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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 quad- Whatever can be proved respecting the properties rilateral prisms. Hexagonal prisms are those having of parallelopipeds, is equally true with respect to those a hexagon for their base; and polygonal prisms are of cubes, because a cube is merely a parallelopiped those of which the base is a polygon. The parallelopiped with square sides. The dice used by gamblers are and the cube are two quadrangular prisms, which, being cubes; and tables, footstools, and roonis often receive of great importance in the arts, require special atten- a cubic form. Just as the unit of measure for plane tion. The parallelopiped is bounded by six four-sided surfaces is a square, so the unit of measure for solids figures, of which every opposite two are parallel. It is is a cube, and the length of each side of the cube is an called rectangular when its base is a rectangle; and inch, a foot, a yard, or any other lineal unit which may when these six planes are all rectangles placed per- be fixed. To measure the solidity of a solid is to aspendicularly to each other, it is said to be right, and certain how many cubic inches, or cubic feet, &c. are oblique when the planes are inclined to each other. We contained therein. The solidity of a parallelopiped is can scarcely look around us without seeing rectangular found by multiplying the length by the breadth, and parallelopipedons. Beams of timber, hewn stones, the the product by the height; the result gives the number box of a gardener's wheelbarrow, the bodies of carts, of cubes contained within the figure. are almost invariably rectangular parallelopipedons ; Let I, t, and h, be the length, breadth, or thickness, and this form is generally given to houses and to rooms. and the height, and v the volume or solid content; It is one of the propositions of the 11th book of Euclid, then
v=lth. that if a solid be contained by six planes, two and two Let AF be a right rectangular parallelopiped. Let of which are parallel, the opposite planes are similar its length AB be 4 lineal units, as 4 inches, its thickand equal parallelograms; thus
ness BC 2 inches, and its
F let the solid CDGH (which is a
height AD 3 inches. The parallelopiped) be contained by
solid can evidently be di- D the parallel planes AC, GF; BG,
G vided into three equal por: CE; FB, AE; its opposite planes
tions by planes through G GTare similar and equal parallelo
and H, parallel to the base grams.
AC; and into four equal H
C Parallelopipeds, when cut by a D
portions by means of planes plane passing through the diagonals of two of the through K, L, M, parallel
А K L M B opposite planes, are formed into two equal triangular to the side BF; and into prisms.
two equal portions by a plane through I, parallel to Let AB be a solid parallelopiped, and DE, CF the BD. Each of the small cubes into which the solid is diagonals of the opposite parallelograms AH, GB, now divided, is a cubic inch; the number of cubic namely, those which are drawn
inches in the lowest portion HC is 4 x 2, or 8, and in betwixt the equal angles in each;
the second and uppermost portion there are as many; and because CD, FE are each of
and in them all, therefore, there are 4 x 2 x 3, or 24 them parallel to GA, though not G
--that is, to find the cubic contents of the solid, find in the same plane with it, CD,
the continued product of the length, breadth, and FE are parallel; wherefore the
D diagonals CF, DE are in the
The solidity of a cube is plane in which the parallels are,
found by multiplying the side and are themselves parallels; and
by itself, and that product again the plane CDEF shall cut the solid
by itself; or, literally, by rais
A AB into two equal parts.
ing the side to the third power. Because the triangle CGF is equal to the triangle Let e = an edge of a cube; CBF, and the triangle DAE to DHE; and that the then v=e. parallelogram CA is equal and similar to the opposite The reason of the rule is one BE; and the parallelogram GE to CH; therefore evident, since a cube is just a A
B the planes which contain the prisms CAE, CBE are parallelopiped, whose length, equal and similar, each to each ; and they are also breadth, and height are equal. equally inclined to one another, because the planes AC, The solidity of all prisms, or of EB are parallel, as also AF and BD, and they are cut any parallelopiped, may be found by the plane CE; therefore the prism CAE is equal to by multiplying the area of the the prism CBE, and the solid AB is cut into two equal base by the height. prisms by the plane CDEF.
Let b denote the base, and h the Def.— The insisting straight lines of a parallelopiped, height; then
v=bh. mentioned in the following propositions, are the sides A pyramid is an angular solid, DI of the parallelogranis betwixt the base and the plane which has a polygon for its base, parallel to it.
