STACK. 
 ANNEX 
 
 SIMPLIFIED 
 MECHANICAL PERSPECTIVE 
 
 FREDERICK

 
 SIMPLIFIED 
 
 FOR THE USE OK 
 
 HIGH SCHOOLS, .TECHNICAL AND MANUAL TRAINING 
 
 HIGH SCHOOLS, EVENING INDUSTRIAL 
 
 SCHOOLS AND ART SCHOOLS 
 
 BY 
 
 FRANK FORREST FREDERICK 
 
 Director of Trenton School of Industrial Arts: Author of Plaster Casts and 
 
 How They are Made", "The Wash Method of Handling 
 
 Water Colour", Etc. 
 
 THE MANUAL ARTS PRESS 
 
 PEORIA, ILLINOIS
 
 COPYRIGHT 
 
 FRANK FORREST FREDERICK 
 1908- 1W)
 
 CONTENTS 
 
 AUTHOR'S NOTE 5 
 
 INTRODUCTION 7 
 
 THK PERSPECTIVE DIAGRAM 9 
 
 To Lay Out Diagram 11 
 
 To Find Vanishing Points 11 
 
 Problems I to XII 13 
 
 THK PERSPECTIVE OF FURNITURE AND INTERIORS 24 
 
 Problems XIII to XX 24 
 
 THE PERSPECTIVE OK CIRCLES 37 
 
 Problems XXI to XXIX 37 
 
 THE PERSPECTIVE OF OBLIQUE LINES 46 
 
 Problems XXX to XXXVII .. .49 
 
 2066125
 
 AUTHOR'S NOTE. 
 
 IN publishing the following notes on perspective 
 which, in substantially their present form, I 
 have used with classes for more than twenty 
 years, I wish to say that I never had a student who 
 learned much about the subject unless he carefully 
 worked each problem and thoroughly understood it 
 before attempting the next; nor do I believe anyone 
 can get much benefit from this series of problems 
 without following the same course. This method, 
 too, even for those who have already studied per- 
 spective using distance-points, etc. is to be rec- 
 ommended in order, that the subject may be made 
 perfectly clear to them. 
 
 The perspective of interiors (Problems XVI- 
 XVIII) may be sometimes be advantageously post- 
 poned until the remaining problems have been 
 worked. 
 
 FRANK FORREST FREDERICK. 
 School of Industrial Arts, 
 Trenton, New Jersey. 
 September 3, 1909.
 
 INTRODUCTION. 
 
 THE sense of color and the sense of proportion are carefully de- 
 veloped by art teachers, but the perspective sense is largely al- 
 lowed to take care of itself. 
 
 By the perspective sense is meant a preception of the relation ex- 
 isting between straight lines their apparent convergence, direction, 
 length and position; a realization of what is meant by systems of lines 
 (lines that have the same direction) ; and the ability to think of sys- 
 tems instead of individual lines. 
 
 The lack of the perspective sense, on the part of a draftsman, is 
 as apparent in his work, to one who has it, as the lack of the color sense 
 is apparent to one who sees and appreciates color. 
 
 Students are prone, in drawing, to draw a line here and another 
 there one line of a system, then a line of another system trusting to 
 their sense of proportion alone to bring the drawing out correct in the 
 end. Asked to draw the interior of a room, or a street scene, the stu- 
 dent sees only a maze of lines because his ability to grasp the perspective 
 of the view r as a whole (his perspective sense) has not been developed. 
 
 A glance at the illustrations in the magazines shows how lacking 
 is the perspective sense even among many of the professional artists 
 whose ability to draw the figure seems almost perfect. We see figures 
 standing upon a floor that is not level, or upon a rug one edge of 
 which only rests upon the floor, or dining from a table upon which a 
 cup could hardly be made to stand. These illustrators, in their school 
 days, probably found the perspective class uninteresting, as do most art 
 students. , 
 
 Is this the fault of the students, or does the fault lie in the manner 
 in which the subject is peresented? , 
 
 All art schools offer courses in mechanical perspective, and in many 
 schools that make a specialty of educating art teachers the students are 
 required to take the course. When they leave school they either forget 
 all about the theory of perspective, or regard it as too difficult to apply 
 to every-day problems, or as taking too much time to be taught to their 
 own pupils, because to them perspective had been a matter of tee-square 
 and triangle only its application to practical problems and freehand
 
 drawing having either never been pointed out, or, if it had, so obscured 
 with rules and methods that the spirit of the thing was entirely missed. 
 The architect and the designer of interior decoration must be mas- 
 ters of perspective: it is a part of their stock in trade. The sculptor, 
 the painter and the illustrator have equal need of this knowledge, and 
 every one who draws should have worked perspective mechanically 
 long enough to have the perspective sense so developed as to make the 
 application of the rules a sort of second nature, even if the theory on 
 \vhich they are based be afterward forgotten. 
 
 Time was when drawing masters believed that a course in mech- 
 anical perspective should be followed before attempting to draw objects 
 freehand. There is something in this old idea. If a student could 
 draw a cube, for example, in mechanical perspective carrying all lines 
 out to their proper vanishing points they believed (and very properly) 
 that his freehand drawing of the same cube would be more likely to be 
 correct. The better plan is to carry on freehand and mechanical per- 
 spective together; for each helps the other, especially if the student is 
 taught to apply in his freehand work the principles illustrated by his 
 course in mechanical drawing. 
 
 This course in mechanical perspective is planned for students of the 
 high school age who have already received some training in mechanical 
 drawing, enough, at least, to understand Plates I and II. 
 
 It is given the title of "Simplified Mechanical Perspective," as the 
 attempt is made to consider the essentials that will develop the perspec- 
 tive sense and enable the student to apply his knowledge to practical 
 problems. 
 
 It is thought that in no other work on perspective is the practical 
 application of the subject to interesting and every-day problems made 
 so direct.
 
 THE PERSPECTIVE DIAGRAM. 
 
 THE PERSPECTIVE DIAGRAM. 
 
 IN Fig. 1, Plate 1, line A-B represents a horizontal plane upon 
 which a spectator is standing. The distance between the point 
 marked E. (Eye) and the point marked S. P. (Station Point) 
 represents the distance the spectator's eye is above the horizontal plane. 
 
 The spectator is supposed to be looking at a cube the plan of which 
 (a-b-c-d) is seen in Fig. 2. 
 
 An imaginary vertical plane, at right angles to the direction in 
 which the spectator is looking, is placed in front of the cube. This plane 
 show r s in Fig. 1 as a vertical line the edge view of the plane and in 
 Fig. 2 as a line upon the horizontal plane its top view, or plan, called 
 the trace of the vertical plane (Tr. V. PI.) 
 
 Visual rays pass from the eye to the corners of the cube and inter- 
 sect the vertical plane as seen in Figs. 1 and 2. If these points of in- 
 tersection should be connected by lines a perspective of the cube would 
 be obtained upan the vertical plane. 
 
 As it is ^ not practicable to draw upon this vertical plane, a nearer 
 plane called the picture plane (P. PL) is placed at any convenient dis- 
 tance from the spectator as the picture plane in Fig. 3, shown in Fig. 
 4 by the picture line (P. L.) the line of intersection of the picture 
 plane and the horizontal plane. 
 
 If the points of intersection of the visual rays with the vertical 
 plane are projected to the picture plane, the perspective upon the pic- 
 ture plane will be the same as the perspective upon the vertical plane. 
 Thus, in Fig. 4, d-e is the same as a-b the apparent length of b-g, and 
 e-f is the same as b-c the apparent length of b-h. 
 
