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Les cartes, planches, tableaux, etc., peuvent §tre filmds d des taux de reduction diffdrents. Lorsque le document est trop grand pour §tre reproduit en un seul clich6, il est film6 d partir de k'angle sup6rieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images ndcessaire. Les diagrammes suivants illustrent la mdthode. 32 X 1 2 3 4 5 6 No. 3. PRICE, 20 CENTS bttni «ioordiiig to Act of the P.tltam.nt of C4iirfm in the <'5 o o n' » y«r OM thou-nd eight hundr«| ,„<■ .ighty^lght, bj Thi Oup Pum,, iini PuiLUBwn Co. of Toronto, at the Department of Agriculture. INTRODUCTORY REMARKS. All work in Linear Perspective requires to be done mechani- cally, except in the case of curves which cannot be drawn by means of compasses. The following are the necessary instruments ; — Pencils — either H or HH, sharpened to a wedge-shaped point, tlie ilat side of which should rest against the ruler in drawing lines. A piece of fine sand paper is about the best thing for keeping tlie point of the pencil sharp, and saves the blade of the pocket knife. Ruler — made of hard wood, at least six inches long, with a straight edge, and divided into inches, and halves, quarters; eighths, and sixteenths of an inch. Compasses — with steel, pencil, and pen points which fit into a socket in one of the legs. The stationary leg should have a needle point if possible, bo that its length maybe altered to corre- spond to whicliever one of the movable points is in use. The .stationary leg siiould be a trifle longer than the other leg when the pencil or pen point is in use, and exactly the same length when the steel point is in use. The penoil used in the pencil point should be a little softer than that used with the ruler, as F or H, and should be sharpened in the same way. In drawing circles its edge should always be perpendicular to the radius. Properly constructed compasses have a hinge joint in each leg, so that when the pencil or pen point is in use, it can be kept perpendicular to the surface of the paper. If tliis is not attended to in the case of the pen point, the pen will not work properly. The joint of the compasses can be tightened or loosened by means of a little metal key which accompanies them. The joint should not be so loose that the legs will change their relative position when the com- passes are being used, nor should it be so tight as to require any exertion to separate the legs. Practice will teach just how tight it should be. The compasses should be held loosely by the joint only, between the thumb and first finger, with the steel or needle point resting on the paper, without any pressure, and the other leg made to revolve around it. The student should practise until he can draw several concentric circles without puncturing the paper with the steel point. It is absolutely necessary that the steel point sliould be as sharp as it is possible to make it. India ink only should be used in the pens, as other inks corrode and spoil the points. The two steel points are used together when it is necessary to measure or to set off distances very accurately. A Drawing Pen for " inking in " straight lines. Its points should be exactly the same length and ground to a sharp rounded edge. In use it should be held nearly vertical, with the handle slightly inclined in the direction of the edge of the ruler, and drawn along the paper at a uniform rate of speed without any stoppages. It should be wiped out with a rag or piece of chamois skin every time it is filled, and before being put away. Protractor, made of either metal, horn, ivory or wood ; used for measuring angles. It is not absolutely necessary, but most boxes of mathematical instruments contain a protractor. Its form and instructions for constructing one are given in an exercise on problem xiii. in book 2, IJigh School Drawing Course. In using it the centre of the semicircle is placed over the point where the angle is to be constructed, with the diameter coinciding with one line of the angle, and a pencil mark made at the circumference opposite the proper number. A line is then drawn through this point from the centre. A S"+ Square, being a triangle of thin wood, will be found useful, .oiigli not necessary, for drawing parallel lines and erect- ing perpendiculars. The ruler is held in position and the set square slid along, with one edge firmly pressed against it. A square about five inches high, having angles of 30°, 60° and 90° will be most convenient. The importance of being able to change the proportion exist- ing between the object and the drawing of it, will be evident when we consider the limited space our paper offers for a picture of a house, a tree, a street, or even of a room. The method adopted for reducing the size of a drawing is called w&rking to a scale, and may be briefly stated as follows : The unit of measure- ment of the object being taken, it is divided into a convenient number of equal parts, and one of the divisions is used as the unit of measurement in the drawing. If an object is 1 2 feet long the unit of measurement is one foot, which is divided into any number of parts, say 1 2. Then one-twelfth of a foot, or one inch becomes the unit of measurement in the drawing, which will be one-twelfth of the natural size of the object, and therefore one foot long. This scale may be expreseed either by the words "scale, 1* to the foot," or by the fraction "y'j." I"* * similar way, if one foot is divid&.l into 16 equal parts and f's ^ ^^^^ ^ the unit of measurement in the drawing, the scale will be one of " j" to the foot,"' or "-f\" It must be remembered that if the scale is expressed by & fraction, it indicates the proportion which every portion of the drawing will bear to the corresponding portion of the object drawn. The sign ' attached to a figure signifies foot or feet, and the sign ", inch or inches .• thus, 1' 6" reads 1 foot 6 inches, and ^ r reads, 2 feet 1 inch, and Scale ^ to 1" reads, Scale 4 feet to 1 incli. HIGH SCHOOL DRAWING COURSE, LINEAR PERSPECTIVE. _ The tern f -erspective, applied to a drawing of an object, indicates tha.r. is a representation of tl,e apparent form of that object when viewed from one point. It has no doubt been noticed, even by the most careless observer, that, except under certain circumstances, objects never appear as they are, and that their appearance changes with every chaiH'o of the spectator's position with regard to them. This dittercnce between app(.araiice and reality is caused partly bv ho convergence of the rays of light, reflected or transmitted l,y the objects to the eye,* and partly by the manner in which these rays cut an imagmary transparent plane interposed between the objects and the spectator. The eye, being opened, admits a flood of light from space, part v Ji' Tv"^- fr'''^ ''■",'" '^^'^^ 'y'-'S ^'^'-i-^ the range of J„n ^ .. ^'f''* Pf":' ""'°"8l> the circular oiumins in the iris, called the pup.l, and the crystalline lens, and exdtes tho optic nerve spread over the inside of tho back of tho eye, thus produc mg the sensation which we call vision. The ravs composing this volume ot light are convergent and meet in thefocal point of the crystallino lenn, forming a cone, the base of which may be sup. posed to be at any distance from the eye. t Only one portion of an object can be seen distinctly at one time In order to obtain a complete and correct idea of the whole, the gaze is directed at diflercnt parts of it until it has all been examined. When tho eyo is (ixed upon one point every- thing about that point in all directions is seen more and more mdistinctly m its distance is increased, so that the angle limiting the lield ot distinct vision is necessarily comparatively small In I>o.spective it is fixed, for tho sake of conve nience, at 60°, and • In i«rspcctiye tho spectator is Bupimscd to bo l,.