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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I CD 9' ELEMENTS OF PLANE AND SOLID FREE-HAND GEOMETRICAL DE AWING, WITH LETTERING; AND SOME ELEMENTS OF GEOMETRICAL ORNAMENTAL DESIGN, XNOLUDINa THB PBIITOIFLBS OF HARMONIO ANGULAR RATIOS, ETC. IN THREE PARTS: Part I.— Plane Drawing, or from "the flat." Part IL— Solid Drawing, or from *»thb round." Part III.^Elsmbnts of Gbombtrio Bbautt. fob draughtsmen and artisans; and tbaohers and students of industrial and meohanioal drawing. SrEDWARD WARREN, C.E., fohineb professor in the renrselaer polttbchnio institdtr, and mass. inst. of tbohnoloot; and boston normal art school; and author of a complktb BBRIBS of text-books ON DBSORIPTiyE OSOMETRT AND INSTRUMENTAL DRAWINQ. J rf ' • . . W' - . « «< • J .^ » •* . NEW YORK: JOHN WILEY ik SONS, 15 AsTOR Plage, 1878 OOPTBIOST, WILEY & SONS. 1878. Trow*s Printing and Bookbinding Ca, 205-2x3 Mast x^th St.f NEW YORK. CONTENTS. PAGl Prefagb yn Pbefacb to thb Sboond Edition xi PART I. PLANE DRAWING. CHAPTER L Exercises on directioDS of straight lines 1 First principles 1 Materials 3 Directions 2 Single lines. 8 Parallels 5 Opposite Unes 5 Double lines 6 Use of the copy-book. Size of fignres 6 CHAPTER II. Elementary and practical exercises on right angles 7 Principles « • 7 Examples of single lines at right angles, vrith sides horizontal and yertioal 8 With sides oblique 9 17 CONTENTS. PAOl Bight angles -. ^ Pairs of parallels at right angles. The pairs horizontal and vertical 10 The pairs in oblique positions. 10 Practical examples. 10 Occasions for free-hand sketching 11 CHAPTER in. Distances, and division of straight lines. 12 Principles Id Exercises in marking off a given distance 13 Division of lines into equal parts 14 Practical applications. 15 Enlargement and reduction 18 CHAPTER IV. Circles and their division 20 Principles 20 Examples. Circles and Arcs 20 ' Division of circles 22 CHAPTER V. Proportional angles. 23 Principles 23 Elementary examples 24 Practical examples 26 CHAPTER VI. Figures bounded by straight lines 27 Principles 27 Elementary examples 28 Practical examples. 31 CHAPTER VII. Rectilinear and circular combinations 33 Principles 33 Exercises 34 CONTENTS. CHAPTER Vin. pAoa GuryeB, and ourved objects in general. Ex. (103-118) 37 CHAPTER IX. Iiettering , 47 General principles 47 Roman capitals 48 Letters in general. Their classification and construction. 51 Practical remarks 54 FART 11. SOLID DRAWING. CHAPTER I. Object, or model drawing. Rectilinear models 57 Definitions and principles. 57 Bxercises 59 Curvilinear models. 60 CHAPTER II. PerspectiTe and projection free-band drawing 63 Definitions 62 Indicated exercises. Properties and treatment of wood 63 CHAPTER III. pictorial projection sketching 66 Definitions and principles 66 Isometrical drawings. 66 Practical applications «... 68 VI OOMTKErrS. PABT III. 9 ELEMEHTS OF OEOMETBIC BEAUTT. CHAPTER L pAoa Elementary ideas. Uni^, Variety, Freedom. , 73 CHAPTER n. Numerical and geometrical ezpreaaion of the elementary ideas 78 CHAPTER HL QenexaX applications of the idea of beanty in ratios. Analogy of form and Bonnd. 84 CHAPTER IV. Application to triangles and rectangles. 90 Triangles 90 Rectangles 98 CHAPTER V. Geometric beanty of polygons. Geometrical design. 96 CHAPTER VL Cnrrilinear geometric beauty. Circles and ellipses. 103 CHAPTER VIL Curvilinear geometric beauty. Ovals 110 Natural and artificial curves. 110 The egg-form, or ovaL Ill Industrial applications 120 The method by co-ordinates 123 CHAPTER VIIL Geometric symbolism. 126 Definitions and general illustrations 126 Geometric illustrations 127 FROM THE ORIGINAL PREFACE. In geometrical, or mechanical drawing, exclusive reliance for accuracy may, in theory, be placed on good drawing instruments, properly used. Practically, these instruments are not absolutely perfect as means to the ends to be accomplished by them, and from this, and momentary negligences of the draftsman, they are not in- fallible in accuracy of operation. But, viewing the eye and hand simply as instruments for executing conceptions of form, they are incomparably more perfect and varied in their capacities in this respect than draft- ing instruments ; and well directed practice should, and will, bring out this capacity. Hence, other things being the same, he will be the most expert and elegant draftsman, whose eye is most reliable in its estimates of form and size, and whose free hand is most skilled in expressing these elements of figure. Accordingly, in harmony with the law of easy gradations and connecting links which pervades nature, we find a special branch of free drawing which is peculiarly well adapted for a prelimi- nary training of the eye and hand of the geometrical draftsman. This training consists simply in drawing various single and combined geometrical lines and figures, of various forms and sizes, by the unassisted hand ; and constitutes a connecting link between ornamental free drawing and instrumental drawing. These brief reflections have resulted from a recent inspection VI 11 I*BEFACE. of a few simple pencil plates of sach drawings forgotten for a long time, having been made by the writer several years since, in connection with the conduct of a short course of exercises of the kind above described. As a further, and I hope not useless fruit of the foregoing views, the following little course is presented to all who, as draftsmen, may promise themselves benefit from the use of it, and for exercises of mingled interrogation and practice in pre- paratory and industrial schools. By means of a love of skill and accuracy in the use of eye and hand, exercises like those of this volume may be made a pastime for the improving (especially if social) enlivenment of numerous odd moments, those times when many subordinate excellencies can he acquired or perpetuated without interference vAth onis la/rger industries. Writing, as merely auxiliary to daily business, is not, in its intention, a branch of drawing. But, as an ornamental art, it is a species of free-hand drawing, not geometrical, however. Hence I have not treated of writing, while ample instructions on lettering have been deemed a due portion of the contents of this volume, since, moreover, the usual small size of letters makes their construction by hand alone more convenient than by the use of drafting instruments. The good tendencies of accurate drawing in regard to mind and character are worthy of notice. Practice in such drawing directly tends to make close and accurate observers, who will thus gain distinct conceptions of the objects of attention, and so of thought generally, and who will then more readily pass on to fidelity in the representation of their observations and con- ceptions. Nbwtok, Mafis., January y 1873. NEW PEEFACE. In distinguishing between artists proper ; and those engaged in the study or practice of industrial design — that of various wares and fabrics — together with those engaged in engineering and mechanical study or practice^ including instrumental drawing; it seems appropriate that both the latter classes should receive a special training — useful also to all — in the free-haThd drawing of regular forms. Hence my former Bnnall work was put forth, partly as an experin:rent. Increasing interest in the subject, and the measure of favor accorded to that less complete volume, has induced the author to studiously revise, remodel, and enlarge his work, adding many new figures, mostly on plates. Of the three Parts, into which the present volume is divi- ded, Part IL is largely, and Part III., almost wholly new. Part III. may interest the general reader. It contains, ap- parently the most appropriate principal extension of the volume, a concise presentation of the elements of geometric beauty, based, in general, on the ingenious and presumably correct theory of D. R. Hay ; but containing principles and applica- tions not found in his " Principles of Symmetric Beauty." Especially, the ovals, or egg-forms, derived naturally, and in unlimited variety, are believed to be new, and an improvement upon his " composite ellipse." The subject of symbolism may, in some aspects, of course^ bo turned into pleasantry. Still, as its use has prevailed for cen- X FBEFACE. taries in some departments, there seems to be no reason for not extending it to others. I have sought to simplify, and to improve as mach as possible, the very little that room could be foand for, on this subject. The figures or patterns in this volume may have a threefold use : Mrst, merely as copies for imitation, in acquiring skill of eye and hand. Second^ as standards, from which to make as many variations by recombination of elements, as ingenuity can invent. Third, as objects to which to apply the principles of beauty developed in Part III. Bearing this in mind, it will be seen, that the appropriate range of use of this volume may extend through Public and Private Preparatory Schools, Arti- zan's and other Evening Schools ; Schools of Design, and the earlier classes in Polytechnic Schools. The pupil's figures may conveniently be drawn in blank copy- books, easily procurable, and, in most cases should be consider- ably larger than the copies, in order to cultivate a broader freedom of movement of the hand. By a new process, enabling the plates to be close imitations of the autograph originals, the rigid straitness of ruled lines, which could not well be imitated by the free-hand, is avoided ; and the copies are such as the pains-taking pupil may reason- ably be expected to equal, and encouraged to excel. Grateful for the favor long accorded to his other elementary works, the author, ever bearing in mind, and recommending joint attention to jprinciples 2inA jpracticey hopes to make them still more acceptable by extending the work of thorough re- vision (for the first time, excepting the Projection Drawing) to all of them. Newton, Mass., August, 1878. FREE-HAND GEOMETRICAL DRAWING. PART I. CHAPTER L EXEECI8E8 ON DIRECTIONS OF STRAIGHT LINES. First Prinovptes. The direction of a straight line is its invariable tendency to- wards some fixed point. The directions of two lines may be alike. The lines are then said to have the same direction, and are called parallel. The drawing of parallel lines ^ or those whose directions are alike, is simpler than that of lines whose di/reo- tions a/re different, BJidi hence is here considered first. A line which is " straight up and down," or perpendicular to the surface of water, like this, when the book is held upright, is called vertical. A level line is called horizontal. The force of gravity acts vertically, hence objects rest with most stability in a vertical position on horizontal surfaces. Likewise, man himself, naturally stands upright, or vertically, and, generally, on surfaces whose lines are level, or horizontal. Hence vertical and horizontal are the simplest, most familiar, or primitive directions of lines, and will be first considered. Lines which are neither vertical nor horizontal, are ohlique. Also lines lying in any flat surface, and not representing either of these positions, are called oblique. Before commencing the succeeding exercises, the learner should be provided with the following materials ; and, through- out his progress, should carefully follow the subjoined general directions. 2 FBEE-HAKD OEOMETBICAL DBAWING. Materials. For the practice of quite young pupils, where Bubstantial ac curacy, rather than fineness of execution is expected, quite cheap paper, or even a slate and pencil will answer. For other pupils, blank drawing books of the usual form, everywhere easily obtained, may be used ; or, drawing paper may be cut into plates of convenient size, and kept in a paper- case such as any one can make for himself. A common semi-circular ^^ protracted ^^ a semi-circular piece of thin material, divided into degrees on its curved edge. A ruler 10 inches long and 1 inch wide. Moderately soft pencils, as Faber's No. 2 and 3. Prepared india rubber, free from grit, of the best kind now known as " Artist's gum." Spare pieces of paper, one, on which to rest the hand and so protect the drawing, and another on which to try the pencils. Also a strip for a measure of distances. A fine file, on which the pencil can, by a rolling rubbing motion, be most neatly and readily sharpened to a round point. When accuracy on a large scale is sought, as a training for bold sketching, plates, 10 ins. by 14 ins. of buff manilla drafting paper, and crayons, should be substituted for pencils, and small plates and figures. In fact, this may be done as a preliminary counterpoise to the somewhat cramping tendency of the mostly minute accuracy required in mechanical drawing. But for direct training in this accuracy, the pencils, and small plates, should be used as above indicated. Directions. Depend on the unassisted eye and hand alone, from the be- ginning. They will, in due time, amply reward the reliance placed upon them. Ruler, Protractor, and Measure may be used to test the straightnessj direction and length of lines al- ready drawn, so that if incorrect they can be re-drawn. But they should never be used to locate, limit, or rule the lines; for thus no education is afforded to the eye and hand, only trifiing skill is gained by them, and so the main object of the exercises is missed. If a line is found incorrect, first consider carefully how it DIKBOnONS OF STRAIGHT LINES. 3 differs from what it was meant to be, then erase it, and study its direction well, and try again. Excellent quality^ and not great quantity of drawing, is to be the chief object of ambition. Avoid the use of the rubber by studying well the position and lengths of the lines hefore drawing them. Mecm to have them appear in a certain way, and then make them so, as truly as possible ; rather than hastily make a careless sketch and then seek how to correct it. Ue sure that a figure is as well done as possible at the time, in obedience to the preceding rules, before attempting a new figure. Hold the pencil between the thumb and forefinger, and rest- ing on the tip of the second finger. It can then be moved both with freedom and steadiness. In drawing lines towards or from you, let the elbow be at some distance from the body. In drawing lines from side to side let the elbow be close to the body. Arrange the seat and paper so as to look at the paper in a direction at right angles to it, without stooping, and let the desk be low enough not to interfere with the elbows. Though all the lines of the following figures are horizontal, when the book lies fiat, yet, for the sake of brevity, it may be understood that all those lines shall be called vertical^ which are so when the book is held vertically. Lines from side to side may be caUed horizontal^ and others, ohliqybe. Kemember especially to sketch each of the figures, first in very faint lines, which can easily be erased if incorrect, before drawing the firm heavy lines of the finished figures. Do not, however creep along the line by short, disconnected, and hesitating steps, thus, ^ * but mark the line by a firm and unbroken movement, first lightly, thus, — and then heavily, thus : Single Lines. Example, 1. Draw vertical lines, beginning at the top^ and far enough apart to prevent each from being a guide to the other, as a parallel. Thus let these lines be drawn at^the middle and ends of the upper half of the plate. See A, PI. I. Ex. 2. The same on the lower half of the plate, but beginning the lines at the hottom. See PI. I. ft FREB-HANB OKOHETTBIOAL DBAWING. Ex. 3. Mark two points bo as to be connected by a yertical line, and then draw a line joining them, beginning a little aJbovt the upper point. Ex. 4. The same, but beginning idow the lower point These, and all the examples, should be varied, by taking lines of various lengths. Ex. 5. Draw . horizontal lines, beginning at the left, and far apart, as at the top, middle and bottom of the left hand half of a plate. See PI. 1. Ex. 6. The same on the right hand half of the plate, and beginning at the right. This will require special care. Ex. 7. Mark two points so as to join them by a horizontal line, beginning to the left of the left hand one, and draw to the right. Ex. 8. The same, only begin to the right of the right hand one. The foregoing constructions not all shown on PL I. will divide the plate into quarters, in which the following may be drawn. Ex. 9 to 12. May consist of the four preceding variations in the manner of drawing, applied to an oblique line, which in- clines ^e>m the body and to the riffhty thus : Ex. 13 to 16. May consist of four similar constructions of lines which incline Jrom the body but to the left. The last two examples should also be practised with the two following variations : First, let the lines be more nea/rly veiiAocH than horizontal, thus : DmEOnONB OF STRAIGHT LINES. Second, let them be more nea/rly horizontal than vertical, thus • Parallels. The following Examples permit so many variations in the order of construction, that each one, as numbered, must be generally understood to include several particular varieties. Ex. 17. Draw two vertical parallels, first drawing the left hand one first; and second, the right hand one first. Also draw each, in the four ways described in Examples 1 to 4. Ex. 18. Likewise draw two horizontal parallels, first, drawing the upper one first ; and second, the lower one first, and each as in examples 5 to 8. Ex. 19. Draw several vertical, parallels, beginning alternately at top and bottom. See a., PI. I. Ex. 20 to 22. May consist of similar variations in drawing two or naore horizontal parallels. See 5., PI. I. Ex. 23 to 28. May consist of similar exercises on two or more oblique parallels situated as in examples 9 to 16, and in- cluding the variations in the amount of obliquity there pointed out. Opposite Lines. These are lines starting at a given point ; and proceeding in opposite directions, thus : + or towards each other from their outer extremities. Ex. 29. Draw opposite lines, one upwards, and one down- wards from the given point. Ex. 30. Do., one to the right, and one to the left of a given point. Ex. 31. Do., in the principal varieties of oblique position. Ex. 32. Is a comprehensive one, consisting of the variation of the three preceding, by beginning to draw the opposite lines in each case from their outer extremities. 6 FBEE-HAND GKOMETKIGAL DBAWINO. Dovhle Linen. All the preceding examples may be made in double lines ; that is, lines as close together as they can be made without touching, and at first of the same size^ and then, of different sizes. Useful practice under this head consists in filling various figures, such as* triangles, squares, polygons and circles, with parallel lines, which should be made equidistant by the eye. Oeneral Example, Construct a series of examples of figures thus filled, each with one, two, three, or four sets of parallels ; which will form an elegant imitation of bold line engraving. See PI. L, Fig. 1. Use of the Copy-hooh — Size of Figures, The figures, many of which are, for convenience, printed of small size, and with the text where they are described, should be considerably enlarged as drawn in the copy-book, by the pupil. The plates give a better idea of the size and style of the figures as they should be drawn. Only, as the pupil's plates may be more numerous and less crowded, his figures may be larger at pleasure, making from one to six to a plate, according to their character. Indeed, where the figures are done with crayons, they should be made much larger, and may each be made to fill a bufE manilla paper plate twice as large as those of the copy-book. Plates IX. and X. contain such figures ; and when so drawn, they cultivate a greater freedom of movement of the hand, com- bined with exactness, than is secured by the finer work with the pencil alone. To avoid injurious wear of the copy-book by repeated trials, it may often be best to draw the figures first on loose waste plates. CHAPTEE IL ELEMENTARY AND PEACmOAL EXEEOI8E8 ON RIGHT ANGLES. Principlea. Beauty of form, considered as residing in certain geometrical properties of regular figures, results from certain proportions between their parts. These proportions may be regarded as arising from the rdatwe lengths of the distinguishing lines of the objects ; or from the rdatwe sizes of their angles. In moving, whether to walk, or to merely draw a line, we must begin each movement at a given point. The direction of our movement is first in our thoughts, rather than its extent, "We firet, if not oftenest, think, or ask, " which way " than " how fa/r?^ Direction is therefore a more primary idea than length. An angUj however, is merely diflFerence between directions from a certain point. Hence angular proportions^ or the pro- portion between the angles of a figure, are more elementary than Unea/r proportions^ or those between the lengths of the lines of the figure, and will be first considered. In doing this it will be convenient to find first some angle as a natural stamxia/rd of comparison for all others, and this we now proceed to do. When, then, two lines are so situated that, in moving on one of them, we do not at all move in the direction of the other, their directions are said to be vndependent. V k m i c f Thus, in these figures, by going from a to J, we find our- selves at the distance ac to the right of a. So by moving from rf to ^ we go a distance equal to df in the direction of the line df But, when the two angles formed hy the meeting lin£s are equalj as at mgh and Jcghj we do not, in moving to any distance on gh^ progress at all to the right or left of gh. Hence the 8 FSEB-HAND OEOMETBICAL DRAWING. directioBB of gh and mk are independent^ and the angle included between them is the natv/ral standard with which to compare all other angles. This angle between independent directions is called a right angle ; and now some of the subsequent exercises are to consist in constructing, by the eye, various proportional paits of a right angle. But, again, it follows from the explanation of vertical and horizontal directions, in Chapter L, that a right angle is in its simplest, most natural, or standard positionjwhen its sides are in the/unda/menM directions oi vertical And horizontal. We there- fore begin with right angles in this position. Observe first, how- ever, that we do not say perpendicular and horizontal, but vertical and horizontal, for a line in any position whatever, is perpen- dicular to another when it is at right angles with it, but there is but one vertical^ or " straight up and down " direction. EaDomples of Single Lines at Jiight Angles^ with Sides Sori- Bontal and Vertical, Ex. 83. Construct one right angle thus ,L_ and thus. and thus, I and thus, I making its sides from one to three inches in length, ea^ih side ending at its inter- section with the other. Slight additions will give these simple elementary figures a pleasing character as designs for geometri- cal borders and corner pieces, thus : ^^^■^HH . • a • designs which it is easy to make evenly by observing the direc- tion to pencil each line faintly at first, while locating it as in- tended. Observing that the beauty of a border depends upon its eoGpressivenesSy as an echo of some characteristic of the work which it encloses, the first design would make SLh agreeable EXBBOISES ON BIGHT ANGLES. 9 comer, for a plate of figures made up of points and straight lines. The second, with its swelled lines, suggests strength in the corner of the border ; which can also be gained by a dia- gonal from the outer corner to, or a little beyond the inner corner. Ex. 34. Construct two right angles, by prolonging one of the sides beyond the vertex of the angle, thus, | and thus, and thus, J and thus, ..J h making the sides of each angle from two to four inches long, in this and the following figures. Ex. 35. Construct ybwr right angles, by prolonging each side through its point of intersection with the other, thus Might Angles with sides Oblique. Ex. 36. Bepeat Ex. 33 with the sides in various oblique posi- tions, and of various lengths, thus : ^ -I r- ^ Ex. J37. In like manner, repeat Ex. 34, thus : -V Ex. 38. Similarly, repeat Ex. 35, thus, but in each case make > the lines from one to three or four ^N^ inches long, from the point of inter- / section. See c and d^ PI. L, for examples of a suitable size for these figures, but which, if drawn on large plates, may be larger still. 