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 axis, 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 quad. altitude; they are to one another as their bases-that I rilateral, having square bases and four similar and
equal triangular sides. An obelisk is also a pyramid, of a cone. The area of the base is multiplied by ibo 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, there base. 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 :
SPHERICAL GEOMETRY. Example.-Find the solidity
A sphere or globe is a solid having one continued of a rectangular pyramid, the
curved surface, and which is conceived to be generated length and breadth of its base
by the revolution of a semicircle about its diamete: being 6 and 4 feet, and its alti
balloons and cricket-balls are spheres. tude 20 feet.
Spherical geometry consists in the investigation of v= bh = } x 6 x 4 x 20
the properties of spheres. = 160 cubit feet.
Every point on the surface of a sphere is equally disBy a regular solid is meant a
tant from a point in the middle of the sphere called solid bounded by regular plane
its centre; any line drawn from the centre to the cit. figures, and its solid angles
cumference is called a radius, and any line dran equal ; that is to say, a solid M*
P through the centre, and terminated at both extremitis in which all the sides are equi.
by the circumference, is termed a diameter. When angular and equilateral. It has
the diameter is perpendicular to the plane of a cirde d been proved that there can be
the sphere, it is termed an axis, and the extremities d only five regular solids; these are often called Platonic the axis are called the poles. Circles of the sphere, bodies, because Plato was the first who investigated whose planes pass through the centre, dividing the their properties. The names of these five regular sphere into two equal parts, are called great circle, solids are the following; and it will be observed that and all others are small circles. By the distance of these names are formed by prefixing the Greek nume- two points on the surface of a sphere, is meant an are ral indicating the number of sides to the termination of a great circle intercepted between them. hedron, from edra (Greek), a seat; that is, a side :- A spherical angle is that formed on the surface di
The tetrahedron is a regular triangular pyramid, the sphere by arcs of two great circles meeting at the whose sides are equilateral triangles. The hexahedron angular point, and is measured by the inclination of is a cube. The octahedron is contained by eight equi- the planes of the circles. lateral triangles. The dodecahedron is contained by A spherical triangle is a figure formed on the surface twelve regular pentagons. The icosahedron is contained of the sphere by arcs of three great circles, called its by twenty equilateral triangles. Each side of a regular sides, each of which is less than a semicircle. solid, except the tetrahedron, has an opposite face A quadrantal triangle is that of which one of the parallel to it, and the edges of these faces are also sides is a quadrant. respectively parallel.
A lunary surface is a part of the surface of the À cylinder differs from a prism in having a circular sphere, contained by the halves of two great circles instead of a rectilineal base; it is contained between two A segment of a sphere is a part cut off by a plane equal and parallel circles and a convex surface. The There are several methods of finding the contents et line joining the centres of the two
solidity of a sphere; perhaps the most simple and the circles is called the axis. When the ci
D most easy to be remembered axis is perpendicular to the plane of
is the following: Find, by the bases, the cylinder is said to be
the rules previously given, right. Steam-boilers, the shafts of
the solidity of the circum- ?, circular pillars, the stone of a com
scribing cylinder, as EABF, n' mon garden roller, the barrel of a
which is a cylinder equal in gun, and many other familiar objects,
diameter and height to the E are cylinders. Combinations of cy- A
diameter of the sphere; twolinders are also very frequently used
thirds of it will be the in the arts; telescopes and opera-glasses, for instance, volume of the sphere, beare merely cylinders fitted one within the other, and cause a sphere is proved to
A of which circular lenses form the base. The solidity be equal to two-thirds of its
H of a cylinder is found in the same way as that of a circumscribing cylinder. prism-namely, by multiplying the area of the base The exterior extent of surface, or contes superficies by the perpendicular height. This rule is founded of a sphere, may be ascertained by multiplying the upon the theorem that a cylinder and a parallelo- diameter of the sphere by its circumference. Thus in piped, having equal bases and altitudes, are equal a globe of 20 inches diameter, and 62-832 circum. to one another. A cone, like a cylinder, has a cir- ference, the convex superficies is 1256-64 inches, be cular base, but it terminates in a point like a pyra- cause 62·832 x 20 = 1256.64. Also, the surface of any mid; sugar-loaves are made in the form of cones. zone of the sphere, as mnn'm', is exactly equal to the The annexed figure is a cone.