<,ki"ng withonly one eye. t^:!:^^;^^^:^ if^'^Clhp- -'"F^th;,c;.o-;;f u;; h™;;;ro;cSy lliw shons, too, that tho opening throngli which the light p.a.Hses, everything lying outside is supposed to be invisible ; therefore, in order to make a picture of the whole circle of landscape, the spec- tator would have to change his position six times, thus dividing his horizon into six difierent parts, each one of which would be contained by an angle of 00°. * This is called the visual ancIe or angle of vision. ^ The word perspective is derived from two Latin words, signify, ing " to look through," and naturally suggests the thought that there is a " something" through which the spectator is looking. This "something" is the Picture Plane (P.P.), or plane of delineation, and is an invisible vertical plane, supposed to be interposed at a given distance between the spectator and the object to be drawn. It is represented by the surface upon which tho drawing is made. A good idea of tho picture plane and its use might be obtoined by placing upright in front of the eye a pane of glass, and tracing upon it the outline of objects seen through it, taking care that the eye is kept in one position. The drawing thus made would be a true perspective drawing, and could easily be transferred to a sheet of paper. The position of the eye is called the Station Point (S.P.). The point towards which the eye is directed, being in the centre of the field of vision, is called the Centre of Vision (C. v.). When looking straight ahead the eye of the spectator is naturally fixed upon the liorizon; the Centre of Vision is there- foie in the horizon. If a circle were drawn with a proper radius upon the pane of glass already referred to as representing the picture plane, its circumference would be a picture of the limit of the field of vision, the circle would be tho picture pkne, its centre would be a picture of the centre of vision, and its horizon- tal diameter would bo a picture of the horizon. This last line representing the horizon is called the Horizontal Line (H.L.). A line drawn from the Station Point to the Centre of Vision represents, not only the distance of the spectator from the picture * This 13 illuatratod in Fig. 5. 1-il m plaufi liiit also t\u'. (liieetioii in wliiuh he is looking, anil is culled the Lino of Direction (L/i).). Fi'^. 1 sliows the rehitivo position and sizo of the picture plane witli regard to the spectator. It will be seen that the picture plane is the base of the cone of liglit ente ing the eye; the apex Jl^'T!>\ of the cone is the station point (S.P.) ; its axis is the Una of direction (L.D.) ; the centre of the base is the centre of vision (C.V.) ; and the horizontal diameter of the base is the horizontal line (H.L.). Fig. 1 also shows the fact that the picture plane may be at such a distance from the eye of the spectator as to be wholly visible, but usually it is supposed to be at such a distance no. 2. that its radius is greater than the height of the eye above the ground and hence a portion of it is hidden by the ground lying between it and the spectator. The line where it cuts the ground is called the Ground Line (G.L.). The portion of the picture plane below the ground line may be visible when the spectator is standing on an elevation, or looking down into an excavation. In sucli a case the ground line is supposed to be in the saiiio place as it would be in it' tlie ground were perfectly level, i.e., It is tlie lino of intersection of the picture piano with a liori- zontal plane upon which the spectator is standing. It is supposed that the lino of direction is parallel with the ground plane or horizontal plane,* hence the distance from the horizontal line to the ground line is always equal to the height of the eye of the spectator above the ground. Tlie lines and points thus far explained are in two different planes— the horizontal line, centre of vision and ground lino in a vertical plane ; and the horizontal line, centre of vision, line of direction and station point, in a horizontal plane — and in order to make use of them in working problems in perspective, they •This is not actually the case, for as the horizon is the line where earth and sky appear to meet, a line from the eye to the horizon will fall the distance of the eye from the ground in traversing the distance between the eye and the horizon, which, in the case of a person five feet high, is about three miles. Therefore the angle formed by the line of direction produced and the grounil plane, is the Tertical angle of a triangle having two of its side* three miles lung and its hue &Te feet long. must be supposed to be all brought into o.io plane without alt^r .ngthe rolatve positions of the centre of X„ lS„tll £ sUtirnoin '"Vh"'°' ^'^-rr of vision, line'of diSiS izXct x:v°"'' r"'.'^p°^'""- in ttwo-lu 0° Z;°°':?' 7 '" *^ ^*'"^''* "'"'^'' tho'sidewalks, the Hues ofwin one another The diflerent points where these lines would if produced, ultimately meet are called VanisMng PoSt8(V P.)! rinl!?n'"T"* ""?• '^*''*^ """^ °"° *'''^' '^" P'^^^'lel retiring ,nnf„rP'^f- ^,^^* '" *^'"' '"""^ P°'"*' ^'"1 t''at all parallel horr zontal retiring lines appear to meet in the horizon. In the illustration of the railway track (fig. 4) where the spectator ,s supposed to be standing on one of the rails. thTrails appear to meet m one point ,n the horizon, and this point is the point towards which the gaze is directed (C.V.). The ra la „ th s case are parallel to the line of direction, and their vanishing ea th " aw. 7 '^'■'"PP.T °" "T"""^ °^ *''" rotundity oh? earth. 1 his can bo proved by standing upon each of the rails in succession, where there are several parallel to each other If the spectator turns either to the right or left until he looks m a direction at an angle of 45' with the tracks, thdr vanishh.g point will not be changed, but will occupy a new position with regard to the spectator and his line of direction; that isrwlla was his centre of vision and the vanishing point of the rails, will stiU be their vanishing point when they form an angle of 45° or any other angle with the line of direction From this it is evident tliat if any horizontal line be followed until It cuts the horizon, it will find there its f>vny vanishing point. "i^iiiiij. Suppose that the circumference of the circle in fig. 5 repre- sents the complete horizon visible to a spectator stationed at I P When looking towards A his centre of vision will be the point A, but when looking towards Ji his centre of vision will be the point Ji. Ihe direction of his line of direction, .ind consequently of his picture plane, which is always perpendicular to the line of direo- t.on, changes with every change of his position, and what vas his oentro of vision when looking towards A becomes, when he looks towards B, the vanishing point f,;r all If k-s running in the mrection i J^ A. In the same way what was liis centre of vision when lookmg towards C, becomes, when he looks towards A the vanishing point for all lines running in the direction .SP C These vanisning points are in the horizon, but wlicn the picture plane IB interposed Ix'tween tliom and the spectator they are represented by the points a, b and c on the respective picture planes (P P ) From what has been said the following rules may be deduced :" I. All retiring lines appear to converge. II. All parallel retiring lines appear to converge in the same point. • }^\ A'' Pa^l'el horizontal retiring lines appear to converce m the horizon, represented by J7Z. IV. AH lines perpendicular to the picture plane appear to converge in the centre of vision. t ^V'^? vanishing point of any retiring horizontal line is found by drawing in the proper direction from the station point, a line to cut the horizon, represented by 7/ Z. If one edge or face of an object, such as a book or a pencil be placed against a pane of glass, and its outline traced upon tiie glass, the drawing of the edge or face will be of Uie same size and shape as in the object itself ; hence it may be inf-rred that mea- surements must be taken upon the picture plane (rcpresentod by the pane of glass), and that all the points in an object which are in the same vertical plane, will, when occupying positions in the picture p ane, be represented by points as far apart as they are in the object. In order to bring any particular point of any object into the picture plane it must be supposed to be moved forward in any direction until it touches the picture plane The point where it touches the picture plane is called a Point of Contact (P.O. ) If it be required to find the point of contact of a point situated above the ground plane and away from the picture plane, the point is supposed first to be dropped vertically to the ground plane and then moved towards the picture plane For the proper working of a problem in perspective it is necessary that we be able to define with great exactness the size the shape and the position of tiie object or objects to be drawn Ihe nosition of an object is usually determined by means of some one of Its principal points, which is compared, as regards position, with the picture plane, the ground plane, and the line of direction 1 hus any point may be required which is 2' to the right of the line of direction, 3' back from the picture plane and 10' above the ground plane ; or, a solid object, such aa a cube, may be required whose edges are 2' long, two of whoso fa«-. are parallel to the picture plane, and having the near left hand corner of its base touching the picture plane 5' to the left and 1' above the ground plane Broadly stated, the position of objects may be- -onfabove or below the ground plane; touching, or lying away from the ! h« A.