S^tate Q^allege of Agriculture At ((acnell iUmnecBitg 3tl)aca. SL f. JCtbratg Date Due Apr74 '5: y *fAyi5 mt 1 Library Bureai Cat. No. 1137 NC 750 kI"™" ""'™''"y '-"'rary 3 1924 014 453 322 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924014453322 HAXD-BOOK l_iF SHADING AND ADVANCED PERSPECTIVE DEAWING. KRtJSrS DRAWING SERIES-PART IV. BY HE E MANN KEUSI, A. M., mSIRUCTOH IN THE PHILOSOPHY OP EDUCATION AT THE NOnUAL AND TRAININO-fiCHOOL, OSWEGO, N. Y. ; AND FORMERLY TEACHER OP DRAWING IN THE HOSE AND COLONIAL TRAINING-SCHOOL, LONDON. NEW YOEK: D. APPLETON AND COMPANY, 549 & 551 BROADWAY. 1876. Entered, according to Act of Congress, in the year 13T6, By D. APPLETON & CO., In the Office of the Librarian of Congress, at Washington. @. ) 7 S' tr3 mTEODUCTIOK This course completes a series of systematic drawing- lessons prepared for use in our schools. Its objects are : to present the laws of light and shade, including those of reflection ; to verify, by means of geometric reasoning, the problems of linear perspective ; and to determine, in a scientific manner, the position of points upon the per- spective plane. It might, therefore, be called Geometric Drawing ; but, since that term is sometimes applied to Industrial Drawing, we have lejected it : and, in order to distinguish it from the previous Elementary Course, which was based on actual observation rather than on scientific deduction, we have called it. Advanced Course of Perspective and Shading. In devising the whole plan of this series, and in carry- ing out its special details, the author has kept steadily in view certain fundamental principles, both of Educa- tion and Art, which have been thoroughly tested during a long experience in the school-room. These principles have been partially stated in the former Manuals of the series ; but their systematic and formal expression has been reserved for this Manual. EDUCATION. Education may be regarded as either general or tech- nical. The objects of general education are development 4 ERUSrS DRAWING. and the acquisition of knowledge. Development means tlie full possession and use of all tlie powers and facul- ties Avith which a human beinsj is. endowed. It means, also, harmony in the action of the faculties, and the true subordination of the executive to the regulative powers. Every human being has need of this kind of develop- ment ; and hence he has a right to every agency and ap- pliance that will accomplish it. He has the need and the right in virtue of his humanity, irrespecti\ e of his condition in society or of his future vocation. The pro- fessional man, the merchant, the mechanic, and the la- borer, are all equal in these respects; and schools are established in order that these needs may be supplied and the rights secured. The second aim of a general Education is the acquisi- tion of useful knowledge. Apart from its effect in devel- oping the intellectual faculties, knowledge is necessary for the preservation and maintenance of life, and for the accomplishment of those purposes in which all human beings have an equal interest. The welfare of the nation and the interests of civili- zation alike demand that education of the nature pointed out should be universal ; and this conviction has given rise to oui' system of public schools. With these vicAvs in regard to the true function of public schools, the author has 'taken upon himself the task of preparing a system of Drawing which shall con- tribute to the general culture demanded, and at the same time shall develop the aesthetic and practical faculties as a part of that culture. He would call special attention to the several methods by Avhich he has sought to ac- complish his task. DEVELOPMENT OF MIND. In order that Drawing may be made an important INTBOJiUCTIOK 5 aid in the development and training of the mind, this course presents the following features : First. Each lesson throughout the entire course has in view the threefold object of training the eye to per- ceive, the brain to think, and the hand to record, so that they may act in unison, and mutually assist each other. Second. The lessons are progressive in character, be- ginning with simple and ending with complex forms. In this respect, they are thoroughly graded to meet the wants of every kind of public scliool. By the u.-^e of the inventive process, the system is made so elastic that, while it is simple enough for the most unskillful, it af fords ample exercise for the disciplined powers of the most cultivated. Third. The inventive exercises are among the most efficient means for the cultivation of the imasrination. The pupil obtains a knowledge of simple forms through jDerception. He then combines these forms into designs, which gradually become more and more complex. This practice arouses into activity the rearranging powers of the mind ; and the work, being concrete in character, often proves efficacious when more abstrjtct illustrations would faiL The creation of a new design brings into use the same powers as the creation of a new poem or ;i new work in any field of literature. The difference is in degree, not in kind ; and the first successes in this direc- tion become sources of aspiration, and powerful incen- tives to further effort. Fourth. The lessons are so arranged that, from the obvious facts directly before him, the pupil is led to an examination of relations and a comprehension of laws. In this way, drawing becomes an efficient means in the cultivation of the reasoning faculties. Further than this, it is a complete illustration of the inductive philosophy, showing the successive steps which the human mind must 6 KSUSrS DRAWING. take in rising from a knowledge of facts to a comprehen- sion of 2irinciples. ACQUISITION" OP KNOWLEDGE. The relations of the Drawing Course to the acquisi- tion of knowledge may be stated as follows : First. Throughout the entire course, the attention of the pupil has been called to real objects, and his mind filled with images of things which are everywhere about him. Vague notions of these things have been changed into vivid ideas, and slight and transitory impressions have been deepened and made permanent. Second. Many of the lessons have been chosen from the different departments of Natural History, thus lead- ing the pupil to an examination of Mineralogy, Botany, and Zoology, which treat respectively of the three king- doms of Nature. The knowledge of science thus carried is made clear, definite, and lasting ; and in this manner drawing is made to subserve science, and science becomes a means for the development of drawing. ESTHETIC CULTURE. While drawing is thus important as an aid to gen- eral culture, it is almost the sole exercise in schools that directly promotes art or aesthetic culture. In his endeav- ors to make drawing contribute, in the most efficient and practical manner, to the growth of the festhetic facul- ties, and to the development of art, the author has been guided by the following considerations : First. All mental operations require materials which are largely drawn from the outward world. Through the avenues of sense, ideas of objects flow into the mind ; and these ideas, abstracted from the objects in which INTRODUCTION, 7 they had their origin, are taken up by the faculties, and used for all the various purposes of thought. • They are arranged in order, thus showing their rela- tions ; they are fashioned into new combinations, produc- ing the effect of new creations ; or they are put into the crucible of induction and made to yield the secrets of truth, power, and beauty, which they contain. Second. In the cultivation of the aesthetic faculty, the same laws will hold good as in the development of other parts of the mind. From the outward world must be obtained the materials out of which all beautiful designs are constructed. A knowledge of the outward world can only be obtained by careful and continuous study ; and here Science and Art meet upon common ground. An examination of natural objects in regard to their qualities and their relations to each other, to the world, and to man, and the general laws derived from such an examination, lead directly to science, while an examina- tion of the forms of objects, the relation of these forms to each other, to the world, and to man, and the general laws derived from the process, lead directly to art. Third. A limited or empiric knowledge of the forms of Nature, whether resulting from imperfect observations or a false philosophy, is a faithful source of the false in art. A design that embodies only a portion of the truth must be distorted ; and a design made up of delusions, assumed as facts, is wholly untrue and grotesque. A school of art founded upon limited observations, or upon untruths assumed to be true, is not only false in itself, but it becomes one of the most formidable obstacles to the perception of truth. Fowrth. In the arrangement of this course of draw- ing, the author has given just enough technical work for the interpretation and expression of Nature; and has then called the attention of the pupils directly to the S ERUSrS DRAWING. beautiful forms which Nature presents. He has done this not only in one, but in every department. He has invaded every field of natural science, and has made each contribute its own lesson of beauty to the general result. By this means, pupils are taught to observe from everywhere ;, and they soon become pos- sessed with all the requirements of the highest art. Fifth. The aesthetic faculty, sometimes called taste, like other mental faculties, is best cultivated by exercise which is guided by intelligence. Having obtained from Nature an infinite variety of forms, it selects those that please it best, and proceeds to put them into new com- binations. The continued comparison, necessary to se- lection, leads to an appreciation of l)eauty ; and the effort at arrangement generally develops ideas of proportion, harmony, and symmetry, which constitute the funda- mental principles of all aesthetic art. The successive lessons of this drawing course have been selected and arranged with special reference to this result. From the very outset, pu})ils are taught to com- bine simple lines, and to judge of the eftect produced. From work actually done, principles are CA'olved. Step by step knowledge increases, the taste is improved, and the ])ower of execution becomes greater, until the mind is fully possessed with a sense of the beautiful in foim. This work of combination continiTes as a part of the ex- ercises throughout the entire course, reaching more and more complex results. The eye is trained to see, and the hand to execute. Beauty is perceived and created. A true and natural taste is developed, \vhich is free from dogmatisms, empiric rules, and conventionalisms. TECHNICAL DRAWING. Technical schools are established mainly as supple- mentary to common schools. When the general educa- INTRODUCTION. <.