 A horizontal plane passing through the spectator's eye, Fig. 1, in- 
 tersects the vertical plane in a horizontal line, the horizon, not seen in 
 Figs. 1, 2 and 3 as it is upon the vertical plane. Its position may be 
 found, however, by projecting, by line 1-2, Fig. 3, its distance above the 
 horizontal plane to the picture plane, and then revolving it (with point 
 R. as center) to coincide with the horizontal plane. It is then seen as 
 in Fig. 4. This revolution of the horizon ( H ) from its position upon a 
 vertical plane to a horizontal plane is necessary in order that the horizon
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
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 10
 
 THE PERSPECTIVE DIAGRAM. 
 
 may be upon the same plane (as a sheet of paper) as the picture line, 
 the trace of the vertical plane, and the station point. 
 
 The point directly opposite the eye on the vertical plane in the 
 horizon is called the center of vision, seen in Fig. 2, at C. V. The line 
 connecting the S. P. with the C. V., Fig. 2, is called the line of direc- 
 tion (L. of D.) 
 
 The points and lines so far found constitute the perspective dia- 
 gram, Fig. 5, Plate II. 
 
 TO LAY OUT A PERSPECTIVE DIAGRAM. 
 
 When laying out a perspective diagram upon which a perspective 
 is to be drawn, the first thing to determine is the scale a quarter, half, 
 or inch to the foot, depending upon the size of the object to be drawn 
 and its distance from the spectator. The first point to locate is S. P. 
 In the problems to follow each S. P. is located in its relation to the 
 margin line of a plate laid out as in Fig. 8.* Thus "S. P. 22 / // to 
 right and 2'0" above" means 22'0" to the right of the left margin line 
 and 2'0" above the lower margin line. After S. P. is located, draw a 
 line to represent the L. of D., Fig. 5, and set off on it, from S. P., the 
 distance C. V. is from S. P., and draw through C. V., at right angles 
 to L. of D., the Tr. V. PI. P. L. is placed wherever convenient gen- 
 erally at or ne#r S. P. H. is placed as far above P. L. as the eye is 
 supposed to be above the level of the base of the object to be drawn, or 
 as far below P. L. as the base of the object to be drawn is supposed to 
 be above the level of the eye. 
 
 TO FIND VANISHING POINTS. 
 
 In working a problem first place the plan at the required angle 
 with Tr. V. PL, as the square A-B-C-D in Fig. 6, and find the van- 
 ishing points of the retreating or vanishing lines. 
 
 To find the vanishing point (V. P.) of any system of lines (lines 
 that have the same direction), follow the direction of any one line (or 
 element) of the system till its point of intersection with the vertical 
 plane is found. This point will be the V. P. of the entire system. 
 The vanishing points of all horizontal systems not parallel to the ver- 
 tical plane are in the horizon. 
 
 *A1I plates are horizontal the 14" edge at the top unless otherwise in- 
 dicated. The 2" margin at the left allows for binding. 
 
 11
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 
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 12
 
 THE PERSPECTIVE DIAGRAM. 
 
 To find the V. P. of A-D and B-C, Fig. 6, draw a line from S. P. 
 parallel to these lines (in this case at an angle of 45 with Tr. V. PI.) 
 to cut the Tr. V. PI. Point X is found to be the V. P. of the system 
 of which A-D and B-C are two elements, and point Y the V. P. of the 
 system of which A-B and D-C are two elements. These points (X and 
 Y) are projected to H. at V. P. R. (vanishing point right) and V. P. 
 L. (vanishing point left) and become the vanishing points to be used 
 in working the problem. They are projected to H. in order that their 
 distance from the horizontal plane may be known. 
 
 After the vanishing points are found, draw the visual rays from 
 the corners of the plan, centering at S. P., but drawn only to Tr. V. 
 PI., in order to obtain the apparent length of lines in the plan as ex- 
 plained in connection with Fig. 4, Plate I. 
 
 PROBLEMS.* 
 PROBLEM I. 
 
 This problem illustrates the method of drawing, at a scale of y$" 
 =1'0", the perspective of a square (6'0"x6'0") that is on the hori- 
 zontal plane, with sides making angles of 45 with Tr. V. PL, 20'0" 
 from S. P. and 9'0" below the eye. 
 
 *NOTE TO TE'ACHERS: Instructors should prepare original problems, to follow 
 each problem here given, that their pupils may thoroughly understand all prin- 
 ciples and methods. Students may be assigned problems to work on the black- 
 board, as problems in geometry are often demostrated before the class. 
 
 The writer has found that some of the strongest students in perspective 
 seem, at first, unable to comprehend the subject working the problems by "rule 
 of thumb" but when real objects or rooms were to be drawn, the whole theory 
 has seemed to dawn upon them, in some cases in a moment. Therefore the 
 teacher should not feel discouraged if some students are slow to grasp the subject. 
 
 It is the writer's opinion that the study of mechanical and freehand perspec- 
 tive should be carried on at the same time. Every problem in this course, with 
 the exception of the first six, should be followed by a freehand exercise. For 
 example: After working Problem VII a similar prism should be placed before 
 the class to be drawn freehand. After Problem XV a similar table should be 
 drawn freehand. 
 
 In the public schools students will have drawn boxes, books, tables, chairs, 
 etc., for years before they are old enough to attempt this or any course in mech- 
 anical perspective. They should, however, combine the study of freehand with 
 mechanical perspective for in their earlier work they have drawn without the 
 sureness and accuracy that is developed by the study of mechanical perspective. 
 
 13
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 The statement of the diagram is: Scale }4"=1'0". S. P. 22 '0" 
 to the right and 2'0" above. C. V. 20'0" from S. P., P. L. 5'0" 
 from S. P., H. 9'0" above P. L. 
 
 After the diagram is complete, draw the square (A-B-C-D), 
 find V. P. R. and V. P. L., and the points of intersection with Tr. V. 
 PI. of the visual rays from S. P. to its corners (points F. A. E.) 
 
 As point A in the plan is against Tr. V. PL, point G of the per- 
 spective will be against P. L. Draw G V. P. R. and project point E to 
 point H. G-H is the perspective of A-B. Draw G V. P. L. and 
 project point F to point J. G-J is the perspective of A-D. Draw 
 J V. P. R. and H V. P. L., and their point of intersection, I, is the 
 perspective point of C. G-H-I-J is the perspective of A-B-C-D. 
 
 The plan may be placed at any angle with Tr. V. PI., and its 
 vanishing points found as illustrated above. In Fig. 7, Plate II, a 
 square is so placed that two sides make angles of 30 and two 60. 
 In Problem II the V. P. R. of lines at 60 and the V. P. L. of lines 
 at 30 must be found as illustrated in Fig. 7. 
 
 14
 
 THE PERSPECTIVE DIAGRAM. 
 
 PROBLEM II. 
 
 Scale #"=1'0". S. P. 3 TO" to the right and 2'0" above. C. 
 V. 14'0" from S. P., P. L. 3'0" from S. P., H. 6'0" above P. L. 
 
 In this problem a 10'0"xl0'0" square is so placed that the right 
 side makes an angle of 60 with Tr. V. PL, and the left side an angle 
 of 30. The perspective, though correct, appears distorted because the 
 
 VF7v-60 
 
 spectator (S. P.) was placed too near so large a square. Distortion of 
 the perspective will also result if the plan is placed too far to the right 
 or left of C. V. It may, however, be placed a short distance to the 
 right or left of C. V. without serious distortion as shown in Problem III. 
 
