r • . ■ ' ' • * ■ % r . ? | 3 ;• ,«*; • '• j- *s 1» « t 1 V 1 rr,l : - C *.s- • V • T*i *»• THE ELEMENTS OE PERSPECTIVE: ILLUSTRATED BY NUMEROUS EXAMPLES AND DIAGRAMS. BY AAEON PENLEY, PBOFESSOB OF DBA WING AND PEBSPECTTVE TO THE HONOUBABLE EAST INDIA COMPANY’S MILITABY COLLEGE AT ADDISCOMBE, AND MEMSEE OE THE NEW SOCIETY OE PA1NTEBS IN WATEB-COLOUES, ETC. ETC. SEVENTH EDITION. NC 15 0 P4I- Mrs prnliai ariifirrm. LONDON: WIN SOU AND NEWTON, 38, RATHBONE PLACE, artists' dolour ftftaUers, bp Sperial appointment, to ??er ftftafestp, anD to Iprinte albert. 1856 . LONDON, PRINTED BY SCHULZE & CO., 13 , POLAND STREET. GETTY CENTER LIBRAS/ PREFACE. -♦- % The very first element of success in the production of a picture, is strictly accurate drawing. The principles of drawing, based upon the laws of geometry, are capable of being reduced to plain and simple rules. Of course the highest results can be attained only by laborious processes and unwearied study; but the elements of perspective, when once clearly and accurately understood, have this great advantage, that they are extremely simple, and, when mastered, form a foundation upon which the subsequent course, however intricate, can be raised with comfort and satisfaction to the student. Entertaining these views, the Author of the following pages believed that his labours might be beneficially ap¬ plied to the elucidation of the Principles of Perspective. The more so, because, from long experience in tuition in this branch of art, he has learnt to see where, in the understanding of the Laws of Perspective, the difficulties mainly lie in the minds of his pupils. What he has drawn up will be found to be short as to the principles; but so VI PREFACE. simple as to be readily and satisfactorily comprehended. The principal recommendation of his book will, however, be found to lie, he believes, in the numerous and carefully drawn examples, which constitute the main body of the work. These examples consist of the drawings of common objects, and they are characterized by this peculiarity, that the position, and the mode of the drawing, of every line are accounted for upon the brief principles first enunciated. The seeming difficulty, resulting from the crowded appear¬ ance of- the lines, (an inconvenience inevitably attending the limited space allotted to the figures,) will be easily surmounted by the diligence of the student, who is recom¬ mended to construct, from the very first, all the figures upon a scale sufficiently large for the full and plain de¬ velopment of the respective objects. The figures in the book will greatly facilitate the successful attainment of this purpose. Nothing remains but to express a hope, that this at¬ tempt to elucidate the early principles of this Branch of Art may meet with the approval of the reader. If this meed of approbation be accorded, the great labour and toil, which have been expended by both the Author and the Editor in the production of this Book, will be amply repaid. E. K. C. LONDON, APRIL, 1851. CONTENTS. —■*— FIRST PRINCIPLES. PAGE Point of sight.10 Vanishing points.11 Extent of picture ..12 Point of station., . . 13 Line of measurement.14 Diagonals.15 Recapitulation.16 EXAMPLES ILLUSTRATIVE OF THE PRECEDING PRINCIPLES. Directions for constructing the diagrams .... 17 Directions for angular and linear perspective. . . 18—19 A square surface in parallel perspective .... 20—21 Parallel perspective. 22 A square in angular perspective.23—25 vm CONTENTS. PAGE Cubes in parallel perspective.26—28 Cubes in angular perspective.29—32 A double house in angular perspective . . . 33—37 Horizontal circles in parallel perspective . . . 38—41 Horizontal circles in angular perspective . . . 42—45 A cross.46—50 A rectangular gateway.51—55 A church and spire.56—61 Conclusion.62 FIRST PRINCIPLES. An accurate knowledge of the principles of “ Linear Perspective” is as necessary in drawing, as that of Gram¬ mar is in the philosophy of language. By it we are taught the proper adaptation of every line, in expressing with truthfulness the parts and proportions of every object that recedes; and hence, without its application, the most beau¬ tifully wrought and the most carefully executed picture, in other respects, would be little else than an assemblage of painful and complicated errors. The first and the most important points to which the attention of the learner must be directed, are : 1. The point of sight: 2. The horizontal line : 3. The vanishing points : 4. The point of station defined by the distance of the spectator from the picture or landscape. The last is of great importance, as by it are determined the true vanishing points in various parts of the picture, 10 POINT OE SIGHT. as well as the lines of measurement, which give the correct perspective width, or depth of the several objects. We will speak first, of the “ point of sight.” The “point of sight” is that spot to which the eye of the spectator is supposed to be directed, when he looks straight before him. If through this point a straight line be drawn across the picture, parallel to the top and bottom lines of it, that line is called the “ horizontal line.” It resembles that horizon, or boundary line, which seems to part the sea from the sky, forming their apparent line of junction. The uses of this horizontal line are various and important. It determines the height of certain vanishing points, and the direction of certain lines of measurement. All horizontal receding lines lying above it, must incline down towards it; as every receding horizontal line below it, will tend upwards to it. If this be constantly borne in mind, many common errors will be avoided. From what has been said, it will be easily seen that the horizontal line, so far from being of the same height in every picture, is to be determined only by the elevation or depression of the point of sight. If this be low in the drawing, so will also the horizontal line be; but if high, in the same degree will the horizontal hue be elevated. This rule is invariable. Besides the point of sight, there are others called “vanishing points.” These points—which are indepen¬ dent of the point of sight as such—all lie in the hori- VANISHING POINTS. 11 I zontal line; and to tliem all the horizontal straight lines of objects in angular perspective are to converge. It is necessary to observe that the point of sight, which is im¬ mediately opposite to the eye of the spectator, becomes,— in what is, in contradiction to the other,—called “ parallel perspective,” the vanishing or radiant point to which all the straight lines of this class converge; so that, in this case, the “ point of sight” becomes a “ vanishing point.” We are thus brought to the two classes of perspective straight lines,—parallel and angular; to the elucidation of which we next proceed. An object is in parallel perspective when one side, or the front, of that object immediately faces the spectator, and lies parallel to his own body, while the lines of the other side recede directly from him; as in the case of a direct view down a passage, or into a street or room. In tliis case the point of sight becomes the vanishing point, and all the straight lines receding from the sight converge in this point. The lines, however, which are opposite and, as it were, parallel to the spectator, must be drawn really parallel, not convergent; because, being parallel, both in reality and in the drawing, they never meet. In respect of the point of sight, it must be observed, that there cannot be more than one in the same picture. The subject of the picture, also, should be confined to as much only of the landscape as the eye can see clearly and 