°t^''^ knowing the size, shape and position of the objects to be drawn we mast also know the lieight of the eye of theVect^ tor above the ground, his distance from the picture plane, and the ,cale on which the drawing is to bo made, or, in other words the proportion winch the drawing will bear to th^ object Keferring again to the illustration of the railway track ^fie 4^ • will bo seen that the ties appear to approach one another,' ai v»ell as to decrease in size, as their distance from the eye is in- rZ i» r*'° ^'^'"'O^ y'-^Pi'lly departing train the decrease in size 18 plainly seen, and gives to the mind the idea that some mysterious contracting force is acting upon the sides of the rear carnage causing them to become shorter and closer together until, at a distance of about three or three and a half miles, the whole IS reduced to a point on the horizon «;,1 .?" '""f ' "P *°*1'° "',^* P"'"*^ '^^^"'^ '* '« necessary to con- sider VIZ. : liow vamshing lines can be measured to any required length or, ,n other words, how the position of any point lying away from the picture piano can he represented. ...J iT^'i"^'"'''*/''*' ^"J?* ""^ f'n^'Ple ^V m^ans of which this is accomphshed is.tluit a hue drawn perpendicular to a line bisect, xngan anylc will intersect both lines of the amjh in points enui- distant from their point oj contact. In fig. 6 ^ ^ is limited, £ C IS unhmited in length. The line A E, perpendicular to the line bisecting the angle ABC, makes n D equal to B A. In the same IZ.Pn'u' i' P^/P^'"''"^"'!^' ■ t° the bisecting line, H L, of the rr^w /f A ' T^"^ {<^cqual to 7//^ and P 7' makes //Ji? equal to //y. Applying this principle to what has been learned con- cerning the drawing of lines in perspective, \e^ a b in L (tie G) correspond to and represent A B. The indefinite line B Cis%J- pendicular to^ B and therefore a lino to represent it in pe.-spec- tive, drawn from b, must vanish in C r(Rule iv.) These two lines arc now represented perspectively, and in order to cut off from ic a part which will represent B £> it is necessary to draw from a a lino which will be the perspective representation ot A E ^y means of Rule v. the vanishing point for A E ia found by drawing from ^ ^ a line parallel to it, to cut HZ in BMP (msht Measuring Point). The line drawn from a to UMPrepVl sents A Am perspective, and, cutting 6 c in rf, makes 6d the perspective representation oi B D ,■ that is, M is the foreshort- pi'cture°''la^ne'''^^''"'''' '^°^*''' °^ "* ^^^"^ perpendicular to the In the case of the line II fT, which is not perpendicular to the picture plane. Its vanishing point and measuring point are found by applying Rule v., that is, by drawing ^P F Pi, parallel to a ' ?u A ^/^^'' ?"'■''"''' *" ^^- ^^^' fi"di"« these points the method of proceeding is just the same as in the other case The original retiring line is A F Pi, and it is measured by mean^ Uif _ From tliis iUustration it may bo spen t! , point flaa ita correspondiru^ me" unnrr nmW i J^, ''^"'^^ing ing point for anv vanki m, ,! ^ ^ V^mt < d .„at the measur- ar^ U the va::fsirg"X a- -tro td" distance from the station noinf T ''"'i '^ ""''"« e<,ual to it3 tho horizontal line i, h1 ^ • ° P°"'*^ '^■'"^'•« ^his arc cuts Crawn w^i^":;!' n rrd"&';'f/>ll'''^'^'^- r " f"" -''' tinued to the left it will H ,7l „ li i. ■ "^ "• '"■^'''"^- ^e con- ^^ ^fP (loft m^surin^ loin T^ '." ''"^'T"*'^' ^^^^ ^ poi-it marked and^J^/^ar«asfarto7hL?; S , JK^^ two pomts /. if P S'Sed^fn'^rtk^^^^^^^^ and are from callingthomeas ing pitf .or C kT/'"' ""?'" '"■'^'' calling the measuring points for otho.v ■ 'i^."*'"'^'°. P^'n^s- a-"! proper name. ^ ^ "^'""^ vanishing points by their tanSr ."^tl^^lUgPHni'^b::^":?"? J•'^*•l« "'''=^"""« °^ d- very careful. Suppose So * I n^' ' ''"''""' ^"' "^^^ *» "^ distance rrmiirwl Ti.„ ""'"/ lo M I t then »• n wdl bo the parin/^h^^C^tarSd ritlu^Utstfth'^r""^^^ °" r* lines marked by capital letters. corresponding m VA TuonLEM 1— Represent properly in perspective the position of a pomt on the ground plana 2' to tho right of the line of direction nnd t' iiway from tho picture piano. The eye of spoc- tator is 6' abovo the gronnd and his distance from tho picture plane is 14'. Scale j'j. (Fig. 7.) The first step ia to draw a liorizontal line across the paper and mark tt point souicwhero near its contre to represent the C V. From C Kdraw a vertical line equal in length to tho distance of the spectator from the /'/', M', and mark it L D, and its lower extremity 6^^ The scale in this problem is ^j of tho natural size, that is, the unit of measurement in tho drawing is to bo J. of 12- or 45" or ^\ so that tlie line of direction will be 'g*" or ir long. With C V as a centre and CV SPas radius draw a semi- circle to find tho measuring points and letter them LMP and R M P. Next on L D measure from C V tho height of the eye of the spectator above the ground, 6', which will be i", and through this point draw the 6' L parallel to U L. The point required in this problem is 2' to the right. On G L measure 2', or \", to the right of Z i) to find a point a which will be the T' C of tho point required, when it is moved forward in a direction parallel to Z Z> to touch tho PP. A line from a to C K will be tho represeniation of a lino on the ground piano perpendicular to P P and 2' to tho right throughout its entire length, and so we know that the point sought will be in it. From n measure the distance of the point from the P P, 4', either to the right or left, as a b, and from b draw a lino to one of tho measur- ing points to cut a C T in c. Then c will bo the point required. Problem 2.— Show the position of a point in tlie ground plane 2' to tho left and 6' boyond the picture plane. The eye of the npectator is 7' from the picture plane and 3' 6" above the ground plane. Scale j'g. (Fig. 8.) Measure on CZ, 2' to the left of ZZ> to find a the position which the point required would occupy if brought forward to the PP. From n draw a line to C V. This will represent the track 8 of the point » on being moved kick along tlio ground plane to the horizon, in a. direction perpendicular to P P. From a mea- sure 6' to the right, to b ond draw 6 LMP to cut o CVin c Then a will bo tho point required. 1' r» j°°''*'' ^■— ^^""^ *•'" position of a point V W to the right, .16 distant from the picture plane and 1' atwve the ground plane Jloight of eye 1'3', distance from tho picture plane 3' 6', and scale j',. (Fig. 9.) Find the position which the point would occupy when in the ground plane 1' 3' to the right and 3' 6' back from P P, by measur- ing r 3" to the right of Z /) to a, and 3' 6" from a to 6, and drawing a C K and 6 iSiV^ to intersect in c. At a erect a '*" '-^_ »j^ perpendicular a d equal in height to the dist.ince of the point required above tho ground plane, T, and from d draw a lino to C V to cut a perpendicular from c in e. Then e will be the posi- tion of tho point required. It is evident that c is tiio proper distance to the right, and iiw.ay from the P P, r.;:,\ thjit- d i- *lie proper diaUriuu to tho right and above the ground plane, so that if a lino be drawn from d parellel to a o it will pass over c at the proper distance. In order to represent it in this direction, it must vanish in C V. (Rule iii.) the picture pInL T oevoof t« ''"" "^ ''"["'^Jion and 8' l,eyon,l isgg-efsss? Heighf 3', disCce 5'; Tale ,\.''''"'' '^'^ ' '^'^^^ ^' «-""'• . ExBRCWB 4.