10 FBEE-HAND OEOMETrRIOAL DBAWING. PAIBS OF PASALLELS AT BIGHT ANGLES. The Pairs HorizontaZ and Vertical. Many variations can be made, and should be, in the order cf drawing each of the following figures. Thus the vertical lines can both be begun at top or bottom, or one in each way ; alsc, the horizontal lines may both be begun at the left end, or right end, or one in each way. Again, both of the vertical lines may be drawn first, or both of the horizontal ones, or one of each in succession. Ex. 39. To give a more ornamental character to these simple elements, after seeking truth of representation, only, in the pre- ceding elementary figures, they may consist in combinations of faint and heavy lines, as shown in a part of the following fig- ures, all of which should be made of lines from a half inch to three or more inches long. J J L n Thejpai/rs in oblique positions. Ex. 40. Eepeat Ex. 39, as follows : Practical Examples. Ex. 41. The preceding elementary examples afford all the operations necessary in forming many simple drawings, either of geometrical designs for surface ornament^ or of objects. EXEBGI8ES ON BIGHT ANGLES. 11 A specimen or two of each is added in this example. The pupil is here again reminded always to make his figures very much larger than those of the book, making from two to four on a plate like those here shown. Ex. 42. Figs. 2, 3, 4, PI. I., exhibit other practical examples of constructions containing only lines at right angles to each other. OcGCbsions for Free-hand Sketching. The travelling student, architect, engineer, mason or builder may often find it desirable to make hasty sketches of neatly contrived details or structures, whether in masonry, wood, or metal. So also may persons of any and all occupations, when seeking to " give an idea " of something which they wish to have constructed ; and, in both cases, drafting instruments and time to use them may not be at command. The examples mentioned, and many similar ones which they may suggest, or which may usefully be collected by observation, should therefore be carefully drawn on various scales by the pupil, as a means of acquiring skill in the useful accomplish- ment of readily making free-hand sketches of geometrical in* dustriaZ ohjecta. CHAPTER IIL DIBTA^CES, AND DIVISION OF STRAIGHT LUnES. Prmctplea. Having considered various directions of straight lines, we are prepared to estimate and represent va/rioua diatancea upon them. Distances are equal or vjiequal. When unequal, we often wish to compa/re them. Distances may be compared, f/rst^ by taking one from the other, and thus finding their difference. This shows how much greater, or smaller, one distance is than the other. Distances may also be compared, second^ by observing how many times one is contained in the other, and thus finding their ratio. This shows Jiow mcmy times greater the larger distance is than the smaller, or what part the smaller is of the greater. When we compare lines in this second way, we speak of them as proportional, or as being in proportion to each other, or as having a certain proportion to each other. An indefinite line is one that has no given limits. In repre- senting distances, we may either mark a given distance several successive times on an indefinite line; or, we may divide a given line into equal parts, and &yfind a series of equal distances. Ecercises in Marking off a CHven Distance. Ex. 43. Draw straight lines in different directions, and mark by the eye, the same distance, once, on all of them, thus : DISTANOBS AND DIYI8I0N OF STBAIGHT LINES. 13 Transfer the distance on the first line to the edge of a slip of paper, and with this, as a measure, see if the distances on the other lines all agree* with this measure. If not, observe whether they are too large or too small, and then, without making any mark on the paper before removing the measure, take away the measure, and correct the distances by the eye. Ex. 44. In like manner, mark a given distance several times, on lines in various directions ; thus : Ex. 45. Draw lines in several directions through the same point, and mark equal distances from the point on all of them ; thns: I 14 FBEB-HAKD OEOMBTKEOAL DBAWIHO. Division of Lines into Equal Parts. Ex 46. Divide lines in Tarions positions, as shown below, into two eqnal parts. This is done by marking the middle point of the line, and is called bisecting the line. Then apply the paper measure, and see if the two parts are eqaaL If they are not, the error found at the end of the line will be double the error in the required half. If three parts had been required, this final error would have been three tunes the error m the single third of the line, and so on. Then make the necessaiy correc* tions, accordingly. To distinguish these figures from the preceding, mark only the ends of the line by dashes extending across the line. Ex, 47. Divide a line into four equal parts. To do this, bisect the whole line, and then bisect each of its halves. A « ' « 1- In each of these exercises, let the given line be taken in various positions, though but one may be shown in the book. In like manner, that is, by bisecting each quarter of a line, we should obtain eight equal parts, etc. Ex. 48. To divide a line into three equal parts, that is, to trised it. Estimate one-third of the line, and bisect the remainder. To divide a line into nine equal parts, divide each of its thirds into three equal parts. Ex. 49. In the preceding examples, we have divided each of the larger spaces into the same number of parts into which tlie whole was first divided. DISTAN0B8 AND DIVISION OF STBAIGHT LINES. 15 Let a line now be divided into six equal parts, for example. Half of a line is more easily estimated than a third, hence di- vide the line &*st into halves. Also, having done this, one-third of a short distance, as the half line, is more easily estimated than a third of the longer whole line, hence divide each half into thirds, giving six equal parts in the whole line. Ex. 50. To divide a line into any prime number, as five, seven, eleven, etc., of equal parts, it is necessary to estimate at once the fifth, seventh, eleventh, etc., part of the whole line. Yet this may be done more readily by dividing the line into two or more parts. Thus, one third of a line to be divided into seven equal parts would contain two and one-third of those parts, and thus we could more easily estimate the size of one of those parts. Practical Applicatio'tis. PL II. gives examples of various useful exercises in distance^ direction^ division. In order to enter upon the drawing of these figures, and many similar ones, with proper ideas and spirit, it is neces- sary to understand that, although they would, in final and finished practice, be drawn with instruments, yet it is highly useful to draw them also first by the eye, and for various reasons, such as follow. First ; it may (see p. 11) often be desirable to make rapid sketches, when instruments are not at hand. Second ; instru- ments are liable to be displaced unconsciously to the draftsmen, giving unequal spaces, if dividers are jarred, or a scale mis- takenly used ; and untrue directions, if a ruler be displaced. In these cases, the eye may be so accurately trained as to readily detect errors, which if long undiscovered would, and do, occa- sion great annoyance. Third ; the eye so trained will often enable the draftsman to make small and simple divisions, espe- cially those requiring only repeated bisection, as halves^ fourths^ etc., as readily and perfectly by the eye, as with the com- passes. All the following figures were originally drawn strictly as here directed, the divisions being tested by marking them on the edge of a s]ip of paper, and the directions, see especially the diagonals in Fig, 6, by tracing theiq at first very faintly. 16 FSEE-HAND GEOMETBIOAL DSAWINO. The pupil should therefore begin with an effectual ambition and purpose to perfect his work without instruments. Ex. 51. Customary hlack-hoard exercises. — Besides figures drawn on the black-board only for varied practice, or some other special purpose, various common subjects of study cus- tomarily require the drawing of numerous black-board diagrams. It should never be supposed, that because these diagrams are temporary, they may be carelessly drawn. On the contrary, on the teacher's part, neat diagrams lend interest to explanations, and naturally stimulate the learner to draw equally good ones; while on the pupil's part, the pains taken in making them helps to form the invaluable habit of doing as well as possible what- ever one does. Accordingly, Fig. 1 is a diagram, connected with the use of instruments, and Fig. 2, is from plane geometry. Fig. 1 represents what is called a diagonal scale, from its diagonal lines at 1, 2, 3. A scale is a contrivance* for repre- senting any actual measure, as a foot, yard, or mile, by some other measure, usually a smaller one. Thus, let the distance from 0, to 1yd., be two inches^ but let it represent one yard. It will then be called one yard. The next lower denomination is feet, hence, making the distance 0, 3ft. equal to two inches divide it into three equal parts, each of which will therefore represent a foot, and will be called a foot. Now suppose we wish to represent fourths of a foot, or tliree-inch spaces. Draw five equi-distant parallel lines, at any convenient distance apart, as shown, and divide the distance 0', 3' on the lowest line, into three equal parts, and draw the diagonals as shown. Then you see that as O'a is one fourth of O'O ; ac is one fourth of 01. But 01 is one foot, hence ao is one fourth of one foot or three inches. In like manner, the distance between the two heavy dots ifi 1 yard, 1 ft, and nr, which is three-fourths of 01, or 9 inches. As a drawing-exercise, the points to be observed are to make those lines straight and parallel, and those divisions equal, that are intended to be so. The pupil can exercise his ingenuity in making other diago- nal scales from the full description given of this one ; as, for example, one of feet, inches, and half-inches ; or one of units, lOths, and 100 ths. DISTANCES AND DIVISION OF STEAIOHT LINES. 17 In Fig. 2, the exercise consists in making ah perfectly paral- lel to AB, and cd to AD ; and in drawing AJ straight from A to h and AeZ, from A to rf. Then, triangles, like AaB, and A5B, having tlie same base, AB, and eqnal altitudes (the perpendicu- lar distance between ab and AB) have equal areas. Also, as the like is true of the triangles AcD, and A(2?D, the triangle Ahd is equivalent to the polygon AaBDc. Ex. 52, Floor and waU decorontion, — PI. II., Figs. 3, 4, and 6, are examples requiring equal divisions of one or more sizes, and parallel lines in different directions. Fig. 3, represents diagonal floor- work in two woods, the con- struction being founded upon the square 0, 6, 6, S. Divide the sides of this square into any desired e\)en number of equal parts, and draw S^ and the parallels to it through the corners and middle points of the sides of the square to form the^araZ- lel hands of flooring ; each band being filled with narrow diag- onal strips. These strips are drawn parallel to the sides of the sqnare, through points of division on an adjacent side ; as pg^ pai*allel to 6, 6, through 5 on 0, 6. Finally, the two woods may be arranged in two ways ; firnt^ touching each other on a common edge, as at ah\ second^ touching only at the corners, as at c? and d. Fig. 4, represents a toothed cornice, the shaded portions giv- ing the effect of shadows. Fig. 5, represents an ornamental band of triangular points, the darker portions indicating a darker color. In dividing, equally, the top and bottom lines, be careful to make the points, as a, of the triangles exactly over the middle points, as J, of their bases. Ex. 53. Fig. 6, gives further occasion for practice in cowr hined eq^iaZ division ^ in the equal bars, and in the larger eqnal spaces between them. Also in the direction of the diag- onal brace. In this, and all similar cases, the divisions^ first made only hy the eye (p. 2, Directions), may be adjusted by the aid of a slip of paper to the edge of which a division of each kind can be transferred and used to test the others. The directions are adjusted by sketching them very faintly, just skimming the point of the pencil along the paper, until they are found to be correct, when the faint traces thus ob- 18 FREE-HAND OEOMETBIOAL DRAWINO. tained can be firmly drawn in heavier lines with a softer pencil. Erdargement and Reduction. PL II., Figs. 7 and 8, and Fig. 9, exhibit two methods of ac- complishing an often desii'able and useful purpose, that of reducing a given figure in any desired i-atio. One of these may be called the method by svhdivision^ the other, that by concentration. Ex. 54. Fig. 7, may represent a frame in the form of a capital "A," carrying a plummet, and standing in a window opening of irregular outline. Fig. 8 is a reduced copy in which each side of the auxiliary enclosing square is five- eighths of that in Fig. 7. By subdivision of this square into the same number of smaller squares in each figure, the por- tion of the figure embraced in each small square will be so small that it can be very accurately drawn by the eye. This method may be applied to the sketching of large ob- jects, by substituting for the subdivided square of Fig. 7, a frame of equidistant threads, crossing each other so as to form squares. Then by setting this frame in some suitable fixed position, and viewing the given object through it, from some fixed point, the portion of the object seen in each thread square can be traced in the corresponding compartment of the similarly subdivided square on the paper. When, however, the ability already exists to draw objects accurately from the original, this method by the thread frame is unnecessary. And where the purpose is to acquire this ability by a sufficiently extended practice in sketching from original objects, the same method might only hinder the re- sult. But in the many cases still remaining, the method will be found usefuL* Ex. 55. Fig. 9, represents a given irregular figure, the outer one, reduced to the smaller one by means of the auxiliary lines which conversje from its ancrles to 0. The truth of the method will be apparent by conceiving to be the vertex of a pyramid whose base is the given figure, and the reduced figure to be a section of this pyramid, parallel to its base. Each side of the ^ See mj K'emftUary PKB^*ECTr?x« Part n., duq^L IV. DISTANCES AND DIVISION OF STHAIGHT LINES. 19 smaller figure will then be parallel to the corresponding side of the larger one, and all of the converging lines will be divided in the same manner. That is, if any one of them be bisected, as in the figure, all of them will be bisected. Thus tlie method, when executed by the free hand throughout, affords practice in three things, ^r*^, the drawing of straight lines in various directions, each joining two given points; secoiid^ in drawing parallel lines ; third^ in the equal division of lines. Pupils can profitably practice extensively on these two methods of copying, with variation of size. The former con- veniently applies to the copying of geographical maps, carpet, paper, and inlaid patterns of regular form, and to letters. The latter method applies better to polygons, regular or irregular, as the boundaries of the map of a field. CHAPTER IV. 0DSCLE8 AND THEIB DIYIBIOK. Prmcij>les. DiBEcnoN is, as before said, tendency towards a certain point. A straight line has but one direction at all of its points. A curve constantly changes its direction. The simplest curve, and the one which will be the natural standard of comparison for all other curves, is the one which changes its direction at a uniform rate. The drde is such a curve, and all its points are at equal distances from one point within called its centre. The circle is, therefore, the simplest curve, and standard of comparison for other curves. Examples. Circles cmd Arcs. Ex. 66. To draw a circle. Sketch, faintly, several lines through a point, taken as the centre of the circle, and, from this point, mark ofiF equal distances on each of these lines. Then through the points thus given draw the circle, thus : Ex. 57. To draw the circle without drawing the lines through its centre. With the paper measure, mark a number of points all at the same distance from the centre, and then sketch the circle through those points. CIECLES AND THEIB DIVISIONS. 21 In both of these constructions, use fewer and fewer guides, ariid at last sketch a circle with no guiding point but its centre. ^Iso practice often in rapidly drawing circles by hand on the black board. The distance from the centre to the circumference of a circle, is called its radius. The distance across the circle, through its centre is its diameter. Parallel circles have the same centre, and are called con- r.e7htric, A portion of the circumference of a circle, is called an arc. Ex. 58. Draw circular arcs in various positions, and of various radii, and length, thus : Ex. 69. Draw parallel arcs and circles, of various radii, and the former also of various lengths and in various positionii, thus ; and then 7na/rk their centres. 22 FREE-HAND GEOMETRICAL DRAWING. Division of Ci/rcles. Circles, or arcs, may, like straight lines, have given distanceE marked ofif upon them, and may be divided into equal parts. The line which joins the extremities of an arc, is called the chord of that arc. When the arc is very short, its length cannot be ordinarily distinguished from that of its chord. It is on this priQciple that any given straight distance may be trans- ferred to a circle or to any curve. Ex. 60. To lay off a given distance on a circle or arc, divide that distance into a sufficient number of small equal parts, and then mark off on the circle, or arc, the same number of similar equal parts, thus, where the straight line is the given distance. I III I Ex. 61. Any diameter of a circle divides it into two equal parts, therefore draw several circles, and one diameter in each ; but in different positions in the different circles, which may also be of various sizes. Ex. 62. Two diameters at right angles to each other, divide a circle into four equal parts. Draw such diameters in various positions. Ex. 63. The radius of a circle applies just six times to its cir- cumference. Then lay off the radius once, as a chord of the circumference, as explained above, and then mark the other divisions, equal to the one thus obtained. Ex. 64. Bisect each quarter circle in Ex. 62, which will give eight equal parts in the whole circle. This bisection can then be continued to any extent, giving sixteenths, etc., of the circumference. Ex. 65. Continue these exercises by trisecting the quarter circles, and bisecting and trisecting the sixth parts in Ex. 63, giving twelfths, eighteenths, etc., of the whole circle. Also make these divisions on circles of various sizes, and on arcs in various positions. The eye will thus be trained to estimate readily any given part of a circumference. CHAPTER V. PBOPOBTIONAL ANGLES, Principl'CS. Afteb acquiring power to draw lines, truly straight, in any direction, and to draw a true right angle in any position, much additional power of the eye to estimate, and of the hand to rep- resent, will result from practice in estimating the values of the angles of objects. But we have seen that the right angle, upon which varied practice has now been had, is the natural standard of comparison for other angles. Hence the new group of valu- able exercises which follow, is designed to train the learner in estimating and representing accurately any fractional part of a right angle in any position. Every circle is considered as being divided into three hun- dred and sixty equal parts, called degrees and marked thus, 360°. Hence a half circle embraces 180° ; a quarter circle, 90° ; a sixth of a circle, 60°, etc. But, as already seen in the last chapter, two diameters at right angles to each other divide a circle into quarters ; hence, as a right angle includes a quarter circle, or arc of 90°, between its sides, it is also called an angle of 90°. In like manner, any angle is said to be an angle of as many degrees as there are in the a^c between its sides, the centre of the arc being at the point or vertex of the angle. In other words an angle is said to be measured by the arc included be- tween its sides. Hence the easiest way to divide an angle into equal parts, or parts having any given proportion to each other, is, to divide the arc between its sides in the manner required, and then to draw straight lines from these points of division to the vertex of the angle. The right angle being, as before ex- plained, the natural angular measure for other angles, a right angle will be taken as the one to be variously divided, in the following examples. 24 FBEK-UAND GEOMETRICAL DRAWING. Elementary Examples, Ex. 66. Bisect a right angle, in each of the positions given in Ex. 33. To do this, sketch carefully a quai-ter circle between the sides of the angle and mark the middle point of this arc. Then join this middle point with the vertex of the angle a^ seen in the figure. To divide the angle into any other number of parts, divide the included quarter circle into the same number of parte. To test the angle thus estimated and drawn, use a " Protractor," as follows : /// The protractor is a semi-circular instrument, whose semi-circn lar edge is divided into 180 degrees. Aright angle is an angle of 90°. Half a right angle is 45°, hence if we place the straight side of the protractor on one side of the angle, and its centre, C, marked by a notch, at the point or verter^ C, of the right angle, as shown in the figure, then the required bisecting line C, 45^, will if correct pass through the 45^ point on the divided edge of the protractor. If it fails to do so, then first carefully esti- mate, by the eye^ the amount of error, and then erase the line and draw it over, remembering to sketch it lightly ^ till found correct. Having found the true direction of the required dividing line of the given angle, ch^aw a numher of pa/raUels to it, in this, and all the following problems of divisions of angles. Ex. 67. Construct a line which will cut oflF one-third of a right angle from either of its sides, thus : PEOPOBTIONAL ANGLES. 