 PROBLEM III. 
 
 Scale }4"=1'0". S. P. 22'0" to right and 1'6" above. C. V. 
 2C'0" from S. P., P. L. 6'0" from S. P., H. 6'0" above P. L. 
 
 In this problem a 7'0"x7'0" square, with sides making angles of 
 45 with Tr. V. PI., is so placed that its nearest corner is 6'0" to the 
 left of C. V. 
 
 IS
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 Ill 
 
 16
 
 THE PERSPECTIVE DIAGRAM. 
 
 PROBLEM IV. 
 
 Scale #"=1'0". S. P. 22'0" to right and I'O" above. C. V. 
 12'0" from S. P., P. L. at S. P., H. 6'0" above P. L. 
 
 This problem illustrates the method of putting into perspective 
 a plan that is beyond the vertical plane. The square is 9'0"x9'0", 
 and is placed 5'0" to the right of C. V. and 3'0" beyond Tr. V. PL 
 Continue side B-A to cut Tr. V. PI. at E. Project E to P. L. at F. 
 Draw F V. P. L., and find the perspective of B-A in line F V. P. L. 
 Having located in perspective one side of the square, the three remain- 
 ing sides can be found as in the foregoing problems. If corner A had 
 been in line with the L. of D. the intersection of line F V. P. L. 
 with L. of D. would have been the perspective of corner A. 
 
 PROBLEM V. 
 
 Scale J4"=1'0". S. P. 12'0" to right and 2'0" above. C. V. 
 17'0" from S. P., P. L. 5'0" from S. P., H. 6'0" above P. L. 
 
 Any point on the horizontal plane may be put into perspective (as 
 point A placed at random) by considering the point as the end of a 
 line drawn to cut Tr. V. PI. at any convenient angle in this case an 
 
 17
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 V.'P.L 
 
 30: 
 
 V.P.K. 
 
 Z2E 
 
 18
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 angle of 60 to the left. Putting the line into perspective we find, of 
 course, the perspective of its end as point A. 
 
 After finding the perspective of point A as described above, find 
 it by considering the same point as the end of a line making an angle 
 of 30 to the right. 
 
 PROBLEM VI. 
 
 Scale J4=1'0". S. P. 22'0" to right and 2'0" above. C. V. 
 16'0" from S. P., P. L. lO'O" from S. P., H. 8'0" below P. L. 
 
 In this problem a square 9 / // x9 / // , is above the level of the eye 
 and therefore H. is placed as far below P. L. as the square is sup- 
 posed to be above the level of the eye. 
 
 When problems are lined-in with ink, the lines drawn at the given 
 angles from S. P., the line of direction, and horizon line may be omitted 
 to avoid confusion, as only the points obtained by these lines are nec- 
 essary for the working of the problems. (See Problems VII-XII.) 
 
 PROBLEM VII. 
 
 Scale %"=1'0". S. P. 23'0" to right and 2'0" above. C. V. 
 17'0" from S. P., P. L. at S. P., H. 9'0" above P. L. In this problem 
 a rectangular solid 6'0"x5'0"xl0'0" rests on one 6'0"xl0'0" face, with 
 sides vanishing to right and left at 45. The plan is placed as in the 
 foregoing problems. One elevation, all that is needed to secure the 
 heighth of the solid, is placed upon P. L. at one side of the problem. 
 
 The nearest vertical edge of the solid rests against the vertical 
 plane and therefore its perspective will be upon the picture plane. The 
 true length of this edge (5'0") may be set off at A-B, or projected 
 from the elevation by line C-A. 
 
 In this system of perspective all heights are secured from eleva- 
 tions; and all heights (as in any system of perspective) must be meas- 
 ured or located upon the picture plane, for there only do they appear 
 of their true length. 
 
 PROBLEM VIII. 
 
 Scale y 4 "=\'Q". S. P. 22'0" to right and 2'0" above. C. V. 
 17'0" from S. P., P. L. at S. P., H. 9'0" above P. L. 
 
 In this problem a square prism, base 8'0"x8'0", altitude 14'0", 
 
 19 *
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 T7TTT 
 
 D 
 
 M 
 
 20
 
 THE PERSPECTIVE DIAGRAM. 
 
 with sides vanishing to right and left at 45 is drawn. The lower base 
 is upon the horizontal plane, the upper is above the level of the eye. 
 Its height is found as in Problem VII. 
 
 PROBLEM IX. 
 
 Scale #"=1'0". S. P. 22'0" to right and 3'0" above. C. V. 
 15'0" from S. P., P. L. at S. P., H. 5'0" above P. L. In this prob- 
 lem a square, 12'0"xl2'0", one of whose sides (A-B) rests against Tr. 
 V. PL, is drawn. Sides A-D and B-C make with Tr. V. PL angles of 
 90. To find their V. P. draw from the S. P. a line parallel to them to 
 
 cut Tr. V. PL This V. P. is found to coincide with C. V. Projecting 
 this V. P. to H. (as other vanishing points are projected) the point 
 marked new C. V. is found. This point is the V. P. of the system of 
 which lines A-D and B-C are two elements. 
 
 The perspective of the square in this problem can be drawn in the 
 usual manner, or by the use of diagonals. As diagonal A-C makes an 
 angle to the right of 45, it will, if put into perspective, cut line H- 
 New C. V. at F and make H-F equal to H-G., and diagonal B-D, if 
 put into perspective, will locate D in perspective at E. 
 
 21
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 3H. 
 
 Y 
 
 Elevated PL 
 ? P,L 
 
 22
 
 THE PERSPECTIVE DIAGRAM. 
 
 In Problems I, III, IV, and VI one diagonal of the perspective 
 will be a horizontal line and the other will vanish at the New C. V. 
 The perspective of the center of any rectangle in perspective may be 
 found by drawing its diagonals 
 
 PROBLEM X. 
 
 Scale y 2 "=l'Q". S. P. ll'O" to right and I'O" above. C. V. 
 8'0" from S. P., P. L. at S. P., H. 4'0" above P. L. In this problem 
 a square, 4'0"x4'0", with two sides parallel to Tr. V. PI., is 2'0" be- 
 yond the vertical plane. Its perspective is obtained by applying the 
 method of finding points beyond the vertical plane illustrated in Prob- 
 lems IV and V. 
 
 PROBLEM XL 
 
 Scale J4"=1'0". S. P. 22'0" to right and 2'0" above. C. V. 
 16'0" from S. P., P. L. at S. P., H. 12'0" above P. L. In this prob- 
 lem an equilateral triangular prism rests on one 8'0"xll'0" face as 
 shown by the plan. The altitude of the triangular base is put into per- 
 spective by projecting it to A-B, drawing A V. P. L., and projecting 
 point C to cut A V. P. L. 
 
 The visible triangular base of the prism is included in a ver- 
 tical plane vanishing to the left. Therefore any vertical distance in 
 this plane must be first set off on the intersection of this vertical plane 
 with the picture plane (as A-B) and then carried into perspective. 
 
 PROBLEM XII. 
 
 Scale ^"=1'0". S. P. ll'O" to right and I'O" above. C. V. 
 8'0" from S. P., P. L. at S. P., H. 6'0" above P. L. In this problem 
 a square plinth 5'6"x5'6", altitude 3'0", with sides at 45, is 2'0" 
 above the horizontal plane. As the lower base is 2'0" above the hori- 
 zontal plane, an elevated picture line, (El. P. L.) 2'0" above P. L. 
 may be used. Whenever a horizontal surface, that is not the upper 
 base of an object on the horizontal plane, is to be put into perspective 
 it is well to use an El. P. L. 
 