surface of the corresponding zone of the cylinder ro'y'
, AB is the base, V is the vertex,
Spherical geometry is of great importance in seren! and the straight line VD joining
of the arts and physical sciences, and more especially the vertex and the centre of the
in astronomy and navigation. base is called the axis. A frustum of a solid is a portion con- E
TRIGONOMETRY-LAND-SURVEYING. tained between the base and a
Trigonometry signifies literally the art of measuring plane parallel to it; thus the
triangles, but with the progress of science the meaning portion EGAB is a frustum of
of the word has been much extended. Trigonometry is the cone. When the axis of a
divided into plane and spherical, according as it is cone is perpendicular to its base,
B directed to the investigation of plane or of spherica! it is called a right cone. Other
triangles. cones are said to be oblique. A
A fixed relation subsists between certain lines dran 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, ares, chords, &c. hare 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 er. the method usually adopted for ascertaining the solidity tremity of the arc perpendicular to the radius pasing
through the other extremity, or it is in fact the half | red handles, or pieces of red cloth, attached to them, of the chord of double the arc. The sine, tangent, and are used for sticking in the ground at the end of each secant of the complement of an arc, are called cosine, chain length when measuring in the field. Ten of them cotangent, and cosecant of that arc. This will be better are commonly used. understood by carefully examining the annexed figure, In measuring land with the chain, two persons are which is a representation of the
F required, one at each end of the chain; the one who various trigonometrical lines.
walks first is, for the sake of distinction, called the BC is the complement of the
leader, and the other the follower. Lines measured arc AB; BMD is the supplement
perpendicularly to chain lines, to the angular points, of AB; angle BOC is the comple
and other points of the boundary of a field, such as to ment of AOB, and BOD is the
crooked hedges, brooks, &c. are called offsets. The crosssupplement of AOB; BE is the D
staff is used in measuring offsets : it consists of two sine of AB; AF is the tangent of
bars of brass placed at right angles, with sights at their AB; OF is the secant of AB; so
extremities, perpendicular to the plane of the bars. BG is the sine of BC, or the
There are narrow slits at 'A and C, to which the eye is cosine of AB; CH is the tangent
applied, and wider openof BC, or the cotangent of AB; and OH is the secant of ings at B and D, with a BC, or the cosecant of AB.
fine wire fixed vertically The following rules for computation are useful in in the middle of them. right-angled trigonometry; that is, in computing the The cross is supported on sides and angles of right-angled triangles.
& staff E, about 4 feet I. When two sides are given, to find an angle. high, which at the lower
| Make a given side radius, then the side made radius end is pointed and shod A is to the other given side as radius to the trigonome- with brass, so that it can trical name of the latter side,
easily be stuck in the II. When one of the three sides and an angle are ground. The sights are given, to find a side.
placed on the top of the I Make any side radius, then the trigonometrical staff, and fixed to any name of the given side is to that of the required side as position by a screw F. the given side to the required side.
A simple cross staff Although in this case any side may be made radius, may be made by cutting it is preferable to make one of the sides concerned two grooves with a saw along the diagonals of a square radius; that is, either the given or the required side, as board, to be fixed on the top of the staff
. It can easily this introduces the radius as a term of the proportion, be ascertained if the sights are at right angles, by and its logarithm being 10, it simplifies the calcula- directing one pair of them, as AB, to one object, and tion.
observing to what object the other pair, CD, are then III. When the two sides are given, to find the hypo-directed; then by turning the sights till the second tenuse.
object is seen through the first pair of sights AB, if the I The sum of the squares of the two given sides is first object is then visible through the second pair of equal to the square of the hypotenuse./
sights, and is exactly in apparent coincidence with the IV. When the hypotenuse and a side are given, to wire, the sights are at right angles; if not, they must find the other side.
be adjusted. 1 The difference between the squares of the hypo- An instrument not less important in surveying is the tenuse and the given side, is equal to the square of the theodolite. This useful instrument, fixed on the top of required side. Or, the product of the sum and diffe- a tripod, consists of two graduated circles perpendicurence of the hypotenuse and a side, is equal to the lar to each other; one of which is fixed in a horizontal, square of the other side.
and the other in a vertical, plane, and is used for meaWhen the square of a side is known, its square root suring horizontal and vertical angles. gives the value of the side.