25 One-thiid of a right angle is 30° — ^measured by one-third of the quarfer circle — Whence in testing the lines after drawing them they should pass through the 30° point of the protractor in the first figure, and the 60° point in the second. In every case con- sider, as above, the number of degrees in the given fractional part of the right angle, and make the test accordingly. In the figure, only two parallels to the required direction are shown. The student should make many more, and in various positions around the original figure. Ex. 6S. Draw a line cutting off one-fourth of a right angle fi-om either of its sides. This can be most accurately done by bisecting half a right angle, thus : Observe, as indicated in these figures, to place the given right angle in any and all of the positions given in Ex. 33. Ex. 69. Construct, successively, angles of one-jiftht and two fifths of a right angle ; i. e., angles of 18°, etc., thus : Ex. 70. Divide a right angle into two parts, one of 40° the other of 50°. This can be most easily done by finding one-third of the right angle, and making the angle and arc of 40°, one-third greater flian the one of 30°, tiius : Ex. 71. Repeat the divisions of the right angle, given in the 96 AMD OBOMKTRIGAL DRAWIKO. preoedii^ ezminpleay upon right angles in varions obliqne po^i tioDsaa in Ex. ^6. Practical Seamples. Ex. 73* A four pointed star, requiring two lines at right angles to each other, and the eqnal bisecting lines of those angles. Ex. 73. A gate. Note that an angle of 24^ is f onr-fifteentha of a right angle. n Ex. 74, An arch, giving practice in parallels, equal distances (each side of the arch, and the heights at the ends) and arcs, of varions sizes, and parts of a circle. CHAPTEE VI. PLANE FIGUEES BOUNDED BY STRAIGHT LINES. Prmeiples, Ajplanejlgv/re is a portion of a flat surface, bounded bylines "When bounded by straight lines, it is called ^polygon. Polygons are of various names, depending on their number of sides. A Triangle has the least possible number of sides, viz., three. It has also three angles, and when one of these is a right angle, the triangle is called right angled, A Quadrilateral, or quadrangle, has four sides, and angles. When both the angles and sides are equal, the figure is a square^ and its angles are all equal. When the angles are right anglefe, bat only the opposite sides are equal, the figure is called a rec- tangle. A Pentagon is a figure of five sides. In a regvla/r pentagon the sides and angles are all equal. Likewise, a riegular Hexagon has six equal sides and angles. The diagonal of a four-sided figure joins its opposite corners, thus: Figures of more than four sides, have more than one diagonal from any one comer. The student is now prepared to sketch such simple objects as depend only on certain proportions between their angles. According to the theory of beauty of angular proportions, briefly alluded to in Chapter II., those regular figures are most beautiful, in which the proportions of the angles can be ex- pressed by fractions whose terms are small numbers. A great many familiar objects have sides of an oblong, that is a rectoAfigxdar form, and these sides are divided by their diagonals into two equal right angled triangles. A triangle is the simplest plane figure, and a right angled triangle is the iimplest triangle^ as a standard for the comparison of angular 28 FBEE-HAND GEOMETBIGAL DRAWING. proportions^ since it contains a right angle, which is the natnial measure with which to compare its other angles. Rectangles, as floors, walls, doors, windows, the spaces be- tween them, etc., are therefore, most beautifully proportioned, wh^n thei/r didgondU divide their right angles into j>art9 beaHng a simple proportion to each other and to a right angle. Thus, if the diagonal of a rectangle divides one of its right angles into anglea of 30° and 60°, the ratio of these is i, and their ratios to a right angle, are i and f . These all being sim- ple fractions, the rectangle will be found to have agreeable pro- portions. Elementary Examples. The construction of regular figures, requires attention to the equality of some or all of the sides, as in Chapter III., as well as to their direction, and the proper size of their angles ; and thus requires the application of examples in all the preceding chapters. Ex. 75. A right angled triangle with equal acute angles of 45° each. This triangle possesses the property of being divided by a per- pendicular from its right angle to its opposite side, into two triangles of the same shape as the original whole. This prop- erty makes its construction easy. Draw this triangle in various positions, and fill it with lines parallel to its longest side, as above. Ex. 76. A triangle each of whose halves is a right angled PLANE FIGUBES BOUNDED BY STRAIGHT LINES. 29 triangle with acute angles of 36° and 54°. Hero |^=f ; M°=f and ^^o=f . Also in the whole triangle ■^^z=i. These ratios being varied, while all of them are simple, the triangle is very pleasing and forms an agreeable end, or " pediment," to a roof, as seen in the figure. Ex. 77. An equal sided triangle. This also, has equal angles of 60° each, and its halves therefore have acute angles of 30° and 60^. Draw several such triangles, and fill each one of some of them with one or more sets of lines, parallel, or perpendicular, to some one of its sides. Ex. 78. Construct squares of various sizes and in various positions, first without their diagonals and then with them. Ex. 79. A figure of four equal sides, but whose opposite angles, only, are equal, is called a Rhombus, thus : This figure is most easily constructed by first drawing its diagonals so that each shall be at right angles to the other at its middle point, and by then joining their extremities. Let rhombuses of various proportions be drawn. A square may also be drawn by its diagonals in the same way. Ex. 80. After the practice thus far had, various designs in plane figures can be executed, such as the following. These examples obviously require the divisions of lines into equal 1 30 FBEE-HAND GEOMBTBIGAL DRAWING. parts. Also, in the second figure, the marking of ($qual dis tances, viz., the semi-diagonals of the little squares. Ex. 81. Embraces a regular pentagon and some applications of it. The external angles of a pentagon formed by producing or extending its sides, are each equal to 72^, or four-fifths of a right angle, and are constructed accordingly. The five pointed star is most agreeably proportioned, by joining the alternate points in order to obtain the direction of the sides of the star points. Also, the middle line of any point, when extended, becomes the dividing line between the two opposite points. Ex. 82. Hexagons. These polygons have angles of 120® at their comers. They can therefore be combined as in pavements, so as to completely fill a given space. It will assist in construct- ing this figure, to remember that each of its sides is equal to the distance from its comers to its centre. Obsei«ve, also, that the longer diagonal is divided into four eqiial parts by the shorter ones, perpendicular to it, and the centre. Ex. 83. Divide a circle into eight equal parts, by diameters at 45® with each other, and jein the points of division by straight lines, which will give a regular octdgouy or eight sided figure. This figure can also be drawn, by considering that its external angles are each equal to 45®, thus : PLANE FIGUBBS B0U17DED BY STRAIGHT LINES. 31 Practical Examples, Ex. 84. Wholly made up of vertical and horizontal lines. ^^^5=11=^1]^ Ex. 85. Embraces oblique lines. a FKEE-HAHD GKOMETEICAI. DBA.WINO. Ex. 86. EnibracoB circular lines. ^ The fif the whole those poi-l be retraci I I CHAPTER Vn. EEOnLINEAR AND OIEOULAB COMBINATIONS. ^ Principles. "We gain, from observation of ornamental figures, and notably from that entertaining instrument^ the kaleidoscope, certain ideas^ which we will call those of unity, symmetry , and variety, in connection with figures compounded of a greater or less number of simple elements. These ideas, all of which are pleasing, we will now proceed to explain, and illustrate. Unity is that property of a figure by which, although com- posed of parts, its parts are so linked together that it addresses the mind as one figure, and not as a collection of separate things. Thus three detached lines are naturally regarded as three separate things ; but, as combined in a triangle, perfectly enclosing a space, they are naturally thought of as forming one thing, the triangle. So also, with the figures in the kaleido- scope, their regular arrangement around a single central point, gives them unity, so that we think of each as one figure. Symmetby is single^ doxible^ or multiple. A figure has single symmetry when it is divisible by only one line into two parts which will coincide , when folded together about that line. Thus a butterfly has such symmetiy, his wings co- y inciding, when folded together about the / centre line of his back. Also any triangle, / two of whose sides are equal, has a single / line of symmetry, as AC in the next figure. / A figure has double symmetry, when it is / divisible by two lines in the manner just de- f scribed. Thus the rectangle ABOD, shown on next page, has two axes or lines of symmetry ah and cd. Figures having mare than two such lines, or axes, of sym- metry, have multiple symmetry. Thus a square has four, two 34 FBBB-HAND OBOMBTBIOAL DBA^WING. of which are its diagonals, and a circle has an infinite number, all its diameters being its lines of symmetry. By VARIETY we here mean precisely what takes place in tam- ing the kaleidoscope, that is, the pleasing result of different combinations of the same given elements. The figures on plates III., IV., and V., will illustrate the ideas of unity^ symmetry and variety^ as here explained, and will afford examples of combination, suggesting many othei-s. Illustration, We have, first, eight different combinations of four equal right-angled triangles. The first, PL III. Fig. 1, has unity in the close union of the triangles, but lacks symmetry, and is thus less pleasing than Fig. 7 which possesses both, though having only single sym- metry. Fig. 3 is without symmetry and is weak in unity^ its parts being only united at a point, and is of inferior beauty. Fig. 2 has double symmetry^ but is weak in unity^ the tri- angles being joined by their shorter sides. It is thus less pleas- ing than Fig. 4, where unity is more strongly expressed, by the union of the longer sides. Figs. 5 and 6 both have double symmetry / Fig. 5 satisfies the idea of unity by means of its unbroken circumference, and Fig. t5 does the same by its solid union of the triangles. Both are pleasing, as is more apparent in case of Fig. 6, by drawing it as in Fig. 8, where the double lines, marking intermediate bands between the triangles, add richness to the figure. ETiercises. From the above full illustration, the learner can proceed to BEOTILINEAB AND OIBCULAB COMBINATIONS. 35 ' invent as many combinations as possible, of the following sets of figures. Ex. 87. Make various combinations of four equal acute isosceles triangles. See one in PI. III., Fig. 9. Ex. 88. Do the same with four equal obtuse isosceles tri- angles. See one in PI. III., Fig. 10. Ex. 89. Do. with four scalene triangles. See one in PI. III., Fig. 11. Ex. 90. Do. with four equal squares. See PL III., Fig. 12, for one. Ex. 91. Do. with four equal rhombuses. See PI. V., Fig. 1. Ex. 92. Do. with four trapezoids ; a figure of single sym- metry with two parallel sides. See PI. V., Fig. 2. General Example. Vary the last six examples in one or more of the following ways : (1) By increasing the number of figures to be combined, still keeping them equal. (2) By making them unequal, but still similar, as in com- bining large and small rhombuses or trapezoids, as in PL V., Fig. 3, of rhombuses, whore the eight angles around the cen- tral point are equal. (3) By making the elementary figures unequal and dissimi- lar, or at least dissimilar, as in PL V., Fig. 5, a pendant composed of right angled, acute angled, and obtuse angled triangles ; Fig. 4 a decorative cross, and Fig. 6, of various rhombuses-like figures of single symmetry (or mono-symmetrical, they may be called). Fig. 5 may be supplemented by additions, like the one shown, at each of the three remaining corners of the square ; and may be varied by other arrangements of the pendant triangle, or by using other figures. Platb IV. shows other examples of symmetry and varied combination based upon a square foundation. Ex. 93. Fig. 1 shows a doubly symmetrical arrangement of four-sided figures of single symmetry. Ex. 94. Fig. 2 shows the variation of Fig. 1, by placing the right angles of its component figures at the centre. Both ar- rangements might alternate in the same figure ; which would then better be placed diagonally. 36 FJUBB-HAKD OSOiOTBiGAI^ OJBAWTirO. Ex.95. In Fig. 3, a combination of squares, the sides each making angles oi 45** with the next, the angles of a iniicht have extended beyond the circumference of tb^ext outer one. Ex. 96. Fig. 4 might be varied by many different ways of occupying the angles between the equal arms of the cross. Exfl. 97-98. Figs 6, 6, each give but one-half of a figure of single symmetry, the whole figure to be di-awn by the student Exs. 99-101. Figs. 7, 8, 9, are mostly composed of circular elements, arranged upon a foundation circle, the Jlrst^ a rosette of six points ; the second^ having the centres of the looped arcs at the angles of the curved quadi-angle ; the W, the centres of the outer arcs of any leaf, at the extremities of the diameter which passes through the next leaf. By means of the suggestions attached to these examples, the learner will readily find the path of discovery leading to new designs, as well as to other variations of those here given. The figures of PI. IV., may well be drawn of such size that four, or even two of them will fill the plate. The use of the segments of centre lines, as in Figs. 2, 8, 9, serves to give vividness to the figures by emphasizing the quali- ties of unity and symmetry possessed by them. This will be- come more evident by omitting them, and then comparing the results, for the same figure. See also, Fig. 3, where the blank centre weakens the expression of unity. Ex. 102. Combine circles by interlacing, as in PL V., Fig* 7. Also with their centres arranged on a circle. PL. iir. P.L.IV. I 1 » t 1 ] " !*^ .i-^fl^' .^v CHAPTER VIII. Curves and Curved Objects in General. Wb have thus far mostly considered circular curves. These, however, are only the simplest among an endless diversity of curves, many of which are of great beauty, as well as common usefulness. When any curve and straight line merely touch at one point, they are said to be tangent to each other, and just at the point of touch, or tam^ency^ they lie m the same dvrection. Hence amy curve can he much more eoMly sketched^ if we know several tangents to it at different points, A circle can evidently be placed, or " inscribed " in a square, so as to be tangent to it at the middle point of each side. A curve similarly inscribed in a rectangle is called an eUipse. Now observe that as all squares are of the same shape, though of different sizes, so all circles must be of the same shape, also. But there is an endless variety in the proportions of different rectangles, and hence there may be an equal variety of ellipses. A right angle being more easily estimated than other angles, it is also a special heVp^ in sketching a curve^ to have one or more lines which the curve must cross at right angles. Hence it will be easier to sketch an ellipse in a rhombus than in a rectangle ; for in the former, the ellipse will be tangent to the four sides^ and will cross each diagonal, at right angles with it, and at equal distances from its extremities. Ex. 103. Sketch ellipses of various proportions by the rhom- ! boidal method, thus : Mark the middle point of each side, as points of tangency of the ellipse; then, make each diagonal of the rhombus, the diagonal of a square containing an inscribed 38 FBEB-HAND OEOMETRIOAL DRAWING. circle, which will croes those diagonals at other points of ellipse. Let this exercise be continued, in the sketching of elli] in rhombuses placed in various oblique positions, and, also, wit tlieir longer diagonals placed yertieally. When an ellipse is inscribed in a rectangle, it crosses the cei tre lines of the rectangle at right angles, at the points of tai gency with the sides of the rectangle. Thus the eight guidii positions afforded by the rhombus, are reduced by union to foi in the rectangle. The ellipse will, however, cross the diagou! of a rectangle at equal distances from its comers, but not in perpendicular direction. Ex. 104. Sketch ellipses in rectangles and other figares, various proportions and positions, thus : An ellipse is a curve of most delicate grace, and should there fore be most faithfully studied and carefully drawn. The mod offensive error in shaping it, is, to represent it as pointed at the] narrow end, which it is not, in the least. i By combining elliptical arcs of various proportions, tangent toj each other, various graceful forms adapted to ornaments, such as vases, may be composed. In doing this the relative propor- tions of the ellipses should not be chosen at random, but so that the angles of their enclosing rhombuses should form simple ratios. Moreover, these rhombuses should be in simple relative positions, and the corresponding angles in the different ones should form simple ratios. Ex. 105. In this design for a vase, all the angles, some of whose decrees are given in the enclosed numbers, are 9°, the sqiuire of 9°, or evm muUiplea of 9°. Also at the base, two rhom- buses have a common vertex ; and at top, two have a side and two OUBTBS AlfD ODBTED OBJECTS HI OENKBAL. vertices in. common. The acT^e rim-rliombus has its r.tdes, perpendicular to a 54 and b 1% its right eide passes through the comer 72, and its diagonal passes through e, the junction of two area, and centre of a 72. Moreover, the diagonal 72-36 coincides with 18-72 produced, and the side 72- 54 is parallel to the diagon- al 18-36. These mostly very simple relations of the rhombuses, and their angles, yield a very pleasing form, each Bide of ■which embraces four differ- ent elliptical arcs, of which the one ninning upward from c terminates on a 54. Ex. 106. In this design, the relations are in part, more, and in part less simple than in the preceding, and the result will hardly be thought more agreeable than before. The principal, and the base rhombuses are of the same proportions, as seen by their angles, and therefore enclose simi- lar ellipses, which gives less decided vari- ety in the outline at the base. The up- per side-rhombus, with its angle of 18", ' side of one in the central rhombus of 60', gives the comparatively complex and unfamiliar ratio -j^. Also its right hand comer is arbitrarily located on a horizontal line through the upper vertex of the central rhombus. In both of these designs the rolling rim might be omitted by terminating the sides of the vases on tlie longer diagonals of the narrow upper side rectangles. Ex. 107. Bv substituting for a rhombus, two dissimilar half 40 FBEl^-HAND GEOMETRICAL DBAWINa. rhombuses, having a diagonal in common, the beautiful egg shaped curve will be formed, thus: In the first of these figures, the acute half angles are 20° and 30*^, whose ratio is therefore f. In the second figure the cor- responding angles are 18° and 36*^, having therefore a ratio of J, and affording a more decidedly egg-shaped curve. Ext 108. An egg-shaped oval may also be inscribed in a regular trapezoid, tliat is a figure having two unequal but paral- lel sides, both of which are bisected by the same line, perpen- dicular to both, thus : Let these ovals be drawn in a great x-ariety of proportions and positions, both in rhombus-like figures and trapezoids, aryi with asJ^reijH4*nt reference a^j>os9iil€ to haves, whicli exhibit a great variety of graceful ovals. CURVES AND CURVED OBJECTS IN GENERAL. 41 Ex. 109. The material of vases, etc., being originally plastic, it may be supposed to settle by its own weight into oval forms before har- dening. For this reason, as well as from the greater stability associated with breadth at base, egg forms are more admired in pottery articles than true ellipses. The annexed de- sign ilhistrates these remarks. Its angles of 36° and 54° ; 54° and 10°-48' (ten degrees and forty-eight minutes) 75°-36' and 18°-54', give the simple ratios f, -J-, ^, J. The student should make a variety of similar designs. Examples 105, 106, 109, are here incidental and preliminary, illustrating a manner of using ellipses and ovals by means of circum- scribed rhombuses, etc. The use of ovals in designs will be more fully and syatematically explained in Part III. _jC Ex. 110. On account of the pleasing associations of stability and decision with horizontal and vertical lines, as indicated in Chap. I., a curve which enters into the composition of any solid and fixed object is most pleasing when it has one or more horizon- tal or vertical tangents. Thus, there is more vigor, as well as variety, in the curve in the second of these figures, than in the first. Ex. 111. As we here propose only such exercises as are more closely associated with geometrical drawing, we only allude to the careful drawing of German text and common writing (script) 42 FBEE-HAND OEOMETBIOAL DBAWTNG. letters on a large Bcale, as an excellent exercise in the close study and varied practice of drawing curves. The German text, and all upright letters should be evenly balanced on each side of an imaginary vertical centre line, in order to give them the most satisfactory appearance. Ex. 112. The varieties of curves being innumerable, a few are here annexed by way of suggestion. The student can devise many others. y The group of four parallel curves affords an excellent ex- ample for practice, each curve being nearly straight in the mid- dle, and sharply curved at the ends, while its left-hand half is convex upwards, and its right-hand half equally so down- wards ; and each with a vertical tangent at its extremities. Of the two spirals, it will be seen that one increases its radius uniformly, giving equal radial distances between its successive turns, while the other expands at an increasing rate. OUEVES AND OUBVED OBJECTS IN GENERAL. 43 The use of tangents in sketching curves is also illustrated in tJiese examples. Ex. 113. An exercise of peculiar utility^ is found in sketching easy curves through several given scattered points. This opera- tion frequently occurs in geometrical drawing, when other than circular curves are to be described. The essential things to be observed in these cases, are,/r«^, to avoid all sudden, irregular, and unnecessary variations in the rate or degree of curvature, 44 FKEEHAND GROMETRICAL DRAWING. and, second^ especially to avoid making an angle at any point in the intended curve. These important requirements can be met by keeping at least three successive points in view at once. Thus, while joining A and B in the figure, keep C in view, and operate likewise in making ail similar figures. The student should practice extensively on this exampU, ^rat taking the points, in many diflFerent relative positions, and then running easily flowing curves through them. Ex. 114. In several of the preceding examples, curves have been drawn tangent to straight lines previously drawQ. We here add an example of drawing tangents to curves already drawn. The tangent may be drawn through a given point out of the curve, as in the first figure, or through a given point on the curve as in the second figure. Ex. 115. Finally, the examples of this chapter close with practice in the very nice operation of drawing symmetrical figures with variously curved outlines. Symmetrical figures (p. 33), are those which are divided in one or more ways, by a centre line, into similar halves, as in this figure. The difficulty in such figures, after forming one side in a pleasing curve, is, to make the other side of exactly the same form, but in a reversed position. This can be done, as in the figure, by drawing lines perpendicular to the centre line, and by marking on them equal distances on each side of the centre line. The following are other examples of symmetrical figures, some of which have two centre lines. The learner can devise many other figures of similar character. Ex. 116. Let PI. IV., Figs. 10, 11, each be taken as one half of a sym- metrical figure, and draw the other half. 0UKYE8 AND CUSVED OBJECTS IN OENEBAIi. 45 J 1 v.^ K r 1. o 2. o 6. 8. 3. Bf . 6. 7. 9 I 10. 11. 12. 46 FSEE-HAND GEOMETBIOAL DRAWING. Ex. 117. Representing a few elementary corner pieces, inns' tratee some of the foregoing principles. 10 is inferior to 9 be- cause its main spur seems weakly placed, or driven in, while the spurred corner, 9, is firmly planted. 