 23
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 THE PERSPECTIVE OF FURNITURE AND INTERIORS. 
 
 IT MAY have occurred to those who are working these problems, 
 that the Tr. V. PI. might be used as H., the points, found by 
 drawing lines at the required angles from S. P., as the vanishing 
 points for the problems, and that the P. L. might be placed as far below 
 Tr. V. PI. as the eye is supposed to be above the horizontal plane. 
 This is true, and Problems I to VI might have been so worked, but in 
 this system of perspective known as the "Plan Method" where the 
 plan of the object to be put into perspective is drawn, this plan and the 
 working lines of the problem would be confused with the perspective if 
 the picture plane, with H., was not moved nearer the spectator. 
 
 Students who have followed these problems should now be suffi- 
 ciently familiar with the subject to be on the lookout for "short cuts" 
 in locating points and lines by means of diagonals, etc, and should learn 
 to distinguish between lines necessary for the complete solving of the 
 problems and lines used only to obtain some required point. It should 
 no longer be necessary to carry all lines out to their vanishing points 
 in order to appreciate their direction, and lines of projection from Tr. 
 V. PI. to the perspective should be omitted if likely to confuse the prob- 
 lem. If the proper degree of hardness of pencil for the paper used is 
 found, all working lines may be made of a delicate lightness that will 
 permit all to be seen without causing confusion. 
 
 Success in working the following problems will depend largely upon 
 a sharply pointed hard pencil. 
 
 PROBLEM XIII. 
 
 Scale y 2 "\'Q". S. P. ll'O" to right and I'O" above. C. V. 
 9'0" from S. P., P. L. at S. P., H. 6'0" above P. L. In this prob- 
 lem a box 3'0"x5'0" and 2'0" deep, with cover open at angle of 60, is 
 placed at an angle of 45 with Tr. V. PI. The box is put into perspec- 
 tive as is the prism in Problem VII. 
 
 The edge of the cover is seen as line A-B in the plan. Drawing 
 a vertical line from C we have a vertical in which the perspective of A 
 will be found. Project point D to E and draw E V. P. L. Where 
 E V. P. L. cuts the vertical the perspective of A is found. 
 
 24
 
 THE PERSPECTIVE OF FURNITURE. 
 
 The same vertical plane includes the end of the box and the edge 
 of the cover as it opens. Any vertical distance in this plane must be 
 first set off on its intersection with the picture plane (as E S. P.) and 
 then carried into perspective. 
 
 This problem will have but a small part of its possible value to 
 students if it is not followed by a practical problem. Bring a trunk into 
 the drawing room, raise the cover to some known angle, and place the 
 student who is to draw the perspective a certain number of feet, as 
 
 wr. 
 
 8'0", from the nearest corner of the trunk. The point on the floor 
 under the student's eye becomes the S. P. The distance upon the floor 
 from S. P. to the nearest corner of the trunk (C. V.) becomes the L. 
 of D. The distance the eye is from the floor gives the distance H. 
 must be placed above P. L. Measuring the trunk to get its dimensions, 
 and turning it so that it makes some known angle with the assumed 
 vertical plane, and adopting a scale for the drawing, a perspective should 
 be drawn. S. P., L. of D., Tr. V. PL, P. L. and H. may be drawn 
 upon the floor with chalk. 
 
 If two students work together, and with tape line find the dis- 
 tances and measurements, problems of this character become more inter- 
 esting. 
 
 25
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 TTV , 
 
 26
 
 THE PERSPECTIVE OF FURNITURE. 
 
 PROBLEM XIV. 
 
 Scale 1"=1'0". S. P. 2'9" to right and 6" above. C. V. 4'6" 
 fiom S. P., P. L. at S P., H. 3'0" above P. L. In this problem a 
 set of steps, rise 6", tread 9" and 2'0" long, make an angle of 30 
 to the right. In the perspective the width of the treads is obtained from 
 the plan and the height of the risers from the elevation, being first set 
 off on the L. of D., and then carried into perspective by lines vanishing 
 at V. P. R. 
 
 Two or three of the lower steps of any stairway provides a prac- 
 tical exercise to apply this problem. 
 
 PROBLENf XV. 
 
 In this problem a table (measurements given on the plan and ele- 
 vation) is drawn as it would appear to a spectator whose eye is 3'6" 
 from the floor, and who is so placed that the point on the floor (S P.) 
 directly under his eye is 4'0" from the point on the floor directly under 
 the nearest corner of the top of the table. 
 
 The statement of the diagram is as follows : 
 
 Scale 1"=1'0". S. P. 5'6" to right and 3" above. C. V. 4'0" 
 from S. P., P. L. at S. P., H. 3'6" above P. L. 
 
 The table top is one inch thick and 2'11" from the floor. First 
 put the top in perspective by using an El. P. L., 2'11" from P. L., and 
 then the legs, using P. L. In the location of the nearest corner of the 
 nearest leg on the floor apply Problem IV. 1 
 
 It is but a step from drawing objects in a room to the room itself. 
 The same principles apply and the method is the same. 
 
 Plate III (not to be worked by students) reproduces a perspective 
 drawing of an inglenook the plan of which is shown by Fig. 9. The 
 drawing is introduced at this point in the course to illustrate the method 
 of approaching a problem of this character. 
 
 'NOTE TO TEACHERS. When selecting furniture for students to measure and 
 draw in perspective, care should be taken to select very simple examples, as, 
 possibly, the book-rack or shelves made by the students in the wood shop. (See 
 accompanying illustrations.) If small these may be placed upon a table which 
 becomes the horizontal plane for the problem. 
 
 After students have become familiar with perspective and accustomed to 
 working problems in other words, when their perspective sense has become 
 developed they should be assigned more complicated and difficult problems, the 
 important lines of which should be found mechanically and the details added 
 freehand. 
 
 27
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 By measuring it \vas found that a point on the floor directly under 
 the spectator's eye (S. P.) was 12'0" from the point marked Y in the 
 plan, Fig. 9. His eye was found to be 3'8" above the floor. 
 
 The first step taken in drawing the per- 
 spective of this inglenook, was the drawing of 
 the plan. The next step was the location, at the 
 same scale, of S. P., L. of D., Tr. V. PL, P. L., 
 and H. in the order named. As the eye chanced 
 to be directly in line with the left wall (Y-Z), 
 Y-Z in the plan was continued 12'0" and the S. 
 P. for the problem was found. S. P. X (the 
 L. of D.) was next drawn, and Tr. V. PL was 
 drawn at right angles to the L. of D. through 
 point Y. The paper was then taken from the 
 board and repinned with Tr. V. PL as a hori- 
 zontal line. P. L. was placed at S. P. and H. 
 placed 3'8" above P. L. V. P. L. was located 
 
 by projecting point Y to H., and V. P. R. by drawing from S. P., par- 
 allel to Y-W, to cut Tr. V. PL, and then projecting the point found 
 
 3/P 
 
 Fto-.S>. 
 
 28
 
 THE PERSPECTIVE OF FURNITURE. 
 
 PLATLffl 
 
 29
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 to H. V 7 . P. R. falls outside the margin line. When drawing per- 
 spectives of interiors, students should use large drawing boards, or 
 fasten small boards to the top of tables, to secure room for the vanish- 
 ing points. These points, when located, can be kept by driving pins 
 into the boards or table. The elevation was then drawn with P. L. as 
 base line. 
 