12 EXTENT OF PICTURE. without effort; and this breadth of the picture is found to be comprised, generally speaking, within an angle of sixty degrees (60°). Now a whole circle contains 360 such de¬ grees; hence, as 360° = 6 times 60°, it would be necessary for a spectator to turn six times before he could sketch a panoramic view; that is, before he could draw the objects round the whole circle of which he is himself the centre. If this be clearly understood, which has been stated about “ parallel perspective,” the difference of “ angular perspective” will be as readily comprehended. In the former, the lines of one side of the given object or objects are parallel to the spectator, while those of the other side recede directly from him; but in the case of “ angular perspective,” neither of the sides is supposed to he immediately before him; but both of them retire,—not indeed, necessarily, in an equal degree, but in a degree depending upon the position of the artist, and upon the angle under which the object is viewed. Sometimes more of one side is seen than of the other; consequently the deviation from the horizontal line, and the inclination of the sides of the object to it, will be more or less abrupt. To apply this to the case of the perspective of a passage, a street, or a room : If the object be supposed to be viewed directly in front, the rules of “ parallel perspective” must obviously be applied in its configuration; and “ the point of sight ” will also be the “ vanishing point.” POINT OF STATION. 13 But if you were standing at a corner of the object, so that both its sides would be seen retiring, then you would manifestly employ the laws of “ angular perspective” in the drawing; and each side will require a “ vanishing point/’ quite independent of the point of sight,—the lines of each side running in a different direction, one set of them on one side of the point of sight in the horizontal line, the other set in the other direction. We have spoken, then, of the point of sight, the horizon¬ tal line, and the vanishing points; it remains hut to speak of “ the point of station,” determined, as was stated, by the distance of the spectator from the nearest point of the proposed landscape or picture. How, it is asked, is this distance to be ascertained ? It is, in fact, somewhat less in length than the width of the picture itself. Hence, if an equilateral triangle were formed upon the base of the picture as a base, the apex of that triangle would be the point of distance. Bor it is the well-known property of the equilateral triangle, that each of its angles contains sixty degrees; and this accords with the rule before laid down, that the width of the picture should be seen under an angle of sixty degrees. It may be added, however, that this rule is not always strictly adhered to ; it being desirable in some instances to stand at a greater distance, and in others not to take in the whole of the subject before you. The “ line of measurement” (laid down in the following examples) is thus determined: 14 THE LINE OE MEASUREMENT. The “ line of measurement” is the distance extending from the “ point of station” to the “ vanishing points.” If a circle be drawn, with a vanishing point as a centre, and with the above distance as a radius, intersecting the hori¬ zontal line, then the “ line of measurement” will be marked off on the horizontal line. The points of intersection of this circle with the horizontal line are called the “points of measurement ■” and, in parallel perspective, these points are always equidistant from the “ point of sight.” It is by the aid of these “ points of measurement,” that the perspective depth or width of any object is deter¬ mined. This is effected in the following manner : The geometrical or actual size of the building or other object of the picture being given according to a given scale, as of inches, or feet, &c.: upon a straight line to be drawn through the base of the nearest corner of the building or object, let a number of feet be marked off upon that line—suppose twenty feet on one side, and fifteen feet on the other, (these lengths being the actual lengths of the respective retiring sides of the building or object). This done, let straight lines be drawn from the nearest corner (above-mentioned) to the vanishing points on each side; these lines will give the proper inclination and direction of the ground lines of the given object. If straight lines be now drawn from the “ points of measurement” to the extremities of given geometrical scales (above supposed to be fifteen and twenty feet respectively), the intersections DIAGONALS. 15 of these lines with the before-determined ground lines, will give the required perspective breadths or depths. The centres of a certain class of rectilinear objects in perspective are easily found by means of diagonals ; i. e. by drawing lines joining the opposite corners. The inter¬ sections of these lines will be, in each case, the centres required—a method as simple as it is correct. For the sake of certainty and accuracy, before entering upon the explanation of the following diagrams, it may be expedient to make a few preliminary explanations, as well as briefly to recapitulate the principles and results obtained in what has been already said. A right angle is the opening of two straight lines meet¬ ing at an inclination of ninety degrees j such is each of the angles formed by the sides of a square. An acute angle is an opening of two straight lines, when they meet with an inclination less than ninety degrees; or, in other words, an acute angle is less than a right angle. In the same way, an obtuse angle contains more than ninety degrees, or is greater than a right angle. The above definitions will be clearly understood if it be remembered that, if the circumference of a circle were divided into three hundred and sixty equal parts, each part would be called a degree. There are, then, five points and lines which have been determined: 16 RECAPITULATION. 1. The point of sight, lying in the plane of the picture opposite to the eye of the spectator. 2. The horizontal line, drawn through the “ point of sight/' parallel to the horizon, and containing the “ vanish¬ ing points/’ and the “ points of measurement.” 3. The vanishing points, lying in the horizontal line; points, to which the straight lines of all objects in angular perspective converge. In the case of parallel perspective, the “ point of sight” becomes the vanishing point. 4. The point of station, which is the extremity of the distance of the spectator from the picture or view to be taken;—a distance generally a little less than the width of the picture. This point may be placed above or below the horizontal line, as may suit the convenience of the artist. 5. The lines of measurement, drawn from the “vanish¬ ing points” to the “ point of station.” They are marked off on the horizontal line by its intersection with a circle having each vanishing point as a centre, and each of the above distances as a radius. These intersections on the horizontal line are called the points of measurement. EXAMPLES ILLUSTEATTYE OF THE PRECEDING PRINCIPLES. —♦— DIRECTIONS FOR CONSTRUCTING THE DIAGRAMS. Throughout the whole of the following examples figures are placed on the several lines, to show the order in wliich these lines are successively to be drawn. This method will enable the pupil to follow the example more easily than he would be able to do, if the plan of defining the lines by letters, instead of by numbers, had been used. Too much attention cannot be paid to the order in which the several parts of the various objects are to'be drawn, for this will ensure a settled and intelligent method of procedure; one that will be less likely to cause confusion than will be the case if the drawing be worked out some¬ times in one way, and sometimes in another. In the several examples of this work, the size of the picture must be accurately and neatly drawn, before any portion of perspective plan is begun. c 18 DIRECTIONS EOR ANGULAR PLATE 1. FIG. 1. This figure represents the whole of the points and lines requisite for working out a drawing in “parallel perspec¬ tive/’ They are: 1 — the point of sight. 