7 is better than 6, because it cuts out less of the interior, and because the grace of the curve is protected by the strength of the square comers at each side of it. Thus the nkdeton of every corner should embody a good idea^ for no rich- ness of detail in ornament can redeem bad governing outlines. Attention to such simple principles as these will guide in the design or selection of bordere, and prevent the necessity of pre- senting an elaborate collection of them here, when they can be seen in such abundance in type-founders' collections, and in engravers' and printers' works, together with various orna- mental devices. Another principle, disregard of which through dispropor- tionate interest in some trivial thing, may spoil a good drawing, is this. Ornamental devices on drawings of solid worth, should never represent anything essentially mean, or rudely comic, or even anything of merely transient interest. Neither should they be attempted unless they are sure to be well executed. Thus, a vignette on a map of a survey may contain a sketch of some pretty view seen from some point. A drawing of an engineering structure or a machine, may contain a pictorial view of the same object ; or of the establishment where it was made, or of the room or building in which the drawing was made ; anything in short, which is agreeable in itself, and not foreign to the subject. Ex. 118. As a concluding exercise upon useful symmetrical figures, various ornamental arrow, or spear heads may be drawn. See PI. v.. Fig. 8. These are useful in iron- work, and as devices for vanes, and as indicating the meridian in maps of surveys. ■J -> J J « > • • •fc CHAPTEK IX. LETTERINO. Oeneral Prmaiples. LetterinGj though not strictly a part of a drawing, is a neces- sary appendage to it, it being generally indispensable to the full understanding and intended use of the drawing. And as, also, there should be uniformity of accuracy and elegance in all parts of the draftsman's work, lettering is properly included among the fundamental operations, which he should be familiar with before applying his art in practical cases. Besides, although geometrical drawings should be principally titled with geometrical letters, yet these letters are, on account of their usually moderate size, as well as variety and curvature of outline, most conveniently made by the free hand. Hence the draftsman's training in lettering appropriately falls among the subjects of free geometrical drawing. Two points should be constantly remembered during the practice of lettering : Jirat^ uniformity of size and proportions, and, second^ beauty and regularity of form in each letter. Ill- shaped letters, if of uniform size, proportions, and distance apart, and truly ranged in a straight line or regular curve, will look tolerably neat. Elegant letters will, on the other hand, appeal badly, if , irregularly sized and located. Both uniformity, and elegance are, therefore, indispensable to perfect lettering. The learner's previous practice, in marking equal and propor- tional distances and angles, should enable him to secure uni- formity in his letters; and his practice on curved and other irregular lines and figures, should enable him to give them elegance of form. All the letters described in this chapter should first be drawn on plates of smooth heavy brown paper, about 11 by 14 inches in size, and with a crayon or soft pencil. They should be made three or four inches high, so as to afford exercise in free and broad movements of the hand, and may afterwards be made of ordinary sizes, on smaller plates, and in title pa^^es. 48 FREE-HAND GEOMETRICAL DRAWING. Roman Capitals, Before entering upon a general discussion of all the varieties of letters, we will make a special study of the common Boman capital letter, which is a sort of standard which all other letters are made to resemble, more or less closely, in certain particulars; and from which, as a starting point, variations are made in de- signing fanciful letters. Plate VI. Tfie Alphabet in Large Roman Capitals, — This alphabet is arranged in three groups, so as to form progressive exercises in the drawing of the letters. The first group em- braces those letters such as I and H, etc., which are composed, wholly or mostly, of horizontal and vertical straight lines. The second group contains all those letters in which oblique straight lines are prominent ; while the third group embraces those let- ters which are largely made up of curved lines. Letters, as large as those of this plate, may be made by in- struments, by observing certain proportions in their form ; but, inasmuch as, in common practice, letters are of such size that they are more conveniently made by hand, it will be far better for the student to make the large letters of PL VI. by hand, at least so far as to sketch their curved lines, and the points through which their straight lines pass ; after which, the lines, if inked, may be ruled. A running commentary on the different letters of PI. VI. will now be sufficient. I, the simplest of all the letters, consists of a vertical column, whose width may pro- perly be made equal to a quarter of its entire height. The caps at the top and bottom project beyond the column a distance on each side, equal to half the width of the column. These pro- portions may be observed in the wide parts and caps of all the lettera. ' We thus have for an I the following complete proportions : Divide its height into sixteen equal parts. Then its height =■ II, its total width -^, width of column -^^ projection of cap 3^, and thickness of cap ■^. These dimensions are to be preserved in the vertical columns of all the letters. Also all wide columns are to be of -^^perpendicular width, and all the caps are to be ^ thick. Having thus fixed upon a proper thickness for the caps, let lines be ruled parallel to the extreme top and bottom lines, to aid in making these caps of uniform thickness on all the letters LETTERING. 49 Each column of the H is like an I. The extreme width of this letter allowing -^ between the caps is equal to -^ of its total height. The height of the arm of the L is ^ of the total height of the letter. The extreme width of this letter, and of F, making the arms -^ longer than they are high, is ^ of the height. The ends of the arms must be -^ thick. F is like an L turned up- side down, with the addition of the middle arm, whose height is half the height of the letter, and whose right-hand line is midway between the right-hand line of the column and the ex- treme right-hand line of the letter. E differs from F only in having another arm. Some designers make this letter a little wider (^4) at bottom than at the top, and also make the height of the top arm a little less than that of the lower one. This method gives variety and an appearance of stability. T, having an arm on each side of a central column, has its extreme width equal ^ of its total height. Notice, on all these arms, that their curved sides are nearly quarter (iircles, giving solidity of appearance to the arms. None of these arms should be short, thin, or pointed. Passing the hyphen we come to letters having oblique ele- ments. V having its average width only equal to half its ex- treme width, since it comes nearly to a point at one extremity, may be made of extra width at the top ; thus, let the total width be such as would be given by two wide columns with ^ between their caps. This width will then be ^^ of the whole height Let the perpendicular width of all narrow columns be -jig-, and the horizontal width of V^ and A at their points ^. Observe, that the left hand column is the wide one, and that in all letters having slanting columns, except Z, the heavy column slants dovmwa/rd towards the right. Similar general directions to the preceding, apply to A. The cross bar of this letter may be half way from the bottom line to the inner angle. In K the under side of the narrow arm may intersect the ver- tical column, a little below the middle, as at two-fifths of its height, so that the wide oblique column may not intersect the vertical column. The extreme width at the top equals the total height, and at the bottom equals ^^ of the whole height. N, having an oblique wide column, but being a square letter, having two vertical columns, does not need the extra width given 60 FRKE-UAND GKOMKTRICAL DRAWING. to V and A. The length of full caps to oblique wide colunine being |^, and to vertical narrow ones ^, the total width at top. allowing ^ between caps, if there were a full cap at the left upper comer, will be y^. There is no cap at the lower right- hand corner. The under edge of its wide column is drawn from the left side of the foot of the right-hand narrow column, tangent to the slight curve which connects the upper left-hand cap with the left-hand narrow column. M has its total width equal to fi^ of its total height. The point of the V-shaped part is on the bottom line, and midway between the inner lines of the adja- cent vertical columns. W, the widest letter of the alphabet, is of an extreme width equal to ^^ of its extreme height. Its oblique lines are parallel to the corresponding lines of Y. The extreme width of Z is equal to \^ of the height. Its arms are lengthened, as there are no caps opposite to them. The lower one is \^ long and -^ high, the upper -^ long and -^ high. The left-hand vertical lines of the left-hand caps of X are in a vertical line. Reckoning from these lines, the extreme width at bottom is equal to ^^ of the total height, and at top it is equal to ^4 of the height. In Y the outer oblique lines intersect the vertical column a little below the middle, as at a distance equal to the thickness of the caps. The whole width at the top equals \^ of the whole height. Passing the second hyphen, we come now to letters in which curves form a prominent part. The total width of J is ^ of its height. Its larger curve, convex downward, has for a chord a hori- zontal line, at a height above the bottom equal to ^ of the height. The extreme width of U is \^, of D ||, of P ^4, and B ||-, of the height. The bow of the P should intersect the column a little below the middle, while the upper bow of the B may properly intersect the column a little above the middle, making the lower bow project -^ beyond the upper one. R is ^ wide at bottom. It diflFers from B so little, as not to need further description. By omitting the tail of the Q it becomes an O. The greatest width of the tail equals that of a wide column, and it extends three- fourths of the same width below the body of the letter. In either case the extreme width equals \^ of the height. The extreme width of C equals \^, The highest and lowest points of its outer curve are in the middle of the extreme width ; and the LETTERING. 6J corresponding points of the inner curve are half way between the inner point of the lower curved arm and the vertical tangent tx) the inner curve. In a letter afe large as this, it is well to let the npper arm set back, a distance equal to the thicknes of a cap, so as to prevent the overhanging look that it otherwise would have. The extreme width of G equals its total height. Its construction is evident from the figure, after the description of C, that has been given. The whole width of S equals that of Q, and its arms are nearly like those of C. Some designers make the lower half higher and wider than the upper half, but as S is, to a beginner, the most troublesome letter, it is here given in its simplest form. To sketch it readily, it is only neces- sary to keep in mind that the outer curve at the top becomes the inner one in the lower half, and so must be carried below the middle of the letter and curved sharply to form the inner line of the lower half. & is less subject to rule than the proper letters of the alphabet. The design on PI. I. is offered as being more pleasing than that in which the wing over the period ends in a rectangular cap. The dotted lines show a niodifica- tron of the design, ending in a large circle. The proportions here given are not absolute, but only relative. Thus an ordinary letter, as an 11, or an E, may be made twice as wide as it is high, or half as wide as it is high, but in that case all the other letters would have similar modifications of their present proportions. Such letters are called, respectively, eao- panded and condensed letters.- Directions, much more minute than the preceding, are some- times given for lettering, but, after affording a few essential hints concerning the general proportions of letters, it is here preferred to leave the details of their design to the taste and judgment of the designer. Letters in General. In examining a type-founder's specimen book, one may imagine, from the exceeding variety of letters therein exhibited, that it must be impossible to reduce them to any system. But a closer examination will reveal a few comprehensive features, according to which all letters may be readily classified iji groups. 52 FBEB-HAND OEOMETRICAL DBAWING. By acquaintance with the distinguishing characters of these groups, and their modes of variation from one another, it will be easy to design uniform letters in any proposed form or style, which is much better than a mere copying of them, without ability to proceed independently of a copy. All lettere may be included in two grand divisions. I. — Oeometrical letters are all those which have a definite geometrical outline, which, when suflSciently large, could be made with drawing instruments ; and — II. — Free-hand letters^ or those of so irregularly varied out- line that they must be made by hand only, guided mainly by the fancy of the designer. Since the lettei*8 called geometrical are the ones mainly used in geometrical drawing, they will chiefly be noticed in this section. The student, by collecting a number of hand-bills, programmes, business cards, sheet-music covers, etc., vrill have materials for a valuable scrap-book of letters, which will be useful for reference, and will contain numerous practical illus- trations of the explanations which follow. By examining such a collection, it will be seen that in all or- dinary letters three things may be distinguished — {a) the essential elements. (5) the complementary additions. {c) the decorations. The essential elements of letters, are those which are neces- sary, and sufficient, to enable one to recognize the letters. The first half of the first and third lines, and the second and fourth examples on the second line of PI. YIL, are letters formed of essential elements onlv. The complementary additions are the caps, and the hanging parts of the arras, etc. The letters of the first five, and the seventh, lines of PI. VII. are, with the exception of those just mentioned, letters having these additions. By the decorations are meant the ornamental shading and filling up of the letters. Thus letters may be represented as if made of wood, stone, or iron ; and of pieces having square or polygonal sections. They may appear as if seen obliquely, or as draped, vine-clad, or casting shadows. In spacing letters, it is a good rule to allow equal areas of blank paper between them. LETTERING. S3 Summing up; the preceding, and other particulars concern- ing letters, are systematically presented in the following table :— IS^ OD 1 3 .B I il tii >■* QQ !zi p; uj -a a o I I Id .a I S & 2 o M i-9 7 OD O 3^1 P50 (So CO I •—I § 'd « 5* '2 ^ .9 I ^'5 P v. 1 S o •3 I .a r5 M s s.a I H |5i lis-ii BHT BHaxjLai 64 FREE-HAND GEOMBIRICAL DBAWINQ. It follows from this, that there cannot be very many radi- cally different forms of letters ; therefore, instead of making a further subdivision of geometrical letters, some of the ways may be mentioned in which varieties of letters are produced by modifications of the elements just giv^n. 1°. By altering the proportions of height and width, forming e^ypanded or condensed letters. 2*^. By retaining or omitting the complementary additions. 3°. By making the wide columns of the letter massiv^e or slender. 4°. By making the letters as if they were flat plates, or as if they were solid, or " block " letters. 5°. By representing the latter as seen directly, or obliquely, so as to show both face and thickness. 6°. By minor modifications in the outlineSj as by rounding the caps into the columns. 7°. By making the usually curved letters polygonal. 8°. Varieties, without limit, may be made, by changes in the quantity and character of the decorations. Practical Remarks. («.) The thickness of the caps is the same as that of the nar- row essential elements. (5.) In pencilling letters, never pencil the ornaments, unless the letters are of extraordinary size, but pencil the outlines only, in very fine lines. ({?.) It is better to do all the pencilling by hand, since instru- ments would perpetually be hiding portions of the letters, and so preventing the eye from judging readily of their proper pro- portions. (6?.) Very small capitals and small letters are better put in off hand, in ink, between parallel pencil lines, to keep them of a uniform height. (^.) The sixth row of PlylV. shows a simple free-hand or " rustic" letter, in two sizes and styles. These are hark letters. Log letters are often seen in handbills, etc. (/!) The sixth row embraces " skeleton" and " full faced " " small " Koraan letters and italics. A common error consists in making the stems of the Vs^p^Sy etc., too long. The total LETTERING. 53 height of such letters need not be more than one and a half times the height of their bodies. (^.) To avoid making letters slightly leaning, stand directly in front of the work, and with the eyes far enough from the paper to be able to see the position of the border of the plate, as a guide. Or, rule vertical parallels at short intervals. (A.) Curves can be more neatly sketched in by a dotting, or very light motion of the pencil, than by a continuous motion with firm pressure. (i.) The ends of the arms of letters like G, C, S, etc., should not be far apart, vertically, but should come nearly together, and should be tangent to vertical lines, in order to give them a plump, finished, square, and stable look. (J.) Even in the most fanciful letters, there is a certain ap- preciable consistency and orderly form. This results from their having an imaginary central skeleton of regular single lines, about which the outlines of their parts are equally balanced. {k.) PL yil. illustrates most of the distinctions of form men- tioned in the preceding table, except the inelegant and unused Italian type. This plate, or one of similar nature, should be constructed by the student. (^.) Polygonal letters may be substituted for curved ones by any who are particularly deficient in free-hand sketching. They may thus be able to secure a desirable uniformity of excellence in their work ; though it is probable that the pains necessary to form an elegant polygonal letter would secure an equally elegant curved one. (m.) In line 2d, example Ist is elegant in being slightly ex- panded, and not heavy. Ex. 2d is neat and easy to make. In titles, the letters should be further apart. Ex. 3<^, of this line and of line 7th, are partly Italian in character, the essential parts being lightest. Ex. i^A, is like Ex. Ist, line 1st, but heavier. In line 7th, Ex. Istj is condensed ; Ex. 2dy shaded, and Ex. Zdj spurred. (n.) After the systematic explanation of letter drawing, with varied illustrations, now given, the student should make an en- tire alphabet of each of the kinds of letters shown on PI. VII. (o.) In combining words to form the titles to maps and draw- ings, the most essential principles are the following. 56 FBEE-IIAND GKOMETRIOAL DRAWING. 1°. To vary the letters in size, or heaviness, or both, accord ing to the relative importance of the different words of the title ; and the heaviness, or blackness, also, according to the general depth of color of the drawing. 2°. To make decided contrasts between the lengths of the different lines of lettering contained in the title, and so that the circumscribing figure, formed by joining the ends of the suc- cessive lines, shall have a pleasing outline. When the spelling of tlie words makes this difficult, the use of " condensed," or " expanded " letters, or the prolongation of some of the lines of the title by long dashes, will afford con- siderable aid. In connection with these principles, study the artistic features of titles, and title pages, critically, with a view to good typo- graphical design. {p.) In the old-fashioned styles of type recently revived, one of the most obvious marks of distinction is, that the arms of the E's, T's, etc., instead of being vertical, are divergent, as in the following letters. E F H L T Z (q.) Finally, remark (i) may be modified by adding that the arms should have vertical tangents, unless plainly meant not to have them, as if 0, for example, in the first half of line Srdj of PL VII., were to consist of three-fourths of a full circle. • ^ PART 11. S03LiII> DRA-TVIN-Q-, CHAPTER I. OBJECT, OB MODEL DRAWING. § 1. — ^Rectilinear Models. Definitions and Prinoijples. This angle made by two intersecting straight lines is called a plane a/ngle. The angular space enclosed by three or more plane surfaces which meet at a point, is called a solid angle. Thus the angle at one of the upper comers, B, of a room, see the figure, where two walls, ABD and CBD, and the ceiling ABC meet, I is a solid angle. In a square-cornered room, such an angle is a solid right angle. This is the simplest of solid angles, and is bounded by three plane right angles, one, ABC, in the ceiling formed by the meeting of two of its edges AB and 3* 58 FEEB-HAND GEOMETRICAL DRAWING. CB, and one in each of the connected walls, as AJBD and CBD. But there are many other solid angles, bounded by three or more plane angles, some or all of which may not be right angles. Thus the angle at the summit of any pyramid is a solid angle and is bounded by as many plane angles as the pyramid has sides. These simple principles being apprehended, no large mis- cellaneous collection of models is necessary in order to obtain skill in making free-hand sketches of geometrical objects from the solids themselves. A few variously proportioned prisms and pyramids each placed in various positions, and. singly, or combined, will afford an almost unlimited variety of practice in combinations of length and direction of straight lines. These objects can be made by any wood-worker, or by pupils for themselves, and may usefully be of wood or pasteboard ; also skeleton forms may be made of wire or light wooden rods.* A set of simple plane or flat-sided drawing models, of con- venient size to be seen across a room, being provided, exercises upon them may be made suitably progressive according to the three following principles. I. Selecting for reference any one of the "oarious solid angles of the body^ let the position of the body be taken, first, so as to show only one of its bounding plane angles ; then two, and so on till the body shall be so placed as to show all the bounding plane angles of the given solid angle. II. Attending to the surfaces of the body^ let it first be placed so that but one such surface shall be visible, then two, and so on till the greatest jx)ssible number shall be visible. III. For each position of the solid or closed model, place by the side of it, and in the same position, a skeleton model of the same body^ which will further vary the exercise by showing lines that are hidden on the opaque model. * When it is desired to purchase manufactured models, Harding's English models, architectural in character, and affording many combinations, may be found useful. So, also, will the excellent elementary sets manufactured at the Worcester (Mass. ) Institute of Industrial Science, and which more closely agree with the principles here stated in the text. Other sets may perhaps be found on inquiry at Art, or School Supply stores. S r"-, VTI ]. ■W p. I \ / D. «. i, 4. 4,^ I, V C fci k *- V c *- *• TK I- U STEAMER 1@ ■aye, the h 18 the 1 BO 'CHEB I LlinrL J ithe 'the dof e at ^h is lof rep- iolid dto OBJEOTy OB MODEL DRAWING. 