— A ball is suspended in the air 8' frn,„ ,u r:t: ft"V,f: '^^^'^r-'hr ^.^-TthrgrSun'dTnd^'^; BepJselftJptit'iL^ottlll'UTa^^L^^^ f"^ «-"^' Problem 4.— Represent in perspective the position of a point 3' to the left, 5' from the picture plane and 7' from the ground. Spectator's eye is 4' from the ground, and 14' from the picture plane. Scale jlj. (Fig. 10.) It will be noticed t'lat the station point has been used thus far, only to show how the measuring points are obtained. Thoy can be found by measuring on the horizontal line +0 the right and left of the centre of vision, the distance of the spectator from the picture plane. In the ensuing illustrations the station point will not be shown. Measure 3' to the left of the line of direction to a and draw aCV. From o measure 5' to 6 and draw 6 Z J/ P. From a draw a i-ertical line a d, 7' long i^nd draw d C V. From c draw a ver- tical line to cut rf C F in e Then . will be the position of the point required. Problem 5. — Show the perspective appearance of a line in the ground plane, parallel to the picture plane. Its left, hand end is r to the left and its right hand end is 4' to the right and 3' back. Position of spectator's eye 3' above the ground plane and 7' 6' from the picture plane. Scale j',. (Fig. 11.) FIS. 11. Find a point, n, on G L V to tlio loft cf LB, and another I'nitit. />, 4' to the right oi L D and draw a C V and h C V. Then th(! li;ft h,\nd end of the line required will bn in a C V, and its nght hand end will be in b C V. From b measure 3' to the left, to c and draw c BMP to cut 6 C V in d. Lines parallel to the picture plane are represented as they are, without any change of direction, and as the lino in this case is in the ground plane, and hence horizontr.l, therefore if from d a horizontal lino be drawn to cut o C r in e, it will be the representation of the line required. As a straight line is the shortest distance between two points, if the perspective position of the extremities of any line can be found, tlio line joining thorn will be the perspective representation of the lino required. PiiOBLEM G.— Represent in perspective a line 6' long, in the ground piano, perpendicular to the picture plane, it.t nearer end being 4' to the loft and 2' beyond the picture plane. Height of eye 5' ; distance from picture plane 10' j scale A. (Fi". 12.) t'"' Hi c^ Fia. 12. OnG L find a 4' to the left of Z D and join a C V. From a measure 2' to b, and from b measure 6' to d and join these poincs with Z M P by lines cutting a C K in c and e. Then e e will be the line required. Prodlem 7.— Draw the perspective view of a Hue in the ground plune having one end 6" to the left and 2' from the picture plane, and the other end 4' to the left and 1' from the picture plane. Height 2' ; distance 5'; scale A. (Fig. 13.) FIG. 13. riu. IJ. Find a C " to tlio lift and d 4' to the left and draw aC V and d C V. Measure 2' from a to b and G" from f< to « and join h L }f P and c L if P. Then c and / will l)e tho oxtroniities of the line r('i|uirr(l, ExKRciSK 5.— Plac tho ground plane 'naralW P'T'"-" " '^™8ht line 3' long in extremity bofrn^rthiiht^^'^f"';'' K^''"' ''' "^^^ ^^-d scale j',. tho appearance of a traoinrnf if J^ "" , '^''' Represent thick ; scale 7j "P°" ""^ S^"'^ ^^'^^ ^ 2' «'i u Probi.bm «. — Place in perfpective a vertical line 14' long, when its lower extremity is in tiie ground plane 12' to the right »nd 30' from t^e picture plane. Height 8' ; distance 36' ; scale A- (Fig- 14-) 1 Hi Cv^^ ^ 4 _______ k HI Fiai4. By means of the points a and b find the point c, which is the position of the lower extremity of the line retjuired. If it be sap- posed to bo moved forward towards the picture plane, in the direction CV c, when in contact with the picture plane it would bo represented by the line a d 14' long. Prom rfdraw a line to C F to cut a vertical line from c in e. Then c e will be the line required. Problem 9.— Place in perspective a square of 3' side, in the ground plane, having two of its sides parallel to the picture plane, and its nearer left hand corner 2' to the left and 2' back! Height 2' ; distance 7' ; scale ■^. (Fig. 15.) As a line may be said to be generated by a point in motion, so a plane may be said to be generated by n. lino in motion in a direction other than that of its length. With this fact in view it will be an easy matter to draw a line in a position similar to that in problem 5, and move it back througli a distance equal to, and in a direction perpendicular to, its length, and thus obtain a square. mp NL e;- "~~~^ y / \. ^^c^ i^. /^^ .^ ^ FI0.I5. ft d In this problem the position of one of the left hand corners of the square is given. In parallel perspective square objects may be said to have near right and left hand corners, and far right and left band con. ,s. This distinction is sufficient to enable the .student to determine accurately which corner is referred to. Ill the present case if the near left liand corner is 2' to the left and the side of the squiins is 3' long, the near right hand corner will be r to the right of L D. Find a 2' to the left and dV to the right, respectively, and from them draw lines to C V. From a measure 2' to the right to b and draw b L M P to c\xt a C V in c. Then c will be the nearer left hand corner of tho square, and a horizontal line drawn from c to cut rf C K in e will be its front side. Referring to what has been said regarding tho measurement of retiring lines it will bo seen that all tinea vaniaMny in the measuring points for C V are at an anyle of JfB" with the P P, hence the line from e to LMP will contain the diagonal of the square required, and, cctting o CFiu/, -will find /the far left hand corner. A horizontal line from ^ to cut dCV in g will com- plete tho square. Problem 1 0. —Show the appearance of a square of 4' side when its plane is parallel to the picture plane, two of its sides beins vertical, and its lower right hand corner on tho ground plane, 2,^ to the right and 3' back. Height 3' ; distance 9' ; scale ' "! (Pig. 16.) <« Measure to the right oi L D 21' to a and draw a CV. From a measure tho distance of the square from the picture plane, 3', to b and draw b It MP to cut a C V i.i c. From a measure 4' to the left to d and draw dCV. From c draw c e parallel to // L to cut dCV in e. This will bo the lower e■> 1 o distance 5' C" ; sciUo 2' to 1". *^ ^'"'K"!' 2' ; it. n,.,;, I.,t h.nd „" rtaCTtufl ,1 "Z »?"" ?»•"•• th. „.e height, dUU... „d"S .'^ifex.tiilo' '"'" "" I I Pkoiilem 11. — !llaki' :„ iicrspc'ctivi! ilrjiwliig of a siiuaro of C side with its plauo vertical ; one of its sides is in tlio ground plane, perpendicular to tlio picture plane ; and its near lower corner is G' to tlin left and 3' from tlio picture plane. Height 4' ; distance 15'; scale J^. (Fig. 17.) f ft ""^•^ imf ^^-..^ ?i^ ^^ J. y <1 ^~"^— - ' b rf (10.17. Having found the point c 6' *o the left and 3' from P P, suppose the square to be moved forward to P P in the direction C V r. Its front side will then be represented by the line a/, and the track of its upper corner will bo in the line / CV. A vertical 1 ine from c to cut _/" C K in jr will be the near side in its proper posi- tion. Ky moans of the measurement b d equal to the side of the square, find \na CV a. point, e, 6' beyond c and from e erect a perpendicular to cut/C V. PnontEM 12. — Place in perspective an equilateral triangle of f>' side, in the ground plane. Its most distant side is parallel to and 5' from the picture plane, and the left hand end of this side is r to the right. Height of eye of spectator 4'; distance from picture plane 11' 9"; scale -^'g. (Fig. 18.) Tlifa triangle in question wiien in this position will have two sides at an angle of 60° with the picture plane and so they will not vanish in the centre of vision. But by means of a slight modification of the rules thus far learned the centre of vision and its measuring points may be used for the purpose of obtaining the position of the corners of the triangle. It is necessary therefore 10 ascertain their position with regard to the picture plane and line of direction, by drawing the triangle and placing it in a posi- tion in regnrd to two perpendicular lines, similar to its position with regard to the picture plane and the line of direction. Tlie ground line, and the line of direction below the ground line may be used for this purpose. Find a point, o, 1' to the right oi L D 14 mill y from G L. From a draw parallel to L, a line, a b, 5' liMig which will be the side of the triangle most distant from PP. I'pon this lino construct tin equilateral triiingle whoso vertex, c, will ljoin(li(! proper position in regard to 7'/' and Jj I). Ifver- tiijil lines bo drawn from a and b to cut 0' L in d and e these piiints will iiidiciito the distance to the right of L D, of a and 6 ; and similarly, a vertical lino from c will cut O fj in a point as far to tlie right of LDo.sc is. If lines be drawn from each of these points in G L, to C V, the corners of the tfianglo will be somewhere in them. But b is the distance h e from P P, there- fore if an arc bo drawn with e as a centre and e 6 as radius it will find on G La point d tliat distance to the left of e. The line d R .