2— the horizontal line. 3— the point of station. 4— the points of measurement. fig. 2. This figure comprises the whole of the points and lines preparatory to beginning a drawing in “ angular perspec¬ tive.” They are: 1 — the point of sight. 2— the horizontal line. 3— the point of station. 4— the nearest corner of the object to be drawn. 5 — the ground line of the building or object, lying on that side, and produced from the nearest corner up to the horizontal line, in order to determine the vanishing point marked 5. 6— a line taken from the vanishing point 5, to the point of station 3. 7— a hue drawn at right angles to 6, and extending from the point of station to the horizontal line, at its AND LINEAR PERSPECTIVE. 19 PLATE I. Tigur* / / • 4 - 2 2 4 ? i « t V % \ t • c 2 20 A SQUAHE SURFACE junction with which the vanishing point, marked 8, is determined. 9— a point of measurement obtained by the use of the vanishing point 8. 10— the other point of measurement, obtained by vanish¬ ing point 5. 11— the geometrical scale of the building or object upon a base line drawn through the nearest corner. These preparatory plans (in Plate I), for parallel and angular perspective, must be thoroughly understood before the pupil is permitted to apply them; as all subse¬ quent correctness will entirely depend upon their correct¬ ness. PliATE II. (The numbers 1, 2, 3, &c., will for the future com¬ mence with the subject to be drawn, and not from the plan to receive the drawing ; and in angular perspective, the “ vanishing points,” and “ lines of measurement,” with their points, will be designated by numbers, as they may occur.) The perspective shown in tliis plate is parallel perspec¬ tive ; and the subject here intended to be represented is a flat and perfectly square surface, such as the floor of a room, a chess-board, or any other such object. It is laid down or drawn as follows : 1—the front edge of the given square. IN PARALLEL PERSPECTIVE, 21 PLATE II. 22 PARALLEL PERSPECTIVE. 2— one side of it receding to tlie vanishing point, which also is the point of sight. 3— the other side receding to the same point. (Hence the object is viewed from a position exactly opposite to its middle.) 4— a line taken from one comer of the front edge, to the point of measurement on the opposite side, and giving the perspective width or depth of the square at the inter¬ section of the hue 3. 5— a line drawn at the above intersection, and parallel to the front edge; this will give the back of the square. The lines 1, 2, 3, and 5, may then be strongly marked, and the figure will be thus completed. The line 6, is taken from the other corner of the front edge to the opposite point of the measurement, showing how exactly either this line, or that marked 4, will give the perspective width of the square. It serves also to find the centre. It will be observed that there are added to the figure upright lines from 5, as well as two slanting ones, and the straight line across them. These are given to show how a room might be drawn from this construction, by making the upright hues to represent the further angular lines of the room; the line across being the end of the ceiling, and the two sloping lines the sides of it. If a chess-board is to be drawn, you would proceed by marking the divisions on the front edge, and taking them A SQUARE IN ANGULAR PERSPECTIVE. 23 towards the vanishing point, as far as the line 5, which would be the back of the board. Straight lines have then to be drawn through the intersection of all these lines with 4 or 6, which will give the number of the required squares in their true perspective. PI.ATE hi. The perspective in this plate is “ angular perspective and the figure it represents is a flat square surface, similar to that in Hate II: its dimensions are supposed to be either twenty feet or twenty inches. To represent such an object you would proceed thus : 1— two lines drawn from the nearest corner of the board, to the horizontal line, and at a distance from each other equal to the thickness of the board; this fixes the vanishing point at 1. 2— a line drawn from the above vanishing point to the point of station. 3— a line taken at right angles to 2, from the point of station, and fixing on the horizontal hue the position of the vanishing point 3. 4— two lines drawn from the nearest corner of the board to the vanishing point 3, similarly to the previously drawn lines 1. 5— one point of measurement, obtained in the usual way, by the distance of 3 from the point of station. 6— the other point of measurement. 24 A SQUARE IN 7—the line of the geometrical scale, being a line drawn across the base of the nearest corner, and marked accord¬ ing to scale, twenty feet or twenty inches. 8, 8—lines taken from either end of the geometrical scale towards the points of measurement, but extending no farther than where they meet the lines 1, 1, and 4, 4. 9, 10—small perpendicular lines drawn at the above intersections, by which the width of the board is ascer¬ tained (see above at No. 1.) 11— the side of the board opposite and really parallel to that marked 4, and therefore tending to the same vanishing point. 12— the back of the board, opposite and parallel to the front marked 1, and consequently tending to the same vanishing point. The lines 1, 1; 4, 4 ; 11 and 12, being strongly marked, the figure will be completed. In 11 and 12 it was seen that, because the two sides and the front and back are respectively parallel, they are to be made to converge, respectively, in the same vanishing points. In the same way, all lines on the same side of an object or building, that retire from the spectator in the same direction, and under the same angle, always converge in the same vanishing point. By due attention to the directions or angles of the various lines in the drawing, many mistakes will be avoided; and it will now be easily seen to which vanish¬ ing point the lines are to retire. ANGULAR PERSPECTIVE. 25 PLATE III. 26 CUBES IN PARALLEL PERSPECTIVE. P1ATE IV. Two upright oblong figures are here represented in parallel perspective. They may be imagined to resemble the sides and fronts of houses, or their blank walls. One of the figures has two others attached to it of equal dimen¬ sions ; and these additions might be similarly multiplied to any extent, by the numbers 7, 8, 9 and 10, in the fol¬ lowing rules. The following is the order in which they are to be drawn: 1— lines forming two complete fronts of two separate and detached oblongs. 2— the geometrical scale at the base, marked twenty feet. 3— the ground lines of the fronts running to the vanishing point. 4— the top lines tending to the vanishing point. 5— lines from the geometrical scale, to the points of measurement, determining the perspective depths of the oblongs. 