\ tion is finished, then the technical education begin?. First make the man ; and then, out of the man, make the farmer, the mechanic, or the engineer. The reverse of this, the effort to make the technical precede the general education — to make the artisan before the man — has al- ways resulted disastrously to those upon whom the ex- periment has been tried. The mind, kept within the bounds of the technical subject, is not only narrowed in its range of thought, but even the technical is not so well understood as though the broader principles upon which it rests had first been mastered. The general law in regard to technical education also applies to drawing. The student, having completed the general work, and received the broad culture resulting therefrom, is prepared to enter the narrow field of tech- nical or professional work with every prospect of success. Wedded to no special school, and confined to no arbi- trary standard of beaut}-, which, under pains and penal- ties, he is required to worship, he approaches his work in the spirit of freedom, and with an intelligent desii-e to attain excellence ; not to defend a theory or a system, but to make truth the goal of his efi^ort. SPECIAL COURSES. The author, however, has not been unmindful of the demands tVn- special or industrial courses. His plan, as contained in his first announcement, comprehended pro- fessional courses in Architecture, in General Designing, in Landscape, and in Mechanical Art. These courses are now in course of preparation, and will in due time lie given to the public. In the general series already published, frequent ap- plications of general principles have been made to the dif- ferent branchi'S of art and industry, and many of the draw- in<'--lessons have indicated the direction in which art 10 KRUSrS BBAWING. may be extended. In tlie special courses, these threads ■of application will be taken up at the points where they have been dropped, affording another proof that the gen- eral course has prepared the wa\' for special courses. The author is aware that, in the present state of pub- lic schools in this country, drawing, for the present and for some time to come, must be largely committed to the charge of teachers who have had little or no opportunity for art education. In the arrangement of his works, in the preparation and succession of his drawing-lessons, and also by means of little hand-books, he has endeavored to aid such teach- ers as much as can be done by books. Many teachers have testified that, without any previous knowledge of drawing, they have been enabled by the use of these means to excite a great interest in it, and that their pu- pils have made rapid and commendable progress. Every teacher, whether he understands drawing or not, should carefully study each lesson before coming to his class ; and, by observing the directions given and adding the results of his own experience, he will make drawing not only successful in his school, but excite a great en- thusiasm upon this subject, and thus realize all the good results which follow from faithful adherence to the laws of Nature and Art. SHADING AND ADVANCED PERSPECTIVE. SHADING AS A MANUAL EXERCISE. By shading is understood the application of a dark tint to portions of a drawing by means of a pencil or brush. In the delineation of real shades, Nature is ex- actly copied ; and the laws which govern the expression will be found in a subsequent portion of this work, the present chapter dealing only with the mechanical execu- tion of shading. Before commencing this work, the pupil is supposed to be able to draw straight and curved lines, of specified length and direction, with so much rapidity and accuracy that the use of the rubber is not demanded. To make shade-lines that will appear easy and deli- cate, the motion of the hand must be steady and rapid, and the pencil of the proper degree of softness. As a preliminary exercise, let the pupil construct squares as in Fig. 1, and cover the faces with faintly- 12 KRUSrS DBAWING. drawn parallel lines, vertical, horizontal, and oblique, as in a^ h, and c, respectively. The beauty of the result will depend upon" the soft- r.ess, regularity, and parallelism of the lines. T invented as directed in the- Analytic Series may be made much more effective by shading the alter- nate spaces, as in a, Fig. 3 ; or, if the lines dividing the SHADINa AND ADVANCED PERSPECTIVE. n spaces have been doubled, the sj^aces between the paral- lel lines may Ije left blank, and the remainder of the de- sign shaded, as in h, Fig. 3. In a and h, Fig. 4, we have curvilinear designs, di- vided and shaded in a similar manner. Fig. 4. ^ An endless variety of designs may here be made by varying the patterns, and by varying the shades in each pattern ; and the pupil should 1)e taiight to exercise his skill here before proceeding further. In these exercises, th6 teacher ■\\dll have an oppor- tunity to cultivate the imagination and taste, giving that practical skill necessary for success in the further prose- cution of the work. At the same time, a great interest may be excited by the production of beautiful forms, and by the emulation produced among the pupils in the class, VANISHING SHADE. In practical work, it will often be found necessary to produce the effect of shades Avhich, from a very dark tint, gradually diminish until they neai-ly or quite disappear. u KBUSrS DRAWING. This is accomplislied by gradually diminisliing the pressure upon the pencil, as in a and b, Fig. 5. Fig. 5. A det^])er shade of the same variety may be made Ijy crossiiio; the lines, as in r. This kind of shading' niay also lie made the basis of inventive designs, as in a and h. Fit!'. 0. Fig. 6. „ I It will be noticed that in a the shading gives the appearance of depth, and that in h the shade alternates with the light portions. In Fig. 7 the shading is almost a necessity in repre- senting the interior sjiaces ; and it also greatly assists in representing the general outline of the object. In Fig. 8 are found other illustrations of these van- SHADING AND ADVANCED PERSPECTIVE. 15 ishing stades. In a we find represented some interior spaces, and in 5 a combination of blocks. Fig. 7. It will not be necessary to niultij)ly instances of tliih inventive work ; l)ut the pupils should be directed to Fig. 8. proceed until they have obtained a number of oiiginal and well-executed designs. SHADE AND SHADOW. Light radiates from all luminous bodies. The chief source of light upon the earth is the sun. The eye is the organ of the body which takes cognizance of light. Without light or without the eye there could be no color, light, or shade ; and all the relations of form, size, IG KRiJSrS DRAWING. and distance, depending upon vision, would at once be blotted out from human consciousness. When luminous rays are intercepted by an object, an obscuration takes place upon the side of the object oppo- site the illumination ; and this darkening is called shade. Shade does not always have visible and easily-defined outlines; yet it always occupies space, and has length, breadth, and thickness. In two cases, however, shade has a visible outline : First, on the surface of the intercepting object turned from the light ; and, second, on the surface of another object iipon which the shade seems to rest. In the latter case, the shade represents the general outline of the intercepting body, and is called shadow. The intensity of light and shade depends upon con- ditions to be discussed hereafter. The most intense ef- fect of light is that of absolute whiteness represented by the paper on which we draw. When this intensity is diminished, tints are required to represent the surfaces of objects thus affected. The term shading, therefore, applies to that gradual diminution and gradation of light from perfect whiteness to perfect blackness. By an observation of Nature, how- over, it will be seen that, owing to the reflection of light, the surfaces of the most opaque bodies have some light tints which modify their darkness, and that. entire black- ness is rarely or never seen. LIGHT -AND SHADE AFFECTED BY DISTANCE. The effects of light and shade, as modified by dis- tance, must be studied directly from objects ; and, to ob- serve all the varying results, the objects chosen should have smooth, white surfaces, and have sufficient size to enable the pupil to discover the modifications of light and shade upon a large scale. SHADINQ AND ADVANCED PERSPECTIVE. 17 The painted wall on each side of the street, as in Fig. 9, may be taken as a good example for illustration. In Fig. 9. a this figure, the sun is assumed to be on the left, so ns to shine directly upon the wall on the right ; while that on the left is in shade, and casts a shadow, which is visible in the picture. With the real object before the class, the questions to develop this lesson should be somewhat in the folloAV- ing order : 1. AVhat do you see? 2. How do the light and shade upon the Avails com- pare ? 3. On the darker surface, how do the different parts compare ? State the differences observed. 