Course in Descriptive Geometry and Stereotomy,
BY S. EDWARD WARREN, C.E.,
PROFESSOR OF DESCRIPTIVE GEOMETRY, ETC., IN THE MASS. INSTITUTE OF
TECHNOLOGY, BOSTON. FORMERLY IN THE RENSSELAER
POLYTECHNIC INSTITUTE, TROY.
The following works, published successively since 1860, have been well received by all the scientific and literary periodicals, and are in use in most of the Engineering and "Scientific Schools"
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I.-ELEMENTARY WVORKS.
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1.-ELEMENTARY FREE- 5.-ELEMENTARY PROJECHAND GEOMETRICAL TION DRAWING. Third edition,
DRAWING. A Series of Progressive revised and enlarged. In five divisions.
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AN
ELEMENTARY COURSE
IN
FREE-HAND GEOMETRICAL
DRAWING,
FOR SCHOOLS, AND FOR THE TRAINING OF THE EYE AND
HAND DESTINED TO MECHANICAL PURSUITS AND ARTS
OF GEOMETRICAL DESIGN. WITH CHAPTERS ON
LETTERING AND ON GEOMETRIC SYMBOLISM.
BY
S. EDWARD WARREN, C.E.,
PROFESSOR IN THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY, FORMERLY IN THE RENSSELAEB
POLYTECHNIC INSTITUTE, AND AUTHOR OF A PROGRESSIVE SEBIES OF WOBKS
ON DESCRIPTIVE GEOMETRY AND GEOMETRICAL AND
MECHANICAL DRAWING.
NEW YORK:
JOHN WILEY & SON, PUBLISHERS,
15 ASTOR PLACE,
1873.
ENTEBED according to Act of Congress, in the year 1873, by
JOHN WILEY & SON,
In the Office of the Librarian of Congress, at Washington.
I'00LE & MACLAOcLAN,
PRINTERS AND BOOKBINDERP
205-213 East Twelfth St.,
NEW YORK.
CONTENTS.
PAGE
PREFACE....................................................... V
CHAPTER I. Exercises on directions of straight lines................... 1
First principles........................................ 1
Materials............................................... 2
Directions...................................... 2
Single lines............................................. 3
Parallels............................................... 5
Opposite lines..........,............................. 5
Double lines............................................ 6
CHAPTER II. Elementary and practical exercises on right angles......... 7
Principles............................................. 7
Examples of single lines at right angles, with sides horizontal
and vertical....................................... 8
With sides oblique...................................... 9
Right angles........................................... 9
Pairs of parallels at right angles. The pairs horizontal and
vertical........................................... 10
The pairs in oblique positions........................... 10
Practical examples..................................... 10
CHAPTER III. Distances, and division of straight lines.................. 12
Principles............................................ 12
Exercises in marking off a given distance. Ex. (42-44)... 12
Division of lines into equal parts. Ex. (45-49)........... 14
CHAPTER IV. Circles, and their divison............................... 16
Principles............................................ 16
Examples. Circles and Arcs. Ex. (50-53).............. 16
Division of circles. Ex. (54-59)........................ 18
iv CONTENTS.
PAGE
CHAPTER V. Proportional angles..................................... 19
Principles............................................. 19
Elementary examples. Ex. (60-65)...................... 20
Practical examples. Ex. (66-68)........................ 2 22
CHAPTER VI. Figures bounded by straight lines....................... 23
Principles............................................ 23
Elementary examples. Ex. (69-77)..................... 24
Practical examples. Ex. (78-80)....................... 27
CHAPTER VII. Curves, and curved objects in general. Ex. (81-94)..... 29
CHAPTER VIII. Lettering........................................... 40
General principles.................................... 40
Roman capitals...................................... 4 41
Letters in general. Their classification and construction 44
Practical remarks.................................... 47
CHAPTER IX. Geometric symbolism.................................. 49
Definitions............................................ 49
Geometric illustrations. The straight line, circle, ellipse,
hyperbola, parabola, spirals, conchoid............... 50
PREFACE.
IN geometrical, or mechanical drawing, exclusive reliance
for accuracy may, in theory, be placed on good drawing
instruments.
