance of two points on the surface of a sphere, is meant an arc of a great circle intercepted between them.

By a regular solid is meant a solid bounded by regular plane figures, and its solid angles equal; that is to say, a solid M in which all the sides are equiangular and equilateral. It has been proved that there can be only five regular solids; these are often called Platonic bodies, because Plato was the first who investigated their properties. The names of these five regular solids are the following; and it will be observed that these names are formed by prefixing the Greek numeral indicating the number of sides to the termination hedron, from edra (Greek), a seat; that is, a side:

The tetrahedron is a regular triangular pyramid, whose sides are equilateral triangles. The hexahedron is a cube. The octahedron is contained by eight equilateral triangles. The dodecahedron is contained by twelve regular pentagons. The icosahedron is contained by twenty equilateral triangles. Each side of a regular solid, except the tetrahedron, has an opposite face parallel to it, and the edges of these faces are also respectively parallel.

A cylinder differs from a prism in having a circular instead of a rectilineal base; it is contained between two equal and parallel circles and a convex surface. The line joining the centres of the two circles is called the axis. When the C axis is perpendicular to the plane of the bases, the cylinder is said to be right. Steam-boilers, the shafts of circular pillars, the stone of a common garden roller, the barrel of a gun, and many other familiar objects, are cylinders. Combinations of cy- A linders are also very frequently used

B

A spherical angle is that formed on the surface of the sphere by arcs of two great circles meeting at the angular point, and is measured by the inclination of the planes of the circles.

A spherical triangle is a figure formed on the surface of the sphere by arcs of three great circles, called its sides, each of which is less than a semicircle. A quadrantal triangle is that of which one of the sides is a quadrant.

A lunary surface is a part of the surface of the sphere, contained by the halves of two great circles. A segment of a sphere is a part cut off by a plane. There are several methods of finding the contents or solidity of a sphere; perhaps the most simple and the most easy to be remembered

is the following: Find, by
the rules previously given,
the solidity of the circum-
scribing cylinder, as EABF, S 'n'
which is a cylinder equal in
diameter and height to the E
diameter of the sphere; two-
thirds of it will be the
volume of the sphere, be-
cause a sphere is proved to
be equal to two-thirds of its
circumscribing cylinder.

A

H

B

The exterior extent of surface, or convex superficies of a sphere, may be ascertained by multiplying the diameter of the sphere by its circumference. Thus in a globe of 20 inches diameter, and 62-832 circumference, the convex superficies is 1256-64 inches, because 62.832 × 20 = 1256·64. Also, the surface of any zone of the sphere, as mnn'm', is exactly equal to the surface of the corresponding zone of the cylinder rss'r'. Spherical geometry is of great importance in several of the arts and physical sciences, and more especially in astronomy and navigation.

in the arts; telescopes and opera-glasses, for instance, are merely cylinders fitted one within the other, and of which circular lenses form the base. The solidity of a cylinder is found in the same way as that of a prism-namely, by multiplying the area of the base by the perpendicular height. This rule is founded upon the theorem that a cylinder and a parallelopiped, having equal bases and altitudes, are equal to one another. A cone, like a cylinder, has a circular base, but it terminates in a point like a pyramid; sugar-loaves are made in the form of cones. The annexed figure is a cone. AB is the base, V is the vertex, and the straight line VD joining the vertex and the centre of the base is called the axis. A frustum of a solid is a portion contained between the base and a plane parallel to it; thus the portion EGAB is a frustum of the cone. When the axis of a cone is perpendicular to its base, it is called a right cone. Other cones are said to be oblique. A A fixed relation subsists between certain lines drawn right cone may be described by the revolution of a in and immediately round a circle; and it is upon this right-angled triangle about one of the sides of the right relation that trigonometry is founded. Most of these angle. It is proved that if a cone and a cylinder have lines, such as tangents, secants, arcs, chords, &c. have the same base and the same altitude, the cone is equal already been mentioned; it only remains to add, that to the third part of the cylinder. From this fact results the sine of an arc is a straight line, drawn from one exthe method usually adopted for ascertaining the solidity | tremity of the arc perpendicular to the radius passing

E

G

D

C

TRIGONOMETRY-LAND-SURVEYING.