In the figure of the theodolite here presented, HRS The first two rules are sufficient for the solution of represents an oblique view of the horizontal circle, all the problems in right-angled trigonometry; but the and mQn a direct last two may sometimes be conveniently employed. view of the vertical
Trigonometry is one of the most useful departments one which extends to in mathematical science. Its applications to practical | little more than a purposes are very extensive, and it is of great impor- semicircle. The vertance in navigation, engineering, and, as we shall im-tical circle is movmediately see, in land-surveying.
able about an imagiLand-surveying is the method of measuring and com- nary axis, coincidputing the area of any small portion of the earth's ing with the radius surface, as a field, a farm, an estate, or district of mo- OQ, which, produced, derate extent. There are three distinct operations in passes through the the art of land-surveying, all of which require the sur-centre C of the horiveyor to possess a competent knowledge of arithmetic, zontal circle. On the algebra, and geometry. In the first place, the several vertical circle is fixed lines and angles must be measured ; secondly, they a telescope W, furnished with a spirit-level I; the must be protracted or laid down on paper, so as to telescope is connected with a movable radius OA, in form a plan or map of the district; and, thirdly, the contact with the opposite side of the vertical circle; whole area of the district must be computed by means and this radius is fixed to a vernier o, movable, by of the foregoing operations. In performing the first means of a screw, along the limb of the circle. When operation, the most useful instrument is a chain called the centre o of the vernier coincides with the middle Gunter's chain, from the name of the inventor, the division Q of the circle, the axis of the telescope is Rev. Edmund Gunter, who lived about two hundred then horizontal, and the instrument thus serves also years ago. It is 22 yards or 66 feet long, and is como as a spirit-level. A vernier to the horizontal circle is posed of 100 equal links, the length of each being 7.92 attached the vertical circle at e, and is movable inches. At every tenth link is a mark made of brass. with it. An acre consists of 10 square chains, or 100,000 square To measure a horizontal angle subtended at the inlinks. There are 80 chains in a mile, and 640 acres in strument by the horizontal distance of two objects : & square mile.
Direct the telescope to one of the objects, and observe Iron pins, about two feet long, called arrows, with the number of degrees at e on the horizontal circle ;
then while this circle remains fixed by means of a | into which a field is to be divided for the purpose of clamping screw, turn the vertical circle till the other taking its dimensions, are called stations, and are de object is visible through the telescope, and in apparent noted by the mark o; thus o, is the first station; 0, coincidence with the intersection of the cross wires, and the second; and so on. The lines joining the staticas, note the number of degrees on the horizontal circle at e; and which are measured by the chain, are called chais then the difference between this and the former num- lines or station lines. ber is the required horizontal angle.
a Divide the field into triangles, or into triangles and To measure a vertical angle: Direct the telescope to quadrilaterals, the principal triangles or quadrilateria the object whose angle of eleration is required; then occupying the great body of the field, and the rest of 's the arc, intercepted between Q and o, is the required containing secondary triangles and trapezoids formed angle. An angle of depression is similarly measured. by offsets from the chain lines. Measure the base add
The plane table is frequently used in surveying. height, or else the three sides of each of the principal This instrument consists of a plane and smooth rect- triangles, then calculate their areas by the rules in angular board fitted in a
Mensuration of Surfaces, and also the offset spares, movable frame of wood
and the sum of all the areas will be that of the entire which fixes the paper on
R the table PT, the ad
Example 1.–Find the contents of the adjoining fel joining figure. The cen- P
from these measurements, A being the first, and B 122 tre of the table below is
second station :fixed to a tripod, having at the top a ball-and-socket joint, so that the table may I be fixed in any required position. The table is fixed in a horizontal position by means of two spirit-levels lying in different directions, or by placing a ball on the table, and observing the position of it in which the ball remains at rest. The edges of one side of the
On chain line.
Offsets. frame are divided into equal parts, for the purpose of
Ag 150 gC = 141 to left. drawing on the paper lines parallel or perpendicular
hE= 180 to right. to the edges of the frame; and the edges of the other
Ai = 597
167 to left. side are divided into degrees corresponding to a cen
Ak = 624 kF = 172 to right. tral point on the board for the purpose of measuring
AB= 769 angles.