4. How do they compare on the lighter surface ? Reference may also be made to the shadows cast by ourselves, which show distinctly some of the features involved in the third question. From the results of these obseiwatiuns, and of others which take into consideration an extended landscape, the followinof facts are derived : 18 KSiisrS DRAWING. First. That shade decreases in intensity as the dis- tance increases. Second. That light also decreases in intensity under tlie same condition. Third. That near objects are distinctly marked both by light and shade. Fourth. That at a great distance all objects appear of a uniform tint, in whicli all differences of light and shade are obliterated. These results may all be expressed in the single for- mula : Light and shade on a plane surface decrease in direct proportion to the distance. INCLINATION OF SURFACES. The Inclination of surfaces has an important effect in modifying light and shade. By reference to Fig. 9, we see that the face of the wall a, although not in the direct rays of the sun, has a lighter surface than h ; while the shadow, s, is darker than either. These differences are evidently not caused by dis- tance, but by the greater or less reflection of light from other parts. The light reflected from c, for example, tends to lighten the shade of 5, but fails to have any perceptible effect upon s, the shadow. In curved surfaces, these differences of shade are more distinctly seen ; and a column exposed to the sunlight is a good ])lace from which observations can be made. In Fig. 10 we have three columns shown in perspec- tive and shaded, with the sun upon the left. The raj-s of the sun, falling more or less directly, illuminate one side; and, where they cease to reach the object, a grad- ual obscuration takes place toward the opposite side. From the centre of obscuration, this shade diminishes to the left, because of the gradual curvature of the cylin- der, which changes the angle of the illuminating rays; SHADIXG AyD ADVANCED PERSPECTIVE. 19 and it climinislies toward the riglit because of distance and the reflection of other objects. Fig. 10. 'l ; 1 • 1 it jjl ii fl 1 II H a The two columns farther back shoAV the general eftect of distance in the modification of light and shade, as has already been stated in the general formula. It will l)e seen from the foreu'oina; that the inclina- tion of surfaces tends, in some degree, to counteract tlie effect of distance in the modification of light and sliade, and that to each must be given its due degree of influ ence. By giving too great distinctness to distant objects, the landscaj)e-painter destroys the etlect of distance, and makes his picture distorted and unreal. A portrait- painter who neglects the shades cansfd by the different inclinations of the several parts of the face, gives to his subject the appearance of flatness and vacuity. The presence of color, as modifying light and shade, is often a great obstacle to the student in the successful representation of a landscape, as he does not know how much shade should 1)e made to represent the green of the trees, or the different colors of ijuildings. In this difficulty he can be greatly assisted 1 )y a good 20 KB USPS DSAWING. photograph, which transmutes color into light and shade, and gives to each color its due proportions. Only from the experience gained by faithful study and by long practice, can excellence be obtained. Study in this direction will also lead to a considera- tion of the deeper science and })hilosophy which indicate and explain all the phenomena of appearance ; and this tends, surely and directly, to the real end of all educa- tion, the development of all the faculties of the mind. CUBES AND FOUR-SIDED PRISMS. In all the succeeding exercises \\here small objects are considered and represented, the work must be per- formed in a room; and, as light from different directions would cause shades which would counteract each other, and so confuse the student, the blinds should be shut upon all but one side, bringing the object directly into the light from the side left open. In many cases, no distinct shadows will V»e visible; but this is of small consequence, as the exercise takes cognizance, primarily, of shades alone. With small ob- jects, the variation of light and shade upon any plain Fig. I I. surface is nearly imperceptible ; and, in practice, it is all- important to remember the facts and principles derived from the observation of larger surfaces SHADING AND ADVANCED PERSPECTIVE. 21 Cubes, or four-sided prisms, may be placed in various positions in regard to the light ; and, in each case, it will be found that the position changes the wliole character, of the shading. In «, Fig. 11, the light illuminates only one of the vertical sides, throvs^ing the opposite side entirely into the shade, and casting one distinct shadow. In i, two vertical sides are illuminated, and two are in shade ; and the shadow cast, coming from two contiguous sur- faces extending in different directions, becomes compound. Only one of the illuminated and one of the shaded sides can be repn'sentc(1. In ^^, Fig. 1l' the liu'lit is dirrctly on tlu' front and Fig. i: P' left vertical surfaoi's ; and the shadow rxteuds liack, and is compound, as in Fit;'. 1 1. In />, Fi'j,-. li\ we have the effect of shade in a hollow cube, a darker shading being necessary to represent the interioi-, as no direct light is admitted. In the shading of objects, great cai'e should be taken not to make the edges more distinct than the shading itself; but rather consider the boundarydine of any face as a portion of the shade. It will be noticed that no two surfaces ha\e precisely the same shade, and that the delineation of the differ- ences will be sufficient to trace the edges distinctly. 22 KRUSrS DRAWINO. In regard to shadows, it will be seen that their direc- tion depends upon the position of the object in relation .to the sun ; and that their outlines tend to the same van- ishing-point as the lines which they represent. For the present, a faitliful C(ij)y of Nature will Ije sufficient for tlie delineation of shadows; Imt, fuithei' along in tlie course, the laws will be developed which will detei'mine their position and outlines under all cir- cumstances. THE CYLIXDER. Tlie'general shading of the cylinder is similar to that of the columns in Fig. 10. It will be noticed that the shade-lines follow the longitudinal direction of the ob- ject, whether it is })laced in a ])er])endiculai', horizontal, or o]di(pie dij'ection. When heavier shading is needed, the effect is best jiroduced l)y means of curved lines crossing the straight lines, wliicli ai'e used for the lirst shade. The drawing of these lines must be done with a great deal of care, Fig. (3. both in regard to curvature and evenness, for an error in either of these directions will give a distorted appear- ance to the oV)ject. SHADING AND ADVANCED PERSPECTIVE. 23 In a, Fig. 13, we have a vertical cylinder and its shad- ow ; in h, Ave have a vertical cylinder open at the top ; and, in c, we have a horizontal cylindrical tuhe open at both en;, color nuist be taken into consideration; A\hite, for exanijde, requiring little shade, while red rtM|uires much more. In all cases of flowers, the touch should lie soft and delicate. In c/, Fig. 16, we have an example of this kind of delicate shading. Fig. 16. In the delineation of rough, iineven surfaces, how- ever, such as the rocks and the trunks of trees, the touch should be bold, producing sharp contrasts in the shad- ing. An example of this kind is given in h, Fig. 16. BEADINO AND ADVANCED PERSPECTIVE. 25 THE SPHERE. The outline of the sphere is represented by a cir- cle; and yet the circle alone never suggests the idea of a sphere. The curvature can only be represented by shadedines drawn in the proper direction, and distributed in the proper manner. A simple circle, as in a, Fig. 17, has no significance except that of inclosing space. By doubling the circle, and a})plying a ft'\v shade-lines, we have a ring, as in h. By the use of parallel lines, we have the disk, as in v. In a, Fig. IS, the sluiding is so disposed as to repre- Flg. 18. sent the inside of a hollow hemisphere, the light coming from the right and above. In b, Fig. 18, we have the representation of a sphere, the light coming from the same direction as in a. It will be seen that the lines of shade are parallel circles drawn 26 KBUSrS DRAWING. close togetter, and made Avith a pencil flattened rather than pointed. The upper jiortion of the right side receives the most light, and this light gradually decreases in consequence of the curvature of the surface. The deepest shade is Ijelow the centre to the left ; and beyond this is a lighter band, caused in part by distance, and in part by reflec- tion. In Nature, a great variety of objects are formed of a general spherical form ; and the student will be amply rewarded by his eflbrt to copy these so as to give the appearance of both convexity and concavity ; and, when he is able to do this well, he has conquered one of the most difficult parts of shading. THE SCIENCE OF PERSPECTIVE. The general laws of perspective were fully developed and applied in Part III. of this Drawing Series. By- means of these laws, all objects in Nature and art may be represented as they appear; and Vty practice, in con- formity to these laws, the eye and hand may be trained to a high degree of excellence in delineation. In all of this work, however, there are certain quan- titive or matheraatic relations, which have not been de- veloped, and whicli are of great importance in giving exactness to the work, and' in abridging the time neces- sary to secure the required results. In the present chapter, these relations are shown and demonstrated, and full directions are given for their ap- plication to all kinds of practical drawing. It will be seen that much of the work is based upon well-known geometric problems. INSTRUMENTS USED. As accuracy is indispensable in this scientific work, instruments become absolutely necessary for drawing and measuring straight and parallel lines, for dividing lines into equal and proportional parts, for drawing and di- viding arcs and circles, and for constructing angles of any ret^uired size. These instruments should consist of a ruler, divided 28 KBUSrS DRAWING. into inclies and parts of an incli ; a pair of accurate com- passes, witli adjustable points ; a parallel ruler ; a right angle ; and a protractor. By the use of these instruments, it is quite possible to mechanically construct figures without understanding the principle upon which they are founded ; but, to be- come independent in thought and in practice, a knowl- edge of the geometric principles involved is indispen- sable. This knowledge also becomes a test, not only of all work performed, but of all appliances and instru- ments used. The demonstration of these problems is left to geometry, from which they are borrowed. In actual work, architects and engineers make use of a variety of instruments constructed upon geometric principles, so that it is unnecessary to resort to geo- metric solutions in every specific case. GEOMETRIC CONSTRUCTION. 