\[ P will cut e C r in h which will be the perspective repre- sentation of b. A horizontal line from h to cut rf C Vink will be the perspective representation of the lino a b which is the most distant side of the triangle. In a similar way the point n may be found as the perspective representation of the point c. Then the triangle n k h will he the perspective representation of the triangle in question. Problem 13. — Show the triangle of tlie last problem when its plane is vertical, the edge on which it rests is in the ground plane, perpendicular to the picture plane, and its near end 6' 9* to the loft, and 2' 11" back. Height of eye 5' 9"; distance from picture pkne 13' 10"; scale -ji^. (Fig. 19.) to fron 6 "B grouna pjane and tha most distant side of it ExERoisR 13. — Heights'- dist 8' — — — V, I md a point a o 9' to the loft of L J), and draw a C V. In it by raeuna of LM P find a point, c, 2' 1 V from P P. This wiU be I«llTrr* J" ^ -^ ,'°^««* «'de of the triangle. Anywhere in G L select a lane cf / o long, and on it construct the equilateral tri- 1 J J- '''^' "• angle rf/ e and fron, its vertex draw a horizontal line to cut a vertical Ime from a in A To the right of 6 measure the distance rf/to h and draw h L MP to cut aCVini wiiich will be the the far end of the lowest side of the triangle. A vertical line drawn from e will bisect the base, df, of the triangle. Bisect '.Aaud .Imwm LAfP. Then n will I,,, the jH-rspective centr<> nt the line t and the vertex of the tiin.igle will be vertically .;.bove It at a distance a k. Draw /.• r. to cut a vortical line Irom n in r and join r c, r i. Problem U._Show the appearance nf a hexagon of .T 0" side which la in the ground nlanc. two of ^ts sides bei,.g parallel tu tLo picture plane and its centr.- being 6' 10" to the right and 7' 9' from the picture plane. Height 10' ; distance 19' G" ; s.alo ^L- (Fig. -U ) l.nd the position of a point, a, T 9" below G L and 6' 10" to the I gilt otL D. W ,tl, this point as a centre and a radius equal to the sico of the hexagon, 5' 9', draw a circle, and in the circle construct a hexagon having two sides parallel to G L. Draw vertical lines 1. e!! 7nr V XvT"' ^ ^' ^ '""' ^'•''"' ^'"'^^ P°'"t« '"''^^ draw lines to V. With m as a cent re and m e as radius draw an arc toc'ut Cri in ^ and from p draw a hue towards E Af P to cut m C >' m r. Then r will be the perspective position of the point fin,I .1 "i""" T'^, ''^ ""'■""' °f ■■^''^^ ^^"^'' "■ radius Id and /y hnd the position of the point o corresponding to d, and the point corresponding to/ Then as d and c are the same distance trom I P,a. horizontal luie from o to cut A CK, will give the FIG. 21. perspective position of the point c, and similarly, horizontal lines from r to cut hCV, and from t to cut kCV, will find the per- spective positions of b and g. The lines joining these six points will represent the sides of the hexagon required. Problkm 15 —Show the appearance of the hexagon of the last problem when its plane is vertical, two of its edges are parallel to the ground plane and perpendicular to the picture plane and the near end of the side on which it rests is 6' 10" to the left and 4' beyo'id the picture plane. Height 9' 9" ; distance 19 6; scale bV (Fig. 21.) Draw a hexagon of W 9" side, with one of its sides in the ground lino Draw horizont.al lines through h k and y / to a perpen- dicular erected at a point on G L 6' 10" to the left. The perspective position of tlio near end of the lowest side is found by a measure- ment 4 to the right of a and a line towards L MP to cut a CV in the point c. The per.spcctive length of the lowest side is n,.>;t touiKl and vertical lines drawn from its ends to meet a lino from M to C ( . The line marked y z will be the top side. It will be seen by the illustration that the distance of p to the right of ,/ ana die distance of o to the left of ., is equal to one half tl,,'. rx^v ?o !"' '"'" "*,"'" '"'•'^■■'K"". tlieivfore points mu.st l„. found in „ ri,2 lOJ nearer tlmnr, ami 2' lO.l," beyond »•. Vertical lino; f,on, rand x to cut n CV in i andj wiiriind t!,- two remaining corners per its a p oft Exercise 14. HeiVhf >■.' • ,r » Of the tnie . . eite:^ ir^L^rJu^ ;e in £S"l4 slow Thf Ii:r"" ''«■«''*- distance and Bcale as therein ^»heniti in the' ro'und^Z^"^*^^ *™°«'^ °'^"«°"'«^ picture plane, and ite vertex tonchi?fA '•''?*" '" P''™"^' *« the 1' to the right. "*"" "'^ P"='"'« Plwie in » point ft •' persper:"nequTlS \ria'n.tts' ^J ^ ^^'^''^ 'V P'- - its base being perSL«h"to^^^^^ ",> *'"« g'-o"nd piano, a point 2' to the left The vpI %^if *"'"''. P''*"''' touching it in of the eye. ^ '^^''*=^ "^ *''« t"a".','l« is to the right n last problen, ,hoJS, JL tdant^^^t' ''"^"? ^"'^ ^<='''« «« ts b,.so is horizontal and pe^pendfcnttT til '*' P'^"^ " ^'"^''=^. Its vertex is in the ground^p^^e v ^^'^, ^ the p.cture plane, and the picture plane ^ *° '''^ "S''* *•"' 5' back from 6 5 5' Pnonr-EM IC— ncpreaent in pprspoctivo n. oircln C in diamtg. (Fig. 22.) c K ^ ^ \ d nc.22. / \ \ / i) \ i • ■ j°"h^ *° *^ *''^ '^"'■^'^ '^'"'^'i ^^i" represent a circle when viewed obliquely, it is necessary to obtain several points in its circumference the perspective position of which can be easily ascertained. For this purpose it is enclosed in a square and tlie diameter and diagonals of the square are drawn, making ei^lit pomts in the circumference of tho circle, viz. : one at each extrem- ity of each diameter, and four others, wliere tho diagonals cut it J.he square enclosing tho circle must be placed in its proper posi- tion below the ground line aa in the case of tho triancle and hexagon. ° In this problem the circle touches the picture plane in a point opposite to the eye. It is evident that excepting when a circle is in a plane, Its circumference can touch the plane in only one point, and that a line drawn in the plane of the circle from the point of contact, perpendicular to the line of intersection of the two planes, will pass through tho centre of the circle. Applying this to the circle in question, as its point of contact with tlie picture plane is opposite to the eye, its centre is also opposite to the eye and therefore in the line of direction. Find on Z2) a point 6 distant from a the length of the IB radius, 3 , of tho circle and with /, ,l, a eentro and i a ns radius draw a circle enclose it in a square and draw the diameters and diagonals of the square. Next place tho square, with its diameters and diagonals, m pi-rspoctiNe. From the points, g h, k and L where tho diagonals of the square cross tho circumference of the circle, draw vertical lines to 6' X and thence towards C K to cut the diagonals of tho perspective .square in points s, t, u and v. through tluso four points, and the extremities, a o and pr of the diameters, draw an elliptical curve which will be the perspec- tive representation of tlie circle. f , 9 \^ ^/ /V"^ P^/S^ LHP /\ /\ / ^^^^Cl' ^^ ^ \ / ] '' / V \J /v. .^\ 1 /"' ^"**'**'*^ -.^^ » t H0.2S. t» Problem ]7..