Elements of plane and solid free-hand geometrical drawing, with lettering; and some elements of geometrical ornamental design, including the principals of harmonic angular ratios, etc. In three parts ... for draughtsmen and artisans; and teachers and students of industrial and mechanical drawing

Front matter

LIBRARV EGRESS.

• MERICA.

INDUSTRIAL SCIENCE DRAWING.

CHAPTER II. Elementary and practical exercises on right angles 7

Examples of single lines at right angles, with sides horizontal and vertical 8

With sides oblique 9

IV CONTENTS.

PAGB

Right angles 9

Pairs of parallels at right angles. The pairs horizontal and vertical 10

The pairs in oblique positions 10

Occasions for free-hand sketching 11

CHAPTER VIII. 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 infallible 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 drafting 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 preliminary 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.

VI 11 PREFACE.

of a few simple pencil plates of such 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 preparatory and industrial schools.

Ey 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 be acquired or perpetuated without interference with one's larger 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 conceptions.

Newton, Mass., January, 1873.

NEW PREFACE.

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-hand drawing of regular forms. Hence my former small work was put forth, partly as an experiment. 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 divided, Part II. is largely, and Part III., almost wholly new.

Part III. may interest the general reader. It contains, apparently 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 applications 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, be turned into pleasantry. Still, as its use has prevailed for cen-

X PREFACE.

turies in some departments, there seems to be no reason for not extending it to others. I have sought to simplify, and to improve as much as possible, the very little that room could be found for, on this subject.

The figures or patterns in this volume may have a threefold use : Firsts 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, Artizan'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 copybooks, easily procurable, and, in most cases should be considerably 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 reasonably 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 principles and practice, hopes to make them still more acceptable by extending the work of thorough revision (for the first time, excepting the Projection Drawing) to all of them.

Newton, Mass., August, 1878.

Chapter I. Exercises On Directions Of Straight Lines

First Principles.

The direction of a straight line is its invariable tendency towards 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 directions are different, and hence is here considered first.

A line which is " straight np 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 ©n 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 oblique. Also lines lying in any fiat 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, throughout his progress, should carefully follow the subjoined general directions.

Materials.

For the practice of quite young pupils, where substantial 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 papercase such as any one can make for himself.

A common semi-circular "protractor" 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 beginning. They w T ill, in due time, amply reward the reliance placed upon them. Ruler, Protractor, and Measure may be used to test the straight ness, direction and length of lines already 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 trifling 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

DIRECTIONS 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 before drawing them. Mean 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.

Be 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 resting on the tip of the second finger. It can then be moved both with freedom and steadiness.

In drawing lines towards or from yon, 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 flat, 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 called horizontal, and others, oblique.

Remember 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 bottom. See PL I.

Ex. 3. Mark two points so as to be connected by a vertical line, and then draw a line joining them, beginning a little above the upper point.

Ex. 4. The same, but beginning below 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 PL I.

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 PI. 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 inclines from the body and to the rights thus :

Ex. 13 to 16. May consist of four similar constructions of lines which incline from 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 nearly vertical than horizontal, thus :

DIRECTIONS OF STRAIGHT LINES.

Second, let them be more nearly 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., PL I.

Ex. 20 to 22. May consist of similar variations in drawing two or more horizontal parallels. See h., PL I.

Ex. 23 to 28. May consist of similar exercises on two or more oblique parallels situated as in examples 9 to 16, and including 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 downwards 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.

Double Lines.

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.

General 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. I, Fig. 1.

Use of the Copy-book— 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 buff 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, combined 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.

Chapter II. Elementary And Practical Exercises On Eight Angles

Principles.

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 relative lengths of the distinguishing lines of the objects ; or from the relative 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 first, if not oftenest, think, or ask, " which way " than " how far." Direction is therefore a more primary idea than length. An angle, however, is merely difference between directions from a certain point. Hence angular proportions, or the proportion between the angles of a figure, are more elementary than linear 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 standard 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 independent.

V h m

Thus, in these figures, by going from a to b, we find ourselves at the distance ac to the right of a. So by moving from d to e we go a distance equal to df in the direction of the line df But, when the two angles formed by the meeting lines are equal, as at mgh and kgh, we do not, in moving to any distance on gh, progress at all to the right or left of gh. Hence the directions of gh and mk are independent, and the angle included between them is the naturcd 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 parts of a right angle.