Trigonometry signifies literally the art of measuring triangles, but with the progress of science the meaning of the word has been much extended. Trigonometry is divided into plane and spherical, according as it is B directed to the investigation of plane or of spherical triangles.

through the other extremity, or it is in fact the half |
The sine, tangent, and
of the chord of double the arc.
secant of the complement of an arc, are called cosine,
cotangent, and cosecant of that arc. This will be better
understood by carefully examining the annexed figure,
which is a representation of the
various trigonometrical lines.

M

N

H

G

B

F

A

BC is the complement of the arc AB; BMD is the supplement of AB; angle BOC is the complement of AOB, and BOD is the supplement of AOB; BE is the D sine of AB; AF is the tangent of AB; OF is the secant of AB; so BG is the sine of BC, or the cosine of AB; CH is the tangent of BC, or the cotangent of AB; and OH is the secant of BC, or the cosecant of AB.

L

I

K

The following rules for computation are useful in right-angled trigonometry; that is, in computing the sides and angles of right-angled triangles.

I. When two sides are given, to find an angle. Make a given side radius, then the side made radius is to the other given side as radius to the trigonometrical name of the latter side.l

II. When one of the three sides and an angle are given, to find a side.

I Make any side radius, then the trigonometrical to that of the required side as name of the given side

the given side to the required side.

Although in this case any side may be made radius, it is preferable to make one of the sides concerned radius; that is, either the given or the required side, as this introduces the radius as a term of the proportion, and its logarithm being 10, it simplifies the calculation.

III. When the two sides are given, to find the hypotenuse.

The sum of the squares of the two given sides is equal to the square of the hypotenuse.

IV. When the hypotenuse and a side are given, to find the other side.

I The difference between the squares of the hypotenuse and the given side, is equal to the square of the required side. Or, the product of the sum and difference of the hypotenuse and a side, is equal to the square of the other side.

When the square of a side is known, its square root gives the value of the side.

The first two rules are sufficient for the solution of all the problems in right-angled trigonometry; but the last two may sometimes be conveniently employed. Trigonometry is one of the most useful departments in mathematical science. Its applications to practical purposes are very extensive, and it is of great importance in navigation, engineering, and, as we shall immediately see, in land-surveying.

Land-surveying is the method of measuring and computing the area of any small portion of the earth's surface, as a field, a farm, an estate, or district of moderate extent. There are three distinct operations in the art of land-surveying, all of which require the surveyor to possess a competent knowledge of arithmetic, algebra, and geometry. In the first place, the several lines and angles must be measured; secondly, they must be protracted or laid down on paper, so as to form a plan or map of the district; and, thirdly, the whole area of the district must be computed by means In performing the first of the foregoing operations. operation, the most useful instrument is a chain called Gunter's chain, from the name of the inventor, the Rev. Edmund Gunter, who lived about two hundred years ago. It is 22 yards or 66 feet long, and is composed of 100 equal links, the length of each being 7.92 inches. At every tenth link is a mark made of brass. An acre consists of 10 square chains, or 100,000 square links. There are 80 chains in a mile, and 640 acres in a square mile.

Iron pins, about two feet long, called arrows, with

red handles, or pieces of red cloth, attached to them,
are used for sticking in the ground at the end of each
chain length when measuring in the field. Ten of them
are commonly used.