59 Exercises. Ex. 119, PL VIII., Fig. 1, represents a cube, selected for illustration, and placed so that but one face is seen. Ex. 120, Fig. 2, represents the skeleton cube in the same position. All the edges \\rill be visible, but in various ways, according as the eye of the observer is above or below, to the right or left of the position indicated by the figure, which is directly in front of E, the centre of the front of the cube. Ex. 121. In Fig. 3, the eye is supposed to be above the level of the top of the cube, which is represented as turned so that three faces, c^ nhy and n, of the same solid angle are seen. Ex. 122. Fig. 4 represents a similar position of the skele- ton cube. Ex. 123. Fig. 5 represents the cube as so related to the eye, tliat only c and n of the three angles shown in Fig. 3 are visible. The eye is here between the levels of the upper and lower bases of the cube. Ex. 124. Fig. 6 represents the skeleton cube situated as in Fig. 5. The pupil can easily satisfy himself by experiment that the nearer he approaches the cube, the more rapidly will the reced- ing lines 2Acd and ef^ Fig. 3, appear to converge, and that the contrary result will take place, the further he removes from the object. Exercises like the foregoing may be arranged for each of the elementary solids. Thus : Ex. 125. Construct a series of six figures, of a solid, and of a skeleton triangular j>yr«mirf. Ex. 126. Do. of a square pyramid. Ex. 127. Do. of a hexagonal pyramid. Ex. 128. Do. of an octagonal pyramid. In each of these four exercises, it may be a solid angle at some comer of the bjise, or tlie one at the vertex which is chosen, for the purpose of guiding the changes of position of the pyramid. Thus, in the two following figures. Fig. 1 rep- resents a square pyramid, in which three sides of the solid angle at a are visible. The pyramid is thus either supposed to 60 FREB-HAITD GEOMETBIGAL DRAWING. be tipped backward bo as to bring its base in sight ; or else, if it stands on a level, to be above the level of the eye which may then be directly in front of some point as E. Fig. 2 represents a triangular pjTaraid, so placed that all of the plane angles at its vertex are visible. Proceeding with these exercises : Ex. 129. Draw six figures, three of a solid, and three of a skeleton triangular ^H«m. In this, and in each of the following exercises, the prism may be successively placed so as to show one, two, or all of the plane angles which bound some one of the solid angles at the lower hdse / or, to show the same for some one of the solid angles at the upper hose. Ex. 130. Draw, as above, a solid and a skeleton square prism^ each in three or more successive positions of increasing complexity. Ex. 131. Do the same with a hexagonal prism,. Ex. 132. Do the same with an octagonal prism. § 2. — Oiirvilinear Models. The three elementary round bodies are the cylinder, the cone, and the sphere. The Bphere can never appear otherwise than circular, hence not much practice upon it is required, with respect to ii^form; though, in studying effects of light and shade^ a series of varied exercises may be founded upon each body, by placing it in t?ie same position in various lights^ as well as in different posi- tio7is in the same light. OBJECT, OR MODEL DBA WING. 61 The cylinder and cone can be placed in positions similar to those already described for the prism and pyramid. That is, with the upper base (or the vertex) inclined directly towards, or from the eye, inclined to the right or left or inclined diagon- ally forward or backward ; that is to the north-east, south-east, north-west, or south-west, the observer facing the north. Not more than half of the curved surface of a cylinder can be seen ; but the entire convex surface of a cone can be made visible by inclining its vertex sufficiently towards the eye. As the exercises proposed in this Second Part are intended to be drawn directly from objects conveniently placed before the eye, without the intervention of drawn or printed copies,* no engraved copies are here given ; but the pupil should draw next, each of the three round bodies in various positions. After sufficient practice in drawing single bodies, a final series of exercises should consist in drawing various combina- tions of them, the combinations consisting of different bodies of the same kind, as prisms; and then of bodies of different kinds, as, for example, a group consisting of a prism, a pyra- mid, a cone, and a sphere, resting upon each other. * If, however, an intermediate stage of work be, in some cases, preferred, large line, or shaded copies of drawings of solids can doubtless be readily provided. CHAPTER 11. PEBSPEOnVB AND PBOJKOTION FBEE-HAND DBAWING. Dejmitions. I. A perspective drawing is one which represents the ob- ject as it would appear when seen at ordinary distances. The most obvious and familiar characteristic of such drawings is, that lines in them, whose originals on the object are parallel, generally converge to a common point. / Thus, in looking down a long stretch of straight railroad track, or through a very long building, as a largo freight depot, the rails in the one case, and the side walls in the other, appear to approach each other as they recede from the eye. For this reason, a perspective drawing is in one sense a dis- torted representation of the object drawn. That is, it does not represent it as the object actually m, but only as it appears.^ when viewed from some given point. For instance, see the six figures of the cube on PI. VIII., the necessary convergence of the receding lines distorts the angles, making those which in reality are all right angles, appear either as acute or obtuse angles, according to their position relative to the eye of the ob- server. If, however, these oblique angles of the drawing impress the mind as being true representations of actual right angles on the object, this result is owing to the modifying effect of our in- timate knowledge of the actual shape of the familiar object. But note, that this effect will not be produced, unless the picture and the object are both viewed similarly, in respect to distance and direction. II. Projection drawings. When, however, the beholder is at a very great distance from the object, as compared with the dimensions of the latter, the lines which are parallel on the object, will be so on the drawing also. Drawings made under this supposition, viz., that the object is seen from an inde- pesbpeghye and pbojeotion free-hand drawing. 63 finitely great distance, are called ^rojeotion drawings^ or simply projectiona. Both of the kinds of drawing just described, and whether made by the free hand or instruinentally, are of use to every one who has occasion to draw. Yet as perspective drawings represent objects only as they appear^ they are chiefly of use to artists ; in other woi'ds, to those who make pictures for our pleasure. But as projection dravmigs represent the forms of objects as they really are^ they are more useful to industrial de- signers, in making patterns or copies, from which things to use are to be made. Thus, of the two figures of a goblet, shown in PI. VIII., Figs. 7 and 8, the first represents it as it might appear standing before one at the table, and is a perspective view. The second is a projection drawing of the same goblet, showing its height, and its diameters at several points on its centre line or axis, on a uniform scale of one third of the full size. The former is therefore vaibY^j pictorial^ while the latter might guide a work- man in making goblets of the particular pattern shown. The final drawing to be strictly followed by the workman might be very accurately laid out by scale, of the full size of the goblet and drawn with instruments. But the preliminary drawings, to indicate the design and its effect, would be drawn by the free hand, whether in perspective, or projection, or both. Hence, as already explained in connection with other applica- tions, the usefulness of the free-hand drawing of geometrical, as well as of natural objects. Indicated JEkerdses, Properties and Treatment of Wood, The following are some of the articles in the drawing oE which the pupil can usefully exercise himself, and which each can generally find at hand in his home without the necessity of collecting models. Firstj hoivsehold wares. Knives, forks, spoons, castors, tea sets, tubs, pumps, stoves, pitchers, bowls, cups, saucers, dishes, etc. Second^ furniture. Under this head a qualifying remark is necessary. Wood may be treated in two radically different ways ; Jlrst^ indepen- dently of its nature as having generally, or, except in root, 64 FEEB-BAMD knot, or crotch pieces, an essentially straight grain ; second, with strict reference to its structure in this respect These different ways give rise to two corresponding parties relative to ornamental design in wood. TUe one, treating it as if it were plastic, or like marble, without a fibrous grain, pro- duces the kind of work often seen, abounding in curved out- lines and carving, as seen in curved and carved table and piano legs (see the annexed %ui-e), sofa and pictui-e frames. The other party claim that wood should be treated strictly according to its nature, that is in straight pieces running with the grain. Thus purely angular, not curved, geometrical work is produced, as seen very simply in the straight backed chaira of our grandfathers, and more elaborately in what is popularly known at present ae the " Eaatlake style." So far, however, as some kinds of wood, as aah, and others, are capable, at least under certain treatment, of being very PEEePECnVK AND PKOJEOTION FEBE-HAND DBAWING. 65 much and variously bent without breaking, as in the " Austrian bent wood" furniture, shown at the Centennial Exhibition, and in many of our light ofBue arm-chairs, it may be fairly claimed that the design does no violence to the nature of the wood. Alai>, 90 far as woods can be found of so close and tena- cions a grain, that the cutting of successive layers of grain, as in a tapering turned table leg, never has the effect of causing a splintering or peeling up of these layers where thus out off, the wood seems to indicate, by its behavior, that no violence 18 done to its natural properties by curving, moulding, or carving it. From this point of view, the rigid exclusiuu of all but straight outlines and angularly geometrical forms from orna- mental wood-work, would seem to be more of a fancy, than well-founded in principle. With these explanations, the pupil will find frequent and use- ful examples for free-hand drawing of geometrical objects in articles of furniture having a regular geometrical form, such as Chests — Clocks — Work-boxes — Book-cases — Tables — Desks— Etc. CHAPTER III. PIOTOBIAL PBOJBOnON 8KBT0HINO. Definitions and Principles* There are special kinds of projection drawing (Chap. II. n.) which combine the exactness of representation of projection drawing, with a good measure of the pictorial effect of per- spective drawings (Chap. IL i.), especially as applied to small objects. They are therefore highly appropriate for the free- hand sketching of such objects. Without going into the principles and details of this subject here, enough will be explained by illustration, to enable the pupil, beginning by imitation of copies, to serve himself suffi- ciently until he studies the subject fully.* In PI. VIII., Figs. 9 to 14, illustrate these pictorial projec- tions by the most elementary examples. The two first, or Figs. 9 and 10, represent a cube already otherwise shown in PL VIII., Figs. 1 and 2. Isometrical Drawing. Here the first figure represents a solid cube, placed so that the three plane right angles, a, J, tJ, which unite to form the solid angle of the cube nearest the eye, are equally exposed, and hence appear of equal size. Hence this kind of drawing is called isometr^ioal drawing, the name meaning egical mea- sure. The second figure represents a skeleton cube of the same size as before. Owing to the entirely regular form of the cube, the equality of all its sides and angles, the most re- mote corner, at which the three edges, ^^,j^, M, meet, appears to coincide with the nearest one, ahc. * See Isometrical Drawing, etc., in my ^^Elbmbntab? Pbojbctioh Drawing." PICTORIAL PROJECTION SKETCHING. 67 Ex. 133. Draw the cubes as here shown, but mucli larger. The angles at a, 5, e, in the drawing, are equal, and of 120** each. The other angles are of 60°, or 120°, as shown. Oblique Projection, — Again, instead of supposing the object to be directly in front of the eye, but inclined as just explained and illustrated, we may suppose the object to be above or be- low the eye, and also at the right or left of it. The object is also still supposed to be at so great a distance from the eye that its parallel lines will appear parallel. The four figures, 11 to 14, of PI. YIII., illustrate this case, by the representation of a half cube seen in as many diflFerent directions. They are called oblique projections to distinguish them from the preceding, and "because the supposed position of the body causes it to be viewed obliquely. All lines in the isometrical figures, in other directions than those shown, would appear less than their real size, as in case of one from a to e, or greater, as one from d to g. The like is true of all lines in the oblique projections, except all lines in the front faces A, B, C, D ; which show their real form as well as dimensions. Ex. 134. Fig. 11, represents the half cube as seen from be- low and to the right. Draw also thiB whole cube. Ex. 135. Fig. 12, the same as seen from below and to the left. Draw likewise any other square prism. Ex. 136. Fig. 13, as seen from above and to the right of it. Draw also the whole cube. Ex. 137. Fig. 14, as seen from above and to the left of it. In each case the line dc^ on the body itself, is perpendicular to the front face of the body, and on the drawing, is made equal to \ ah. The edges which are parallel to dc on the block itself, are parallel to ac in the drawing, on account of the supposed very great distance of the eye from the object, although they would appear to converge in a true perepective drawing (Chap. II. i.). Nevertheless, these^figures, as well as the isometric ones, have a pictorial effect wmch makes them more intelligible to those workmen and others who have little familiarity with drawing. 68 FREE-HAND GEOMETRICAL DRAWING. than ordinary projections are (Chap. II. ii.) ; while they possess the practical advantage of showing the three dimensions of the object in their real size. - Practical Applications. Suppose, now, that a person wishes to have made for him, a covered rectangular tank, with a raised square covered opening near one end. The figures, 1-4 of PI. IX., afford a connected illustration of the kinds of drawing already explained. Ex. 138. Fig. 1, is 2i, perspective view of the tank, not show- ing any of its dimensions tr\\\y. To appear not distorted, the eye should be about four inches directly in front of a point a lit- tle above E. Ex. 139. Fig. 2, shows a pair of projections^ together rep- resenting all the dimensions on a uniform scale ; the lower one. A, is a top view, called a plan^ and shows the length and width of the tank ; the upper one, B, is a front view, called the elevation^ and shows the length and height. Ex. 140. Fig. 3, is an isometrical figure, where all the angles are right angles on the object, but appear as angles of 60° or 120° in the drawing. Ex. 141. Fig. 4, is an oblique projection of the same tank. Both of the last two figures have nearly as much of intelligible pictorial character as the first figure, with the added practical advantage of showing all the dimensions as truly as the projec- tions in the second figure do, and more intelligibly to ordinary eyes. The remaining figures of PI. IX., show a variety of applica- tions of the projections now explained. Ex. 142. Fig. 5, shows a nut A, and bolt B. The nut is six-sided, and, according to the properties of a prism, of six equal sides, the lateral faces, C and D, each appear just half as wide as the middle one. By experiment with a regular six- sided prism, the pupil can easily find how the nut would ap- pear if only two of its faces were visible. Ex. 143. Fig. 6, shows a partial plan, B, and elevation. A, of a square nut, the plan or top view being made first, its PICTORIAL PROJECTION iSKETCHING. 69 comers being, as shown by the dotted lines, a guide to the posi- tion of the vertical edges of the elevation. Figs. 7 and 8 show how to make correct isometrical drawings of objects whose lines are not all at right angles to each other. These lines, as indicated by the faint lines and the letters, are located from lines parallel to those in Fig. 3, since only such lines show their real size. Ex. 144. Fig. 7, represents a frustum of a square pyramid ; drawn, as indicated, by inscribing it in a prism of the same base and altitude. Ex. 145. Fig. 8, a fire-place. Draw Fig. 8 first as shown, and then with the longer edges, as MN, in the direction MO, which will bring the opposite end of the fire-place in sight. Figs. 9, 10, 12, illustrate the isometric drawing of curved objects. Each face of a cube, Fig. 9, being a square, a circle can be inscribed in it, touching each side of the square at its middle point. Ex. 146. Fig. 9 shows the appearance of the isometrical drawing of a cube with circles thus inscribed in its three visi- ble faces. Note that the diameter of the circle is equal to the side of the circumscribing square. • Ex. 147. Fig. 10 shows a frustum of a cone, isometrically, its bases being drawn by means of their circumscribing squares which are placed in the same position as ctbcd in Fig. 9. The axis, MN, joining the centres, M and N, of the bases of the frustum, might have been in the direction of cJ, Fig. 9. Then the circumscribing squares of the bases would have been in the position cdef^ Fig. 9. MN could also have been in the direc- tion of cdj Fig. 9, and these squares would then have had a position like iceg in that figure. The pupil should draw the frustum in the three positions here described. Ex. 148. Fig. 12 is the isometrical drawing of a pipe with a flange at each end, the axis of the pipe being in the direction of erf. Fig. 9. According to the explanations just made, draw this pipe as it would appear with its axis lying in the direction (A or ce^ Fig. 9. Ex. 149. Finally, Fig. 11, is an oblique projection of a pipe 70 FBEE-HAND GEOMEl^BIOAL DRAWING. similar to the last. Tlie figure is more easily drawn, since the circular ends are circular in the figure. Ex. 150-153. Draw Figs! 7, 8, 9, and 10 in oblique projection. By careful comparison of Figs. 5 to 12 with each other, and with 2, 3, 4 as standards, the pupil can learn to make plan and elevation, isometrical, and oblique projection sketches of many common objects; such as boxes with compartments, etc., etc. At this point, the pupil can also profitably, with reference to the exercises to follow, practice the drawing of various curved objects in isometrical and oblique projection. Such objects may be a Cylinder, a Cone, the Frustum of a cone, Rings, both of square and circular cross-section. Vases, etc. These may be taken, first, separately, and then in combination ; placing them first in the simplest positions, and thence advancing to more irregular positions.* The drawings in these cases should be on plates twice as large as those of this book, and the models should be large, from ten to sixteen inches high. Figs. 3 and 5, PI. X., are examples of the application of oblique projection to the drawing of models of problems in space, either on the black-board, or for book illustration. Ex. 154. Fig. 3 represents two planes, H, horizontal, and V, vertical, so placed that V is in the position of the front faces of the blocks shown in PI. VIII., Figs. 11-14. ABO is then a square cornered block, whose front face is parallel to V, and top, ABD, parallel to H. The figure ahd^ equal to ABD and directly under it, on H, is then called the '' horizontal projection," or ^^plan " of the Mock. Likewise the figure a'Vc' ^ equal to ABC, and directly be- hind it, on V,is called the "vertical projection," or ^^ elevation^^ of the block. See PI. IX., Fig. 2. Ex. 155. PI. X., Fig. 5, represents similar planes, H, and V, in the positions parallel to the right hand and top faces of the blocks, shown in Figs. 11 to 14 of PI. IX., and viewed as in- dicated in Fig. 13. Then PQP' represents a plane, as of paper or tin, meeting H in the line (" trace ") PQ, and meet- * See note on p. 58. PICTORIAL PROJECTION SKETCHING. 71 iiig V in the line P'Q. This plane is then pierced at C (whose projections on H and V are o and c') by the line AB, whose projections are ab and a^b\ Figures like these are highly useful in representing combi- nations of forms in space, such as would not be so readily intelligible from ordinary diagrams. Most of the figures in " solid geometry" can advantageously be drawn in this way. The remaining figures of PI. X. give examples, but of de- tails, only, and on a suitable scale for practice, of the stricoture and machine sketching^ which is one of the several practical applications contemplated in the more elementary drill afforded in the earlier chapters of Part I. Fig. 1, as indicated by the braces, shows three views of the assemblage of parts at one of the joints in the floor of a kind of bridge. Ex. 156. The lower figure is a sketch, as seen in a side view of a bridge, of the structure of the floor at the point where one of the main cross-beams is suspended by iron rods, as R, from the supporting fi-ame, or truss above. This beam is composed of the four pieces whose ends are shown. The lateral ones, A, B, bolted to the deeper central ones, form rests for the floor- joists, rrij which lie lengthwise of the bridge and support the floor planks^. The irregular cast iron block below, is shaped with wings (3" x 8'') wiiich support the iron links LL, which tie the bridge together like a bow-string ; and form a bearing below for the nut on the lower end of the suspension rod R. Directly over this, the " plan " shows only the chain links L and the under side of the cast iron block, C. At the right of the plan is shown the view of the several parts as seen in looking at the end of the bridge. This figure should be viewed, looking to the right, while facing the left hand end of the plate. The same parts bear the same letters in the three views embraced in the figure. Views of different sides of the same thing should be drawn on the sa/nie scale, and placed on the same level, when not ar- ranged as in Fig. 1. The eye can then pass readily from one view to the other, and trace the corresponding views of the same parts. Ex. 157. PI. X., Fig. 2 shows a method of firmly binding 72 FBBE-HAND GEOMETBIOAL DBA WING. together two timbers, M and N, at right angles, without cut- ting either of them. This is here done by harpoon bolts, one of which is A, A', where A' is a view at right angles to that shown at A, and shows that the upper end of the bolt is beaten out flat. The bolts pass through N, the hook hh! gives them a hold on M, which is secured by the small cross bolt, J. Ex. 158. PI. X., Fig. 4 shows a plan, or top view of a joint of the iron truss of the same bridge to which Fig. 1 belongs, and at the upper end of one of the rods, as R, Fig. 1. Ex. 169. Fig. 6 shows the stout casting, called a shoe^ which receives the ends of the last pair of tie rods, T, lying in the direction of LL, Fig. 1 ; also the foot of the iron truss. Ex. 160. Fig. 7 is a plan and elevation of a " shaft coup- ling," or, one form of the many contrivances for locking to- gether two pieces, A and B, of a line of shafting. In this example, this is done by means of a stout collar, C, in two pieces bolted together, and a key, ^, partly lot into the shafts and partly into the collar. On the preceding figures, the measurements indicate by arrow-heads the distances or dimensions to which they refer, and denote feet by single accents, and inches by two accents. Thus, 3' : 6" is read, three feet and six inches. Guided by these examples, the learner should exercise him- self in making neat pencil sketches, after the manner of those on Plates IX. and X., but larger, of such mechanical objects as are accessible, such as railroad chairs, frogs, and switches ; grindstones, hay-cutters, bridge joints ; roof -framings, as found in barns, attics, etc. These sketches should be large enough to exhibit the small- est parts, and contain the recorded measurements without con- fusion, or obscurity. By practice in thus carefully shetching^ and neatly and com- pletely measuring the parts of any structures accessible to him, of wood or iron, or masonry work, or machines, or parts of them, the learner will not only learn to make such sketches readily and neatly, which will often be a serviceable accom- plishment, but will, by degrees, collect an album of valuable examples of construction, the exact knowledge of which may be useful. PUVll tog» ting of V 8ho\ oat' holfl E oft and E. reoe direi E- ling, geth exan piec aud O arro' and ThiM Gi aelfi on I as ar grin, in ba Til est pi ftiBio: By PL.IX 73 ting of • 8flO> out hr.1 I oe • and rec* dire B linr get ex»* pie* and C am and Th" O 1 PL.X,-. 1 L. 01 « PART III. B3X.K]\CESNT?S OF GS-B^OMIKTRIO BKATJTY. CHAPTER I. ELEMENTARY IDEAS. UNITY, VARIETY, FREEDOM. 