-Show the circle of the last problem when its plane is vertical, perpendicukr to the picture plane, and its cir- cumference touches tho ground plane and the picture plane in points 4 to the left. Height 4' ; distance 1 1 ' 9" ^ scale j'g. (Fig. When in this position the centre of the circle will be 3' from the picture plane, 3' above the ground plane and 4' to the left find o 4' to the left. From a measure 3' to b, at b erect a perpendicular 3' long and with « as a centre and c 6 as radius draw a circle Enclose tlio circle in a square aghk and draw it^ diameters and diagonals. Place the square with its diameters and diagonals in perspective. From the points marked I, m, n and o draw horizontal lines to the perpendicular o A and thence towards C F to cut the diagonals of tlin sqij^ire in the points tttt. Through the points d, t, v, i, x, t, iv, t draw an ellipse Problem 18.— A circle 10' in diameter stands upright on the ground plane, parallel to tho picture plane at a distance of 5' beyond it. Its centre is 4' to the left of the eye. Show its appearance. Height 6' ; distance IC ; scale ^\. (Fig '>4 ) ai LxKncs.: 10._n.,.l,t r,' ; distnnco 11'; s.alo ,' A l.o.- agou of 4 sido stands on tho ground plane ^vitll two aides vor o^d nndparalk-l to tl,o picture piano, and its piano perSiouiS tho picture piano. Tho point on which it ^rosts L S^to the rTgl t and 4 back. .Show its perspective uppoaranco. '^ Exercise 20.-Heisht 4' 6'; di.stanco 9'; soalo ,«- Placo and touching the picture plane in a point 3' to tho left EXERCSK 21.-He,ght4'G"; distance 9'; scale,-. Show the appearance of a circle 5' in diameter, its plane beii^g perpen- dicular to the picture piano and ground plane, and its circu.nfer- enco touching the ground plane in a point 8' from tho picture piano and 3' to the right. ^'i-i-uii. i'i I 19 Find a point r, on tlie ground f to the left and .V bovond tli,- I>i-tM.;o ,,l:,Me. This will bo tl,., point of .■„„taet of tlw"drclo in 'inos .on wall th<. ground, and its centro will bo directly over this |,o,nt at a d.staneo of 5'. At a em.t a i-erpendieular .V lon^ and draw d C y to cut a vertical line fro.n ,,■ in «. Then e c will be the i.erspe>Hvo length of ono of the radii of the circle. With e us a centre and « -■ aa a radius draw u circle. FIG. 24, Problem 19._Show the circle of the last problem when it is removed to a distance of 20' from the picture plane, its centre -"""/t- *°oJ'f '^^* °^ *'"' ''y^' Height C ; distance 20- ; scale sV- (i''g- 25.) If the usual method of measuring vanishing lines were adopted, in order to hnd a point on a C V, 20' from tho picture plane, it would be necessary to measure from « to tho right, a distance of -,, , and the point on the ground line thus found, would bo be- yond the Inmt of the paper. Tho most convenient method of ineasunng great distances is to use a IM/ Mmsuriiuj Point found Dy Ijisectmg tho distance between any vanishing point and its measuring point. When the half measuring point is used, ono- lialt ot the niea.surenient roiiuired is taken on the ground line. FIG. 25 kself Iv n. T *',? *°.''° *■■'''•'" ''"y""^ a point which is tins point must be f. and by a lino from the half me.asurin- point and the hall measurement on the ground line must bo taken froni the point of contact. That is, if the position of the circle™ question were given as 15' beyond the circle of the last proWem fiom J MP through c to obtain a point of contact on a L, and diZ:eVi"?:s:c." (Fig:°26.r^ "'""^^ p'^""- «^^«'" «" »r „ flC. 26. ferrng to wh.at has been said in connection with problem IC as to tho position of the centre of a circle with r ird to t lo noint ."Which Its circumference touches a plane, it wi 11 be s o7t° a the Imo of intersection of the plane of the circle uifl. +) j plane is in a (7 Fand thatalin^ ?^n-S:nd:S ^'J^f^Zn he centre of the circle. This perpendicular line will be paraTlel o r:r7Jir P-^- P'^- pleasure fr^l^^ *S^ ^f^^ t^chesthe^rounrV^'i;^-^ tho san. way as the circle in problem 1 7. On eachtileofT measure 4 to /and jr, and from these points draw lines to RuTl the points through which to draw the curve ^ '" ""'"'"^ tXEHClSE 22.-noi«ht 16'; digtftnon 40'; scale 16' to 1' allelto the picture plane, and itn centre being .JO' l.ovond the picture plane, 10' above the ground pk„o and ,V to the HgS ExERcisK 24-He.ght 5' 6'; distance 9'; scale ," Pkce .n perspective a hexagon of 2' 6' side, when perpendlcSf^r to the ground plane, two of its sides being perpendicular to le picture plane, and its centre is 2' to the leftfa' back from t e picture plane, and 2' 6' above the ground plan^ ExKRc.SE 23._HoiKht 1' 6'; distance 4'; scale ,' . Place llleTtTtI ni : ^'"r" "^ l"" '"^•'' ^" °f ''« «*•««« are pTr allel to the picture plane, and two of them are parallel to tho th^'l.^^ ^^Tv '' ^"".""^ '" "' '"'''^ ^^"■^ t''« picture plane 2' to the left and 1' above the ground. r""". - w Exercise 25 -Show the hexagon of exorcise 24 when its plane is perpendicular to the picture piano ind ground pllne two is 4 o tt%T;hr''?Tror'r.'''°*''"' ""''"'r "''-- -dTcentT: Pno < ,KM 21— 'Inoo m prrHp.-ctivo a tuIh, of 4' 10" r.lRe stand- UK on tljo Krn„,„l ,,|,.,.„ w„h two of its furps i.ir„II..l U, tl,u ».turo ,,l,u,o ami ll„, ,„.,.,• Ini haiul conuT of i.s bus,. tou..hin« lio ,„ctu.o ,,!.„„ iV "''^''. ' ^' '''^'' ^^•'''^•'' ^^"' -"t""> '^ ''i"«"""l <-f tl" las.; and w,ll tl„.r..foro cnt aCY in a point c r..pr,...scntini,' t'"' far left hand corner of the base. ] )raNv ,• d parallel t,. a /,. On a A eon- Ktruct tl,o s,,Mare «/y6, and from /and y draw linos to CT to cut vertical lines from c and ,/ in /, ,ind k. Join h k. Puoi.r.KM 2i._Pla.o iu perspoctivo a block C square and 3' to « ^f^ will pass through »- an.I i and «. From this it mrtv b« nterrod tnat either the C V or the measuring pointg may )«, Gsod n Measuring verti-.d distances. In this c.vso tho meiusuring point ho bet ,.r ono to use, a., by means of it only one vortic^a line n..|U.r..d for oMainin« the hei„d,t of tho block and tho height of the pole, while ,t C Y bo use.l, at least two vortical linos mSst be drawn, ono tion. o or ,, to lind tho height of tho block, and another Irom / to iind tho height of the polo. sto,rH"t''«' -^iTT''*';""',*'"'' P«'-«P'"ctive appearance of a block of stone M x « X IG standing on tho groun.l piano with ite axis o.t.cal and two of its large faces parallc.| to tho picture plane. Il'o .joar right hand corner of tho base is 1 U' to the right and 4' beyond tho puturn plane. The eye of spectator is ,V (i* fron tho ground, and 30 from the picture plane. Scale ,V (Fig "8 ) LMP ¥L ^ / X ' ^ ""-/- -- y *^^ • s^ 6L • Q thick, resting on the ground piano on ono of its large faces, and having two of its small faces parallel to tho picture pliu. The centre of tho baso is 5' to tho right and 0' away from th, picture fj "^\ /" *, '''■"*'" "^ *'"' *"P ^^"^ !>'■'>«•" "■ vortical polo 5' hi-h. ileight 6 ; distance 14' 9" ; scale :^\. (Fig. 27.) Find the point n 5' to tho right and C beyond /' /'. As this is tho centre of tho baso mea.suro on each side of / 3' to o and p and draw o CFand ;; CT to cut m li M P in r and s. Draw re and _« I parallel to (/ L From o and ;, draw vertical linos o t and pu 6 long and join their o.vtremities with VV. Ihnw veitieil lines from tho corners of the base to cut t C Y and nVY in w y z and r yhich points will bo tho corners of tins top of tho hljek iTom I draw a vertical lino 8' long, and from its upper oxtreiiitv draw a line to C Y to cut a vertical line from n in e. Then » e ' bo tli(! pole required. If a vertical lino bo drawn at m and me, Buremeuts of 3' and 5' taken on it, lines drawn from these po'nts 22 r t!\t U~n t'"" ''"'".'> ^-' *°,/',';' "S''t °^ ^' D, from b measure 1 i.d^t .?, 1 eV ' Tl'^T ' V^'- T'"^" '' ^^i" •>« the near 1 ifilit 1,„ ml corner of tho base of the block. Tho near loft hand cornor ot the base will bo 8' to the left of d. Find a 8' o the W n I. and draw „ C Y From ,/ draw a hori^ontalin; to cV 'v . e. Then rf« wdl be the fro f side of tlie base of the block n « draw a lino to Jt .^n' to ..t 6 C Y in f, and draw' '' UKl to d,: Ihen e./Jd will bo tho baso of the block -' ,1„ > ock bo supposed to bo moved forward until its fron face touc e, vertical lino e,ua. iii 1^^ ::1^: ,:;j^ ^^^ ?s:^ ^iHz: : aiir^ifS'^ trf T *° ^ f -^ perpidfczl ^:: oenlkMlar f;om /'in ^ -''" " ''""'""*''' !"'