In measuring land with the chain, two persons are
required, one at each end of the chain; the one who
walks first is, for the sake of distinction, called the
Lines measured
leader, and the other the follower.
perpendicularly to chain lines, to the angular points,
and other points of the boundary of a field, such as to
crooked hedges, brooks, &c. are called offsets. The cross-
staff is used in measuring offsets: it consists of two
bars of brass placed at right angles, with sights at their
extremities, perpendicular to the plane of the bars.
There are narrow slits at A and C, to which the eye is
applied, and wider open-
fine wire fixed vertically
ings at B and D, with a
in the middle of them.
a staff E, about 44 feet
The cross is supported on
high, which at the lower
end is pointed and shod
with brass, so that it can
easily be stuck in the
ground. The sights are
placed on the top of the
staff, and fixed to any
position by a screw F.

A

E

B

A simple cross staff may be made by cutting two grooves with a saw along the diagonals of a square board, to be fixed on the top of the staff. It can easily be ascertained if the sights are at right angles, by directing one pair of them, as AB, to one object, and observing to what object the other pair, CD, are then directed; then by turning the sights till the second object is seen through the first pair of sights AB, if the first object is then visible through the second pair of sights, and is exactly in apparent coincidence with the wire, the sights are at right angles; if not, they must be adjusted.

An instrument not less important in surveying is the theodolite. This useful instrument, fixed on the top of a tripod, consists of two graduated circles perpendicular to each other; one of which is fixed in a horizontal, and the other in a vertical, plane, and is used for measuring horizontal and vertical angles.

I

A

In the figure of the theodolite here presented, HRS represents an oblique view of the horizontal circle, view of the vertical and mQn a direct one which extends to little more than a semicircle. The verable about an imagitical circle is movnary axis, coinciding with the radius OQ, which, produced, passes through the centre C of the horizontal circle. On the vertical circle is fixed

W

m

n

T

S

L

R

a telescope W, furnished with a spirit-level I; the telescope is connected with a movable radius OA, in contact with the opposite side of the vertical circle; and this radius is fixed to a vernier o, movable, by means of a screw, along the limb of the circle. When the centre o of the vernier coincides with the middle division Q of the circle, the axis of the telescope is then horizontal, and the instrument thus serves also as a spirit-level. A vernier to the horizontal circle is attached to the vertical circle at e, and is movable with it.

To measure a horizontal angle subtended at the instrument by the horizontal distance of two objects: Direct the telescope to one of the objects, and observe the number of degrees at e on the horizontal circle;

623

DRAWING-PAINTING-SCULPTURE.

of

DRAWING is an imitative art, by which the forms, posi-ing, many of even those who travel for the purpose
tions, and relations of objects are represented on a flat informing the public, are unable to commemorate such
surface. The faculties employed in this, as in other objects, or, at the best, can give only a few rude scratches
imitative arts, are possessed in a certain degree by all in outline, which a professional artist has afterwards to
persons. Some possess these faculties in so high a fashion into shape-a shape, of course, in which correct
degree, as to become fitted to exercise them as a pro- representation is not to be looked for. In this point of
fession, for the gratification of mankind at large. In view, drawing takes its place, as a useful art, by the side
others, they are manifested so moderately, that a pro- of writing, being, like it, a means of description, and
tracted effort to make such persons become tolerable one which may occasionally be even more serviceable
draughtsmen would only be labour thrown away. The than that art, though certainly not capable of so gene-
majority of mankind, however, are so far endowed, as ral an application.
to be able, when instructed, to delineate any simple
object, and to enjoy much pleasure from higher deli-
neations produced by others.

The practice of elementary drawing at school, hitherto greatly overlooked, is calculated to produce the most beneficial results. As regards those who possess the faculties for design in a high degree of excellence, early practice will awaken those faculties, and furnishing them with stimulants to progress, secure the benefit of their ultimate exercise for the community. Lesser degrees of excellence will also be developed-such as would in vain perhaps essay excellence in the higher walks of art, but might become of incalculable value in connection with certain branches of manufacture.