6— perpendicular lines raised at the intersection of the lines 3 and 5, and giving the farthest upright corner lines of the oblongs. The two figures will thus be completed. The remaining lines inserted in the figure are intended to give two other oblongs (or rather their retiring sides) attached to the first, and supposed to be of the same dimensions. They are determined first by finding the CUBES IX PAKALLEL PERSPECTIVE. PI/ATE IV 28 CUBES IN PAR ALLE L PERSPECTIVE. centre 7 of the near corner line 1. From 7 a line is drawn to the vanishing point. A line marked 8 is then drawn from the near extremity of 1 through 6, where it is cut by 7; at its intersection with the bottom line 3, the perpendicular line 9 is raised, and another oblong front is completed. A line 10 is drawn, and determined as the line 8 was, from the top of 6, and by crossing the lines 7 and 3. The lines 13, 14, and 15, are inserted merely to show the inner side and back of the other ob¬ long, as they would be seen were the object made of glass. Thus 13, 13 are lines for the top and bottom of the back; formed by drawing them to the vanishing point ; 1.4, 14 are the top and bottom lines of the farthest side, found by straight lines being drawn from both ends of 6, until they meet 13, 13; at which point of meeting the upright corner hue 15 is raised. Thus the corners of the oblong are completed. In the next plate, the size of the picture has not been drawn to allow of three figures of a similar • character being given, instead of one only. The perspective of the first two is in consequence rather sudden, which here, as in all cases, arises from the shortness of the distance of the point of station from the horizontal line. PLAT£ V. The object shown in Figure 1 is a cube, having, there- CUBES IN ANGULAR PERSPECTIVE. 29 fore, all its faces of equal dimensions; and as both sides recede. “ angular perspective” is employed. The point of sight, horizontal line, and point of station, having been fixed upon, the line A is first to be drawn, touching the bottom of the nearest corner, and is for the geometrical scale or height of the cube, which, in this instance, will be called twelve feet ; that is, twelve feet must be marked on the scale from the comer on either side. 1— the ground line of the square, taken from the centre of the geometrical scale line to the horizontal line ; by its junction with which is determined the vanishing point for that side. 2— a line drawn from the above vanishing point to the point of station. 3— a line drawn at right angles at the point of station to the line 2, as far as the horizontal line, its intersection with which will give the correct vanishing point to the other side. 4— the ground line of the cube running to the last vanishing point. 5— the nearest corner of the cube, twelve feet in height, being equal to the width. The points of measurement are next to be ascertained, and to be marked in the usual way; and the lines B drawn from the ends of the geo¬ metrical scale towards the point of measurement give the perspective width or depth of both sides. This is found 30 CUBES IN at their cutting of the ground lines 1 and 4. The line 6 represents the top line of one side of the cube, and runs from the nearest corner to the vanishing point. 7— the other top line; and it is drawn to the other vanishing point. 8— the far corner line raised vertically from the crossing of the lines B and 1. 9— the other corner line raised vertically from the in¬ tersection of the lines B and 4. The lines 1, 4, 5, 6, 7, 8, 9, being strengthened, the figure is complete. In Figure 2, the cube is shown as in Figure 1, with the addition of four others of equal dimensions. This is effected by first drawing the cube in the order as described for Figure 1, and then finding the centre of the upright line 5, that being the nearest corner hue of this first cube. The centre being found at 10, take the line 10 to the vanishing point for that side of the cube; this will give the centres of all the other upright fines of that side of all the added cubes. The fine 11 is drawn from the top of the corner fine 5, through the intersection of 8 and 10, until it meets the ground fine 1, at its junction with which the upright fine is raised for the far corner fine 12 of the second cube. The three other cubes are described precisely in the same manner, being found by the diagonal lines traversing each pair of the cubes, through the intersection of the centre fine 10, with each perpendicular fine raised from Fig 2 ANGULAR PERSPECTIVE 31 PLATE V. 32 CUBES IN ANGTJLAH PEESPECTTVE. the meeting of the previous diagonal line with the ground line 1. It will be perceived that a further distance of twelve feet is added to one side of the geometrical scale, and marked A. This is done merely to prove the correct¬ ness of the first diagonal hue 11, passing through the centre line 10, to determine the perspective depth of the second cube. Tor if a line is taken from the end of the geometrical scale A to the point of measurement on the horizontal line, it will be found to meet the ground line 1 at exactly the same point: thus proving the truth of both modes of drawing. The former mode, however, is more convenient where a number of cubes are to be drawn ; as the geometrical scale might extend far beyond the limits of the paper, and consequently give much more trouble. Figure 3 differs from the others in this respect that they are solid cubes, while these two in Figure 3 are trans¬ parent, so as to show all the surfaces and angles of the cubes. And further, the geometrical scale is used for the two cubes, because, being only two, it will be found in this way that fewer lines will be necessary, leaving the figure less intricate and confused. The construction, in tliis case, is as follows: The two front sides of the cubes are produced in the same way as in Figures 1 and 2, so far as fine 10, which is the farthest corner line of the second cube. A DOUBLE HOUSE IN ANGELAS PERSPECTIVE. 33 11—the line is drawn from the extremity of 10 to the vanishing point of 7, the two hues being really parallel. 12 is drawn from the top of 9 to the vanishing point of line 6, these also being parallel. 13 is drawn from the top of the upright centre line 8, to the vanishing point of 11 and 7, these being all really parallel to each other. 14 is the far ground line taken from the lower extremity of 9 to the vanishing point of 1, these lines being also parallel. 15 and 16 are lines drawn from the corner end of 10 and 8 to the vanishing point of 4, the three lines being really parallel. 17 and 18 are upright lines raised at the intersection of the lines 16 and 15, with the ground line 14, being the far comers of the cubes; they respectively will meet the inter¬ sections of the lines 11 and 13 with 12. These lines will complete the figure. PLATE VI. The subject of the present plate is a double house with two. gables, all in angular perspective. The walls of the house are treated in precisely the same manner as the cubes in the previous examples. The geometrical scale is first taken, with the plans of the windows and doors marked upon it in their proper dimensions. This last proceeding must be carefully attended to. D 34 A DOUBLE HOUSE 1— the ground line of the building taken to determine the first vanishing point on the horizontal line. 2— the line from this point to the point of station. 3— a line at right angles with 2, at the point of station, and determining the other vanishing point. 4— the other ground line taken to this last vanishing point just found. 5— the nearest upright line. 6— the top of the wall, tending to the vanishing point on that side. 7 —the other top line tending to the second vanishing point. 8— the farthest corner line of the front raised verti¬ cally at the intersection of ground line 1, with the line B drawn from the geometrical scale to the point of measure¬ ment. 9— the other corner line of the front building, and which is, in fact, the centre line of the gable side. 10— from the centre of the line 5, a line 10 is drawn to the vanishing point 2. 