 The important lines in the perspective of the inglenook were ob- 
 tained from the plan and elevation, but the details were drawn by guess. 
 If the perspective had been drawn at a larger scale (1"=1'0" was used) 
 these could have been drawn accurately. Architects and designers for 
 interior decoration often, for convenience, draw perspectives at small 
 scale and then enlarge them to any size desired. 
 
 If the lines to obtain widths become confusing on account of their 
 number, they can be drawn in sets and each set erased before the next 
 
 Tr.V.Pf 
 
 30 
 
 Fig'.ll. 
 
 is drawn. In Plate III the set used to obtain the seat was erased before 
 the set used to obtain the fire-place were drawn. 
 
 In working perspective problems much experiment is often nec- 
 essary to get the perspective upon the paper where wanted. If the plan 
 and elevation be drawn and then cut apart and pasted or pinned upon the 
 paper upon which the perspective is to be drawn, much time may be 
 saved as their position may be changed without going to the trouble of 
 re-drawing. 
 
 PROBLEM XVI. 
 
 In this problem, Fig. 10, a spectator, whose eye is 4'0" from the 
 floor, is supposed to be standing in a door at the end of a room 19'0"x 
 
 30
 
 THE PERSPECTIVE OF FURNITURE. 
 
 XVI 
 
 XV I L 
 
 26'0". He assumes the vertical plane to be 18'0" from his eye. The 
 perspective represents that part of the room (8'0") lying beyond the 
 vertical plane. The statement of the diagram will be: Scale %"=1'0". 
 S. P. 22'0" to right and 3'0" above . C. V. 18'0" from S. P., P. L. 
 at S. P., H. 4'0" above P. L. 
 
 31
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 Put the room into perspective according to Problem IX, and draw 
 the door and windows 3'0" wide. The height of the room, the door 
 and windows is shown by a section of the wall of the room. 
 
 XVIII. 
 
 VPL -; 
 
 H YPR 
 
 PL 
 
 PROBLEM XVII. 
 
 In this problem, Fig. 11, a spectator whose eye is 4'0" above the 
 floor, is standing 15'0" from the corner of the room. The Tr. V. PI. 
 is assumed to be lO'O" distant. The drawing represents the perspective 
 of that part of the room (5'0") lying beyond the vertical plane. The 
 diagram is drawn as follows: 
 
 32
 
 THE PERSPECTIVE OF FURNITURE. 
 
 Scale J"=1'0". S. P. ll'O" to right and 6" above. C. V. lO'O" 
 from S. P., P. L. I'O" from S. P., H. 4'6" above P. L. The door is 
 3'0" wide and the floor is divided in 2'0" squares. 
 
 When drawing interiors, where the distance from S. P. to C. V. 
 is less than the height of the room, P. L. may be placed wherever con- 
 venient above the plan or below S. P., Fig. 12. If H. is kept the 
 required distance from P. L., and the widths are obtained from Tr. V. 
 PI, the perspective will be the same as if drawn upon a P. L. placed at 
 or near S. P. (See Problem XVIII.) 
 
 PROBLEM XVIII. 
 
 (For this problem the 8" side of the margin line is considered as 
 the top.) 
 
 Scale ^"=1'0". S. P. 8'0" to right and 4'9" above. C. V. 
 9'0" from S. P., P. L. 2'0" above lower margin line, H. 4'0" above 
 P. L. The spectator, whose eye 
 
 is 4'0" from the floor, is sup- 
 posed to be standing 16'0'' 
 from the corner of a room. The 
 vertical plane is placed 9'0" 
 from his eye. The ceiling of 
 the room is lO'O" from the floor 
 and therefore P. L. is placed 
 below S. P. The doorway is 
 7'6" high, the fire-place open- 
 ing is 2 / 0" high, the tiles are 6" 
 square and the picture mould- 
 ing l / 6 // from the ceiling. A 
 rug 3'0" square is placed in the 
 center of the door and extends 
 through into the adjoining room 
 The shelf projects 4" from the 
 chimney breast and is on the 
 level of the eye. Point A is 
 located as point A in Problem 
 V. Part of the hearth is found 
 ^ ' ' to P r J ect m front of the verti- 
 
 cal plane. This frequently happens when designing interiors, and if the 
 projection is not too great the drawing is not seriously distorted. To 
 
 P.L. 
 
 33
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 find the perspective of lines extending in front of the vertical plane, find 
 the perspective of the ends of the lines that are beyond the vertical plane, 
 and continue the lines until they meet in front of the picture plane. 
 
 After students have worked Problems XVI, XVII and XVIII, 
 they should be assigned ends and corners of rooms and halls to be drawn 
 in perspective as practical problems to apply the knowledge gained. 
 Before students undertake the remaining problems in this series they 
 should be able to draw in perspective any view of any room from 
 any point. 
 
 The practice and experience gained in this work is not limited to 
 the actual drawing of the perspective. To measure a room and draw 
 its plan and elevation or section to decide upon the angle the sides of 
 the room shall make with the vertical plane to locate to the best ad- 
 vantage, the station point and the vertical plane, gives an all-round 
 training in mechanical drawing that is quite as valuable to the student 
 as the development of his perspective sense. 
 
 PROBLEM XIX. 
 
 A student whose perspective sense has not been developed always 
 has trouble when drawing the gables and eaves of houses, and the lines 
 of intersection of chimneys with roofs. Some of our best landscape 
 
 34
 
 THE PERSPECTIVE OF FURNITURE. 
 
 painters seem unable to handle this really simple problem in perspective. 
 
 Scale #"=1'0". S. P. 22'0" to right and I'O" above. C. V* 
 12'0" from S. P., P. L. at S. P., H. lO'O" above P. L. 
 
 After drawing the roof, draw the top of the chimney from an ele- 
 vated picture line as far (5'0") above P. L. as the top of the chimney 
 is above the horizontal plane. 
 
 To find the line of intersection of the chimney with the roof, con- 
 sider the right side of the chimney as included in a vertical plane cutting 
 the roof. The line of intersection of this vertical plane with thereof 
 is found as follows: 
 
 Continue the side of the chimney to point A. Project A to B. 
 Draw B V. P. R. to find C. Draw C-D, a vertical line, and connect 
 
 I2'o" 
 
 D with V. P. R. Project the width of the right side of the chimney to 
 D V. P. R. and the intersection of the right side of the chimney with 
 the roof is found. 
 
 The line of intersection of the left side with the roof is found by 
 projecting the point of intersection of the ridge of the roof with the 
 chimney to its perspective and E-F is found a short line but quite as 
 important as any line in the problem. 
 
 35
 
 THE PERSPECTIVE OF FURNITURE. 
 
 PROBLEM XX. 
 
 Scale *4"=1'0". S. P. 22'0" to right and 2'0" above. C. V. 
 14'0" from S. P., P. L. at S. P., H. 7'0" from P. L. 
 
 Draw, first, the rectangular prism representing the body of the 
 house and the triangular prism representing the roof. Then draw, 
 from an elevated picture line, the triangular prism, which, with the 
 bases omitted, becomes the projecting eaves and gables. 
 
 36
 
 (.J 
 
 THE PERSPECTIVE OF CIRCLES. 
 
 THE PERSPECTIVE OF CIRCLES. 
 
 A CIRCLE may appear as a circle, a line or as an ellipse. It appears 
 of its true shape when it is at right angles to the direction in 
 which the spectator is looking, as a line when included in a plane 
 that passes through the eye, and as an ellipse when the circle is in any 
 other position in relation to the eye. 
 