1. To hisect a given line. Let a b, Fig. 19, be the given line. From a and h as Fig. 19. C centres, with a radius greater than one-half of the line, draw arcs intersecting at c and d. A straight line con- necting G and d will bisect the given line at e. THE SCIENCE OF PERSPECTIVE. 29 2. To erect a perpendicular from a given point in a given line. Let a, Fig. 20, be the given point in the given line h c. With a as a centre, describe an arc cutting the given Fig. 20. d ^ 1 line at h and c/ then, with a radius greater than e( c, from h and c dravs^ arcs intersectina; at el. A line di'avv^n from a to cl will be the perpendicular required. 3. From a given point outside of a given line, to draw a perpendicular to that line. Let a b, Fig. 21, be the given line, and c the given point without the line. From the point c, ^vith a radius greater than the shortest distance to a b, draAV an arc. Fig. 21. cutting the line at a and b. Bisect a b at d according to Problem L A line drawn from c to d will be the per])endicular required. By the use of the rule and the right angle, the last t^vo jiroblems can be easily solved mechanically; liut the pupil should be taught to observe the geometric rrlations involved in the work. 30 KBUSrS DBAWINQ. 4, From a given point, to draw a line parallel to a given line. Let a b, Fig, 22, be the given line, and c the given point. From c draw the perpendicular c d (Prob. III.), Fig. 22. ei- H6 and from the point c, in the line c d, erect the perpen- dicular c e (Prob. II.). The line c e will be parallel to a h. 5. To constritct similar triangles having one angle in common. Let a h r- rangements for the admission and exclusion of lio-bt, so that just the necessary amount only is received ; for the refraction, reflection, and absorption of light, so that dis- tinct images may be made; for the adjustment of the parts to perceive differences of distance ; and for the di- rect transmission to the brain of the different impressions made. In Fig. 26 we have a representation of a section of the eye, giving the relative position of its several parts. Fig. 26. The eye is inclosed by a strong, fil)rous membrane ((/) called the sclerotic, which holds all the interior por- tions in jiosition, and serves as a foundation for the in- sertion of the muscles which move the eye. THE SCIENCE OF PERSPECTIVE. 33 Inside the sclerotic i.s a thin and finer membrane (h) known as the choroid. The inside of the choroid is cov- ered with a black pigment, which absorbs the rays of light not needed in vision. The sclerotic, in the front part of tlie eye, is modified so as to form a transparent coat (V) known as the cornea. The cornea is a convex lens that collects and concentrates rays of light. Back of the cornea is situated a movable curtain ((/), which gives color to the eye, and is called the ii-is. A circular opening in the iris (e), through which light is admitted into the interior of the eye, is called the pujiil. By the expansion and contraction of the muscles of thr iris, the pupil is enlarged or diminished, admitting more or less light into the eye. Just back of the pupil is a transparent fibrous l:)()(ly, convex on both sides, and known as the crystalline lens. The outer convex surface of this lens has the power of adjustment to suit differences of distance. Through a small opening in the back part of the ej^e, the nerve (g) known as the optic nerve passes, conveying impressions'of vision to the brain. This nerve spreads out in an exceedingly delicate net-woi'k over the interior of the eye (Ji), where it is called the retina. Upon the retina the visual image is formed. The opening (n) between the crystalline lens and the cornea is filled with a thin, transparent fluid called the aqueous humor; and the opening (m) between the crys- talline lens and the retina is filled Avith a transparent, semi-fibrous and semi-fluid substance known as the vitre- ous humor. When we say that an object is seen by us, we mean that a portion of the rays of light Avhich fall upon the object are reflected so that they enter the eye through the pupil ; that the direction of the rays is so changed 34 KBUSrS DRAWING. by the lenses and humors of the eye that an image is formed upon the retina ; that an impression of this image is conveyed through the optic nerve to the brain; and that, through the brain, it enters our consciousness and becomes a part of our being. VISUAL ANGLE. The angle formed by two rays of light proceeding from the extremities of an object and entering the eye is known as the visual angle. The size of the visual angle depends upon the size of the object and its distance from the eye. Without considering distance, the apparent size of an object depends upon the visual angle under which it is seen. For examjile, in Fig. 27, the angle c o J, which deter- Flg. 27. o- ----4 mines the apparent dimensions of the object c d, is greater than the angle a o b, which measures the object a h. It has been ascertained that the greatest visual angle possible is that of about 60°, or two-thirds of a right angle. In Fig. 27, let cod represent the largest visual angle. The angle e of, made by the object ef, is greater than 60°, and hence the light from e g and h f cannot enter the eye, and therefore these portions of the objects cannot be seen. THE SCIENCE OF PERSPECTIVE. 35 In common language, an object held in the hand so that it exactly covers a more distant object, is said to appear as large as the latter object. In Fig. 27, the object /' s appears to be as large as a h, the visual angle being the same. An object so far away as to form no visual angle ap- pears as a point, or vanishes altogether. By the use of optical instruments like the telescope, objects which ap- pear as a point to the naked eye are often so magnified that a visual angle is formed, and they appear to have dimensions. By following the direction of the rays of light which form the visual angle to the retina, it will be seen that they cross each other, and that the image formed upon the retina is reversed. The angles in o n and^ o g with- in the eye, are the same as the corresponding angles cod and a h within the eye, except so far as the rays of light are refracted by the lenses of the eye. DISTANCE. The apparent size of an object depends on its dis- Fig. 28. e s r X ^"' i 'ni. ,,■■-'' ^- ----'"'-^:^ tanco from the eye. In Fig. 28, let the objects a h, c d, and e f, he of the same size, and placed at the same dis- 36 ERU8P8 DRAWING. tance from each other as a h is from the eye, and let o s 1)iseet them all. The triangles cox and g o y are similar because their sides are parallel and the angles all equal, and hence the sides are proportional. But o y is oiie-half of o a; by con- struction, and hence ^ y is one-half of c x, and ^ A is one- half oi c d; therefore, the object a \ placed at a certain distance from the eye, appears twice the size of c t? at double the distance. In the triangle e of and c o J, n m : e f :: o x : o s, ov n m\e f :: 2 : 3. Hence, e f will appear two-thii-ds as great as c d, but c d appears only one-half as great as a h ; hence, ef apjiears only one-third as great as a h. The same result is shown in a more general way by reference to Fig. 29. Let a h ha an object placed at any Fig. 29. convenient distance from the eye, and c d another object of equal size, placed farther away and behind it. Then, in the two triangles, g o h and c o d., g li'.e d wo m:o n. But c d-^d i. Hence, g h -.(ihwom-.o n. In this figure, g ]i represents tlie apparent size of the farther object, and a h the apparent size of the nearer one ; so that the ap- parent size of the farther object is to the apparent size THE SCIENCE OF PERSPECTIVE. 37 of the nearer one as tlie distance of the nearer object is to the distance of the farther one. Expressed in general terms, we have: The apparent length of ohjuts of the same length is in inverse ratio to the distance from the cijt. This law applies equ;illy to the two dimensions of all superficial areas ; to horizontal as well as to vertical measurements. The area of surfaces is found by multiplying the length Tjy the breadth, and, as both length and breadth diminish inversely as the distance, for the more general law in regard to apparent size, we have: The apparent area of oiijeds having tJic same area is incersehj as tlit square of the distance fjvni the eye. For example, a surface of one square inch one foot from the eye, will completely cover a surface of four square inches two feet from the eye ; of nine square inches, three feet from the eye; and of sixteen square inches, four feet from the eye. Expressed in other terms, the visual angle that is necessary for the coanizance of an area at twenty feet distance, will only embrace one-fourth of the area at ten feet distance, and one-sixteenth of the area at five feet distance. DETERJIINATION OF THE POINTS OF DISTANCE. In this exercise, it is proposed to apply tlie laws de- rived from the observation of objects at different dis- tances to the exact drawing of a square surface in ])er- spective, as viewed from difl:erent distances. Case I. — The observer., standing in front of the square, at a distance equal to the length of the square. In Fig. 30, letahcd represent a square ; E the place of the observer, with the distance E o equal to a h. Let /S' represent the j)oint of sight. By a comparison of the visual ano-les, under which a h and c d respectively are 38 KRtJSrS BBAWINO. seen, we find that g J/, one-half of a b, represents the apparent size of c d. In accordance with the laws of perspective already known, the lines a c and b d receding directly back ap- pear in the direction of a S and b S. By erecting upon a b the perpendiculars g m and h /?, and continuing them until they intersect with a s and b s respectively, and drawing m n, we have the exact position of the line c d Fig. 30. :::=--D \ as it would appear in perspective under the conditions named, and a m n b is the perspective representation of a c d b. The triangles a b S and m n S are similar ; and, as mn\s> one-half of a b, m S must be one-half of a S', and n S one-half of b S. Through S draw the horizon-line S D parallel to a b, and from a through n draw an D until it intersects Avith the horizon-line a D. THE SCIENCE OF PERSPECTIVE. 