1. The human mind everywhere contains among its posses- sions, the idea of heauty. Thus, it is familiar to every one that there are a multitude of objects which give us great pleasure, independently of any practical use that they have. That quality of objects, by rea- son of which they afford this pleasure, is called their heauty, 2. It may not be possible yet to give a definition of beauty that shall include every possible case. But it seems highly probable that it consists in expressiveness of good^ where good is defined as perfection of structure, or heing ; of action, or doing ; and of consequent acquisition, possession, or having. 3. Examples. — The beauty of a goblet, as distinguished from a tin cup — though the latter, being more durable and capable of many more uses, has far more utility — lies in part in its ex- pressiveness of the fitness of things, which is a kind of good. That is, water being transparent and colorless, yet sparkling, there is beauty in the idea of drinking it from a material like itself in these respects, from a vessel which seems as if made of water that had become permanently solid. Again, the beauty of a statue of perfect youth, so life-like that it makes even marble seem flexible, consists in its clear expres- siveness of the suppleness, readiness, abounding life, and varied capability which are so many perfections of being. Also, in its expressiveness of the mastery of mind over matter, shown in the genius of the sculptor. 4 74 FBEE-HAND GEOMETRICAL DRAWING. Once more, the beauty of a greyhound consists in the evident expressiveness, in every line and conformation of his body, of the word, " go," whispered in his ear at his creation, as seen in the evident delight with which he runs a race with every fleet horse that passes his gate. 4. But tltere are many orders ofheauty^ corresponding to as many departments of thought. One of these relates to beauty oiform. Beauty of form is of two kinds; the one free, flexible, mobile; the other rigid, precise, fixed. The beauty of a flower, or a statue, is of the former kind ; that of a building or a pavement of geometrical inlaid work, is of the latter kind. A statue is placed under the first head, not because it is flexible, but because it represents flexible beauty, and, in pro- portion to its perfection, represents it so perfectly as to seem flexible ; so that the climax of the sculptor's art lies in such completeness of triumph of mind over matter as to make rigid material seem flexible. 5. Of these two species of beauty, one, as we have said, relates to regular, or as it may therefore be called, geometric heauty. This alone concerns us now, as appropriately connected with a course oi free-hand geometrical drawing, auxiliary to a course of instrumental geometrical drawing. Now the exactness of everything geometrical renders it cer- tain that, if definite principles and resulting rules can be found anywhere, by following which beautiful forms can certainly be produced, it will be in the department of geometric beauty. There is reason to suppose that the ancient Greeks possessed and used such rules, unless their marvellous genius for such beauty, as well as for purely free or non-geometrical beauty, made them infallibly, though unconsciously, conform to them. 6. Let us next examine this idea of heauty and seek to find some of its parts or elements. Unity. — Passing by a large vacant city lot at a certain time, we may notice in process of collection a large amount of scat- tered stone — ^iron — lumber — ^lime. Passing the same spot a year or two later, we find that aU ELEMENTARY IDEAS. UNITY, VARIETY, FREEDOM. 75 these materials, with other finer ones, have been combined to form a grand temple of art, science, or religion. The human mind, of itself, feels and knows a difference between these two cases, and expresses the difference in appro- priate speech. The first is an assemblage of materials for a purpose not yet realized. This assemblage is, to the mind, no one thing, and takes no distinctive name as any one thing. The finished building embraces the same materials, but com- bined for one purpose. This one purpose, governs the orderly arrangement of the before scattered materials, and makes each piece contribute in some way towards the fulfilment of that purpose. This fact makes the result one thing to the mind, to which one name can therefore be given — a museum, or a church. Hence we say, that unity is one of the primary ideas of the human mind, and that a principle of unity pervades and binds together things otherwise thought of as separate. 7. Illustrations. — It is this same idea which makes the dif ference between anarchy and civil order ; between chaos and creation, and which makes the mind always conceive of the sum of all things as being really one thing, because the product of one mind, for some one all-embracing purpose, and hence called the universe. Finally, there is no stronger proof of the permanent reality of this idea of unity, than the existence, always and everywhere, of systems of philosophy. For philosophy, as is plain from many definitions of it, is an attempt to discover the central or initial thought, purpose, or idea, from which all things visible and invisible spring. In other words, it is an attempt to see the universe as from its centre ; in a word, to place ourselves in the position of Deity, with the intelligence of Deity — as Plato said 2,000 years ago, " a resembling of the Deity so far as that is possible to man." 8. Kinds of Unity. Uniformity, — But unfiy is of two kinds, simple and compound. To continue a former illustra- tion, one brick, one board, in the building is a simple unit, as being of uniform substance througliout, and not formed by put- ting together separate pieces. On the other hand, the entire building is a unit, because made for one purpose, and evidently 76 FSEB-HAND GEOMBTBICAL DRAWING. made for one purpose, in that every part of it contributes to- wards the attainment of that one purpose. Yet it is a highly compound unit, because composed of many separate parts. When the separate parts are like the whole, or when, if the nature of the case renders this impossible, the like components are equal and similarly placed, the principle of uniformity enters, and the unity, though still compound, is simplified, or more nearly approaches simple unity. Hence we have not only unity, but unity in variety^ with, or without uniformity also, as an element in the idea of beauty. As each part co-operates with the others to form the unity of the whole, we will call this compound unity, Tvai^mony. 9. Freedom. — But there is variety in a higher sense. There may be twenty great buildings all built for the same purpose. Yet they may all be well adapted to that purpose and hence beautiful, though built in twenty different waj's, or from the plans of twenty different designers, none of whom ever saw the plans of any of the others. This kind of variety may be a part of what is meant when writers on Art speak of the " freedom of the domain of art " as compared with the precise rules by which we are bound in mathematical operations ; and when they speak of the freedom of the mind over matter, when the mind seeks to express its ideas and purposes by material forms. This variety indicates the principle of freedom^ since it re- sults from such action as is most imconscious of conformity to rulesj and is most diflScult, if not impossible, to bring under the operation of rules. 10. Primary relation of fundamental proportions to acces- aories, — In analyzing the variety just described, nothing is more familiar than the habit of distinguishing between the propor- tions and the decorations of a structure. And no principle is, or should be, more familiar than that no kind or amount of de- coration can conceal or compensate for deformity of the naked proportions, skeleton, or framework, which supports those de- corations. Now the very idea of proportions, is, that they are something of a definite geometrical character, having precisely measura- ble, or numerical relations between them. This being true, it follows that the skeleton — that is, the figure composed of the ELEMENTARY IDEAS. UNITY, VARIETY, FREEDOM. 77 principal lines of any structure, or other object, should form a geometrical figure governed by the laws of agreeable geometri- cal proportion ; supposing, as in Art. 4, that it is possible to discover these laws. 11. Summary, — We are now prepared to enter upon the study of geometrical beauty, guided by the three principles of Unity ; Harmony ; ^xA FreedoTn^ such as maybe exercised upon an underlying frame or skeleton having beautiful geome- trical jyroportions. These terms, harmony and freedom, are not inconsistent, as should here be understood, with the terms, symmetry and com- bination, before used (see Part I., Chap. VIL) ; for symmetry is but one species of harmony, and freedom consists partly in the multitude of combinations which can be made from the same elements. CHAPTER 11. NUMERICAL EXPRESSION OF THE ELEMENTARY IDEAS. 12. ViTRUVius is the one ancient writer on architecture and its details, to whom modem writers refer ; and his knowledge of the principles of Greek art seems to have been traditionary and incomplete. Yet he says, " The several parts which con- stitute a temple ought to be subject to the laws of symmetry / the principles of which ought to be familiar to all who profess the science of architecture. Proportion is the commensnra- tion of the various constituent parts with the whole, in which symmetry consists." And he then describes the details of the proportions of the human body, as being the most beautiful created thing ; and says " the laws of symmetry were derived by the artists of antiquity " (the Greeks of whom he wrote) " from the proportions of the human body." 13. But how these laws of symmetry were derived from the human body he does not show. Nevertheless we have the following clue to their possible discovery. 1°. The eye and the ear are sometimes spoken of as the nobler senses, the especial senses of the soul. 2°. There are accordingly provided, to satisfy these senses, heauty of sound for the ear^ and beauty of form, for the eye. 3°. The laws of concordant sounds, harmonious to the ear, are well known. If then, taking the human body as an illus- tration of the utmost beauty of form, its proportions should be found subject to the same laws as those of harmonious sounds, the principles of unity and harmony already explained would make it seem highly probable, that the true principles of exact or geometric beauty oiform were the same as those of exact or harmonic beauty of sound. For, moreover, the principle of freedom. f as well as of mathematical precision, enters the do- main of both eye and ear. This is seen in the beauty of oratory NUMERICAL EXPBES8I0N OF THE ELEMENTARY IDEAS. V.l and elocution, as distinguished from music, properly so called, which corresponds, it may be, with the beauty of natural ob- jects as distinguished from geometrical ones ; or more exactly, in the fact that various pieces of music may yet be, in some evident manner, appropriate to the same words. 14. Direction^ the primary element of Form,, — The two elements ot form are direction and length. The following con- siderations seem to show that direction is the more fundamental of these. Firsts practical considerations. If the inquiries of a hun- dred travellers seeking their way to an unknown location were noted, it would probably be found that they would ask " which way," before they asked " how far." Again in describing a survey^ it has long been customary to describe the direction, called the bearing, of each line, before stating its length. Once more, in making any ornamentals design^ whether of regular or free outline — but especially in the latter case, as in sketching fruit and flower forms or scroll work — the mind is more occupied with the direction to be given to the pencil at each point of its progress, than to the size of the sketch to be produced. Second, But a precise geometrical reason may be found in the fact, that the triangle, the fundamental figure into which all others may be decomposed, may have an infinity of sizes, all of one shape, but cannot have any variety of form with a fixed length for each side. That is, in similar triangles, similarly placed, the correspond- ing sides may be of different length, but have the same direc- tion. 15. It may be objected that there can be many triangles of the same size or area, but of different forms, as well as manj^ of the same forms but of different sizes, and thus that the ideas of direction of sides, and length of sides are equally funda- mental. But it is to be noticed that the similar forms of dif- ferent sizes can be instantly perceived to be similar, while the equal sizes, of different forms, could only be known to be equal, by measurement and computation of their areas. Also, if one were trying to sketch a symmetrical, that is an 80 FBEE-HAND GEOMETRICAL DRAWING. isosceles triangle of pleasing form, he would do it by trying varions angles between the base and the adjacent sides, as in- dicated at A, Fig. 1, until he found a triangle of pleasing proportions ; rather than by trying various given lengths, as c and d, of those sides, placing them together by means of dividers, as indicated at BO and BD. Tr» 11 A ^ B Fig- ^« 16. Summary. — Distance and Directio7i, are two radical geometrical ideas. Distance lies at the foundation of sizCj as large or small. Direction lies at the foundation of form^ or the shape of things ; and it is the form of objects rather than their size, which determines their beauty. Hence direction appears as the root idea in geometrical forms. 17. An angle is difference of direction, or the measure of relative direction. Hence it seems natural to look for the principles of geometric beauty in the ratios between the angles of figures, rather than in those between the lengths of their sides. Some remarkable numerical properties of the circle, as the measure of angles at its centre, will confirm this view. 18. The circle has for ages been divided into 360 equal parts, called degrees, for the purposes of angular measurement. Whether the selection of, this number was the result of acci- dent, or experiment, or of abstract reasoning, may not now be known ; but its relation to the principles of unity, variety, and harmony, as expressed by numbers, is very striking. First. Its prime factors are 1. 2. 2. 2. 3. 3. 5 = 360. Now, 1°. These factors are the abstract unit, 1, and its first three prime multiples. NUMERICAL EXPRESSION OF THE ELEMENTARY IDEAS. 81 2°. Of these factors, 2, the ji/rst even numhei\ consists of the two equal halves 1 and 1. It is therefore expressive of the principle of uniformity, as in the division of a body into two equal halves. 3°. Next, 3, is \\iq first odd number. It can be separated only into the equal parts 1, 1, 1, or the unequal ones 1 and 2. It is thus the first and simplest numerical representative of the principle of variety. 4:°. Next, 5, is the second simple or prime multiple of 1, and the first and simplest which combines in itself the numerical representatives of uniformity and variety, 2, and 3. It is there- fore the simplest numerical representative of the combination of ^iniform,ity with variety. 19. The factor 2 is the foundation of the series 2, 4, 6, 8, etc., made by taking 1, 2, 3, etc., successively as multipliers. The f aetor 3 is likewise the foundation of tlie series 3, 6, 9, 12, etc. ; and 5, is likewise the first term of the series 5, 10, 15, etc. The numbers afforded by these series can, however, be better exhibited with regard to the details of their dependence on the piimary numbers 2, 3, 5, as follows, where the exponent of each primary number is the multiplier for the series begun by that number. 2', 4, 8, 16, etc. 2% 6, 18, etc. 2*, 10, 25, etc. 3", 6, 12, etc. S\ 9, 27, etc. 3', 15, 75, etc. 6', 10, 20, etc. 5', 15, 45, etc. 6'^ 25, 125, etc. 20. The factors of 360, properly combined by multiplication, yield all of the nine digits except 7, viz., 1 ; 2; 3 ; 2x2=4; 5; 2x3=6; 2x2x2=8; 3x3=9. The same numbers are also found in the foregoing series. But seven is peculiar, in containing none of the radical num- bers 2, 3, 5, as a factor, and as a sum, 3 -f- 2 -f- 2, it is redundant as compared with 5, in containing 2, twice. To be sure, 8 as a sum, 2 + 3 + 3 is redundant, though differently, but can be re- solved into 2's, it being 2 x 2 x 2, as 9 can into 3's, while 7 is impracticable. 4* 82 FREE-HAND GEOMETRICAL DRAWING. We conclude therefore that 2, 3, 5, are the primary numeri- cal representatives of nniformity and variety, and of their com- bination without superfluity ; and thence that 360 may have been chosen to express the divisions of a circle on account of its containing all the digits but 7.* 21. It is interesting to note in passing, that the remaining digit, 6, is the fivBt perfect number^ that is, one which is equal both to the sum^ and to \!ciQ product^ l-f-24-3, and 1 x 2 x 3, of its factors, and thus numerically represents, both as a sum and a product, the principles of unity, uniformity and variety and their combination. 22. Having thus arrived at the significance of the numbers 2, 3 and 5, the number of ways and the manner in which 360 contains them is quite surprising, as is distinctly shown to the eye, in the annexed table. 360=2x2x2x45. (1) 360=3x3... x40. (2) 360=5 x72. (3) 72=2x2x2x9. (3) 45=3x3... X 5. (1) 40=5 x8. (2) 8=2x2x2x1. (2) 9=3x3... xl. (3) 5=5 xl. (1) That is : Firsts 360 contains, as before seen, 2 thrice as a fac- tor; 3, twice and 5 once, with quotients, after successively di- viding out all these factors, of 45, 40, and 72, in the three cases respectively. Second^ these quotients contain the same factors in like manner, as shown in the second group. Thirds the like is true again, taking the quotients of the second group in the order seen in the third group, where the final quotients are each, 1. In this curious result, we see again, exhibited in numbers, the principle of uniformity. But the order in which the first two groups of quotients are used as the next dividends, is found as indicated by the paren- * The number 7 is also excluded generally in the formation of musical ratios, but is said to be employed in Chinese music, and it enters into the composition of some peculiar theoretical systems, not in actual use. /^ N / \ \ f \ 1 ^*S^ f \ y^ "'^^^ / i< '^ P >^ V "^^- — '' NUMERICAL EXPEE88ION OF THE ELEMENTARY IDEAS. 83 theses and the adjoining circle, by combining 1, 2, 3 in every possible order taken in rotation, in the direction of the arrow. Here again we have likewise the principle of variety. 23. Ratio the principle of combination, — Numbers can be compared in two ways, by difference and by ratio ; diflFerence being obtained by subtracting one number from another; and ratio y\>y dividing one by the other. Aristotle (as quoted by Hay) defines harmony as " the union of contrary principles " (as those of uniformity and variety) " having a ratio to each other." \nhe2Mt\i\3\ forms ^proportions constitute harmony; and Vitruvius defines harmony as " the commensuration of the various constituent parts with the whole ; " that is, each part bears a certain ratio to the whole. 24. The reason why ratio^ rather than difference^ should be the combining principle of parts into a whole, seems to be that difference belongs to the domain of things^ and ratio ^ to that of pure thought. That is, this is so in this sense, that things of a kind can be added or subtracted forming a greater or less number of things of the same kind, as 6 pounds + 3 pounds ai'e 8 pounds ; 10 feet — 2 feet are 8 feet, etc., processes which seem to imply after-thought, or the putting together, or put- ting apart of things complete in themselves. But, on the other hand, we cannot multiply 5 pounds by 3 pounds, or divide 10 feet by 2 feet, and get any real result. We can however multiply 5 pounds by the abstract number, 3, giving 15 pounds, and divide 10 feet by the abstract num- ber, 2, giving 5 feet. Addition thus seems related to the miscellaneous assem- bling of the materials supposed in Art. 6, but ratio to their combination, primarily in the mind, and then realized by the bands, in a thought-out system^ which is a unity, and not an assemblage ; though a compound unit, in which however each component contributes in its proper degree to the intended use of the wholej as when the windows of a building are suflScient for its light, and its entrance doors suflScient for ingress and egress. CHAPTER III. OBNESAL APFLICATIONS OF THE IDEA OF BEAUTY IN BATIOS. Analogy of Form and Sound. 25. Linear and superficial heauty. Since ratios, rather than differences, determine harmonious proportions, beauty arising from marked divisions of a line, will consist in ratios between the parts of the line, or between the parts and the whole; ratios which, according to the last chapter, are derived from the numbers 1, 2, 3 and 5. Such ratios will be furnished to any desired extent by the several series given in Art. 19. Again, coupling the principle of ratios, with that of direc- tion, as more fundamental than distance, the ratios between the angles of a plane figure would determine the beauty of its proportions, rather than the ratios between the lengths of its sides. It only remains then to determine the natural unit of angular measure. 26. I'he angular unit. This unit must be simply an angle that is naturally, not arbitrarily fixed. Recurring then to the figures in Part I., Chapter II., p. 7, we see that when kg is moved either way from its position perpendicular to mkj the angle on one side of it is acute, and that on the other, obtuse. Acute angles may vary in size, infinitely between 0° and 90°, and obtuse angles may vary indefinitely between 90° and 180°. But the equal angles of 90° on each side of hg^ when it is per- pendicular to m^, are the limit between all acute and all obtuse angles. That is, the right angle, being of necessarily fixed size, is the standard of comparison for all other angles. 27. Beginning now with known principles, from which to proceed to the unknown, the first illustration of beauty sensi- bly derived from the divisions of a line, sh.iU be the beauty of GENEBAL APPLICATIONS OF THE mSA OF BEAUTY IN EATIOB. 85 soundj occasioned by the notes given out by the divisions of a vibrating string. By showing that the long and well established laws of musi- cal harmony, or beauty of sounds are based upon the numbei*s 2, 3, 5, and their multiples, it will be more readily apparent that the abstract principles of beauty already described in con- nection with these numbers, are also as truly the bases of geo- metHcal harmony, or beauty of geometrical for^n. That the musical terms employed may be better understood, a sketch of a portion of a piano or organ keyboard is here given. Fig 2. B^ Q P E F C A B C, D, E, F, q, A| 28. A note occasioned by a certain number of vibrations per second (fixed upon by agreement), whether of a stretched string, as in a viol, harp, or piano, or of a column of air, as in an organ pipe, is adopted as a standard of comparison, and designated by the letter C. A string one-half as long. Fig. 3, vibrates twice as fast, and gives a note designated as C„ which is described as an octave above C. In other words, the difFer- ence in sound between these notes is called the interval of an octave. h '/x Fig^S Again : f of the length of the string will vibrate three times while the whole string vibrates twice^ that is f times while the whole string vibrates once^ and will yield the note designated as G, and described as being at an interval of a fifth above C. 29. By continuing to take different simple fractional parts 86 FSBE-HAND GBOMETBIOAL DBA WING. of the string, based, according to preceding principles, on the numbers 2, 3, 5 and their multiples, we find the results given in the following table. The jlrBt column contains the order of the notes and the letters which designate them ; the second, the number of vibra- tions made in producing each note, during one of the vibrations made in producing the note C ; the third, the number of vibra- tions made in producing each note, beginning with D, during one of the vibrations belonging to the next preceding note / the fourth, the name of the interval between each note and the first / i\\Q fifth, the name of the interval between each note after the first and the next preceding, these intervals being those which are expressed numerically by the ratios of the second and third columns respectively. The ratios in the second column are ob- tained experimentally. Those in the third are found as follows. Taking for illustration G and A, if we call f =1, what would \ become \ Ans. f : 1 : : | : lx| = \ x f = V"- f I. 11. III. IV V. 1. C 1 1 1st Ist, or unison. 