 _ If a circle is inscribed 
 
 within a square there are 
 four points, 1, 3, 5 and 7, 
 Fig. 13, where the circum- 
 ference of the circle is tan- 
 gent to the sides of the square, 
 and four points, 2, 4, 6 and 
 8, where the circumference 
 of the circle intersects the di- 
 agonals of the square. 
 
 When a circle is put into 
 perspective it is usual to draw 
 a square about it, find the 
 eight points just mentioned, 
 put the square into perspec- 
 and draw the 
 
 (the perspective of the circle) through the perspective of these points. 
 
 PROBLEM XXI. 1 
 
 Plate vertical. Scale 1"=!' 0" S. P. 4' 0" to right and 3" above. 
 C. V. 3' 9" from S. P., P. L. at S. P., H. 2' 0" above P. L. 
 
 In this problem a circle, 6' 0" in diameter, is directly in front of the 
 spectator. A square, with sides tangent to the circumference, is drawn 
 and points 1, 3, 5 and 7 found. By drawing the diagonals of the square 
 points 2, 4, 6 and 8 are found. Putting the square into perspective 
 
 'NOTE TO TEACHERS: When students begin the study of the circle in per- 
 spective it is well to draw circles upon the floor, or upon the blackboard, as 
 problems to be worked by the class. These should be drawn large (six or eight 
 feet in diameter) and the students placed near them, otherwise their drawings 
 will be too small to be worked accurately. 
 
 37
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 t 
 
 
 \ 
 
 t 
 
 4- -4- 
 
 38
 
 THE PERSPECTIVE OF CIRCLES. 
 
 according to the method explained in Problem IX, the perspective of 
 the circle the ellipse may be drawn through the perspective of the 
 eight points 1 to 8 inclusive. 
 
 PROBLEM XXII. 
 
 Plate vertical. Scale X"=1'0". S. P. 16'0" to right and I'O" 
 above. C. V. 15' 0" from S. P., P. L. at S. P., H. 12' 0" above P. L. 
 A square, 19' 0" x 19'0" on the horizontal plane, has its sides van- 
 ishing to right and left at 45. Within the square is a circle 19'0" in 
 diameter. 
 
 To find the perspective of the circle draw the perspective of the 
 square with its diagonals and diameters, and find the perspective of the 
 points 1 to 8. 2, 4, 6 and 8 are found at the ends of the diameters, 1 
 and 5 on the diagonal parallel to the picture plane, as points are usu- 
 ally located, and points 3 and 7, on the diagonal at right angles to the 
 picture plane, according to problems IV and V. 
 
 PROBLEM XXIII. 
 
 Plate horizontal. Scale Y 4 =V 0". S. P. 22' 0" to right and V 0" 
 above. C. V. 20' 0" from S. P., P. L. at S. P., H. 8' 0" above P. L. 
 A vertical circle, 14' 0" in diameter, vanishes to the right at 45. 
 
 39
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 PROBLEM XXIV. 
 
 Plate vertical. Scale #"=!' 0". S. P. 16' 0" to right and 7' 0" 
 above. C. V. 15' 0" from S. P., P. L. 2' 0" from lower margin line, 
 H. 19' 0" above P. L. 
 
 In this problem a circular plinth, base 16' 0" in diameter, altitude 
 ll'O", rests on its side in the horizontal plane which is 19'0" below 
 
 the level of the eye. The axis of the plinth vanishes to the right at 45. 
 A lowered picture line is used, to prevent confusion of lines, because 
 the distance from S. P. to C. V. is less than the diameter of the base 
 of the plinth, and also because the distance from P. L. to H. is greater 
 than the distance from S. P. to C. V. P. L. can be placed wherever 
 convenient providing it is at a greater distance from Tr. V. PI. than H. 
 is from P. L. 
 
 40
 
 THE PERSPECTIVE OF CIRCLES. 
 
 Put the visible base into perspective (as in Problem XXIII). Find 
 the eight points through which the ellipse representing the invisible base 
 is to be drawn by carrying lines from the eight points already found 
 on the visible base to V. P. R., and projecting to them, from the plan, 
 the corresponding points on the invisible base. Only the working lines 
 used to find the invisible base are lined-in. 
 
 The entire ellipse, in both freehand and mechanical perspective, should 
 always be drawn even if but a small part of its circumference is after- 
 wards lined-in. In no other way can the true elliptical curve be ob- 
 tained. 
 
 10' 0" 
 
 PROBLEM XXV. 
 
 Scale }4"=1'0". S. P. 33' 0" to right and 1' 0" above. C. V. 
 18'0" from S. P., P. L. at S. P., H. 5'0" above S. P. 
 
 In this problem a plinth 16'0" x 20'0" and 3'0" thick rests on one 
 3' 0" x 20' 0" face and vanishes to the left at 30. A circular arch 
 is cut from this plinth as shown by the elevation. For convenience in 
 working, a circle of the size of the arch is drawn upon the picture plane 
 with its diameter coinciding with the nearest vertical edge of the plinth. 
 
 PROBLEM XXVI. 
 
 Scale ^"=1'0. S. P. 6' 3" to right and 6" above. C. V. 8' 6" 
 from S. P., P. L. 1' 6" from S. P., H. 4' 0" above P. L. 
 
 41
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 XXVM 
 
 t 
 
 4- 
 
 42
 
 THE PERSPECTIVE OF CIRCLES. 
 
 From a plinth vanishing to the right at 45, base 5'0" x lO'O", alti- 
 tude 2' 0", a semi-circular arch, with radius of 4' 0", is cut as shown by 
 the elevation. From the drawing it will be seen that the diagonals of 
 the enclosing square may be put into perspective and used as tests in the 
 location of the eight points through which the ellipse is drawn. 
 
 PROBLEM XXVII. 
 
 Scale %"=l'Q". S. P. 22' 0" to right and 1' 0" above. C. V. 
 20' 0" from S. P., P. L. 2' 0" above S. P., H. 10' 0" above P. L. 
 
 Any area may be put into perspective by enclosing it within a rec- 
 tangle and locating points in its perimeter by means of known lines. In 
 this problem a pointed-arch opening, as a door or window, is drawn. 
 Lines A and B are placed at random. 
 
 43
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 PROBLEM XXVIII. 
 
 It is hoped that students have noticed that while the short diameters 
 of the ellipses drawn have coincided with a diameter or diagonal of the 
 square used in finding the perspective of the circles, the long diameters 
 have not, but have in every case appeared to be nearer the spectator. In 
 
 this problem the method of finding the long diameter of an ellipse repre- 
 senting a circle upon, or parallel to the horizontal plane, is explained. 
 
 In practice this diameter is located by guess as an ellipse is seldom 
 drawn large enough to permit the exact location of the long diameter 
 to be found. 
 
 Plate vertical. Scale Y 4 "=\' 0". S. P. 16' 0" to right and TO" 
 above. C. V. 15' 0" from S. P., P. L. at S. P., H. 10' 0" above P. L. 
 
 The circle is 2V 0" in diameter. 
 
 44
 
 THE PERSPECTIVE OF CIRCLES. 
 
 After the ellipse is found, as in Problem XXI, draw from S. P., to 
 right and left, lines tangent to the circle. Find the exact points of tan- 
 gency by drawing through the center of the circle lines at right angles 
 to the tangent lines. Points X and Y will be found to be the tangent 
 points. Put X and Y into perspective, according to Problem V, and 
 the ends of the long diameter of the ellipse will be found. 
 
 PROBLEM XXIX. 
 
 Plate vertical. Scale ^"=1' 0". S. P. 8' 0" to right and I'O" 
 above. C. V. 9' 0" from S. P., P. L. at S. P., H. 8' 0" above P. L. 
 