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.
Ience, 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.
These brief reflections have resulted from a recent inspection
vi 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
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 be acquired or perpetuated without interference
with one's larger industries.
My work may be thought deficient in the number and style
of its illustrations; but it is meant to be supplemented by a
free use of large plain lithographic, or other copies of ornamental devices; and of the blackboard.
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 a closing notice, and indeed need
hardly more. 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,
PREFACE. Vii
and who will then pass on to fidelity in the representation of
their observations and conceptions, and to growth in truthfulness of spirit.
It is proper to add that the theory of beauty of form as dependent, more on angular than on linear proportions, which is
briefly recognized in this volume, is due to D. R. Hay, author
of several ingenious and attractive works on geometric beauty.
The short chapter on the largely unwrought subject of geometric symbolism, may serve, in the absence of anything better,
as the seed of richer and more abundant conceptions of beauty
of thought linked to expressive forms.
NEWTON, Mass., January, 1873.
CHAPTER I.
EXERCISES ON DIRECTIONS OF STRAIGHT LINES.
First Principles.
THE direction of a line is its tendency towards a certain
point.
The directions of two lines may be alike. The lines are then
said to have the same direction, and are calledparallel.
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 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 in space, not vertical or horizontal, are called oblique.
Also lines lying in any flat surface, and not at right angles to
each other 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.
1
2 FREE-HAND GEOMETRICAL DRAWING.
MATERIALS.
For the practice of quite young pupils, where substantial ac
curacy, rather than fineness of execution is expected, quite cheat
paper, or even a slate and pencil will answer.
For more advanced practice, heavy flat wove writing paper,
Whatman's, or the German cartoon paper (drawing paper), may
be employed, cut into plates of convenient size, as 7- by 10
inches,-the dimensions afforded by paper of the "demy" size.
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 comparatively large scale is sought, as a
training for bold sketching, large plates of coarse 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.
DIRECTION S.
Depend on the unassisted eye and hand alone, from the beginning. They will, in due time, amply reward the reliance
placed upon them. Ruler, Protractor, and Measure may be
used to test the strcaightness, 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 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 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.
Ex. 2. The same on the lower half of the plate, but beginning
the lines at the bottom.
4: FREE-HAND GEOMETRICAL DRAWING.
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.
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 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 right, 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 -nore nearly vertical
than horizontal, thus:
DIRECTIONS OF STRAIGHT LINES. 5
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.
Ex. 20 to 22. May consist of similar variations in drawing
two or more horizontal parallels.
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.
6 FREE-HAND GEOMETRICAL DRAWING.
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 diferent
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.
CHAPTER II.
ELEMENTARY AND PRACTICAL EXERCISES ON RIGHT 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.
tf \1
Thus, in these figures, by going from a to b, we also move as
far as ac in the direction of the line ac. 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, move at all in the direction of gm or gk. Hence the
8 FREE-HAND GEOMETRICAL DRAWING.
directions of gh and mk are independent, and the angle included
between them is the natural 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, when its sides are in
thefundamental 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 RIGHT ANGLES, WITH SIDES HORIZONTAL AND VERTICAL.
Ex. 33. Construct one right angle thus, L_ and thus,
and thus, r 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. 9
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, or progression, as in the shading, difficulty, or importance of the enclosed figures.
Ex. 34. Construct two right angles, by prolonging one of the
sides beyond the vertex of the angle, thus, and thus,
and thus, T and thus, ~
Ex. 35. Constructfobur 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 posi.
tions, and of various lengths, thus:
Ex. 37. In like manner, repeat Ex. 34, thus:
Ex. 38. Similarly, repeat Ex. 35, thus, but in each case make
the lines from one to three or four
XJ^~-* <^ inches long, from the point of intersection.
10 FREE-HAND GEOMETRICAL DRAWNG.
PAIRS OF PARALLELS AT RIGHT ANGLES.
The Pairs Horizontal 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; also,
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 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.
iL iL
The pairs in oblique positions.
Ex. 40. Repeat 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. 11
A specimen or two of each is added in this example.