2. D I I 2d Major 2d. 3. E I i/ 3d Minor 2d. 4. F ^ -^ 4th Diatonic semi-tone. 5. G I I 5th Major 2d. 6. A I ^ 6th Minor 2d. 7. B J/ I 7th Major 2d. 8. 0, 2 ^ 8th Diatonic semi-tone. 30. Illustration of further application of ratios, founded directly or indirectly on the primary numbers 2, 3, 5. The intervals from to E, from F to A, and from G to B each consisting of a major and a minor 2d — and = \ — are called major thirds. Those from E to G, from A to C, and from B to the next higher D, each consist of a major 2d, and a diatonic semi-tone, and are each = -J. Thus G, f-r-E, |- = |x^=|. These intervals are called minor thirds. Again, |-h-|- = f-J-, an interval called a chromatic semirtone. Once more, F, f-f-D, | = f x| = if and ff-=-| = |f x^ = GENERAL APPLICATIONS OF THE IDEA OF BEAUTY IN BATIOS. 87 Also, 1 : 1 :: V : V-^f = ¥ x I = f ?• This interval, f^, between the odd interval (D — F) and a minor third, or be- tween a major and a minor 2d (remembering that f^ means 80 vibrations belonging to one note, simultaneously with 81 vibra- tions belonging to another note) is called a comma. 31. Sharps and Flats. — ^A note so much higher than a given note that 25 of the vibrations which make it take place while 24: of those which make the given note occur, is called the sharp of that note, and is marked Jf. Also a note so much lower than a given note that 24 of the vibrations which make it occur dur- ing every 25 of those^which make- the given note, is called the fiat of that note, and is marked JZ. Batio. Thus we have, C — cS — If Djz — H D I H Elz |v I E i^ Continuing this process for every note of the scale, we shall find, in place of the eight notes, C to Ci inclusive, twenty-one notes, with an exceedingly complicated variety of intervals, arising from the various combinations of major and minor tones and the diatonic and chromatic semi-tones. 32. Keys and their muturt7W^7^ rectangles are those made by placing the halves of these triangles together by their hypothenuses. These rectangles will be the square ; the rectangle whose diagonal makes angles of 30° and 60° with its sides; and the rectangle whose diagonal makes angles of 18° and 72° with its sides. Second. We have found (Arts. 40, 41) that from these pri- mary triangles others are derived, and from these, in turn, a cor- responding series of rectangles proceed. 48. It is interesting next to note that most of the regular polygons, yield simple harmonic ratios, by means of their sub- division into equal triangles, by radii from their centres to their comers. 49. Thus the equilateral triangle, Fig. 10, divides into three Elg^ lOi Fig. ^». triangles of cO® and 120®, giving the ratios { and J. The square, Fig. 11, divides into triangles of 45° and 90°, giving the GEOMETKIO BEAUTY OF POLYGONS. GEOMETBICAL DESIGN. 97 i-atios |- and ^. The regular pentagon, Fig. 12, is composed of triangles of 54° and 72°, giving the ratio f . D>N Ex. 5. In like manner note the angles and ratios afforded by the other reg- ular polygons, np to the dodecagon or polygon of twelve sides. 50. Proceeding with polygons, as before with rectangles, the only equal and regular polygons which can combine without leaving unfilled spaces between them, are the equilateral tri- angle, square and hexagon. Indeed, as the hexagon is itself composed of six equal equilateral triangles, we might say that the equilateral triangle and the square are the only indepen- dent figures that can so combine. Only, the equilateral tri- angle can combine in other ways than in hexagonal groups, so that, practically we may admit the three figures as separate. 51. While combined rectangles constitute the moi-e essential or useful members of many objects, combinations of various regular polygons, or other polygonal pieces founded upon them, mav be made, as in PI. XL Ex. 6. PI. XL, Fig. 1, a five pointed star. Ex. 7. PI. XI., Fig. 2, a dover-leaf pattern, where the centres of the circu- lar compartments are the yertices, a, d, c^ of the equilateral triangle (ibc, Ex. 8. PL XI., Fig. 3, an equally four armed cross. Ex. 9. P]. XL, Fig. 4, an eight pointed star. 52. An immense number of decorative designs, wholly or mostly geometrical, can be based upon th-Qsquare^ divided asia 5 98 FREE-HAND OEOMETRICAL DBAWING. PL XI., Fig. 3, into nine equal squares, or, as in Fig. 5, into sixteen / the latter division being founded on the number 2, the former on the number 3. 52. In many current systems, the lines composing these designs would be located by considerations of distance^ in some obvious systematic manner. But according to the principles of direo- Hon, and of simple ratios between angles^ as properly govern- ing geometrical design, these lines should have simple angvZar relations to each other. Thus, wishing, in PI. XI., Fig. 3, to place a four-pointed star behind the cross, lines may be drawn from each corner of the large square, as at A, where the lines run to B and C, comers of the furthest arms of the cross. Or they may be drawn, as at D, to cornei-s, E and F of the central small square, or again, so as to divide the distance C E or C H, in any given manner. These methods may seem sufficient, be- cause, after a fashion, they are definite and systematic, though the angles at A and D have no simple ratios to each other or to an angle of 90°. 63. But it happens that definiteness and system can be had in another, and, according to the principles before established, a better way. Thus, the lines at d make angles of 30° with each other, giving, with the right angle at flj, the ratio J. Again, the star lines at a include an angle of 45°, thus forming with 90° the ratio ^, while the external angles at a and c are of 22° : 30' and 67° : 30' respectively, giving the ratios with 45° of \ and f , and with each other of \. The point at a is less clumsy than that at A ; that at d is more decidedly acute than that at D, if de- cision as to the acuteness be wanted ; or, if something nearly like the point D be desired, with harmonic angular ratios, it can be had by substituting 36° for the angle of 30° at d. This will give angles of 27° each side of it at rf, and thus the ratios Here it is especially interesting to note, that if the star-point at A be considered well proportioned for a stout one, it is not so, on account of its principle of construction, but because, as shown by calculation, its angle is very nearly one of 54°, which bears the simple ratio, \ to 90°. Likewise if D be preferred to c?, the secret of its superiority lies not in the manner of drawing its sides, but in the fact that its angle is very nearly one of 36°, which would bear to 90° the simple ratio, \ ; and would divide GEOMETBIO BEAUTY OF POLYGONS. GEOMETMOAL DESIGN. 99 the right angle at D into the varied parts, 27°, 36°, 27°, instead of the three uniform ones, of 30° as at d. 54. Again, PI. XI., Fig. 5, may serve as a guide to many designs based on a sixteenfold division of the primitive enclos- ing square, and made on the principle of simple ratios among the angles. The angles of the corner points are 36°, giving adjacent and alternate angles of 27°, and thence a ratio of f , and the ratio of 36 to 90 or f. The outer angles of the inter- mediate points are 60°, and their inner ones 90°, affording the ratio f . The obtuse lateral angles of the same points are 105° each, thus introducing, and with evident good effect, ratios of ^5^ and -^^ or ^ and f in which the hitherto excluded num- ber, 7, appears. (See Note on Art. 20.) The angles of 120° and 153°, adjacent to each other, give the ratio ^, so near ^ = f that it may be called a tempered |, while the three angles of 105, 120, 135 give the ratios }, i, and f. Assembling now the ratios found, in progression, we have — i- f- 1- i-^=T-^ *- f- ■i i- h in which we miss only ^ from the continuous series, while we have in its place i^, bearing to \ the previous ratio | ; and ^, bearing to | the previous ratio |. The modification of PL XI., Fig. 5, shown in Fig. 13, is pri- marily afforded by enlarging the small corner squares, until they are exactly embraced, as at d^ and c, by the sides of the 100 FBEE-HAKD GBOMBTBIOAL DBA WING. comer points. The arrangement thence suggests other lineSj which may or may not be pi*eferred to the less elaborate Fig. 5, on PL XL Ex. 10-13. Draw PL XI., Fig. 3, uniformly; H. XL, Fig. 5; and com- plete Fig. 13 aa here begun. 55. Ilere again, having fully stated and explained the guid- ing principles, we will, in place of an extended series of copies, all essentially alike in being founded on fanciful relations of distance, ask the pupil to exercise himself fully on tlie follow- ing— Ex. 13. General Example. — Prepare several groups of squares, with three slightly separated squares in each group. Divide one square of each group into/<9t^r equal squares, another into nine, and the other into sixteen ; in order to secure a regular or symmetrical figure in each case. Then, as in PI. XI. „ Fig. 3, place in like compartments of each square some combination of straight lines, located with reference to simple ratios between the angles which they make with each other, and between these angles and a right angle. 56. PI. XL, Fig. 6, shows how pentagons combine, leaving rhombus-formed openings between them. But the angles have simple harmonic relations, and give an agreeable figure. The angles of the pentagons being 108° each, the acute angles of the rhombuses are 36® each, giving the ratio \, and the obtuse ones are 144°, with which 36® makes the ratio J, and with which 108® makes the ratio f . 67. Further geometrical decoration of each pentagon may be made in various ways, which will readily suggest themselves, under the guidance of our uniform principle of simple angular ratios. Thus, by drawing all of the diagonals joining alternate points of a pentagon, a five-pointed star will be formed, PI. XI., Fig. 1, having a pentagon for its central body, on which stand the points. It is interesting to notice also, that the tri- angle, ABC, of 36°, 72°, 72°, and described in Art. 39 as re- presenting the third order of geometric beauty, founded upon the number 5, is not arbitrarily so described, since it is natur- ally derived from the regular pentagon, which by its five equal sides and angles, represents in geometry the numeral five in arithmetic. 58. Kegalar hexagons combine without leaving vacancies be- GEOMETRIC BEAUTY OF POLYGONS. GEOMETBIOAL DESIGN. 101 tween them, as will be readily seen on trial. They may be easily drawn by means of the equilateral triangles found by dividing the sides of a large equilateral triangle into the same number of equal parts, and drawing parallels to the sides, through the points of division. Ex. 14. Oonstruct a group of eqnal regular hexagoiui. Octagons, whether regular or with alternate sides smaller than the intermediate ones, combine so as to leave square spaces between them, PI. XI., Fig. 7. Here the angles of 90° and 135° at any of the corners, give the simple harmonic ratio f. The size of the comer squares. Fig. 7, will be properly deter- mined by harmonic division of the sides of the original squares from which the octagons are formed. That is, the corners of the small squares should (25, 35) divide the sides of the original squares into parts having simple ratios to each other. We will here leave rectilinear combinations, having given the guiding principles and illustrations which may enable the learner to make any designs founded upon rectangles and re- galar polygons, so that they shall possess geometric beauty of form. 59. The principal field for rectangular work will be found in the main divisions of buildings. That of various triangular and polygonal work will consist in subordinate features, bay- windows, summer houses, etc., and in geometrical decorations, such as wood or tile inlaid work, and other geometrical surface decoration. Ex. 15. Gonstmct PI. XI., Fig. 6, and inscribe in each pentagon a five- pointed star, as in Fig. 1. Also make the central small pentagons, as dbc^ Tig. 1, black, and the star points lightly shaded. £x. 16. Gonstmct PI. XL , Fig. 7, with any additional geometrical decor- ation. Ex. 17. In Ex. 15, make the comer squares smaUer, and a square on each side of each. A group of five equal small squares vdU thus be uniformly placed at each comer of the octagon, and oblong hexagons will be included by the combined squares and octagons. Ex. 18. Construct PI. XL, Fig. 8, as it is, then inverted, then turned right for left, and then inverted again ; making it two or three times as large as shown. This exercise, often repeated, both on paper and blackboard, will greatly aid in aoourately estimating all the most important angles occurring in geometrical decoration, and in various positions. CHAPTER VI. OtIBVILINEAB GEOMBTBIO BEAtTTT. Circles and Ellipses. 60. The principle of unity, already defined (Arts. 6-8), to- gether with that of temperament (34), will here be remarkably exemplified by considering curves, not separately from the pi-e- ceding rectilinear figures, but as combined with them. 61. The circle is a curve, all of whose points are at a uni- form distance, called the radius, from a fixed point within it, called its centre. The ellipse is a curve, such that the sum of the distances of each of its points from two fixed points within it is uniform, and equal to the longest line which can be drawn in the curve. These fixed points are called the fod of the curve. According to this definition, the curve, Fig. 14, may be de- scribed by a point P, moving so that the sum of its distances B from two foci, F and F', is always the same, and equal to the longest chord, or transverse axis, AB, of the curve. Many differently shaped ellipses, wide or narrow, may thus be formed according to the distance apart of F and F', while AB remains of fixed length. Pins being fixed at F and F', and a firm OUBTIUBEAB thread of the length PFF'P being placed around them, and a pencil point at P, this point, moved so as to keep the string Btratched, will trace the ellipse. CD, perpendicular to AIJ at its middle point, O, is the shorter, mitior, or conjugate axis. 62. Harmonic relatione qfthe triangle, square, and circle. — Fig. 15, represents a square, 2, 6, 10, 14, each of whose sides is divided into four equal parts. Lines parallel to adjacent sides of the square, through these points of division, divide the area of the sqoare into sixteen eqnal sqnare parts, as tlie peri- meter is already divided into sixteen linear parts. If, now, a circle be inscribed iu this square, the lines just described will divide its circumference into twelve equal parts, as numbered in the figni-e, and radii from the points of divi- 101 FBBE-nAND GEOMETRICAL DBAWDSG. sion will divide the area of the circle into twelve equal parts. Here we have the ratio f between both the linear and area divisions. The inscribed figures are also remarkable. Remembering that an angle at the circumference of a circle is measured by half the arc of that circumference between its sides : 1°. The inscribed triangle, 3, 9, 0, and therefore, also, its half, 9, Cj 0, is one of 45*^, 45°, 90°, that is, one of the first order (37). 2°. The triangle 8, 4, is an equilateral triangle, whose half, 8, rf, 0, is therefore one of 30°, 60°, 90°, that is, one of the second order (38). 3°. The triangle 10, 2, is one of 30°, 30°, 120°, whose half, 10, J, 0, is therefore again one of 30°, 60°, 90°. The foregoing are the most remarkable, in connection with the subsequent figures, though it is interesting to note the fol- lowing, also. 4°. The triangle 11, 1, is one of 15°, 15°, 150°, whose half, 11, a, 0, is one of 15°, 75°, 90°, giving the ratios |, |, | (40). 5°. The triangle 7, 5, is one of 30°, 75°, 75°. whose half, 7 i? 0, therefore gives again, but as in (2°) with the longest base verti- cal, a triangle of 15°, 75°, 90°, whose ratios are \, -J-, |^, as before. Finally, drawing the chords 1, 11 and 5, 7, we have the rect- angle 1, 5, 7, 11, belonging to the second order of symmetry (38) in that its half, the triangle 1, 5, 7, is one of 30°, 60°, 90°. 63. Harmonic relations of ellipses. The ellipse, being bo to speak a somewhat monotonous curve, owing to its double sym- metry (Part I., Ch. VII.) which makes its four quarters alike, it is less valuable for many decorative purposes than the egg- 1 formed curves, which will be described further on. We shall therefore here treat this curve and its relations, less fully than Hay has done, but more completely, so far as the treatment goes ; and with much more elementary demon- strations of the properties noted. In Fig. 16, the rectangle 2', 6', 10', 14', is of the same pro- portions as the inscribed one, 1, 5, 7, 11, in Fig. 15, but larger, its longer side being equal to aside of the circumscribing square in Fig. 15, while its half 2', 6', 10', is a triangle of 30", 60°, 90°. CURVILINEAR GEOMETRIC BEAUTY. 105 By the definition of the ellipse (Art. 61), describe arcs with 4' and 12' (on the rectangle) as centres, and the half side, 4', 2', as a radius, and they will meet on the line 0' c' 8' at the foiei, F and F'. Having the foci, and axes, 0', 6', and 3', 9', the curve may be drawn by Art. (61), or otherwise, as most convenient. Fig. '6. So much being done, divide the sides of the rectangle, each into four equal parts, and join the points of division as in the square, which, as before, will divide the rectangle into sixteen equal parts, and the circumference of the ellipse into twelve parts, which, however, are not equal. 64 . We shall now fiiid^ hy applying a prot'^actor^ the fol- lowing remarkable results; naming the inscribed triangles in fche same order as for the square. 1°. The right angled triangle 9, c of 45°, 45°, 90°, is * A graduated semi-circle for measuring angles in degreeii. 6* 106 FBEB-HAND OEOKBTBIGAL DBAWING. transformed into the semi-equilateral triangle, 9' '7v*i — ^ -Arc;^ s^-- 4^3 t ^ \ r 'X 81. Results of the variations, — 1. The more acute the angles at a and J, the longer, and more nearly pointed, the oval be- comes. 2°. The greater the difference between the angles at a and 5, the greater is the difference in the form of the two ends of the oval. 3°. But when the shorter segment be of the base ah is in- scribed in the larger of the two unequal angles into which AOB may be divided, the more unequal these angles, the more acute is the oval at the end at the acute angle, and obtuse at the opposite end, as is successively more and more strikingly shown by the oval Oa o^ b^ in Figs. 23, 24, 25. 82. To make a flat oval— ohq broader than it is long — the 118 FBKE-HANB OEOMBTKIOZ./. DBAWING. / U^r jH — V -^-1 I \ 1 \ *^:r J M \ '\»^^^ Fig.g7. ]* I xA \ "^v L \ a.\ OTTBVILINEAB GEOMETBIO BEAUTY. OVALS. 119 base ab^ Figs- 29, 30, of the transfonned triangle, must be shorter than the greatest width, 2 O C, or 2 8 ^, of the oval. The axis, a^ 5i, of the oval, being thus brought nearer to O, the »3) 36 Fig. 29. j semi-circle A O B should be larger, to give an oval ot the desired size. 83. Only half ovals are shown, in order to avoid confusing 120 FBBB-HAND OEOMETBIOAL DBAWING.* the figures. They will exhibit themselves better by coi them ; the two halves on each side of a^ b^j or a^ ^29 ^^ alike in all the figures. With these explanations and illustrations, pupils an< cal designers will be put in possession of the means structing ovals of every possible variety of form, an^ upon definite angular relations. Any oval can then, if desired, be enlarged or diminij size, without altering its form, and without the labor of reconstructing it, by either of the methods explained I., PL II., Figs. 7, 8, 9.* InduatriaZ Applications. 84. T?ie industrial applications of the ovals now d< both to objects of utility and beauty, are so numerous want of space, we can only name some of them, in c< with the principles governing their design, and illusti few sketches. But more tlian this is unnecessary, for signer is furnished with the necessary principles to gui he is made independent of multiplied copies to be mei tated without further thought. Among the most important of such objects, are pottery and glass; as pitchers, bowls, vases, goblets, fruit gas-shades, etc. ; and of metal, as tea-sets, butter dish< lamps, urns, etc. ; also architectural ornaments, moi railings etc. ; and church, school, hall and house f urniti 85. Free application. — The ear is the best instrum( testing the musical beauty which results from the combi of musical chords, separately agreeable, into a piece of Likewise the eye is the best instrument for testing the * The advanced student, if acquainted with Descriptive Gkomel recognize the surface from which these ovals are cut as a right conoit directrices are the semi-circle ACB, and a vertical at O, equal to whose elements are all parallel to the paper. Besides the conoid, e^^ forms can be cut from the ** annular torus/ of circular or elliptical section, by planes parallel to its axis, and cut two curves. 4 f^L W W kr C , OUEVILINEAIt GEOMETETO BEAUTY. OVALS. 121 of compound forms, made by combining separately pleasing elementary forms. Moreover, as we have seen, very many ovals can inscribe the same harmonic triangle. Thus, in various ways, it is evident, that, in the higher forms of curvilinear beauty, we depart further and further from the domain of rules alone, and that the principle of freedom (Art. 11) or intuitive perception of beauty prevails; just as in music, a piece may be composed, strictly according to rules, and yet be utterly destitute of beauty, while another maybe very beautiful, and yet the secret of its beauty be inexpressible by any rule. Hence, for all purposes of application of the ovals here de- scribed, it is suflScient to direct the designer to construct a large number of them on card-board, classifying them according to the distinctions given in Art. (78), and then to combine them tangentially to each other, or intersecting each other also, as in PL XII., until the outline formed by their combination sat- isfies his eye ; or is pronounced beautiful by persons of taste, independently of each other, and, better still, unbiassed by a knowledge of the method by which the design was pro- duced. And here note, that angles formed by the meeting of curves, ' are, for purposes of comparison, to be estimated by the tan- gents to those curves at the point of intersection of the curves. 86. Application^ governed hy rules. The ovals here described are so graceful, that they combine together in graceful forms almost as readily as different plant leaves do in a bouquet. Still, if ornamental forms, like those of PI. XII., are to be de- signed in a strictly systematic manner^ instead of by merely satisfying the eye hy trialy three points may be kept in mind while making the design. First ; If different ovals are to be used in the same design, those may be chosen in which the angles of each of the inscribed triangles form harmonious, or simple ratios with those of the others. Second ; T j angles made by the axes of auxiliary ovals, with the vertical axis of the entire ligure, as in the neck and foot of Fig. 1, may have a simple ratio to 90°. Third ; According to Art. (35) the axis of the figure may bo 6 122 FREE-HAND GEOMETRICAL DRAWING. divided by the diflFerent members of the entire design, into segments having simple ratios to the whole height 87. The designs on PI. XIL, were mostly formed by the first method (85) that of satisfying the eye by triaL Yet the seg- ments of the heights are, generally, very nearly if not exactly simple ratios of the whole heights. These designs are not offered as models of oval composition, bnt only to indicate the manner of combining ovals in forming regular objects with curved outlines. Fig. 1, for example, is composed of the oval a, s^ J, of Fig. 25, for its body, with a part of a, «, J„ Fig. 23, for its neck, and of the end at a^ of a^ *, i„ Fig. 22, for its foot. From this begin- ning, innumerable minor modifications can be made by the pnpil or designer, by taking variously proportioned ovals for the body, and various portions of different ovals for the neck and foot, until the most satisfactory forms shall be attained. In like manner, each of the following examples may be vari- ouslv modified. 