 In this problem the perspective of a circular plinth (diameter 8'0", 
 altitude 8' 0") as seen by an eye 9' 0" distant and 8' 0" above the hori- 
 zontal plane is found. Any object so large, seen from so near, will ap- 
 pear when put into perspective, as distorted or in "violent perspective," 
 as Problem XXIV, but from this problem two facts relating to the ap- 
 pearance of cylindrical objects of great assistance in freehand drawing 
 can be learned: 
 
 First: The short diameter of the farther ellipse is, compared with 
 the long diameter, proportionately greater or longer than the short diam- 
 eter of the nearer ellipse. 
 
 Second: The long diameters of the ellipses representing the bases 
 are at right angles to the axis of the solid. This is but approximately 
 true in this problem on account of its violent perspective, but is near 
 enough to enable the student to realize that tests for the drawings of 
 cylindrical objects are lines drawn to represent the axes of the objects 
 and lines at right angles to the axes to represent the long diameters of 
 the ellipses. 
 
 If the point of intersection of the long diameter of an ellipse repre- 
 senting the circular base with the axis of an object should be taken as 
 C. V. there would be no distortion and the right angle would appear 
 as such. 
 
 45
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 UV.PL 
 
 THE PERSPECTIVE OF OBLIQUE LINES. 
 
 SO FAR, in this course in perspective, we have found and used only 
 the vanishing points of retreating horizontal lines all of which 
 have been found in the horizon. 
 
 It is now time to use the vanishing points of oblique lines, for all 
 lines, whatever their direction, that are not parallel to the picture plane, 
 
 have vanishing points. 
 
 In Fig. 14, not only do lines 
 1-4, 3-5 and 2-6 vanish at V. P. 
 R., and lines 1-2 and 4-6 at V. P. 
 L., but lines 1-3 and 4-5, which 
 are parallel, vanish at U. V. P. L. 
 (Upper Vanishing Point Left.) 
 and lines 3-2 and 5-6 at L. V. P. 
 v/p-p L. (Lower Vanishing Point Left.) 
 If these upper and lower van- 
 ishing points can be found in ad- 
 vance, as V. P. R. and V. P. L. 
 are found, their use will make the 
 solving of many problems much easier and simpler. 
 Line 1-2, Fig. 14, is a retreating horizontal 
 line, and 1-3 a 
 retreating ob- 
 lique line, but 
 they both lie 
 
 in the same vertical plane. Planes 
 have their vanishing traces as lines 
 have their vanishing points. H is 
 the vanishing trace of all horizon- 
 tal planes. A vertical drawn thru 
 V. P. L. is the vanishing trace of 
 the vertical plane of which the tri- 
 angle 1-2-3 forms a part. All lines 
 lying in or parallel to a plane have 
 their vanishing points in the van- 
 
 ..1VRL 
 
 uKjd si|j jo 
 the vanishing point of 4-5, as well as that of 1-3, must be in this vertical 
 
 46
 
 TTo.16. 
 
 THE PERSPECTIVE OF OBLIQUE LINES. 
 
 trace above V. P. L. and the vanishing point of 3-2 and 5-6 must be 
 below V. P. L. 
 
 If the vertical plane containing triangle 1-2-3 be revolved to co- 
 incide with the V. PI. the true angle made with the horizontal plane by 
 1-3 may be seen and the vanishing 
 point of 1-3, and all lines parallel 
 to 1-3, found. It is known that the 
 vertical plane containing 1-2-3 
 makes an angle to the left of 45, 
 therefore, Fig. 15, line X-S. P. is 
 revolved to Tr. V. PL and the an- 
 gle made by 1-3 with the horizon- 
 tal plane (in this case 45) is con- 
 structed and point A found. If Tr. 
 V. PL was used as the horizon 
 point, A would be the vanishing 
 point of all lines making angles of 
 
 Tr.V.71. 
 
 H. 
 
 EL, 
 
 A 
 
 45 to the left with the vertical 
 
 plane and angles of 45 upward 
 
 with the horizontal plane. 
 
 As has been explained (Plate I) Tr. V. PL is not used as the horizon, 
 
 point to be used as the U. V. P. L. 
 ( B ) is placed as far above H. as A 
 is above Tr. V. PL That is, dis- 
 
 ^ i. tance C-B is made equal to dis- 
 
 tance X-A. 
 
 A diagram for a problem con- 
 taining lines vanishing to right or 
 on acount of the confusion of 
 lines, but a new line (H) is drawn 
 the required distance above P. L. 
 which is placed wherever con- 
 venient. 
 
 As A, Fig. 16, is found to be 
 distance X-A from Tr. V. PL, the 
 
 B 
 LVPL< 
 
 V.-RR. 
 
 I! 
 
 im 
 
 left at 45 with the vertical plane and up or down at 45 with the hori- 
 zontal plane is constructed as in Fig. 17. Distance V. P. L. U. V. P. 
 L. is made equal to distance X-A. Distance V. P. L. L. V. P. L. is 
 made equal to distance X-B. Distance V. P. R. U. V. P. R. is made 
 
 47
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 48
 
 THE PERSPECTIVE OF OBLIQUE LINES. 
 
 equal to distance Y-C. Distance V. P. R. L. V. P. R. is made equal 
 
 to distance Y-D. 
 
 In Fig. 14, line 1-2 is included in two planes. It is in a horizontal 
 
 plane and also in a vertical plane. Its vanishing point is V. P. L. the 
 
 point of intersection of the traces of the two plances containing the line. 
 
 Every retreating line lying in 
 two planes has its vanishing point 
 at the intersection of the traces of 
 the planes. 
 
 In Fig. 18, U. V. P. L. V. 
 P. R. is the vanishing trace of the 
 plane containing 1-2-4-3, and U. 
 V. P. R. V. P. L. is the vanish- 
 ing trace of the plane containing 
 1-5-6. Therefore line 1-6, which is 
 common to both planes, will have 
 
 its vanishing point at X, the point of intersection of the traces of the 
 
 two planes containing the line. 
 
 PROBLEM XXX. 
 
 Plate vertical. Scale %"=1' 0". S. P. 16' 0" to right and IT 0" 
 above. C. V. 13'0" from S. P., P. L. at S. P., H. 8'0" above P. L. 
 
 In this problem a square 13" 0" x 13'0" rests on an edge that van- 
 ishes to the right at 45. The square makes an angle of 45 with the 
 horizontal plane. The square does not, of course, appear as such in the 
 plan. V. P. R. is the vanishing point of the two edges parallel to the 
 horizontal plane. U. V. P. L. is the vanishing point of the two edges 
 making angles of 45 upward to the left. 
 
 In laying out the diagram for a problem containing lines making 
 angles with both the horizontal and vertical planes it is always advisable 
 to find the four upper and lower vanishing points that their location 
 may be known before the problem is worked. 
 
 PROBLEM XXXI. 
 
 Scale ^"=1'0". S. P. 8'0" to right and 5' 6" above. C. V. 
 6' 6" from S. P., P. L. at S. P., H. 4' 0" above P. L. 
 
 A box 4'0" x 6' 0" and 3' 0" deep stands on the horizontal plane 
 as shown by the plan. The cover is open at an angle of 30. 
 
 49
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 PROBLEM XXXII. 
 
 Scale #"=1'0". S. P. 32' 0" to right and 22' 0" above. C. V. 
 26' 0" from S. P., P. L. at S. P., H. 16' 0" above P. L. 
 