But, again, it follows from the explanation of vertical and horizontal directions, in Chapter I., that a right angle is in its simplest, most natural, or standard position,wh.en its sides are in the fundamental directions of vertical and horizontal. We therefore begin with right angles in this position. Observe first, however, that we do not say perpendicular and horizontal, but vertical and horizontal, for a line in any position whatever, is perpendicular to another when it is at right angles with it, but there is but one vertical, or " straight up and down " direction.

Examples of Single Lines at llight Angles, with Sides Horizontal and Vertical.

Ex. 33. Construct one right angle thus, | t and thus,

and thus, and thus, making its sides

from one to three inches in length, each side ending at its intersection with the other. Slight additions will give these simple elementary figures a pleasing character as designs for geometrical borders and corner pieces, thus :

L designs which it is easy to make evenly by observing the direction to pencil each line faintly at first, while locating it as intended.

Observing that the beauty of a border depends upon its expressiveness, as an echo of some characteristic of the work which it encloses, the first design would make an agreeable

EXERCISES ON RIGHT ANGLES.

corner, 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 diagonal from the outer corner to, or a little beyond the inner corner.

Ex. 34. Construct two right angles, by prolonging one of the

and thus, sides beyond the vertex of the angle, thus, and thus, and thus, — I

making the sides of each angle from two to four inches long, in this and the following figures.

Ex. 35. Construct four right angles, by prolonging each side through its point of intersection with the other, thus

Right Angles with sides Oblique.

Ex. 36. Repeat Ex. 33 with the sides in various oblique positions, and of various lengths, thus :

Ex. 37. In like manner, repeat Ex. 34, thus :

-l -r y y-

Ex. 38. Similarly, repeat Ex. 35, thus, but in each case make

the lines from one to three or four inches long, from the point of intersection.

See c and d, PL I., for examples of a suitable size for these figures, but which, if drawn on large plates, may be larger still.

FREE-HAND GEOMETRICAL DRAWING.

PAIRS OF PARALLELS AT RIGHT ANGLES.

The Pairs Horizontal and Vertical.

Many variations can be made, and shonld be, in the order of 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 pi representation, only, in the preceding elementary figures, they may consist in combinations of faint and heavy lines, as shown in a part of the following figures, all of which should be made of Lines from a half inch to three or more inches long.

L

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The pairs 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.

EXERCISES ON RIGHT ANGLES.

A specimen or two of each is added in this example.

| >ijgfP jgi, IN*!

il

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, PL I., exhibit other practical examples of constructions containing only lines at right angles to each other.

Occasions for Free-hand SJcetching.

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 accomplishment of readily making free-hand sketches of geometrical industrial objects.

CHAPTEK III.

DISTANCES, AND DIVISION OF STRAIGHT LINES.

Principles.

Having considered various directions of straight lines, we are prepared to estimate and represent various distances npon them.

Distances are eqtial or unequal. When unequal, we often wish to compare them. Distances may be compared, first, 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 how many 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 representing 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 so find a series of equal distances.

Exercises in Marking off a Given Distance.

Ex. 43. Draw straight lines in different directions, and mark by the eye, the same distance, once, on all of them, thus :

DISTANCES AND DIVISION OF STRAIGHT LINES.

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 ; thus:

Division of Lines into Equal Parts.

Ex. 46. Divide lines in various positions, as shown below, intc two equal 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 equal. 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 times the error in the single third of the line, and so on. Then make the necessary corrections, 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.

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 trisect 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 the whole was first divided.

Let a line now be divided into six equal parts, for example. Half of a line is more easily estimated than a third, hence divide the line first 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 Applications.

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 necessary 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 ; instruments are liable to be displaced unconsciously to the draftsmen, giving unequal spaces, if dividers are jarred, or a scale mistakenly 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, occasion great annoyance. Third ; the eye so trained will often enable the draftsman to make small and simple divisions, especially those requiring only repeated bisection, as halves, fourths, etc., as readily and perfectly by the eye, as with the compasses.

All the following figures were originally drawn strictly as here directed, the divisions being tested by marking them on the edge of a slip of paper, and the directions, see especially the diagonals in Fig. 6, by tracing them at first very faintly.