88. PI. XII., Fig. 2, represents a glass fruit dish, the body composed of the lower portion of Fig. 30, ending a little above its line c, ^„ of greatest width ; and the foot composed of an arc of a^s^b^, Fig. 24, in the vicinity of «,. Slight variations in the fashion of the body would make its brim just on the line, Cj (?i, of greatest width, or a little below that line. The top of the dish, being of less diameter than the body below, expresses reserve. When, as in the last modification mentioned, the top diameter is greatest, generous freedom is expressed. Or, leaving sentiment for utility, the first form is better adapted to carry fiuid contents without spilling, and the second, to supporting a pyramid of fruit. 89. PI. XIL, Fig. 3, represents a garden vase. The outline of the upper member is an arc of the oval in Fig. 22, from Oj towards a^, and is superior in two ways to an outline composed, as is sometimes done, of a straight line tangent to a circular arc. First, it is wholly curved. Second, the curvature con- stantly varies, instead of being uniform, that is, monotonous, as in a circular arc. The middle member is composed of the larger segment of a smaller oval of the same form as that of Fig. 29, and is much OTJEVILINEAB GEOMETRIC BEAUTY. OVALS. 123 finer than a simple semi-circle, though the difference is not great on the scale of the figure. The lower member is composed of various mouldings, all the cnrved portions of which would, when of full size, be composed of arcs at the tip of some of the more pointed ovals. 90. PI. XIL, Fig. 4, is a two-handled jug, in which, to secure flowing combined outlines, and an absence of straight lines, the form of the actual piece of pottery from which the figure was taken, was modified by giving a very slight curvature to the handles, taking for this purpose the straightest portion of the oval fl52 '^2 ^2 <^f F%- 24. Also the decision expressed by an exact right angle is secured by taking the top outline of the oval a^ s^ h^ of Fig. 26, for the curve of the top of the figure. A portion of a new oval, not shown in any of the diagrams, forms the body of the jug. It was formed from a circle of a diameter equal to the greatest diameter of the jug, with the angle A O B bisected, and the in- scribed triangle of the half oval having base angles, a and J, of 10° and of 60°. 91. PI. XII., Fig. 5, represents a portable gas-light, in which the outline of the shade consists of a little more than the most flattened half, o^ 8^ 5i, of the small oval similar to Fig. 29, the exact half, ending at the double lines near the top. The stand- ard is formed of an arc of tfre acute oval used in Fig. 4, but broken, to secure shadow and variety, by the two rings. Strictly, and when of full size, the moulding of these rings should be in ovals. 92. PI. XII., Fig. 6, represents a fruit dish wholly composed of arcs of the oval, of Fig. 30. The flatter half, enlarged a tri- fle, is taken for the body of the dish, terminating on the line c^ Si of greatest width of the oval. The standard is composed of nearly the whole of the outline of the more convex half, a^Oiy of the same oval. The Method by Co-ordinates. 93. Compound, or waving curves may be sketched by a method suggested by that of finding the location of a stream, or other irregular line, in a survey, viz., by distances to the given 124 FBEE-HAKD OEOMETBIOAL DBAWING. line, measared perpendicularly from points at given distance apart on a fixed straight line of reference, bs ah in Fig. 31. But in applying this method to the free design of some ideal line of beauty, we are no longer bound by given distances either on, or from ab, but can, according to the principles of this Part III., substitute angular ratios for them. Thus, in Fig. 32, the left hand profile. A, is determined by co-ordinate distances. The height 04 is divided by trial into T'g- ^^- four equal parts, then 11 equals one of those parts, the next equals half of 01 ; the next, one-fourth of 01 ; and the top width is three-fourths of 01. This seems simple and systematic, but the result is less pleasing than B, or than Fig. 33, which it somewhat resembles. And we venture to say that any design, made like A, Fig. 32, by related distances, will only happen to be pleasing, as a result of repeated trials, while forms like B, and Fig. 33, composed of arcs of ovals, (85, 86) will almost always be graceful. CTJEVILINEAB GEOMETBIO BEAUTY. OVALS. 125 Fig. 33 is composed of an arc of the side of cD^ O^ Jj, Fig. 22, from A to B, and of the pointed end of a^ 8^ J^, Fig. 26. It be- longs to that class of vase forms, which have a tapering neck and a somewhat sharply curved body, and is decidedly superior in configuration to like forms composed of arcs of composite ellipses. Ex. 2S. Apply the prinoiples of this chapter to the designing of a cap and sancer. Ex. 24. Design, likewise, a garden nm or yase. Ex. 35. Do. A fountain. Ex. 26. Do. A sammer-hoose, applying the prinoiples of Chapters IV. and v., to the rectilinear parts. Ex. 27. Do. A library table. Ex. 28. Do. A parlor stoye. Ex. 29. Do. A flower garden. Ex. 80. Do. A pulpit. CHAPTER VIIL OEOMETRIO SYMBOLISM. Definitions, and General lUtcstrations. 9i. Among the elements of geometric beauty, but of a very diflFerent kind from those of harmonious proportion thus far explained, is the symbolism of geometrical figures, or the anal* ogies between some of their properties and certain elements of life. Examples of such analogies may here form an appropriate conclusion. They are generally expressed by the words, type, emblem, symbol, of which the last only will be particularly con- sidered. 95. A Bymbol is anything apparent to sense, which yet, of itself, naturally expresses, represents, or suggests to the mind some truth of life; the natural counterpart in the world of matter, of something corresponding to it in the world of mind. In this natural correspondence, a symbol is quite diflFerent from ah emblem, or a type, as may be sufficiently seen by. re- flection on the common use of the words. Thus every one says, " the national emblem^'' speaking of his country^s flag, but not the national symbol. Here, the connection between the thing and the thought is dependent on association, and mutual agree- ment, and not on inherent natural correspondence and may be equally strong, whatever the thing chosen may be. A type belongs to the same general form of existence as the thing typified. It is a part, taken as a representative of the whole ; a specimen, as the representative of a class ; a lower form, as a representative of a higher form of being or action of the same kind. 96. To illustrate : The mingled verdure and bloom of spring, are symbols oi the freshness, modesty and promise of un per- verted youth. The tints and fruits of autumn, or a sunset in OEOMETBIO SYMBOLISM. 127 crimson and golden light, are symbols of the close of a worthy, or a splendid career. A monument is an emblem of departed greatness. A broken monument is a symhol of a broken life. The American flag is an emblem, of the nation's life. Its rivers are the symhol of the scdLe of its life^ its ideas, and its actions. Its best, and its woi-st, treatment of the Indians, are types of its highest and of its lowest humanity seen in all other relations of life. Again : Water, by its properties, is a type of fluids generally. The ocean is of itself, because of its apparent boundlessness, a symbol of eternity. The oak, with its mighty and horizontal arms, is a symbol of Belf-suflScient rough and rugged strength, and independence. The elm is a symhol of united strength and grace, and thus of culture. Hence, apart from practical convenience, the avenues of cultured towns are appropriately lined with elms, rather than with oaks. With the idea of symbols thus awakened, the following ex- amples of geometric symbols will suflSce to lead the mind into action on the subject. Oeometric lUustrations, 97. A straight line is the symbol of repose, monotony, per- manence and deadness. It is so by reason of its monotony of forni, in having but one unchanging direction. It is therefore adapted to situations where repose, in the shape of fixedness or permanence, is natural or desirable. Thus, in the fervent tropical heats of a land like Egypt, where vigorous activity is to be dreaded, and the repose of utter inac- tion courted, the main outlines of the buildings, naturally and forcibly express these facts by the free use of straight lines, and these^ as the boundaries of most massive and heavily pro- portioned forms. Stout and short vertical columns, mile-long avenues of bolt upright figures, with folded arms and all facing alike, and the immense horizontal bases of the pyramids, and the lines of the immense stones which compose them, all illus- trate this. Also, in foundations generally, where permanence is most desii'able, the main lines are mostly straight and horizontal. 128 FREE-HAND GEOMETBIOAL DBAWING. But in a church, the multitudinous flowing and uptending lines should onl}^ express the endlessly varied, yet only beauti- ful and elevated, individual and associated life, that should, visibly, centre in, and flow from the stirring exercises and ac- tivities within it. 98. The cirde is a symbol of monotonous routine, and hence, as a symbol of eternity, represents only a dormant, unprogres- sive one. It is thus, by reason of its single centre and uniform distance from that to the circumference, and its consequent uni- form rate of variation of direction at all points, and its per- petual return to the same point of beginning. Hence it is peculiarly appropriate that a nation fallen into a state of decay or lethargy, and whose earthly life might then be largely expressed by the stiff, dead straightness of a right line, should adopt the circle as its symbol of eternity, an eter- nity of endless dull repetitions of one unvarying round, *'One unvarying round," is just what the circle sensibly is, and it is therefore the natural symbol of a life made up of routine in one unvarying round. Fig. 84. Fig. 35. Again, life is either sensual or spiritual ; and, in a given amount of it, as the one prevails, the other is wanting. Kow monotony of life indicates absence of thought-activity, and hence, secondarily, the circle as the symbol of monotonous rou- tine, unenlivened by varied thought, is also a symbol of sensu- ous, more than of intellectual existence. Hence the Romans, who were a grosser, and more materialistic people than the Greeks, spontaneously as it were, made great use of the circle in their architecture, while the Greeks rejected it. Thus the coarseness of the compound circular moulding, Fi^. OEOMETBIC STMBOLISM. 129 34, is apparent in contrast with that of the freely varied prin- cipal curve of Fig. 35, whose quick terminal curves, with the more uniform portion, included between them, readily express early entrance upon a prolonged career of excellence, promptly closed where its work is done. 99. The ellipse being only the general form, of which the cir- cle is a particular case, it is not expressive of anything radically different from what is symbolized by the circle. Its continually varying rate of curvature expresses more of varied life than the circle does. Also its two foci, representing a two-fold govern- ing purpose, or idea, or all-engaging pursuit, give more of life to it as a symbol. As contrasted with a circle, for a window, its compression in one direction may make it expressive of partly constrained or contracted, rather than of full-orbed and equally all-embracing life and character. Hence elliptical topped windows, for ex- ample, are less f reqaent and pleasing than semi-circular topped ones. 100. Quite otherwise from the foregoing is it with the hyper- hola^ which is sufficiently defined for present purposes by say- Fig. 86. ing that it consists of two equals opposite^ and infinite branches, ADF and GEB, Fig. 36, to which a pair of straight lines, M and N, crossing at the centre C, are tangent only at an infinite distance from C. Such lines are called dsymptotes. The fixed points R and K are called it^foaiy each one, ^ focus. The complete symbolism of this line is remarkable for its ready and striking truthfulness. 6* 130 FREB-HAND GEOMETRICAL DRAWING. The general idea of the infinite approach of a carve to a straight tangent, as a symbol of an infinite progress towards perfection, or to the absolute ideal, never actually attained, has long been familiar ; but is realized in the case of any of the many different curves which have asymptotes. The distinc- tive symbolism of the hyperbola may be more precisely stated. Firet, there is material civilization, as that of peoples who excel more in material arts, than in personal or national virtues, and there is moral civilization, as of peoples or communities eminent for truth, justice, pure patriotism, and philanthropy. Also there is material barbarism ; and there is moral barbar- ism, illustrated by the injustice or cruelties practised upon weaker, or savage, peoples by nations who were far advanced in many material arts. Now, in the hyperbola, one asymptote, as M, may represent material perfection, or material degradation ; the former, for example, to the right, and the latter, to the left of C. The other asymptote, N, may then represent to the right of C moral perfection, and to the left, its opposite. Thus the two branches of the curve, each tangent to both asymptotes, naturally repre- sent the opposite possibilities of indefinite progress towards good or evil, either material or moral. 101. Spirals are, as compared with the circle, noble symbols of immortal life, with growth and progress, inasmuch as, unlike the circle, they do not return into themselves, but ever proceed in wider and wider circuits, expressive of the expansive progress of all noble lives. They may, therefore, well enter into the composition of the decorative parts, at least, or the seals, or heraldic devices of the buildings whose uses are representative of human progress. And they could hardly appear otherwise than in the ornamental details, because the visible representative of the inspiring idea should be, like the idea itself, over and above the working rooms which must be merely adapted to the work to be done in them. 102. Imagine now a curve, such that the positions of all its points should be governed by one fixed point and one fixed line. Together with such a curve, imagine any organization, the GEOMETSIO SYHBOLISIC 131 various branches, or departments of whose work, should be gov- erned by some one central idea, and some one executive body, representing, so to speak, a certain line of policy. Such a curve would be a symbol of such an organization ; and, if, in future times, attention were paid to symbolism between the inward idea, or purpose, and the outward material agencies through which the idea was put in operation, in all departments of activity, as it has already been in some, nothing would seem more natural than endeavors to realize this symbolism. 103. Thus, for a long time it has been customary to build churches in the form of a cross ; to decorate heroes with jewelled, and hence brilliant stars; to mark a court-house (temple of justice) by a statue holding a balance ; to crown a building which is the property of a nation, or, in some sense, even of mankind, by a vast dome, expressive of the firmament under which all live. 104. With equal propriety, apparently unthought of only be- cause the field of application is much more recent, might sym- bolism enter the field of education. It does so on a small scale, when, for instance, a quill is made the device for the vane of a school-house, or an engineering instrument the device used for a breast-pin by the students of a school of engineering, or when the iron fence-posts around a military academy are in the form of cannon, and the pickets in the form of spears. But, on a larger scale, the buildings for the general and special pur- poses of B.uy large educational establishment, together with the residences for its teaching body, might, if its grounds were sufliciently exten- sive, be easily arranged in a symbolical b manner. 105. Thus, returning now to Art. (102) there are two curves, at least, | which agree with the definition there suggested. These are the parabola, Fig. 37, each of whose points, as a, is at equal distances, a F and a b, from a Jlxed point, F, the focus, from a Jlxed line, D b, the directrix. Also the conchoid, Fig. 38, a curve of two branches, and all of whose points are at the same fixed distance from a given line, Pg. 3?- 133 FBEB-HAND OBOMETBIOAL DBAWTNG. measured on lines drawn from 2k fixed point. When this point . is nearer the fixed line than the fixed distance, one branch of | Fig. 38. the curve will be looped. Thus EE is the fixed line, and A the fixed point. Then dC = de; ba^= be, etc. GEOMETBIO SYMBOLISM. 133 106. Turning now from these curves to a complex educa- tional establishment, we find for its corresponding fixed ele- ments, 1°, a foundation course of study ^ alike for all, 2°, a teaching and governing body. There would then be depend- ing upon these (a) the various specialties to which the institu- tion might devote itself ; (J) the incidental features of its life, lodgings, gymnasium, etc. ; and finally {c) a select group of struc- tures, devoted to the most refined purposes of the institution. 107. Parabolic plan. The several elements just stated might find their material organization on a parabolic plan^ as follows. The collegiate or general building in which the foun- dation course (1°) should be given, would naturally stand at the focus. The residences of the teaching and governing body (2"^) would be located on an avenue marking the directrix. Then, (a) the schools for the several specialties or professions, would be arranged at intervals on the curve, and with paths to them located as at F a and b a. The axis of symmetry, D P, being therefore a special line, and D and V, special points upon it, a chapel, D, library, V, a museum, and observatory may be built upon it. Subordinate structures might be located at convenience on lines within the curve, and parallel to D F. 108. The conchoidal plan. The conchoid^ when laid out on a grand scalis on the ground, permits the symbolical expression of the ideas, stated in Art. (106), in the matei^idl organization of an institution, as its published curriculum exhibits them in the printed expression of the logical organization. I*'. A grand building, surmounted with a dome, as the sym- bol of comprehensiveness, and with lofty porticos facing the four cardinal points of the compass, as the symbol of its equal openness to all, should stand at A, and contain instruction rooms for all the general subjects. 2^. Professors, as the immediate personal determining ele- ment in the life and work of an institution, should have resi- dences ranged along the fixed determining line E E. And d e may be 1000 feet or more. 3°. D D, being the superior branch of the curve, should be allotted to the series of buildings devoted to the several profes- sional schools, and reached from A by paths on the radial lines as a J 08ition8 of lines have a significance, as well a their forms. Thus a prevalence of vertical lines symbolizes a< GEOMBTBIO SYMBOLISM. 135 piration, npward-tending thought and purpose ; and hence gives noble meaning to a lofty gothic cathedral interior, where the prevailing direction of the lines is vertical. The same idea gives effect to the humblest village spire. Hence the betrayal of offensive vain consciousness, or of obtuse- ness, either in the maker or beholder, in adding an up-pointing hand to the top of a spire, as if the spire were made to say, " See with what beautiful expressiveness I point to heaven ; " or, more likely, as if the mind could not understand the upward pointing of the spire without this Explanatory addition, which robs the imagination of its dues in being left free to give mean- ing to what it sees. A prevalence of horizontal lines, is expressive of a clinging to the earth, as in the popular life of the Greeks, most, or all of whose gods were but exaggerated men, crimes and all ; and then^ set over this world's woods and fields, seas and skies, wars and passions, rather than over a universe of life, to be moulded into enduring forms of living beauty by them. Hence the marked pi edominance of the horizontal in the Greek temples, with their flat roofs and horizontal mouldings, and flat doors and window tops. Once more, and in a derivative manner, horizontal lines ex- press firmness^ decision^ stahilityj and hence are the proper characteristic lines of foundations and supports. The repose, Fig. 39. Fig. 40. or fixity, which they primarily signify, leads to the secondary meanings, unchangeableness, and thence decision, or stability, as stated. Hence the curved outlines of mouldings on support- ing parts best flow into the horizontal top and bottom surfaces of such parts. Thus, Figs. 39 and 41, show a better relation of the curved contour, as tangent to the bases, than Fig. 40, does. 1S0 FBEB-HAND QEOMeTBlCAL DRAWISa. 113. Carvings. Work becomes 8o costly ae soon as straight ontliiies are abandoned, and especially as carved work begins to be employed, that its consequent difficulty of attainment makes it symbolical of the giace and beauty that can only be had under the beet conditions, or, as the result of man's best aspirations; while the plain lines of ordinary work represent, Fig. 41. by comparison, humbler human industries. Hence a bit rf choice carving to crown, or tip, or face a piece of otherwise plain work, happily symbolizes the cheerful co-operation of hap- piness and honest industry, the meeting of truth and beanty. It is in the light of such reflections that the real vulgarity of mere flat sawed scroll work, on which no elevated intellectual or artistic thought or fond purpose has been exercised, is fully shown. Being purely mechanical products, they can serve no high thought or purpose. 113. An entirely d-^erent principle, however, governs the employment of ornamental castings from really rich and beau- tiful designs. Here, the thought is the nobly generous one of bringing to every humble home, by means of a beneflcent art of multiplication, beauties of decoration which could not other- wise be had. The " preciousness " of the immediate products of the skilled and refined hand becomes only their hatefulness when they are prized mainly because none but one wealthy pur- chaser can own and enjoy them. The "ginger-bread" products of the scroll saw, from inch boards, are mean in origin, material, and execution, and are therefore to be discarded fortheir iuherent demerits ; but good castings, from beautiful designs, inherit and palrtake of the OEOMETRIG BYMBOUSM. 137 characteristics and associations of their original, and are, by all means, to be commended, where originals cannot be had. Somewhat in the same line of thought with the remarks on carvings ; broken pediments, as in the annexed figure, and con- taining a carved bust or other form of life, may be mentioned as symbolizing the escape of the spirit from the hindrances and imprisoment of the body. 114. Without further illustration, it may now be enough to add that the foregoing somewhat numerous, and widely varied examples may serve to set the thoughts in motion upon the line indicated, so that the student may be aided in his efforts to give to all his works an attractive and elevating Tneaning^ at the same time that they fulfil the bare physical conditions required of them. Apply the principles of this, and the preceding chapters, in designing the following : Ex. 31. An altar. Ex. 32. A pulpit. Ex. 83. A book-case. Ex. 84. A parlor organ case. Ex. 86. A church porch. Ex. 87. A district school -house. Ex. 88. A sideboard. Ex. 89. A public library entrance. Ex. 40. A mantel-piece. THE END.