 A triangular prism, size and location shown by plan and elevation, 
 is drawn using upper and lower vanishing points. 
 
 PROBLEM XXXIII. 
 
 Scale J4"=1'0". S. P. 16' 0" to right and 11' 0" above. C. V. 
 13' 0" from S. P., P. L. at S. P., H. 8' 0" above P. L. 
 
 A cube, faces 8' 0" x 8' 0", rests on one edge as shown by the plan 
 and elevation. All lines used to obtain the perspective are lined-in to 
 show how few are necessary to work the problem. 
 
 50
 
 THE PERSPECTIVE OF OBLIQUE LINES. 
 
 X 
 X 
 X 
 
 51
 
 SIMPLIFIED MECHANICAL PERSPECTIVE. 
 
 52
 
 THE PERSPECTU'E OF OBLIQUE LINES. 
 
 PROBLEM XXXIV. 
 
 Scale J4"=1'0". S. P. 12' 0" to right and 6' 0" above. C. V. 
 lO'O" from S. P., P. L. at S. P., H. 5' 0" above P. L. 
 
 A cube, faces 7' 0" x 7' 0", rests on one edge as shown by the plan 
 and elevation. First find the perspective of the edge of the cube resting 
 in the horizontal plane. 
 
 PROBLEM XXXV. 
 
 Scale #"=1'0". S. P. 32' 0" to right and 22' 0" above. C. V. 
 26' 0" from S. P., P. L. at S. P., H. 16' 0" above P. L. 
 
 XXXVII 
 
 This is a problem that frequently presents itself when drawing 
 roofs. All dimensions necessary for working the problem are given on 
 the plan and elevation. The intersection of the roof vanishes at the 
 intersection of the traces of the roofs as explained in Fig. 18. It is as 
 
 53
 
 THE PERSPECTIVE OF OBLIQUE LINES. 
 
 necessary for a draftsman making a freehand drawing of roofs to know 
 where the line of intersection vanishes as it is for the architect who is 
 drawing a perspective mechanically. 
 
 PROBLEM XXXVI. 
 
 Scale i4"=l'0". S. P. 16' 0" to right and 11' 0" above. C. V. 
 13' 0" from S. P., P. L. at S. P., H. 8' 0" above P. L. 
 
 In this problem a chimney, 4' 0" x 4' 0", extends from the center of 
 a roof 16'0" x 16'0". The top of the chimney is 12'0" from the hori- 
 zontal plane. 
 
 PROBLEM XXXVII. 
 
 Scale j/S"=l' 0". S. P. 32' 0" to right and 22' 0" above. C. V. 
 26' 0" from S. P., P. L. at S. P., H. 16' 0" above P. L. 
 
 In this problem a dormer-window extends from a roof 28' 0" x 
 30' 0". Other dimensions necessary for the working of this problem 
 are given on the elevations. 
 
 54
 
 Books on the Manual Arts 
 
 Essentials of Woodworking. By IRA s. GRIFFITH; illustrated 
 with numerous pen drawings by Edwin V. Lawrence. 
 
 This is a comprehensive textbook on woodworking tools, materials and processes, to supplement, 
 but not to take the place of, the instruction given by the teacher. The book contains three parts: 
 I Tools and elementary processes, including laying-out tools and their use, saws, planes and 
 their use, boring tools, chisels, grinding and whetting, form work, laying out duplicate parts, 
 scraping, sandpapering, and fastening parts. II Simple, joinery, including directions for making 
 the common joints, elementary cabinet work involving drawer construction, paneling, rabbeting, 
 and door construction. Ill Wood and wood-finishing, including a great amount of information 
 that should be given to a student along with his work in wood. The book does not contain a course ' 
 of models. It may be used with any course. Price. $1.00. 
 
 Beginning Woodwork. At Home and in School. By CLINTON 
 SHELDON VAN DEUSEN; illustrated by Edwin Victor Lawrence. 
 
 A full and clear description in detail of the fundamental processes of elementary benchwork 
 in wood. This description is given through directions for making a few simple, useful articles 
 suitable either for school or home problems. Even without a teacher a bright boy, by following 
 this book faithfully, may acquire considerable skill. It is a safe guide for farmers' boys as well as 
 for city boys, and is especially well suited for use in rural and village schools in which the teacher 
 has had but little experience in the use of woodworking tools. The book is illustrated by more 
 than one hundred figures, including ten plates cf working drawings. Each of these figures is an 
 original drawing made expressly for this book. Price. $1.00. 
 
 Problems in Woodworking. By M. w. MURRAY. 
 
 A convenient collection of good problems ready to place in the hands of the pupils. It consists of 
 forty plates bound in heavy paper covers with brass fasteners. Each plate is a working drawing, or 
 problem in bench work that has been successfully worked out by boys in one of the grades from 
 seven to nine inclusive. Many of the problems can be worked out in various ways according to 
 the individual ability, interest and taste of the puvil. Price, 75 cents. Board covers, 20 cents extra. 
 
 Problems in Furniture Making. By FRED D. CRAWSHAW. 
 
 This book consists of 32 plates of working drawings suitable for use in grammar and high schools 
 and 24 pages of text, including chapters on design, construction and finishes, and notes on the 
 problems. Price, in heavy paper covers, $1.00. Board covers. 20 cents extra 
 
 Problems in Mechanical Drawing. By CHARLES A. BEN- 
 NETT. With drawings made by Fred D. Crawshaw. 
 
 This book consists of 80 plates and a few explanatory notes, and is bound in heavy paper covers 
 with brass fasteners. Its purpose is to furnish teachers of classes beginning mechanical drawing 
 with a large number of simple, practical problems. These have been selected with reference to the 
 formation of good habits in technique, the interest of the pupils, and the subjects usually included 
 in a grammar and first-year high school course. The book covers simple projection straight 
 lines and circles, problems involving tangents, planes of projection, revolution of solids, develop- 
 ments, intersections, isometric projection, lettering and working drawings. Each problem given 
 is unsolved and therefore in proper form to hand to the pupil for solution. Price, $1.00. Board 
 covers, 20 cents extra.
 
 Woodwork for Schools on Scientific Lines. By JAMES 
 
 THOMAS BAILY and S. POLLITT. 
 
 This is the American edition of an English book containing many problems designed to cor- 
 relate mathematics and physical science with manual training. Price, 75 cents. 
 
 Clay Work. By KATHERINE MORRIS LESTER. 
 
 This book covers the whole range of clay work for the elementary school technique of clay 
 modeling, study of plant forms, human figure, story illustration, simple architectural ornament, the 
 making of tiles and ornamental pottery. Price, $1.00. 
 
 Classroom Practice in Design. By JAMES PARTON HANEY. 
 
 A concise, up-to-date, richly illustrated booklet on the teaching of applied design. Very sug- 
 jresthe. Price, 50 cents. 
 
 The Wash Method of Handling Water Colour. By FRANK 
 
 FOKRKST FKKDKRICK. 
 
 "This little book is a helpful guide and affords a stimulus to the use of water-color as practiced 
 by the earlier painters, whose beautiful work is unexcelled." Price, 50 cents. 
 
 Manual Training Magazine. 
 
 An illustrated, bi-monthly publication devoted to the interests of the Manual Arts in Education. 
 Subscription price, $1.50 a vear: single copies, 35 cents. In foreign countries, including Canada, 
 $1.75 a vear: single copies, 40 cents. 
 
 The Manual Arts Press 
 
 Peoria, Illinois
 
 UC SOUTHERN REGIONAL LIBRARY FACILITY 
 
 A 000 085 439 8