39 As we have already shown, the point where the di- agonal of a perspective square meets the horizon is called the Point of Distance. The triangles a m n and a S D are similar, having equal angles and parallel sides. "We have already seen that a m is one-half of a 8, and hence m m is one-half of s D. But in n is also one-half oi ah and oi E o. Hence 8 d\9, equal to E o. Case II. — The observer., standhig in front of the square, at a distance equal to twice the lenatli. of the square. In this case g h, Fig. 31, which represents the appar- :^i3 ent size of c d, is two-thirds of a Z», as has already been shown. 40 KRUSrS DRAWING. In the similar triangles S in n and S ah^ m n, equal to g h, is two-thii'ds of a b, and m S and n S are equal to two-thirds oi S a and S h respectively. In the similar triangles a ra n and a S D, a m is one- third of a S, and m n is consequently one-third of S D. But m 11 is two-thirds of a h, and « h is one-half of ^ o. Hence in n is one-third oiE o andS D respectively, and E is equal to S D. Case III. — The observer, standing at any distance from tlie square. From the demonstrations of the foregoing examples, we find that, in Fig. 32, the ratio oi E o to a b will always equal the ratio of a m to a S, and hence will equal the ratio of n m to S D. But, as n m-.Eown TO : s D, it follows that ^0 is equal to 8 D. Hence the general \si\y : The perpendicular distance of the observer from the THE SCIENCE OF PERSPECTIVE. 41 hase-line of the view, and the apparent length of the hori- zoTirline, between the point of sight and that of didance^ are equal. COROLLARIES. The following corollaries may be drawn directly from the foregoing illustrations of the visual angle : First. The relative proportions of the sides of a square, seen in j)erspective, vary according to the dis- tance of the observer. Second. In squares, with tw^o sides parallel to the base-line, all diagonals, and all lines which recede at an ano-le of 45°, tend toward a Point of Distance. Third. Since every square has two diagonals, there are two points of distance equally distant from the point of sio'ht, to the right and to the left. Usually in draw- ing, however, only one of these points is needed. Fourth. The points of distance cjinnot be chosen ar- bitrarily, but must always be made to conform to the distance of the observer. Fifth. In drawing any object, a better effect is pro- duced by choosing a point of observation at a consider- able distance rather than one riear by. In a, Fig. 33, we Fig. 33. have the appearance of an ohject from a near stand-point, and in h from a distant one. It will be seen that the 42 KBUarS DRAWING. proportions of the latter are mucli more pleasing to the eye than those of the former. tSixth. In any picture, if we know the size of any regular object, we are able to judge of the distance which the artist has taken for his point of observation ; and conversely, if we know the distance, we can deter- mine the size of the object. Seventh. When, in representing an object, the point of distance is taken without reo^ard to the distance of the observer from the object, the whole picture will appear distorted and unnatural. It will be seen that each of these corollaries ex- presses a truth or law in regard to appearance and the manner of expressing it. These several laws or prin- ciples should be thoroughly comprehended by the stu- dent, as their observance is a necessary condition of suc- cess in future work. TO DETERMINE THE DEPTH OF RECEDING SQUARES. In Fig. 34, l^t a b c d represent a square viewed at a Fig. 34. \h d> - distance of three times its length, a h. But c c? is one- third farther away from the observer than a b, and hence represents one-fourth of the distance of the observer, and consequently of S D. Drawing the diagonal c D, the point of intersection / will give the depth of the second THE 8CIENGE OF PERSPECTIVE. ■13 square. For similar reasons, the point Ji will determine the depth of the third square. These distances may be determined in another man- ner, as follows : In Fig. ;j.j, let ah d c represent a large square to be divided into smaller ones, say four in each direction. Fig. 35. ■0 •■ ' d.---- / / o\ - \ / / . ...-■ \ -\ / 4- 1 \ A J---"\ \ \ Divide a b into four equal parts by the points e,/, and g. From these points, draw lines to the point of sight. From a, draw a Dto the point of distance ; then lines drawn parallel to a b through the points of intersection, m, n, 0, and d, will give the depth of four times four squares into which the large square is divided. In Fig. 36, we have the representation of a perspec- Fig. 36. .<#?K , \ tive square, ^ith each square divided into eight equal parts, or the whole divided into sixty-four equal squares. 44 EBU8P8 DEAWING. By shading the alternate square divisions, we have the appearance of a chess-board in perspective. CUBES IK PERSPECTIVE. Rows of equal-sized cubes are represented in Fig. 37. Fig. 37. .■■;T^''-. y — ; ■■ '. ^' ^ ^ ^-^ y 7 •■■■ 71 / \ / \ \^ /;.,.-- 7 \ \ ^^^ P \ The depth of the successive cubes may be found ac- cording to the method of finding the depth of squares already shown ; and the cubes may be completed by the general laws of pers])ective. A flight of steps may be easily represented in the manner already indicated for the representation of squares and cubes. In Fig. 38, the line representing the front of each of Fig. 38. "~~ \ i * \ Vii \ \m - ^4- ., -- --s..^ -\ , ■ ^ "1 c'^' ,^_ _ ^'x^ c the two flights of steps is assumed to be three times the depth of the step. THE SCIENCE OF PERSPECTIVE. 45 By dividing the line a c, and drawing the diagonal a 0, the point of distance as in Fig. 35, we obtain a d as the depth of the first step, d e of the second, and e Jt of the third. The height of the back edge of the first step is indicated by d m, and, by continuing the vertical line upward, making m n equal to d m, we have the front edge of the second step. In the same manner, g s gives the back edge of the second step ; and s I; equal to (/ s, the front of the third step. The flight to the left can be best drawn by continu- ing the horizontal lines of those to the right, and by placing each step within the limits of lines drawn from the extremities to the point of sight. A culje may l)e so placed that three pairs of its reced- ing lines tend to the point of distance. The question may arise, how can the depth of this cube be determined ? In Fig. 35, it was shown that a diagonal, di-awn to the point of distance, determined the width of the receding squares; and, in the present case, Avhere the diagonals coincide with the size of the cube, as c a and a b, Fig. 39, the depth of the cube may be determined by draw- Fig. 30. f/ / ! iuo' m n through a point parallel to the base, and by making a n and a*m each equal to the side of the square of which a h [a the diagonal; then, from the points m 46 KBUSI'8 DRAWING. and n, draw the lines m s and n s to the points of sight, and the intersection of these lines with the diagonals at c and b will give the required depth of the cube. For example, let the diagonal lie 12 feet, then, calling each of the shorter sides w, and, since the sum of the squares on these sides is equivalent to the square on the longer side: 2 a?" = 12'; hence the side of the square would be \/^^-^ = 8.5 nearly. By making m a and n a each 8.5, we get the depth of the diagonal in any position in which it may be placed. In Fig. 40, we have three square bases in different Fig. 40. positions, completed by determining their depth as be- fore, and by drawing a c and h c toward the points of dis- tance until they intersect each other. In Fig. 41, we have three cubes completed. The height and width of these, c I and h d, are represented as THE SCIENCE OF PERSPECTIVE. 47 10, and the projection on the base by the length o^ a h, which can be ascertained by the formula \ \ \ 1 1 \ \ 1 1 secting S a and 8 b respectively ; and this line will indi- cate the depth required. To save the trouble of measuring the visual angle, the same result may be obtained as follows : Make S D equal to E o; then, upon a b, from b meas- ure the distance b h equal to d b. From h draw h D to the point of distance, and the intersection of Ic D ^vith ^S' b will indicate the depth of the perspective represen- tation. In Fig. 44, \/ b to h, making b k equal d b. Then TEE SCIENCE OF PERSPECTIVE. 49 draw h D. From the intersection of Jc D witt 8 h, m n drawn parallel to a h will give the apparent depth of the rectangles. Fig. 44. K d % s D , / \ " ^' u^/ -i.'^--' /'-^ \ .