______l8L~~~~~~~~~~~~~IHiiI lI11
The pupil is here again reminded always to make his figures
very much larger than those of the book.
CHAPTER III.
DISTANCES, AND DIVISION OF STRAIGHT LINES.
Principles.
HAVING considered various directions of straight lines, we are
prepared to estimate and represent various distances upon them.
Distances are equal 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. 42. Draw straight lines in different directions, and mark
by the eye, the same distance, once, on all of them, thus:
1-. j- I -
DISTANCES AND DIVISION OF STRAIGHT 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. 43. In like manner, mark a given distance several times,
on lines in various directions; thus:
Ex. 44. Draw lines in several directions through the same
point, and mark equal distances from the point on all of them;
thus:
14 FREE-HAND GEOMETRICAL DRAWING.
DIVISION OF LINES INTO EQUAL PARTS.
Ex. 45. Divide lines in various positions, as shown below, into
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. 46. Divide a line into four equal parts. To do this,
bisect the whole line, and then bisect each of its halves.
I ~ I... I~
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. 47. 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. 48. 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.
DISTANCES AND DIVISION OF STRAIGHT 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 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. 49. 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.
CHAPTER IV.
CIRCLES AND THEIR DIVISION.
Principles.
DIRECTION 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 circle 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 AND ARCS.
Ex. 50. To draw a circle. Sketch, faintly, several lines through
a point, taken as the centre of the circle, and, from this point,
mark off equal distances on each of these lines. Then through
the points thus given draw the circle, thus:
Ex. 51. 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.
CIRCLES AND THEIR DIVISIONS. 17
In both of these constructions, use fewer and fewer guides,
and at last sketch a circle with no guiding point but its centre.
Also 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 concentric. A portion of the circumference of a circle, is called
an arc.
Ex. 52. Draw circular arcs in various positions, and of various
radii, and length, thus:
Ex. 53. Draw parallel arcs and circles, of various radii, and
the former also of various lengths and in various positions,
thus; and then mark their dentres.
) KY
18 FREE-HAND GEOMETRICAL DRAWING.
DIVISION OF CIRCLES.
Circles, or arcs, may, like straight lines, have given distances
marked off upon them, and may be divided into equal parts.
The line which joins the extremities of an are, 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 principle that any given straight distance may be transferred to a circle or to any curve.
Ex. 54. 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.
E.r
Ex. 55. 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. 56. Two diameters at right angles to each other, divide a
circle into four equal parts. Draw such diameters in various
positions.
Ex. 57. The radius of a circle applies just six times to its circumference. Then lay off the radius once, on the circumference, as explained in Ex. 54, and then mark the other divisions,
equal to the one thus obtained.
Ex. 58. Bisect each quarter circle in Ex. 56, 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. 59. Continue these exercises by trisecting the quarter circles, and bisecting and trisecting the sixth parts in Ex. 57, 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.
PROPORTIONAL ANGLES.
Principles.
AFTER 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 represent, 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 valuable 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 hundred 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 arc 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 between 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 explained, the natural angular measure for other angles, a right
angle will be taken as the one to be variously divided, in the
following examples.
20 FREE-HAND GEOMETRICAL DRAWING.
ELEMENTARY EXAMPLES.
Ex. 60. Bisect a right angle, in each of the positions given in
Ex. 33. To do this, sketch carefully a quarter 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 as
seen in the figure. To divide the angle into any other number
of parts, divide the included quarter circle into the same number
of parts. To test the angle thus estimated and drawn, use a
" Protractor," as follows:
0
180 C^
The protractor is a semi-circular instrument, whose semi-circu
lar edge is divided into 180 degrees. A right 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 vertex, C, of the right angle,
as shown in the figure, then the required bisecting line C, 45~,
will if correct pass through the 450 point on the divided edge
of the protractor. If it fails to do so, then first carefully estimate, 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, draw a number of parallels to it, in this,
and all the following problems of divisions of angles.