11— a diagonal line traversing the side through the intersection of 10 and 9, until it meets the ground line 4. 12— the farthest upright line turned vertically on 4, from the last-named point, up to the top line of the wall 7. 13— diagonal lines drawn across the upper parts of the IN ANGULAR PERSPECTIVE. 35 sides of the house as far as the line 10. These determine the perspective centre of each gable. 14 a perpendicular line raised from the intersections of the first pair of diagonal lines, as far as the line 15; which last line is drawn from a point taken in the nearest corner line of the house, produced upwards; this point being higher or lower according to the height of the roof. 16 a perpendicular line raised from the centre of the second pair of diagonal lines, as above. 1 7—the lines forming the gables, drawn from the upper extremities of the centre lines 14 and 16, to the upper extremities of the lines 5, 9, 12, at the line 7. Two of the lines 17 would, if produced, run out of the picture, until they meet the line C, drawn vertically upwards from the vanishing point 2. 18 and 19 are for the tops of the roof, drawn from each gable point to the vanishing point 1, both tending in the same direction. 20—the remote sloping line of the roof drawn to the point where the line C is intersected by the lines 17; it being really parallel with these latter; and all parallel lines must, in perspective, converge to the same point. 21 are lines raised for the width of the windows and door, as well as the chimney; determined by the junction of the points in the scale A with the point of measurement, and cutting the ground line 1 in proper proportions. 22 are points placed on the nearest corner and upright 36 A DOUBLE HOUSE line 5, being the correct height of the window-frames and doors. From each of these points lines are drawn to the vanishing point 1, and their intersections with the lines 21 give the perspective height of the windows and door. 23 is the line continued from the last of the lines 21 through the roof to the top of the line C, in the same manner as the line 20 was drawn. 24—the near upright line of the chimney, brought down to meet the line 23. The lines 25 and 26 are the top lines of the chimney, taken to the vanishing points respectively. 27—the other upright line of the chimney taken from 29, to meet line 25. 21—the far corner line of the chimney taken from 26. 29—the bottom line of the chimney, running to the vanishing point 1. This done, the house is finished. Obs. —As it frequently happens that a multiplicity of lines are required in working out many perspective draw¬ ings, it is desirable to rub out, as we proceed, those por¬ tions of the lines running to their several vanishing points, which he beyond the building or object itself, as we thereby prevent a confusion of parts and lines. This figure is placed on one side of the paper, to allow of one vanishing point being much more distant than the other. IN ANGULAR PERSPECTIVE, 37 PEATE VI. 38 HORIZONTAL CIRCLES IN PLATE VII. The figures of this plate represent three circles put into horizontal perspective, each varying in a slight degree from the others, in the manner of its treatment. The most simple mode of proceeding is as follows : We will first take Figure 1. The size of the circle being given, and the place of its position, let a square 1, 1, 1, 1, be drawn upon and below the geometrical scale, having its side equal to the diameter of the given circle. In this square the circle must be struck, by first drawing the diagonal lines 2, the intersection of which is the centre of the circle. The circle 3 being made, draw the perpendicular lines 4 through the circle’s intersections wkh the diagonal lines 2. These points are marked in all the figures by a black dot on the several crossings. The lines marked 5 are lines drawn horizontally and perpendicularly through the centre. This done, draw the side lines 2 from 1 to the point of sight or vanishing point. Then take B from the geometrical scale to the point of measurement, to determine the depth. The line 3 must then be drawn at the intersection of B and 2, parallel to 1. The lines marked by the upper numbers, 4, 5, 0, are lines running to the vanishing point, from the corresponding lines below. PARALLEL PERSPECTIVE. 39 The lines 7 are diagonal lines, drawn from each corner to find the centre, as well as to intersect 4 and 6. 8 is a straight line drawn through the centre, parallel to 1 or 3. The circle in perspective is produced by these several lines, by placing a small dot at each crossing of the diagonal lines with those running to the vanishing point, correspond¬ ing precisely with those in the square below. The circle has then to be drawn through each dot, curving gradually towards them. It will then be found that both the upper and lower forms appear exactly similar. The upright lines, marked 9, being taken from the several dots forming the circle, show how a pillar or a column may be raised: or they may be made to give the width and position of any circles that may be required, either above or below the horizontal line. Thus, in the construction of the upper circle, by the aid of the lines 9, the two equal lines 9a ar.e raised to the proper height, and the straight line 1 drawn from the one to the other. The side lines 2 are to be taken to the vanishing point and the line 3 to be drawn from the point of measurement on the left, to give the position of the straight line 4. (By taking the two vertical lines marked 9b, it will be found that they meet the side lines 2, at precisely the same point in which the lines 3 and 5 meet them, thus proving the correctness of the pan.) The lines 3 and 5 are the diagonal lines. Two vertical 40 HORIZONTAL CIRCLES IN lines are then to be drawn from 4, 4 in the square below, to the line 1 above, giving the points 6 and 7, from which lines must be taken to the vanishing point. Take also the horizontal line 8 through the centre. It now only remains for the dots to be placed on the intersections of the diagonals with the other lines, and the circle can then be drawn, gradually curving round to each. Figure 1 is now completed in all its parts. Figure 2 is drawn in the same way as Figure 1, as far as the diagonals and the straight lines through the centre. The lines 5 are two upright lines, placed in the square at pleasure, and at their intersection with the diagonals, straight lines are drawn across the square, parallel to the sides. Where these cut the inscribed circle, two other lines marked 8 are drawn, and the parallel straight lines 9 across the square are taken from the intersection of the lines 8 with the diagonals 2. The perspective of this figure is treated from all these lines, as in Figure 1. Lines are to be drawn to the vanishing point, and the dots for the circle placed at then’ intersections with the diagonal lines, through which straight lines are to be drawn, corresponding to those in the plan below. Figure 3 differs from the other two in this respect, that only half a square, containing a semicircle, is given below. This answers the purpose equally well, and is often much 42 HORIZONTAL CIRCLES IN more convenient than the method of the whole circle. The perspective part is treated as in the other cases. The figures being given in succession, the pupil may, as an exercise, work out the lines from them without the aid of farther explanation. PLATE VIII. AYe have here two circles with upright or vertical lines raised from the points; the perspective employed in the drawing of them is “ angular perspective.” They may be supposed to represent the shafts of two columns, and their square bases. In the squares under the geometrical scale, the outer square is for the base of the pillar, while the inner one is for the column itself. It will be seen from the figure, that the size of the square and circle is always drawn from the geometrical scale in angular, as well as in parallel perspective. They are, in fact, both treated alike. The circular segments, drawn from the side of the square to the geometrical scale, carry the dimensions of the side of the square up to the geometrical scale by means of the compasses, one foot of which is kept on that corner of the square which touches the perspective plan, and the other is placed on the points of the side of the square in succession. This is a very neat way of carrying ANGULAR PERSPECTIVE. 