The double of the areas of the component triangle Å magnetic compass-box C, is fixed to one side of and trapezoids are found, in order that there may be the table, for determining the bearings of stations and only one division by 2-namely, that of their sun. other objects, and for the purpose of fixing the table in gi = Ai — Ag = 447, ¿B = AB— Ai= 172, 914 the same relative position in different stations. There hk = Ak -- Ah = 301, Bk : = AB - Ak = 145. is also an index-rule of brass IR, fitted with a tele
Twice the area of the scope or sights, one edge of which, called the fiducial triangle AgC Ag 9C: 150 x 141
= 21:59 edge, is in the same plane with the sights, and by which trapezoid Cgil = gi (Cg + Di) = 447 x lines are drawn on the paper to represent the direction (141 + 167)
= 18;675 of any object observed through the sights. This rule triangle DiB = Bi' ¿D = 172 x 167
= 28721 is graduated to serve as a scale of equal parts.
triangle AhE = Ah:hE = 323 x 180
58146 A principle of measuring by triangles, which is trapezoid hEFk = hk (h E + kF). alike cominon to land-surveying and the trigonometri- (180 + 172)
10192 cal surveys of engineers, may be comprehended from triangle BkF BkkF = 145 x 172 the following figure. We wish to find the distance be
Twice area = 37637 tween two objects that are either invisible from each And area = 188291 = 1 acre 3 roods 21-26 poles. other, or inaccessible in a straight line from each other. These admeasurements, instead of being written cat
as above, are generally registered in a tabular fost. A field-book, which is used to enter these measurements, is divided into three columns. The different
distances on the chain line are written down in the A А
middle column, and in the right and left-hand column the offsets are inserted, with any remarks that may be made. The measurements on the chain lines are #Tit.
ten in order upwards in the middle column, the first B
being written at the foot of the column, as the surrera
can thus more conveniently compare the measurements Let A and C be the two objects inaccessible in a with the imaginary lines in the field. straight line from each other, on account of a marsh. In surveying a whole country by trigonometria Measure two lines AB, BC to the objects and the measurement, or in engineering plans for canals, ricontained angle B. In the triangle ABC, two sides ways, and roads, it is necessary to make allowance ft AB, BC, and the contained angle B, are known; hence the earth's convexity in all the calculations of levels AC may be found.
The degree of convexity, or departure from a true level, Such a problem as the above is common in measur. is reckoned to be about 7 inches and 9-10ths in the ing heights and distances; and it will be understood, space of a geographical mile. (See HYDROSTATICE, NC that the principle of throwing the area of any given 15.) In land-measuring, the scale of operations is field or set of fields into triangular spaces, is that pur- ordinarily too limited to require any such allowance sued in all processes of land-measurement. In most for difference of levels. instances fields are irregular in form; their outlines We have now, as far as our limits would admit, prebeing often bent, with a greater width at one place sented an outline of the methods pursued in land-surthan another. In such cases, after measuring the veying; and to those who design following out the study areas of the triangles, the odd pieces at the sides re- of this, as well as other branches of theoretic and prac quire to be measured, and their aggregate area added tical mathematics, we recommend a regular course to the whole. We may illustrate the process of sur- of instruction from Mr Bell's excellent treatises in veying as follows:
CHAMBERS'S EDUCATIONAL COURSE-Works so cheap, as The angular points of the large triangles or polygons, I to be within every one's reach.
DRAWING is an imitative art, by which the forms, posi- , ing, many of even those who travel for the purpose of 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 genemajority of mankind, however, are so far endowed, as ral an application. to be able, when instructed, to delineate any simple Referring to the volumes on Drawing in our Educa object, and to enjoy much pleasure from higher deli- TIONAL COURSE for a methodic series of instructions, our neations produced by others.