« *« ™t a per- f em • .rT •"' ""''v'"'" '" ^'^^ Cr by a line cutting a !• . emb.tiii:ir from i^ in «. Join »i/, v-utunj, a IvVKUCHB 5C._Tn tlio illustration M.nv, A /I C D J^ F U a ho^•^so^ ,!• ,.,.,. on a sculo of ,V tiivo in ^our own w.'ds iu [>o.lt„>n and 8uo, and ti.e position of tl,„ spcctnlor. T'i, t ' mako It tho baao of a pyramid whoso altitudo W'ill bo T. Exercise 2S._neigl,t 4'; distnitco R'; Rcnlo '. Uoprc-sonfc p operlym porsprctivo a cubo of 3' od^o r,-sti„."^n tho Lr C « ounJ:? ^°"'" f-/'^ -'S'-r-llel to^.oth piotuto pla^o a,;,! ground plane, and its near r.glit corners 1' to tlio left and 2' back loZ *''<;.r"'-'"''" P''-'""- Represent also a triangular prism 4' holm, pT°" ♦''^K■•r"'' P'"^"" "P"" ""o «f it« oblonK faces, their 1 i. '''I °*,^'''"'^'' »■■« P'"''«l'''l to tho pictu.0 piano with cube. The edges of tho . uls of tliis prism are all 2' 6" long. RxRiinsR 2, -Plueo ,•„ porspeelivo a cirde, 8' in dl.,„ •..,• when ,ts piano ts n,elined ..pwards to tho left nt an «n«lo f ,■' .„ 1 i "\'l","/""P'« '""'-■''^» the ground plane in a • - tho left ond 5' back from tho picture piano SS ^ liOULE.M H I.- Show the block mentioned in tlio last problem wlicMi It IS lyiM;Lf Oil the jrmund upon one of its obloni; faces, itj two rnds bemg pan.llel to Iho picture plane, and its near ridit hand corners being 5' to the left and i' beyond the picture plane ifeight 5' G'; distance 30'; scale j'j. (Fig. L>8.) I'ind a point /•, 5' to the left and l' be)"jnd the picture plane. iJleasure 8' from o to s, draw s C )' aud a hori^iontal line from r to at .m angle of 45° with dc and tlierefore contains one diagonal of the base, and will cut d C V in the near right iiand corner, p of the base. Complete the base by drawing horizontal lines from c and e cut to a C V and d C V. Before proceeding to measure the height or thickness of the b ock. It will be well to consider that the centres of the three objects under consideration are in the same vertical line, and that cut It. At s erect a perpendicular, s.r, S' long and draw rCV Draw a vertical line from the left hand end of the horizontal line from r, to cut a: C V in y. Complete the ne.ir end of the block by a horizontal line from y and a vertical line f:„m r, to intersect i., z I'lnd tlio lower riglit hand corner of the f»" end of the block bva measureiiient of l»i' to the right of p to t, and a line t L MP cut- ting o C) in V. Draw a line i-ro parallel to (J L. This will be the lower edge of the far end of the block. On vw construct a square, and join its upper corners with the upper comers of the square representing the near end of the block, o, ■^!?°'^';f'*' .-•"'•— Represent in perspective a block of atone ^ X .} X 1 lying on the ground plane upon one of its square fac^s two edges of which are parallel to the picture plane, its far loft hand corner being 2' to the right and 4 from tlie i.iotiire plane. ' Centrally upon this place a cube of 2' edge whose sides are panille to the .■orre.sponding sides of the block on which it rests Make the top face of the cube the base of a pyramid 2' hish Height of the rye 8'; distance from the picture plane T G"- scale j',, (Fig. 29 ) i v ■ Find „, 'J' to the right of /. /), to the right of a measure 4' to 6, and draw « C V and b L MP intersecting in r which will be the far left h/vnd corner of the l)ase of the block. Measure 3' from a to d and .iraw d C V. The line through c vanishing m LMP is ■u their 8.de,s are parallel. Therefore a diagonal of the base of the block will pa.s vertically beneath two corners of the base and two corners of the top of the cube, also two corners of the base and the vertex of the pyramid. Prom this it is manifest that if lines be drawn parallel to this diagonal and at proper distances vertically above it, one will pass through two corners of the top will nJ= H ^ *r "°™"'' °^ *''" '^"•'"' "* *•'« <^"be, another will pass through two corners of the top of the cube which are ;tn„Xtr . " }T "* t'-^.Py^mid, and another will pass through the vertex of the pyramid. One of the diagonals, c e, of the base of the block is already produced to cut the ground line in h. At b erect a perpendiculai on It me,isu.e 1 to ,j and draw g L MP. Vertical lines from e and e to cut this will find the near right hand and the far left hand corners of the top of the block. A vertical line from tiie near left hand corner of the base will cut a horizontal line from hm the near left hand corner of the top, and a line from it to C \ to cut a vertical line from <• will be the left hand edge of the top. Having obtained these points and lines, the block can easily be completed. As the edges of the cube are 1' shorter than the edges of the top and bottom of the block, its right and left hand faces will be 6" to the left and right of the corresponding faces o the block therefore measure 6" to the right of a, to k, and 6" to the left of d, to I, and draw A CF and iCF cutting the diagonal ExKRcisB29._Height 6'; distance 10' .aoale ' VU ■ long. °* ^^^ *°P ^""^ P'^'^e an upright pole 4' •iS^!±z^!:«'''«'^:^'^?--'«■ porp;^;;;p;;;^::«^? I \^^^^l '^' ; -i^,^. . show ti. with its axis vertical a.n,i u.,1T(\ \\ '° standing on end picture plane The centre othl f obong faces parallel to the back. ^"'^ °^ "'^ ^«^° " « to the right and .O' -Height 8' ; distance 15' ; scale ,'. ExEacisR 31 , edgo of the block. ^''- ^"^"'^ **'° "^PPearanco of a vertical polo 10' hZ rZ'ZlLJll'' '"""■° °^ **"-' "^8° "" ^^'>i'=l' ig rw..ufe against the centre of the far horizontal ecmp and ?!. These points will be tho far left hand and the n-.AT right hand comers of a square roprosentinjj tho base of tho cube when rcstii\g on tlie ground. Find the other two corners of this square, and from tho points 7)i, p, o and n, draw vertical lines to cut tho diagonals of the top of tho block in r, i; t and h. Join these points and thus obtain tho base of tho cube. The top of the cube wlien it is in tho position mentioned will be on a level with the eyo,_and therefore its top face will be represented by a straight line in II L. From tho points r, r, t and .«, draw vertieal linos to cut // A. These lines will complete tho cube. Next, from w, measure on the perpendicular erected at 6, 2' to x, and draw x First finil a point, D, on the ground plane 1' to the right and .">' from the pictur' piano. Measure .")' to the left of A to //, and draw II CV. From D draw a horizontal lino to cut li CV in E. Then D E will bo the top edge of tho near wall of the excavation. From what lias been .said in connec'tion with tho explanation of tlio picture plane and its use, and from the statement of the fact that the portion of the picture plane below tho ground line can be rendered visible, it may bo inferred that me.asurenients on the picture piano can be taken below tho ground line aa well as on it or above it, and conseiiuently, that if vertical lines U' or J' long bo drawn from A and B, and their lower extremities /''andtf CMP '/iMP 67 A MP to cut a vertical line from tho centre of tho base of tho block, in J. This will bo tho vcatex of the pyramid. Join it with tho points representing the corners of the top of the cube. Piiom.KM 26.— Tho spectator is looking into an excavation Ti' wide, 1 2' long, and .1 ' f)" deep. Its long sides are perpendicular lo thn picture phine. the near top eoriicr nf tlie ri^ht liainl face being 3' fi'om th(^ picture plane and 1' to tho right. "(Fig. 2!».) Show tho appearan<'o of the excavation, and represent by a lino the position and size of a man .">' (i' high, standing in "the excavation midway between the side walls and 4 from the far end. Height 3' ; distance 7' G" 30 scale j'^ . (Fife'. 20.) brqund />'• fee icIlcS, «J ft-S»u*UfSl. rig. 29. bo joined, tho oblong BFGA will represent tho appearance of tho near wall of the excavation if it were moved forward to touch tho picture plane. Therefore lines from /'and G to C V will tind the lower edges of the left and right hand walls. In order to measure the distance of tho far wall from the picture plane, 1,")', it will ))e niu.st coiiM-iiirnt to »:;(■ .a half nic.i.iuring point, found us explained in problem I'J. Produce F G indetinitely to the right, and on it measure fioni // one-half of 1.") , or ,S;]", and from A' draw a lino to \ ^f I' to cut // CV in L. ' The lino 11 CV is u.sed instead of /' CV or (! CV, because by means of it the position of the man in the problem can be ascertained, as well as ExERcisB 32.