43 the size of one part to another, provided the two corners meet, as they do here. We proceed with the perspective thus : Line 1 is the first ground line to fix the vanishing point 1. The other vanishing point is obtained in the usual way, by a line drawn at right angles with the first from the point of station. Line 2 is the other ground line tending to vanishing point 2. The lines B are then drawn from the ends of the geometrical scale to the points of measurement. It may be useful to recapitulate, as illustrated in this example, the way in which these points of measurement are found. These points are always determined by placing one foot of the compasses on the vanishing point, as a centre, and extending the other to the point of station. A segment of a circle is then drawn, meeting the horizontal line. This point of intersection will be the point of measure¬ ment. Both points are found in the same way. If the compasses do not extend far enough, the distance from either vanishing point, to the point of station, may be measured with a piece of paper, and the length marked off on the horizontal line. Lines 3 and 4 are drawn from the intersection of the ground lines 1 and 2 with the lines B, from the geometrical scale. 5—is the centre line dividing the two perspective squares. The whole of the lines are now to be drawn towards the / 44 HORIZONTAL CIRCLES IN points of measurement, as far as the ground lines 1 and 2, from whence they are to converge to the two vanishing points respectively. The inner square (below) will be determined by the meeting of the two first fines marked 1. This square should be strengthened at once, to separate these fines of which it is composed, or rather to distinguish them from the others. The diagonal fines in the perspective circles are then to be drawn, and dots to be made at the junction of the fines, tending to ihe vanishing points. It now only remains to curve hues gradually through each intersection or dot, to form the perspective circles. The upright fines forming the pillar are raised from the dots, as was done in Plate VIII. Obs. —Several fines in this example have not been num¬ bered, to prevent confusion, the explanation given being sufficiently clear. The fines (with their respective num¬ bers) of the two original squares and circles have not been noticed, from their being treated in the same manner as those in Figure 1, Plate VII. PLATE IX. The subject of this plate is a cross raised upon a square base of two steps, and having its sides equally inclined to the plane of the picture. ANGULAR PERSPECTIVE 45 PLATE VIII. V 46 A CROSS. We proceed thus, in its projection or drawing : Draw the geometrical scale in its proper place, that is, through the foremost corner of the base; on which set off, first, the actual width of the first step; then that of the second; and then the width of the bottom of the square base of the shaft or pillar, bearing the cross near the top; then the width of the second and first step again in suc¬ cession. After this, place one foot of the compasses on the centre point of the scale, and turn round the other foot from each point to the other side of the produced scale; both sides will then be divided equally. The height of the pillar must then be represented by a vertical straight line raised from the centre of the geometrical plane, with the actual vertical dimensions of the cross and base marked upon it. (For the sake of easy reference, the drawings of the base of steps and of the cross itself are given separately ou a larger scale; and the point of distance is placed above instead of below the horizontal line, for the same reasons.) The line 1 is the ground line of the first step running to the first vanishing point on the horizontal line. 2—the line from the above to the point of station. 8—a line at right angles with 2 at the point of station, to determine the second vanishing point. 4— the other ground line running to vanishing point 2. 5— a vertical line of the height of the first step. 6— its edge drawn from 5 to the vanishing point 1. A CROSS. 47 7—the upper edge of the other side drawn from the same point to the vanishing point 2. The lines B are lines drawn in succession from the points on the geometrical scale towards the points of mea¬ surement, and intersecting the ground lines 1 and 4. The lines 8 and 9 are the remote comers of the step, raised vertically at the meeting of the outer lines B, with the ground lines 1 and 4. The lines 10 and 11 are the farther edges of the top drawn to the two vanishing points from the upper extre¬ mities of 8 and 9. The first step is then completed. The second step is begun thus : Raise the vertical lines 12 from the lines marked Bd cutting the ground lines 1 and 4. Produce them all from their intersections of the edges 6 and 7, to their respec¬ tive vanishing points ; and, where they intersect each other, there raise the upright lines 15, 18, and 19; the line 15 being made equal to the thickness of the second step. The lines 16 and 17 are the upper edges of the second step, drawn to the vanishing points; 13 and 14 are por¬ tions of the lines 12, strengthened or made darker to form the ground line of the second step. The lines 20 and 21 are the farthest edges of the upper step taken from the upper ends of 18 and 19 to the vanishing points respectively. The second step is now finished. 48 A CROSS. The width of the base of the pillar with its proper position is found just as the second step was. The lines marked 22 are raised from the intersections of the inner lines B with 1 and 4, and are carried on the upper surfaces of both steps towards their respective van¬ ishing points. The intersections of these lines give the position of the vertical lines 23, 26, and 27, being the tliree corners of the base of the shaft, while the lines 22 give the ground lines of that base. The lines 24 and 25 (in the principal figure) are taken from the upper end of 23, towards the vanishing points, and determine the front upper edges of that base. The equal lines 28 and 29 are drawn to the vanishing points from the small line slightly inclining inwards to the right at the top of the base, laying down the width of the main shaft itself. The two other small lines are then made. The lines 30, 31 and 32 are the three visible corners of the pillar, and raised vertically as far as 33 and 34. 33— is the upper edge of one side of the cross, found by taking the line to the vanishing point 1, from the top of the perpendicular geometrical scale. 34— is the upper edge of the other side, drawn from the corner of 33 to vanishing point 2. The construction of the cross is as follows : 1— is the upper edge of the cross-bar, drawn towards the vanishing point 1. 2— is its lower line, drawn towards the same point; A CROSS, 49 PLATE IX. TjS/ IX E 50 A CROSS. both these lines are taken from the two points marked on the vertical scale. 