Ex. 61. Construct a line which will cut off one-third of a
right angle from either of its sides, thus:
l,.1_/ //
PROPORTIONAL ANGLES. 21
One-third of a right angle is 30~-measured by one-third of the
quarter circle-hence 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 consider, 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. 62. Draw a line cutting off one-fourth of a right angle
from 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. 63. C( nstruct, successively, angles of one-tfftl, and two
fifths of a right angle; i. e., angles of 18~, etc., thus:
Ex. 64. 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 than the one of 30~, thus:
Ex. 65. Repeat the divisions of the right angle, given in the
22 FREE-HAND GEOMETRICAL DRAWING.
preceding examples, upon right angles in various oblique posi
tions as in Ex. 45.
PRACTICAL EXAMPLES.
Ex. 66. A four pointed star, requiring two lines at right
angles to each other, and the equal bisecting lines of those
angles.
Ex. 67. A gate. Note that an angle of 24~ is four-fifteenths
of a right angle.
/////n g p i p //dista
Ex. 68. An arch, giving practice in parallels, eqnal distances
(each side of the arch, and the heights at the ends) and arcs, of
various sizes, and parts of a circle.
CHAPTER VI.
PLANE FIGURES BOUNDED BY STRAIGHT LINES.
Principles.
A plane figure is a portion of a flat surface, bounded by lines.
When bounded by straight lines, it is called apolygon.
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 angles,
but only the opposite sides are equal, the figure is called a rectangle.
A Pentagon is a figure of five sides. In a regular pentagon
the sides and angles are all equal.
Likewise, a regular 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 corner.
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 expressed by fractions whose terms are small numbers.
A great many familiar objects have sides of an oblong, that
is a rectangular 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
simplest triangle, as a standard for the comparison of angular
24 FREE-HAND GEOMETRICAL DRAWING.
proportions, since it contains a right angle, which is the natm al
measure with which to compare its other angles.
Rectangles, as floors, walls, doors, windows, the spaces between them, etc., are therefore, most beautifully proportioned,
when their diagonals divide their right angles into parts
bearing 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 angles of 30~ and 60~, the ratio of these is i, and
their ratios to a right angle, are - and i. These all being simple fractions, the rectangle will be found to have agreeable proportions.
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. 69. A right
angled triangle
with equal acute
angles of 45~ each.
This triangle possesses the property of being divided by a perpendicular from its right angle to its opposite side, into two
triangles of the same shape as the original whole. This property makes its construction easy. Draw this triangle in various
positions, and fill it with lines parallel to its longest side, as
above.
Ex. 70. A triangle each of whose halves is a right angled
PLANE FIGURES BOUNDED BY STRAIGHT LINES. 25
triangle with acute angles of 36~ and 54~. Here 3 — _ 36~W
and 4 —-. Also in the whole triangle yBg —Q. 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. 71 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,
y\ X \ Xor perpendicular, to some one of its sides.
Ex. 72. Construct squares of various sizes and in various
positions, first without their diagonals and then with them.
Ex. 73. 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. 74. 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 equaal
26 FREE-HAND GEOMETRICAL DRAWING.
parts. Also, in the second figure, the marking of equal distances, viz., the semi-diagonals of the little squares.
Ex. 75. 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 720, 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. 76. Hexagons. These polygons have angles of 120~ at
their corners. They can therefore be combined as in pavements,
so as to completely fill a given space. It will assist in constructing this figure, to remember that each of its sides is equal to the
distance from its corners to its centre. Observe, also, that the
longer diagonal is divided into four equal parts by the shorter
ones, perpendicular to it, and the centre.
Ex. 77. Divide a circle into eight equal parts, by diameters
at 45 with each other, and join the points of division by straight
lines, which will give a regular octagon, or eight sided figure.
This figure can also be drawn, by considering that its external
aIgles are each equal to 45~, thus:
PLANE FIGURES BOUNDED BY STRAIGHT LINES. 27
PRACTICAL EXAMPLES.
Ex. 78. Wholly made up of vertical and horizontal lines.
~ III I IIIIIIIIIIIIIlilm i!
a3 ~~1 t' ll/ i i/{{11{i
Ex. 79. Embraces oblique lines.
28 FREE-HASD GEOMETRICAL DRAWING.
Ex. 80. Embraces circular lines.