-In the illustration below ABCDi&thc persoec- W°h"rcel'reT" °' '' ^'f ^' "^''"» °^ '* ^^^ *"« '™- H^«tan.i , ! '» "• ""'^ measuring points, and the height, tra Iv nn7h ' m"", ^Y^'' '*' *'"' ^"^^^ °^ ^ "ock 3' thick. Cen! 3J. t' ^'""'^ P'^'f * "''^ °^ 5' «^dge whose edges will bo parallel to the corresponding etlges of the block. ExERCisK 33._The lino EF is the lowest edge of the left hand square ace ot a b ock 2' thick. Ascertain and state in !our own words Its si?.e and position and show the appearance of hole 6' quare passing horizontally through it fro.n face to face. The ho e passes thi^ough the centre of the square faces, itstopa,,d bottom edges being horizontal ^ )i .mi ii w the position of the far wall of the excavation, and thus a line is Trv ■ r"?rl "'''n'^ '■' ''"'•'^""t'-'l li"° to cntF CV in M and vation. From V and ^ dra«- tho vertical lines MP and .VO VrL! 7 ' "'ll' <^°'"I'l^'t" the drawing of the excavation. JnTnf ;, '"'''"™ ^' ^' *'■" '"■''""^'' °f *•'« '»'^'» f'-o>" the far end of the excavation, 4', and draw Ji ], M P. Then .S' will HT\K-'f '""^ ^V' ^^-^"'""K- At // e-rect a perpendicular • Then ^T'"' ■n",'' '^/r ^^^ *" <="* ^ ^'"•^''^''' '■"« f^°'" '^' in I . Ilien .S y will bo the representation of the man nn»!^r?-''i'-"'^r;.^'''""' '" V^'^-m'^Uy^ a flight of five steps, each one ot which ,s 5' 6" long, 1 1" :.igh and -'' wide. The front f^ce .f IMP When the ends are parallel to the picture plane. The steps ascend towards the left from a line 1' 10' to the left, the near end of which '" /P ■^n'j" P"'"'''' P^^"^- "«'g'»t 6' C; distance 13' ; scale rT-^'n^, °''''''" Vf P°«i*'°n of the corners of the oblong space J 6 X 9 . covered by the steps. Divide the space o h into five equal parts, and draw lines from 2', 4', 6' and 8' towards C T as far as n t At o erect a perpendicular 4' 7" high, divide it into five equal parts, and from the points of division draw lines towards C K as ara.s;,,- From the points of division in «r draw horizontal lines to intersect vertical lines from the points of division in »/ 1 he points of intersection, .,, f, v, w and x, of these lines connected by vertical and horizontal lines wiU give the profile of the near MMP '■■■• t Ol each step :3 parallel to and facing the picture plane, and the near nglit hand corners of the lowest stop are 'J' 3" to the right, and - 9 beyond the picture plane. Height of eye C G"; distance from picture plane 1 3' ; scale f to the foot. (Fig. 30.) FimU the near right hand corner of the bottom of the front, step. On 6- Z, from b measure five distances of 22' each, and draw lines from the points 2, 4, 6, 8 and 10 towards RMP to cut a /k *'"f "'"y "'° perspective width of each step is obtained on a I . At a erect a perpendicular 55 " long, divide it into five e;iuai parts and from these points of division draw lines towards C I to cut the vertical lines shown, drawn from a C V. These lines |il give the j)erspoctive appearance of the right hand end of the lliglit nt steps. 1- rom a measure 5' 6" to the left to c, draw o C V and from *" one another, ground plaT The centre n/r' ^.Y^ P'"*""-" P'ane and above the ground plane 2' from M •'?''-'•'* ^-^^ of the cross is 6" Height rrrdTstevf:^,e';^"*""P^^'^^'^^ w;th^™:,'^7he "ounfn^^^^^^^ Py--''^ «^-ding of the base are 18" lon^u and if<= . f -^"^ o-' T^''^'*'' ♦''o edget of the base ia ] -T to L lef nnH 1'". u . '^ '°"8- ^l-o centre are paralld to the pLut plane ' *"" •"^«'=' °' '^' »^^ Exefcirarand^-syaTJ'^V^ir,^-^''* -'^ '^-t-- as in Exercise 40.— Hei"lit T n" ■ ,!,•■,* i. lies on the ground withThe axis' n •^'"''. t ' .''^'° i'^- ^ -^--oss pondicular fo the picture Jlane^ts sh^ftlsl' r''T"l'^' ^"-^ P^ arms are cubes of r 4' side att.c Imo H t I >*• '■^''° fw^es 1' 4" from tho top of ttsJa 1 T « f '"'.''' ."'"'-^ *°P the shaft is 3' 6" to the VanVS' JLTH: STet^ ^^^ ''^ J IJ n fpntro of tho spliorp. Tliig is illustrated in thu small drawing to tho right, showing tho lolativo position of the sjihoro and llio spectator, drawn to a scale of j\. Tho lino y s shows the diamntcr of the circle which will represent the sphere, and the point x, its rratre, is tlio distance x x nearer than the centre of tho sphere. Measure twico x' x from on P, and represent tho point S in perspective at s. With » as a centre, and a radius equal to its distance from either of tho extremities of the ellipse represent LMP It will bo noticed that this square is in a different position with regard to tho picture plane than tho objects treated of by previous problems ; that its sides are neither parallel nor perpen- dicular to the picture plane. However, it can be treated in the same way as tho triangle or hexagon, althouga in this caao this is not tho most convenient method. First draw the square A li C D with one corner touching G L at A, 3' to tho right. From the corners draw vertical lines to cut the ground line in d, A and b, and from these points draw lines to C V. From d measure, by means of an arc, the distance of D from tho a L, and draw A VPi to cut rf C K in d. In a simihir way, liy means of tho arc C e, iind tho position of the point c. From d' draw a horizontal lino to cut b C V in b', and join d' c and A b'. It will be seen that the sides of the square Ht4P Gi rin- 3S- Th, iiig the flat face of tho hemisphere, dr.iw a semicircle, spectivo centre of the hemisphere is shown at o. PnonLf;M 33. — Place in perspective a square of T)' side in the i;round plane. Its sides are at .'Ui an,'lo of 4.3" with the picture plane, and its nearest corner touches the pict.iro plane 3' to the right, ileight 5' ; distance It'; scale j'^. (Fig. 33.) 83 vanish in VPi and VPi, which points are also the measuring points for C V. Uut from tig. 0, and tho remarks made thereon v,o h.ivi! learned that every vanishing point has its corresponding measuring point, and we have also learne and np. Find the itre r of the base of tiie cube by means of the diagonals, from ; draw a vertical line to intersect a CVxtit. Join to,tm,tn and Ip. Pkoblbm 36. — A model of an obelisk 8" square at the base, C" square at the top, stands on the ground plane with its axis problpms 3.3 and 3 J, and draw its diagonals. Centrally between D uiid /<, uikI C and E, tj>ke the niHasureniont of the edges of the top of the shaft and transfer these measurements to the front edges of tlie base of the shaft, in Z, M, A' and 0. From these points draw lines towards VP\ and V Pa to cut the diagonals of the large square, and thus obtain a smaller square representing tiie top of the shaft when in the ground plane. At P erect a Fig. 34. vertical and the nearest corner of the base 12" to the right and perpendicular, and on it measure 2' to W, and from W meaaut« 6* to R. Draw W CV and R C V. Draw vertical lines from the 4' from the picture plane. The edges of the base are at an angle of 45" with the picture plane. The top of the shaft of the obelisk is finished with a pyramid 6' square and 6" high. The total height is 2'. Show its perspective appearance. Height r 3" ; distance 2' 3' ; scale ^. (Fig. 34.) Find the perspective position of the corners of the base, as in 85 corners of the smaller square, two of them to cut R CV and the other two to cut S VP\ and S V Pi. In this way the appear- ance of the top of the shaft is obtained. Join its comers wiih A, F, H and G, and also with the point X where a vertical line from A"" cuts W CY. ^ is H t !! ( i t ' I. 1 THE HIGH SCHOOL DRAWI NG C OURSE. THE FOLLOWING ARE THE BOOKS IN THIS COURSE : . 1. — Freehand. 3.— Linear Perspective. 2. — Practical Geometry. 4 — Object Drawing. 5. — Industrial Design. These Books are fully illustrated, and printed on heavy drawing paper. They are sold at 20 cents each, at all bookstores. 7^/ie Mechanical Drawing Course, THIS COURSE WILL CONSIST OP THE FOLLOWING BOOKS : I. — Projection and Descriptive Geometry. 3. — Building Construction. 2. — Machine Drawing. 4. — Industrial Design. S- — Advanced Perspective. In both of the above Courses, the trade will be supplied by Toronto Wholesale Dealers in School Books. THE GRIP PRINTING A^ f UBLISHIJ{G COJ\IPANY, publishers, 26 & 28 FRONT STREET WEST. TORONTO.