As a means of elevating tastes and desires, and thereby embellishing what might be otherwise a routine of commonplace existence, drawing appears in its most interesting light. The person who has acquired a knowledge of botany, feels a new pleasure in examining the parts of a hitherto unseen plant; he who has acquired a knowledge of geology is interested in passing along a road, the side of which displays a deep section of strata, or from which he may view various granitic elevations; he who has acquainted himself with the principles of machinery, experiences an enjoyment in contemplating the intricacies of some great engine which another knows nothing of; and in the same manner he who has studied the art of drawing, discovers a source of new and innocent gratification in the innumerable forms and tints of external nature. Things formerly passed with a careless eye and a vacant mind, then assume a character which arrests attention and awakens thought. Those faculties of the mind which perceive and appreciate the figure, colour, and arrangements of objects, and trace in all a natural and appropriate beauty, spring up from a dormancy which might have otherwise known no interruption; a new association of our mysterious being with the physical world around us is practically established; and the value of existence becomes by just so much enhanced. Not surely that it is desirable that an absorbing interest should be created in all minds respecting the outward aspect of nature to the neglect of the more serious affairs of life. All that can be contended for is, that as many as possible should be rendered capable of looking with pleasure, instead of indifference, upon the beauties of nature, so that they may realise the benefit of this part of the intellectual and sentimental powers which have been conferred upon them; a portion of their nature which, like others, may be abused, but in its moderate use is not only a source of innocent pleasure, but may become the means of anticipating and supplanting many pursuits of a less worthy character. Nor, while the art is perhaps chiefly acquired with these views, may it be without some results of a more directly useful kind. In many situations-when wandering in our own, or roaming in foreign countries-we may see objects of which we would be glad to carry away some memorandum, and of which the slightest pencil sketch would be sufficient to awaken a recollection at any other time. And yet, for want of a few elementary lessons in drawNo. 90.

Referring to the volumes on Drawing in our EDUCA design on the present occasion is to present a popular TIONAL COURSE for a methodic series of instructions, our view of what may be done by comparatively unlearned persons to acquire a knowledge of the art, including perspective, which is the foundation of all pictorial delineation. It will be understood, then, that our observations are addressed directly to the pupil.

DRAWING.

Drawing is effected by various materials, as chalk, black-lead, or coloured pencils, Indian ink, &c. Drawor Bristol board, but also sometimes on tinted papers, ings of a simple kind are made principally on white paper in which case the lighter parts are brought out by white crayons. Commence a study of the art by acquiring ease of hand, and in fact learn what the hand can do by its different evolutions. For this purpose, drawing lines with chalk on a black board is perhaps the best exercise. Either, therefore, on a board with chalk, or on paper with a pencil, learn to make drawings of lines, straight, curved, or a modification of either. Observe how much more beautiful is the appearance and effect of a curve in comparison with a straight line; nature delights in this waving of forms, of which we have examples in the bending of boughs in trees, the serpentine winding of rivers, and the curvilinear outlines of animals.

and how

You may begin the drawing of objects by copying other drawings; but this species of exercise can only make you familiar with the manner in which lines answer the purpose of representation. To be master of the art, you must throw aside all drawings or copies, and learn to draw by your own ingenuity from tangible objects in nature and art. In this study the hand is taught to obey the conceptions of the mind. When, for example, we see a chair standing on the floor, we observe its shape or figure, its line of back, seat, legs, and all other parts about it. We then take a pencil, and bending the mind intensely on the form of the chair, try to define all the lines of the object on the paper or board. The more perfectly that the hand can obey the direction of the mind, while bent in thought on the the object, so will the drawing more resemble the original in all its details.

up

The pupil, therefore, must be taught to call whole power of his mind respecting the appearance of an object, when he wishes to represent it by a drawing. On this, indeed, may be said to rest the fundamental principle of delineation in all its branches. No one can possibly attain even a mediocrity in the art, who does not, less or more, possess this power of recalling images to the mind, and of training his hand to act in strict conjunction with its dictates. The hand is only the instrument employed by the imitative faculty, and can do nothing useful without the strong concurrence of the art in which design or figure is an element, is founded will. Accomplishment in penmanship, and every other on this fundamental basis.

A plan of instruction in drawing has been lately 625 introduced from the Normal School of Versailles into