3— is a centre line between the lines 1 and 2, drawn also towards the vanishing point 1. 4— a diagonal line drawn from the intersection of 2 and 31, through the centre on the upright line 30 to meet the upper line 1. At the meeting of 4 and 1 the perpendi¬ cular line 6 is drawn, giving the perspective width or depth of the first projection of the cross. The far pro¬ jection is found in the same way; that is, by the diagonal line 5 traversing the pillar from the line 1 through the centre 3 to the line 2; at this intersection the vertical line 7 is raised. 8— is the lower liue drawn to the vanishing point 2. 9— is the corresponding line to 2, drawn towards vanish¬ ing point 1. 10— is the lower line of the front of the projection, drawn towards vanishing point 2. 11— the upper hue verging to the same point. 12— the far corner hue, raised vertically at the meeting of 9 and 10. 13— the lower line of the farthest projection drawn to¬ wards vanishing point 2. Thus one side of the cross is completed. The other must be worked out in the same manner; the vanishing point 1 being substituted for 2 ; and on the contrary 2 for 1. The entire figure is then finished. A RECTANGULAR GATEWAY. 51 PLATE X. This figure represents a rectangular gateway, with three arches of equal dimensions; having the left hand ground line more inclined to the plane of the picture than the right hand one. First, take the geometrical scale through the nearest point of the structure, and mark the breadths of the several arches and spaces between them on that scale, and also (on the right hand) that of the side of the gateway. Line 1 is the ground line, determining, by its inter¬ section with the horizontal line, the vanishing point 1. 2— a line from the above to the point of station, which, for convenience, is placed above the horizontal line. 3— a line at right angles with 2, determining the position of vanishing point 2. 4— the ground line of the side, drawn to the last van¬ ishing point 2. 5— the nearest upright corner raised from the point on the geometrical scale to the full height of the building. 6— the highest front edge drawn from 5 to vanishing point 1. 7— the corresponding edge of the side taken to vanish¬ ing point 2. The lines B are then to be drawn towards the points of measurement, cutting the ground lines 1 and 4. E 52 A RECTANGULAR GATEWAY. The lines 8 and 9 are the far corners, raised vertically at the intersection of the line B and the lines 1 and 4, to meet 6 and 7. The lines 10 and 11 are three lines verging to the vanish¬ ing points from the corner 5 of the building 5 ; they represent the cornice of the gateway and its direction. 12 is a geometrical scale drawn through the highest point of 5, and having the size of the circle of the arches, and the space between the corner and the arch, set off upon it. 13— the semicircle struck from the centre of 12, and of the size of the real arch of the gateway. 14— three lines, forming half of a square round the semicircle. 15— diagonal hues from each corner of the half square to the centre; a, two vertical lines drawn through the intersection of the diagonals and the circle, and also to the centre of the line. The lines 16 are taken from the lower extremities of the last-mentioned lines towards the point of measurement 1, and meeting the line 6, from which intersections perpen¬ dicular lines are let fall, giving, on the face of the gateway, the perspective breadth of the arch above. The lines 17 form the perpendicular sides of each arch; they are raised from the lines B at their intersections with the ground line 1. 18—a line taken from the height of the spring of the arch set off on 5, to give the height of the lines 17. It is A RECTANGULAR GATEWAY. 53 also above a line taken from 5, to determine the top of the arches. Both these are drawn towards the vanishing point 1. The lines 19 are diagonals drawn from the top corners of each perspective half to the centre, as in the plan placed above. 20—a line drawn to the vanishing point 1, through the intersection of the two lines 16, and corresponding to intersections of the diagonal in the first arch. This line 20 will intersect all the other diagonals at those points through which the circular arches must, at the next step, be drawn. The lines 21, 22, and 23 are the ground lines of the side of each archway, drawn to the vanishing point 2, from the intersections of the lines B with the ground line 1. 24 is a line drawn from B, where B intersects the ground line of the side 4. Its direction is towards van¬ ishing point 1, and it determines the depth of the inner side of the first archway, the whole of the inner sides of the others not being seen. The line 24 is consequently the ground line for the back of the building. 25 is the vertical line for the back of the first archway, commencing at the intersection of 21 and 24, one being the ground line of the inner side of the arch, and the other the ground line of the back. The lines 26, 27, and 28 all verge to vanishing point 2, from the upper ends of the perpendiculars 17. They 56 A CHURCH AND SPIRE. 35— a line similar to 18, drawn from the intersection of 34 and 33, to vanishing point 1. 36— diagonal lines drawn to the centre, drawn from the insersection of 29 and 31. 37— a line drawn from the intersection of 29 and the first diagonal 36, towards the vanishing point 1 ; this gives the intersection on the other diagonal line. It now only remains to curve the arch carefully through these respective points. With this the whole of the figure is completed. PI.ATE XI. A church, in the form of a cross, with steeple and spire, is the subject of this plate. How is the drawing of it to be commenced, and what part of the building is to be attached to the geometrical scale ? In all buildings where there are projections, as in the present figure, it is necessary to bring such projections forward, or rather towards the geometrical scale, as being nearer to the spectator than the main part of the building from which they spring. Therefore the ground line, No. 1, is brought forward to the geometrical scale, meet¬ ing it in a point from which a geometrical perpendicular line, marked 1, is raised, with the several heights of the walls, the steeple, the base of the spire, and its apex, set off upon it. They are marked on the line by dots. Some points on the geometrical scale are marked B a, l»Ii.\TK XI. [to face page 56. A CHURCH AND SPIRE. 57 and they are intended to set off the ground dimensions of the tower or steeple ; and these lengths are the same on both sides of the geometrical scale, the tower being square. The precise measurement of all the parts, both horizontal and perpendicular, and the ground line 1 being thus set off, the lines 2 and 3 must be made at right angles to each other at the point of station, to determine the vanishing point 2. 4 is the other ground line from the geometrical scale to the vanishing point 2. 5 is the ground line of the main part of the building tending to vanishing point 1, and determined by the inter¬ section of the hue from b in the scale (to the point of measurement) with the ground line 4. The ground line for the side of the projection is also marked 5, and is drawn to the vanishing point 2, that line being de¬ termined by the intersection of the line drawn from c, to the point of measurement with the ground line 1. Lines 6 are two points drawn from the first point on the upright geometrical scale, to the vanishing points on either side. These are to give the height of the walls up to the gables. Lines 7 are the corner lines of the walls of the building, found at the meeting of b and 4, c and 1, B and 4, B and 5, 5 and 5 ; the latter being the line at which the projec¬ tion joins the body of the church. Line 8 is the lower line of the roof, drawn through the 58 A CHURCH AND SPIRE. intersection of 6 and 7, and inclining towards the vanish¬ ing point 1. It is carried beyond the projection to show the corresponding line in the farthest end of the building. 