-■""^ 7 \ h PERSPECTIVE OF ROOMS. In any practical application of the principles of per- spective to the delineation of rooms, careful measure- ments must be made and taken as the basis of the work. In Fig. 45, let a I c d represent the ground-plan of 4 50 KBUSrS DRAWING. the I'oom, and ah m n a. vertical section. We will sup- jiose that a c, the length of the room, is eighteen feet ; (/ h, the width, fourteen feet ; and a m, the height, ten feet. The diagram is drawn to the scale of one-eighth of an inch to the foot. ' In Fig. 46, M-e have the room with the above di- mensions shown in perspective. The line a h represents fourteen feet, and a c ten feet. The length of the room, eighteen feet, is represented by o ^, and a line drawn Fig. 46. c \ h d s 9 e ' -\ ----- — — — " a from to D intersects s ^ at /, and gives the apparent depth of the receding lines b f and a e. The rectangle, e f g h, is the perspective representation of the equal rectangle abed at the distance of eighteen feet. Since rooms in most modern buildings exceed the height of ten feet, the ];)oint of sight should be nearer the floor than the ceiling, as is shown in Fig. 47. In this figure, ■\\e have two windows in the right wall, each six feet high and three feet wide ; the space from the front line to the first window is three feet ; the distance between the windows, six feet ; and the eleva- tion of the windows from the floor, three feet. On the left wall is a door seven feet high and three wide, and placed six feet back of the front line. In the back wall is a fireplace six feet wide and four high. The distance, b c, is three feet. At the intersection THE SCIENCE OF PERSPECTIVE. of c D with s b at 1, we have the position of the nearest side of the front window. The distance b d is six feet, and 2 gives the position of the farther side of the first window. B e is twelve feet, and 3 represents the po- sition of the nearer side of the second window. B g \i^ fifteen feet, and 4 represents the position of the farther side of the second window. The line b o is eighteen feet, and 5 marks the lower, right-hand corner of the room. Fig. 47. T- \ \ ^ / / / / p ^ 1 1 o '' ■ .....■-■■'"■' —■— >. K \ The distance m n is six feet, and lines drawn from these points to the point of sight, intersecting the line which gives the depth of the room, give the width to the fire- place. The distance b h is three feet, and bj> is nine feet, and lines drawn from h and p to the point of sight respec- tively give the line for the bottom and to]^ of the win- dows. The position and widtli of the door can be deter- mined in the same manner as those of the windows. The distance a t is four feet, and a v seven feet, and lines from t and v to the point of sight determine the height of the fireplace and door respectively. OCTAGONS m PERSPECTIVE. To represent a regular octagon in perspective, we first draw the octagon within the square, as in (/, Fig. 48. It Avill be seen that the points a, b, c, and (/, are situated 52 KBUSrS DRAWING. upon diagonals of the square ; the points e and / are to the left and right of the centre respectively ; and the points g and h are directly above and below the centre respectively. Through the points a c and 5 d draw vertical lines, and continue them io mn upon the line which consti- tutes the upper boundary of the square which circum- scribes the octagon, and at the same time forms the base of the perspective square. From the points »?, ?;, o, and J9, draw lines to the point of sight, and complete the per- spective square by drawing the diagonals 8 p and o t, and the line s t. The intersection of the diagonals with the lines m x and n y at 1, 3, 5, and 7, marks the respective positions of the points a h c dm the original octagon, as shown in perspective, and the points 2, 4, 6, and 8, correspond to the points ^,/, Ji^ and e. By connecting the eight points numbered, we have the perspective octagon as in h, Fig. 18. CIRCLES IN PERSPECTIVE. By reference to Fig. 48, it will be seen that a circle can easily be represented in perspective by connecting THE SCIENCE OF PERSPECTIVE. 53 tlie eight points that mark the angles of the octagon l>y a curved line. The result will be a perfect ellipse, as rep- resented in Fig. 49. In tliis fiij;ure, a a represents the long axis of the ellipse, l)ut d d is tlie ])ers])eetive repre- sentation of the horizontal diameter of the circle. The centre of the circle c is represented l)y w;, in the centre of d d. At first sight, it would seem that the diameter, being the longest straight line which can be drawn within a circle, should coincide with the long axis of the ellipse. Fig. 49. ■i ,,.-- 1 ~---._ •■. V. /ii\ / _,. n\ -"""^ .2 ^iC '■■■■•■i-'"' ■'■■■"' 1 ^ which represents the circle in perspective. By reference to Fig. 50, however, it Avill be seen that certain chords in perspective appear longer than a diameter. When a circle recedes directly toward the horizon, as in Fig. 50, the apparent size of its several parts is deter- mined by the relative size of the visual angles, as shown in Figs. 27, 28, and 29. The diameter d d gives the visual angle '/ E r/, which is measured on the base by r .s. Tangents from e touch the circle at the extremities of the chord c .c, forming the angle c E c, Avhich is measured on the base by m n. It 54 Knusrs DRAwma. will ba seen that m n is greater than r s, and lience the chord c c in perspective will api:)ear greater than the Frg. 50. S \ c 1\ \ ^ :: J \\ , .■'' >;■■.; / '■, v'-^ ^ ■^ s ■ In ''•■ E diameter d d. It will also be obvious that the angle formed by the visual tangents will be the greatest that THE SCIENCE OF PERSPECTIVE. 55 can be drawn, and that the line representing it in perspec- tive must correspond with the longer axis of the ellijisc. From n, draw the vertical line n c' until it intersects a line drawn through the centre of A 5? at a. This line c' c' is parallel to 1 2, and forms the longer axis of the ellipse. The position of the line d' d\ representing the diameter of the circle, may be determined by erecting the perpendicular s '/', or by the intersection of the diag- onals 2 3 and 1 4. The method of drawing a perfect ellipse has already been indicated in the Hand-Book of Analytic Drawing, page 57. We will only add that half of the long axis, a c, when laid from one extremity of the short axis 7i to/, one of the foci, will indicate the means by which the ellipse can be correctly traced. PERSPECTIVE OP ANT REGULAR POLYGON. In a, Fig. 51, ^yt have a regular hexagon inscribed in a circle, the length of the sides being found by making Fig. SI. them equal to the radius of the circle. In b, Fig. 51, through the points and 2> draw vertical lines. Con- tinue the line c e drawn from these divisions to the point of sight. The divisions of a h and a d are the 62 KRUSrS DBAWING. perspective representations of diagonals of rectangles, and hence it is obvious that the. divisions o^ m n cannot cor- rectly measure the height of the prism. To find this ex- act height, v^^e can erect the square a o upon one of these divisions of m n, and the diagonal a o will be the unit of measure for the height. Hence a c is measured by two such units. BUILDING IN PERSPECTIVE. To draw a building in perspective, we need to know its exact dimensions, and these may be obtained from the plan of an architect, or from the measurement of some familiar building, as the pupil's OAvn home. The ground-plan is next constructed, as in Fig. 62, Fig. 62. the parts drawn to an exact scale, to which all the work must conform. Architects often draw what is known as a geometric elevation of a building, that is, the appearance of each side giving the exact altitude of all its parts, without considering appearance or perspective. THE SCIENCE OF PERSPECTIVE. iVi In Fig, 63 we have a side elevation of the plan repre- Flg. 63. sented in Fig. 62, and in Fig. 64 we have the front ele- vation of the same plan. It will be noticed that the lines Fig. 64. giving the outline of the roof are vertical, and not reced- ing, as they would be in perspective. By the measurement of the ground-plan and eleva- 6i KBUarS DRAWING. tions, it will be found that the building is 40 feet by 25 ; the piazza, 11 feet by 8; front door-step, 5 J feet by 5; the front door, 4J feet broad, and the windows 3 feet broad. The height of the -walls is 17 feet ; of the roof, 12 feet ; of the piazza, 12 feet ; of the door, 9 feet ; of the windows, 7 feet ; of the front steps, 4 feet. The roof projects 2^ feet, and the chimneys are 5 feet broad, 3 feet thick, and 7i feet high. Drawing the same building in perspective, we have Fig. 65, with all the parts complete in outline. Fig. 65. -IS 40 ..J In Fia:. 65, the scale at the bottom can be used for the measurements of the length and breadth, and that at the right for all vertical lines. REFLECTION OF LIGHT. A body from which light is emitted is called lumi- nous. When rays of light, proceeding from a luminous body, encounter an opaque body, they are turned out of their course or thrown off by reflection. The angle formed by a ray of light striking the surface of an opaque THE aCIENriC OP PERSPECTIVE. C5 body, and a line perpendicular to the point of contact, is called the (iu(ile of incidence ; and the similar angle made by the reflected ray is called the cDKjIe of refection. When a ray of light strikes a reflecting surface per- pendicularly, it is reflected directly back; and, when it strikes obliquely, it is reflected obliquely in the opposite direction. In a, Fig. 66, let a h represent a reflecting surface, and c a ray of light striking it perjiendieularly at m. The ray will then be reflected directly back toward c. Fig. 66. a c -*l \ \ .. In b, Fig. 66, the ray of light c strikes the surface obliquely at m, and is reflected obliquely and in the op- posite direction toward c/, c m n forming the angle of in- cidence, and d mn the angle of reflection. From this example may be infei'red the first law of reflection : The angles of incidence and refection ctre cd- ways efjucd. ' The rays of light from a luminous body are not all reflected in a regular manner; but a portion are ab- sorbed, and a portion are reflected many times and dif fused through space. In consequence, the intensity of light is always diminished by each reflection ; objects are made visible by the innumerable reflections from all the objects that surround us ; and the abrupt transitions of light and darkness are softened into infinite gradations of light. 