design on the present occasion is to present a popular The practice of elementary drawing at school, hitherto view of what may be done by comparatively unlearned greatly overlooked, is calculated to produce the most persons to acquire a knowledge of the art, including beneficial results. As regards those who possess the perspective, which is the foundation of all pictorial delifaculties for design in a high degree of excellence, early neation. It will be understood, then, that our observapractice will awaken those faculties, and furnishing tions are addressed directly to the pupil. them with stimulants to progress, secure the benefit of their ultimate exercise for the community. Lesser
DRAWING. degrees of excellence will also be developed-such as Drawing is effected by various materials, as chalk, would in rain perhaps essay excellence in the higher black-lead, or coloured pencils, Indian ink, &c. Drawwalks of art, but might become of incalculable value in ings of a simple kind are made principally on white paper connection with certain branches of manufacture. or Bristol board, but also sometimes on tinted papers,
As a means of elevating tastes and desires, and in which case the lighter parts are brought out by white thereby embellishing what might be otherwise a rou- crayons. Commence a study of the art by acquiring tine of commonplace existence, drawing appears in its ease of hand, and in fact learn what the hand can do most interesting light. The person who has acquired by its different evolutions. For this purpose, drawing a knowledge of botany, feels a new pleasure in exa- lines with chalk on a black board is perhaps the best mining the parts of a hitherto unseen plant; he who has exercise. Either, therefore, on a board with chalk, or acquired a knowledge of geology is interested in pass-on paper with a pencil, learn to make drawings of lines, ing along a road, the side of which displays a deep sec- straight, curved, or a modification of either. Observe tion of strata, or from which he may view various grani. how much more beautiful is the appearance and effect tic elevations; he who has acquainted himself with the of a curve in comparison with a straight line; and how principles of machinery, experiences an enjoyment in nature delights in this waving of forms, of which we contemplating the intricacies of some great engine which have examples in the bending of boughs in trees, the another knows nothing of; and in the same manner he serpentine winding of rivers, and the curvilinear outwho has studied the art of drawing, discovers a source lines of animals. of new and innocent gratification in the innumerable You may begin the drawing of objects by copying forms and tints of external nature. Things formerly other drawings; but this species of exercise can only passed with a careless eye and a vacant mind, then make you familiar with the manner in which lines assume a character which arrests attention and awakens answer the purpose of representation. To be master of thought. Those faculties of the mind which perceive the art, you must throw aside all drawings or copies, and appreciate the figure, colour, and arrangements of and learn to draw by your own ingenuity from tangible objects, and trace in all a natural and appropriate objects in nature and art. In this study the hand is beauty, spring up from a dormancy which might have taught to obey the conceptions of the mind. When, for otherwise known no interruption ; a new association of example, we see a chair standing on the floor, we obour mysterious being with the physical world around serve its shape or figure, its line of back, seat, legs, and us is practically established; and the value of existence all other parts about it. We then take a pencil, and becomes by just so much enhanced. Not surely that bending the mind intensely on the form of the chair, it is desirable that an absorbing interest should be try to define all the lines of the object on the paper or created in all minds respecting the outward aspect of board. The more perfectly that the hand can obey the nature to the neglect of the more serious affairs of life. direction of the mind, while bent in thought on the All that can be contended for is, that as many as pos- object, so will the drawing more resemble the original sible should be rendered capable of looking with plea- in all its details. sure, instead of indifference, upon the beauties of nature, The pupil, therefore, must be taught to call up the so that they may realise the benefit of this part of the whole power of his mind respecting the appearance of intellectual and sentimental powers which have been an object, when he wishes to represent it by a drawing, conferred upon them; a portion of their nature which, On this, indeed, may be said to rest the fundainental like others, may be abused, but in its moderate use principle of delineation in all its branches. No one can is not only a source of innocent pleasure, but may be possibly attain even a mediocrity in the art, who does come the means of anticipating and supplanting many not, less or more, possess this power of recalling images pursuits of a less worthy character. Nor, while the to the mind, and of training his hand to act in strict art is perhaps chiefly acquired with these views, may conjunction with its dictates. The hand is only the init be without some results of a more directly useful strument employed by the imitative faculty, and can do kind. In many situations--when wandering in our own, nothing useful without the strong concurrence of the or roaming in foreign countries—we may see objects will. Accomplishment in penmanship, and every other of which we would be glad to carry away some memo- art in which design or figure is an element, is founded randum, and of which the slightest pencil sketch would on this fundamental basis. be sufficient to awaken a recollection at any other time. A plan of instruction in drawing has been lately And yet, for want of a few elementary lessons in draw- introduced from the Normal School of Versailles into No. 90.