9 is also the lower line of the projecting roof, taken from the intersection of 6 and 7, down to the vanishing point 2 until it meets 8. The central point 10 must next be set off on the per¬ pendicular geometrical scale for half the height of the walls, from which point the lines 10 are drawn to the two vanishing points, to give the centre by which to determine the gable, and which will be found at the intersection of 10 and the diagonals 11. From this intersection a vertical line is raised to meet one of the lines marked 13,—these lines, 13, being drawn thus ; from the second point in the vertical scale, a line is set off, on that scale, equal to actual height of the gable 6 ; and from the third point, so de¬ termined, the lines 13, 13, are drawn to the vanishing points. The lines 14 are drawn from the lower corner of the roof, at the extremities of the lines 7, through the centre or crown of the gable. The lines 15 are the tops of the roof from the points at 13 to the two vanishing points, as far as the middle of the sides of the tower; 16 being the small line of the side of the roof, from the upright line not numbered as far as the tower, to meet lin e 20. 17—the four ground lines of the tower, from the points A CHURCH AND SPIRE. 59 Ba on the geometrical scale, at their junction with 1 and 4, give the exact position of the tower in the centre, and of the interior of the building, by being taken to the two vanishing points, and intersecting each other. The lines 18 are perpendicular lines raised from each of the corners of this ground plan of the tower, to the lines 13, 19 ; the meeting of the lines 21 showing the height of the tower’s farthest corner, which of course cannot be seen. The vertical hue 20 has now to be drawn up the side of the gable end to the roof, 14, from the meeting of B« with 4 (below); from which intersection a line is to be drawn to the vanishing point 1. The intersection of this line with the centre upright 18 of the tower, will show at what part of the roof the tower is seen rising. The small side line of the tower is to be drawn from the above comer to the roof line 15. The lines 21 are lines intersecting each other, and drawn to the vanishing points from the far corners of 13 and 19. They form the ceiling of the tower. The interior of the tower is shown in order to find the centre of the spire, which is ascertained by the diagonal lines traversing the ceiling, and intersecting it at 22. From this point a vertical line is raised as high as 23. The lines 13 and 19 being the highest lines of the tower, tills height, as also that of the spire, is determined thus : The third point on the perpendicular scale, or hue of 60 A CHURCH AM) SPIRE. heights, is for the top of the tower, from which point 3 line has to be drawn to the vanishing point 2. The upright line 20, from B a (that being the measurement of the tower) at its intersection with 4, is to be carried upwards through the gable line 14, until it meets the line from the third point. At this intersection a line is to be taken to vanishing point 1; this is marked 13, and wherever it cuts the centre upright line 18, that will be the top of the steeple, or tower. The top line, numbered 19, is found by drawing a line to vanishing point 2, from the intersection of the line 13 and the vertical central line 18, of the tower; and thus both the visible top lines of the tower are deter¬ mined. The lines 24 are the three lines next to be drawn; they are taken from the three top corners of the tower to the point of the spire, which point is found in the same manner as was the top of the tower, namely, by drawing a line from the extremity of its measurement on the vertical geometrical scale, to the vanishing point 2, intersecting the centre line 12 running vertically through the top of the gable on the right. From this intersection another has to be drawn to the vanishing point 1; and the point, where this line cuts the centre line 22 of the spire, will be the highest point of the spire in perspective. It appears that the line 23 must be taken from the in¬ tersection of the centre line marked 12, and the line A CHURCH AND SPIRE. 61 drawn from the extremity of the vertical scale to vanishing point 2. The reason is, that the top of the line 12 would be the height of the spire, if the centre of the spire were situated over that gable point; but as it is in the centre of the building, and consequently much farther from the spec¬ tator, therefore a line must be taken from thence to the vanishing point 1, the tower being placed in that direc¬ tion. The height of the upper and hexagonal part of the steeple is found in the same way, namely, by taking the line 27 first to vanishing point 1, and then to vanishing point 2. 25 is a line with the geometrical size of the side of the hexagonal base set off upon it, and drawn through the nearest comer of the top of the tower. Prom the dots in the scale are drawn lines to both points of the measure¬ ment (1 and 2 below), until they cut the lines 13 and 19. From these intersections, lines are then taken upwards to the apex of the spire, and marked 26. The line 27 is the base of one side of the upper part of the spire (as explained above), taken to the vanishing point 1. A dot is made on the centre line 24, where it cuts 27, and the line 29, taken from it to the vanishing point 2, determines the base of the other side. The lines marked 30 are four sloping lines taken from the three corners of the tower to the intersections of 26 with 27, and those of 26 with 29. 62 CONCLUSION. 28 is drawn from 27 to 29, giving tlie base of one of the six sides of the spire. This completes the whole. —♦— CONCLUSION. The above examples illustrate all the principles which were laid down in the beginning of this work. These principles and examples, if thoroughly comprehended by the student, will form a nucleus round which he may be constantly gathering fresh information and skill, both as regards knowledge and practice. Let him, however, be sure to be able to understand every word that is stated here, and to be quite perfect in the delineation of all the examples contained in this book. For this purpose, he would be wise, not in being content with one attempt only, even though that should be a tolerably successful one, but in making such repeated trials as will enable him to produce a good, clear, and satisfactory performance in each case. We will add one important caution: that he should have well-constructed instruments, (compasses, scales, and rules); and that he should determine the posi¬ tion of each point, and draw each line, with the greatest accuracy and care. This, besides confirming habits of neatness and method, will save him a vast amount of trouble; as an error in an early stage of drawing, not only entails errors in the subsequent processes, but ex- CONCLUSION. 63 tends those errors, and thus produces dissatisfaction and disgust. This done, he will find it tolerably easy to apply what he shall have learnt to the drawing of actual objects— from models or from nature; and he will have, too, the power of appreciating, with much greater delight and satisfaction, the beauties of the immortal works of the Great Masters of Art. THE END. London: Printed by Schulze and Co., 13, Poland Street.