66 KRUSrS DRAWING. WATER REFLECTIONS. Ill determining the laws of water i-eiiections, a mirror may l;)e used for all purposes of experiment and investi- gation. In Fig. 07, let d h I'epresent an oljject rising perpen- dicularly from the water at />, and vie^ved from e. A ray of light from the point a strikes the water at c, and is rellect('(l toward the eye, making the angles a c g and (- c <] e(pial. The reflection of the 2:)oint a is seen in the direction of e c. The angles e c li and a c h, being com- plements of equal angles, are equal ; the angles e c li and h c d, 1 >eing vertical angles, are equal ; hence, the angles a c l> and Ij c <1 are equal. In the triangles a c I> and h e J, the side <:■ h is common, the angle a c h is e(j^ual to the angle 1> <■ d, and the angle a I c is ec|ual to the angle J' h c; and hence the triangles are equal, and the side d b is e(|ual to tlie side <( />. Therefore, the }»ointr/wiU appear as far lielow the surface of the water as the point a is above. As rays from every visible part of the oliject are re- flected, all following the same law, the reflection ^\•ill ap- pear to the eye r/weiitd, and of the sajne size as fJie ohjert. THE SCIENCE OF PERSPECTIVE. PERSPECTIVE OF REFLECTIONS. fi" In Fig. 68, let a 5 c (/ be a perspective representa- tion of a mirror, placed in a horizontal position, and e m Fig. 6B. an ol)ject 2i('rpt'n(licular to it. AYe have already shown that the reUcction iii ii wouhl lie inverted and of the same size as e m. Now, if wci raise the ndrror to a ver- tical position, ^vithont changing the relative position of the object, as in Fig. 60, we find tliat e m, representing a line retreatinc: at riirht anu'les, tends to the point u. In general, it may be said that the reflection of any point appears as far behind the base of the mirror as the point is in front of it. ns KBUSrS DRAWING. In Fig. 70, let a h c dhe sl horizontal surface reflected by tlie mirror from c J. The lines a c and h d will tend toward ^S', tlie point of sight. To determine the reflec- tii 111, liisect r (I at r, and extend k e and h d to X Through tlie ]ioint e dra\\' ti n and h ;«, and the figure c d on ri will l)e the ]ierspective representati( m of the surface a I c d. Fig. 71. In Fig. "il, lei ah c d represent the side of a rectan- gular box, with its opening turned from us, and standing THE SCIENCE OF PERSPECTIVE. 69 at some distance in front of a mirror. The position of the front of the reflected image is found as in the hist example, and the depth by drawing lines to the points of distance. It is obvious that the reflection will reveal the in- terior of the bi IX, which, from the position which the eye occupies, is invisible in the real object. To become familiar with these reflections, the pupil should draw articles of furniture, such as tables, chairs, and the like, and their reflections in the mirror. In each case, the pupils should rt-preseiit the actual ajjpearance, and observe how this conforms to the principles above given. SHADOWS IN CONNECTION WITH SOLAR LIGHT. Shade and shadows, as they appear in Nature, have already been noticed, and there only remains the expla- nation of the law which determines their position and outline. In Fig. 72, let a J c d represent a Avail, with the sun Fig. 72. h , d.^ to the left. The lines (( c and 7i d are parallel, and, as will be seen by actual observation, the lines c m and (/ n of the shadow are, and appear j^arallel also. Again, as 70 KRiisrs DBAwma. the sides a h and c d are parallel, the corresponding sides of the shadow c d and m n are so in reality, although, when represented in perspective, they appear to tend toward the point of sight. The reason of this parallelism in the outline of the shadow will be given in the follow- ing chapter. THE RAYS OF THE SUN. The sun, as a luminous body, radiates light in all directions ; and these lines of light, when considered in the aggregate, are widely divergent. Of the rays which radiate from the sun, only a lim- ited number can reach tlie earth. In Fig. 73, let >S' represent the sun, and ^the earth. Fig. 73. As the sun is larger than tbe earth, it will be seen that the lines a c and b d, marking the limits of all the rays which reach the earth, are convergent, and will meet at a point somewhere behind the earth. At this point the earth's shadoAv will terminate. The distance of the sun from the earth is so great that the angle of convergence is very small, and, indeed, so small that it can hardly l)e noticed or measured in shadoAvs of objects upon the earth's surface. In the delineation of shadows, the rays of the sun may be considered as pairdlel to eadi otlier. TEE SCIENCE OF PERSPECTIVE. 71 SHADOWS IN CONNECTION "WITH ARTIFICIAL LIGHT. When a luminous body, like a candle or lamp, is close at hand, the rays intercepted by an opaque body are widely divergent, making the boundary-lines of the shadow divergent also. In Fig. 74, let I represent the flame of a candle, and ah G d Sin. opaque surface. The divergent rays, I a and Fig. 74. I b, if not interrupted, would fall on the ground at m and u. The shadow follows the same direction, and the line a b \s represented in the shadow by m n. But m n in reality is longer than a b, and hence the lines m c and n d are also divergent. On further investigation, it will be seen that, the nearer the object is to the candle, the greater will be its shadow; and that many of the shadows produced by ob- jects very near the light are apparently limitless. As pupils are seldom called upon to delineate objects when illuminated by artificial light, these brief hints in regard to shadows will be found sufficient for practical purposes. 72 ERUSrS DRAWING. SHADOWS OP FOUR-SIDED PRISMS. When the observer is so placed that the shadow is thrown directly toward him, the delineation of the ob- ject and its shadow must follow the general laws of per- spective. In Fig. 75, let a h c d represent a four-sided prism, and D the position of the sun. The lines a D and h D, Fig. 75. // JJ A / / / TR representing rays, are in reality parallel, but, as they are shown in perspective, they vanish at D. This makes them appear divergent ; and the extremity of the shad- ow will appear to be in n, the lines m c and n J con- verging toward A', the point of sight. TH-E SCIENCE OF PERSPECTIVE. 7,3 The length of the shadow will depend upon the al- titude of the sun, or the distance between 8 and D. When the shadow is thrown directly back from the observer, the lines which bound it on the right and left, though parallel, apparently converge to the point of sight, and their delineation follows the same law as that of real objects under like circumstances. Pupils should here have a considerable amount of exercise in drawing objects and their shadows from dif- ferent stand-points, both in regard to the light and to the position of the objects. COMPOUND SHADOWS. Compound shadows are those made by two or more faces of a solid. Let Fig. 76 represent a four-sided prism, with the sun illuminating the sides a b g li and n d (/ e, while the sides dec f and h c h f are in the shade. Frg. 76. The limits of the shadow may be determined by draw- ing parallel lines from the direction of the sun through 5, c, and (/, until they meet horizontal lines from /;,/', and e at /«, ?i, and 0. The figure m n efh shows the out- line of the one shadow cast by two faces. The pupils should now draw the shadow of the same 74 Kit USPS DRAWING. kind 0^ solid with the sun to the right, with the sun in front, and with the sun behind the object. We have last to consider the method of determining the shadow, when the sun is in any position, in relation to the object. Let Fig. 77 represent a four-sided prism, with the sun situated on the left in the direction indicated by g J. The shadow must lie in the opposite direction from the Fig. 77. sun ; and g d prolonged \vill touch the horizon at x. The other lines, e m and / o, which mark the outline of the sliadow, will vanish at the same point. The length of the shadow depends upon the height of the sun ; and, having ascertained this height, we fix the position of y upon the vertical line x y, toward which all the rays passing over the points a^ b, and e, seem to tend. The intersection of these t^vo sets of lines at wi, n, and o, gives the length of the shadow. It Avill be noticed that m n of the shadow tends to the same point on the horizon as a h of the object, and n as e h. THE SCIENCE OF PERSPECTIVE. lO From the foregoing examples and principles, the pu- pil should delineate the shadows of triangulai-, hexag- onal, and octagonal prisms, iu various positions in re- gard to the sun. PYRAMIDS AND CONES. As the shadows of these objects, like the objects themselves, terminate in a point, their delineation is not a difficult matter. In a. Fig. 78, \&t a l> a d e represent a four-sided pyra- mid, with the apex at a. The sun is supposed to be Fig. 78. in the direction of m n, which line would pass through the centre of the base at o. The line determines the direction of the shadow. Its length, depending upon the height of the sun, is determined by the point n, where the line n a, representing a ray of the sun, passes over a, and intersects m n. The outlines of the shadow are marked hj c n and e n, drawn from opposite corners of the pyramid. In b, Fig. 78, a b e represent a cone, with the apex 76 KRtTSrS DBA WING. at a. The sun, situated behind and a little to the right, throws the shadow in the direction of m n. The inter- section of the line a n, representing a ray of the sun, and 7n 11, at n, gives the length of the shadow; and the two tangents, b n and c n, mark its outline. It will be noticed, either by actual experiment or by geometric construction, that', if the light of the sun falls upon a cone or pyramid at an angle approaching 90°, Fig. 79. ^y more than one-half of the surface of the solid is illumi- nated, and the shadow is correspondingly diminished. With the sun directly overhead, the ^vhole surface will be in the light, and the shadow will disappear alto- gether. The pupil should next determine the shadoAVS of a hexagonal and an octagonal pyramid, and of a truncated P3'ramid and a truncated cone. THE SCIENCE OF PERSPECTIVE. 77 THE CYLINDER. The shadow of an upright cylinder is formed accord- ing to the method given in Fig. 77. Fig. 79 represents a cylinder standing upon one of its circular bases. The line g e, passing through the cen- tre of the base at o, gives the direction of the shadow. The point x upon the horizon is the vanishing-point for d n and f m, the lines which mark the lateral bounda- ries of the shadov,\ Lines from the direction of the sun, drawn through a, I, and c toward y, determine the length of the shadow and the shape of its farther extremity. The pupil should next draw cylinders with the shad- ows in front, and to the right and left. BROKEN SHADOWS. This name is given to the shadows which are in part cast upon the ground, and in part upon objects lising from the ground. The delineation of these shadows re- Flg. 80. quires only an application cf the laws already developed, and can be left to the pupil. 78 KRUSrS DRAWING. In Fig. 80, let m represent a vertical object, as a pole or the trunk of a tree, and n a wall at no great distance. The direction of these shadows will vary at different times of the day according to the position of the sun in the direction of a, b, or c. In Fig. 81, we have the pole m leaning against the Fig. 81. wall n. In this case also, the shadows vary with the time of day, differing from those in Fig. 80 by meeting at the iipper extremity of the object. Good examples for practice may be found in the shadow of a leaning, four-sided prism or cylinder, and a ladder leaning against a wall. SHADOW OF A SPHERE. In order to define the position of different parts of the sphere, we will suppose it to be inclosed in a hollow cube. In Fig. 82, we have a sphere, a c b d, inclosed in the cube X //. It is evident that the sphere must touch each of the six faces of the cube at the centres a, b, c, d, e,f\ THE SCIENCE OF PERSPECTIVE. 70 which points may be easily ascertained by drawing di- agonals. Assuming that the sun is to the left, the shadow will be cast to the right. As one-half of the sphere is illuminated, the boundary of the light and the shade is a circle. Let efgJi represent that circle passing through the centre of the sphere at right angles to the sun's rays. Fig. 82. Rays, forming tangents in the direction of the sun, drawn through e,f, g, and h, will strike the ground at ;', 8, m, and n. The perspective circle drawn through these points will give the boundary of the shadow of the sphere. APPLIED COURSE. The exercises of the Ai)]ilie(l Course are arraiiuecl accoriling to the principles developed in the ^lauual. A full appreciation of tliese principles will i)revent the pujiil from copying figares in the \»u>\i in a mechanical,, thoughtless manner, and induce him t;> reflect on the laws which modify their appearance. Tliese drawiiio-s, with hardly an exception, have been copied from Na- ture, with special regard to the laws of light and shade, and the appearance of the texture which characterizes the surface of particular objects. BOOK I. The drawings of this book represent shaded prisms, pyramids, cones and cylindei-s, sphere and hemisphere, together with applications suggested by these forms. In order to obtain the necessary variety, some of these solids are represented truncated, and others hollowed out, as on pages 1 and ?>. Pages 2, 4, 5, 7, and 8, con- tain appropriate applications to common objects. • BOOK II. This l)Ook enters a new path in the delineation of foliage. In the mechanical copying of trees, an indefi- G i-2 KMUSrS DRAWING. nite scattering of dots and crocliets seems intended^ to give an idea of foliage. Such an arbitrary application of a conventional style prevents, of course, every at- tempt to represent faithfully the characteristic features of any species. There is necessarily an indistinctness of outline in the delineation of a whole boua;h or tree when viewed from a distance, which differs someAvhat from the ap- pearance of individual leaves seen close to us. Xever- tlieless, the foliage in different species of trees presents characteristic outlines. For instance, the drawing to the left — first page — with its light, feathery appearance, suggests tlie elm, while the more compact appearance, of the next figure reminds us of the beech or sycamore; tile next sugi;\'sts the poplar, and the last one the oak or butternut. In the lower row, the bough in the mid- dle, with its parallel lines, bears resemblance to the willow^ while the sharp-pointed trrfts on either side represent grass. The more easily and gracefully the undulations pertaining to each kind of foliage are exe- eutt'(l, the more natural will be its appearance. On i^ige 2 the foliage is accompanied with some shading, and grouped into forms, which, while they rejji-esent certain outlines of trees, exemplify the pecul- iar modifications of light and shade of jiyramidal or cylindrical ()bjeets. On page 3 the shading of some trees, esjiecially of evergreens, is still more minute, so that the eliaraeter of the spruce can l)e easily distin- guislied from that of the pine. The principle underlying the drawing and shading of foliage has already been discussed. The pupil will see the necessity of defining first the proper i^osition of tlie trunk and branches before ajiplyino. i]^^, foliage. The character of the bark must also receive due atten- tion. APPLIED COURSE. gf. As these drawings have all been copied from Ameri- can scenery (mostly from the Catskills), and include trees which are met with everywhere, an appeal to reality in Nature will be found botb easy and profitable. BOOK III. This book contains more finished landscapes and water-reflections, a proper delineation of Avhicli requires considerable skill and perseverance. The teacher ought not to allow pupils to attempt these studies until they have had the necessary preparation by means of the preparatory exercises in perspective and shading. A vast amount of mischief is often done by passing oif an incorrect " daub " of a difficult study as a specimen of art. More special directions are given in the separate books. BOOK IV. The remarks made for the drawings of the two jire- ceding books apply also to those of Book IV. The shading of the cylinder, the cone, etc., finds its applica- tion in the delineation of numerous plants and flowers, which form the subject of the third book. Observe, for instance, the cylindrical shape of the fringed gen- tian (page 6), and the conical shape of the calla (page 5 ). In the shading of flowers, the texture of which is very delicate, there must be a skillful blending of the shade with the outline. The leaves, however, require more vigorous touches, occasionally suggestive of veins. THE END. Youmans's First Book of Botany. Designed to cultivate the Observing Powers of Children. By Eliza A. Youmans. 12mo. 183 pages. $1.00. TnlB little book tiae proved a wonderrul success, and Is empbatlcally a step in the riffht direction. Although it has been Issued but a short time, It has been adopted for use in the cities of Ohlcago, Si Louis, New Tork, Columbus, Nashville, Milwaolcee, etc., etc., and in the States of Maryland, 1111- uol*, and Arkansas. It is to be speedily followed by the Second Book of Botany and six large and 'beaatlfaily-colored Botanical Charts, alter the plan of Henslow. Hon. Siiperintendent Bateman. of Illinois, says : " As a sample of the true method of teaching the elements of science in primary schools. Miss Youmans's book is deserving of the highest praiee. In this respect I have seen nothing eqnal to it. The same method, pnrsned in all the natural sciences, would soon give us a generation of accurate aod intelligent young observers of natural objects and phenomena, and change for the better the whole tone and character of com- mon-BChool instrnction. Newton Batbtlkh, Sup*t Public Instruction^ IjOOkyer's Astronomy, accompanied with numerous Illustrations, a Colored Representation of the Solar, Stellar, and Nebular Spectra, and Celestial Charts of the Northern and the Southern Hemispheres. American Edition, revised and specially adapted to the Schools of the United States. 12mo. 312 pages. Teachers are delighted with this American Edition of Lockyer's Astronomy. The following oneollcited testimonial from Rev. Dr. J. A. Shepherd, Principal of St. Clement's Hall, Ellicott Olty, Md., will show the estimation is which it Is held : " I have been a teacher of Astronomy as a specialty for more than twenty-five years, and beg to say that, in my opinion, your Edition of Lockyer's A^-tionomy Is, without dovbt, the best text-book on tfu subject that has ever appeared in this country. Tlie peculiar arrangement is one which none but a maater-mind would have ventured to make. The illnstratlons are, in their way, perfect little gems. 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The author's aim throughout the book has been to give a connected view of the whole subject rather than to discuss any particular parts of it, and to supply facts and ideas founded tliei'con, to serve as u basis for subsequent study. He grasps this fascinating science in its eiiiirenoss, and reyeals the wonderful secrets which man has wrested from the bosom of the beavens, with a straightforward clearness of statement and a felicity of illustration which reach the understanding and impress the memory. The arrangement adopted is new. The Bun's true place in the Cosmos is shown, and the real movements ut' the heavenly bodies are carefully distinguished from their apparent movements, which greatly aids in imparting a correct idea of the celestial sphere. Astronomy has made great strides within a few years past ; and Lockyer, who has him- self been the pioneer in many important researches, has incorporated in the present volume all recent discoveries. 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