*I~ X^~I
those portions of each line, which are meant to be visible, can
be retraced in firm and heavy strokes.
CHAPTER VII.
CURVES AND CURVED OBJECTS IN GENERAL.
WE have thus far considered only 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 tangency, they lie in the same direction. Hence
any curve can be much more easily 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 ellipse.
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 help, 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. 81. Sketch ellipses of various proportions by the rhomboidal method, thus:
We may mark the middle point of each side, as the points of
30 FREE-HAND GEOMETRICAL DRAWING.
tangency of the ellipse, the sketching of which will then be
quite easy.
Let this exercise be continued, in the sketching of ellipses
in rhombuses placed in various oblique positions, and, also, with
their longer diagonals placed vertically.
When an ellipse is inscribed in a rectangle, it crosses the centre lines of the rectangle at right angles, at the points of tangency with the sides of the rectangle. Thus the eight guiding
positions afforded by the rhombus, are reduced by union to four,
in the rectangle. The ellipse will, however, cross the diagonals
of a rectangle at equal distances from its corners, but not in a
perpendicular direction.
Ex. 82. Sketch ellipses in rectangles and other figures, of various proportions and positions, thus:
An ellipse is a curve of most delicate grace, and should therefore be most faithfully studied and carefully drawn. The most
offensive error in shaping it, is, to represent it as pointed at the
narrow end, which it is not, in the least.
By combining elliptical arcs of various proportions, tangent to
each other, various graceful forms adapted to ornaments, such
as vases, may be composed. In doing this the relative proportions of the ellipses should not be chosen at randomn 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. 83. In this design for a vase, all the angles, some of whose
degrees are given in the enclosed numbers, are 9~, the square
of 9~, or even multiples of 9~. Also at the base, two rhombuses have a common vertex; and at top, two have a side and two
CURVES AND CURVED OBJECTS IN GENERAL. 31
vertices in common. The
acute rim-rhombus has its
sides perpendicular to a 54
and b 72, its right side
passes through the corner 72,
and its diagonal passes
through c, the junction of
two arcs, and centre of a
72. Moreover, the diagonal
72-36 coincides with 18-72/ /
produced, and the side 7254 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 side of
which embraces four different elliptical arcs, of which _
the one running upward from c terminates on a 54.
Ex. 84. In this design, the relations
are in part, more, and in part less simple
\- 6 A^ ~ than in the preceding, and the result will
hardly be thoht m ar thought more agreeable than
before.
The principal, and the base rhombuses
are of the same proportions, as seen by
X \\ their angles, and therefore enclose similar ellipses, which gives less decided vari60/ ety in the outline at the base. The upper side-rhombus, with its angle of 18~,
/60 30 side of one in the central rhombus of
)\30 60A30, gives the comparatively complex
and unfamiliar ratio -3. Also its right
I[_ _____ __ hand corner 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 the longer diagonals of the
narrow upper side rectangles.
Ex. 85. By substituting for a rhombus, two dissimilar half
32 FREE-HAND GEOMETRICAL DRAWING.
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 -. In the second figure the corresponding angles are 18~ and 36~, having therefore a ratio of
i, and affording a more decidedly egg-shaped curve.
Ex. 86. An egg-shaped oval may also be inscribed in a
regular trapezoid, that is a figure having two unequal but parallel sides, both of which are bisected by the same line, perpendicular to both, thus:
Let these ovals be drawn in a great variety of proportions and
positions, both in rhombus-like figures and trapezoids, and with
as frequent reference aspossible to leaves, which exhibit a great
variety of graceful ovals.
CURVES AND CURVED OBJECTS IN GENERAL. 33
Ex. 87. 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 design illustrates these remarks. Its angles of
36~ and 540; 540 and 10~-48' (ten degrees
and forty-eight minutes) 75~-36' and 18Q-54',
give the simple ratios i, a, A,.
The student should make a variety of similar designs.
Ex. 88. 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 horizontal or vertical tangents.
Thus, there is more vigor, as
/J <~ ~ well as variety, in the curve in
the second of these figures, than
in the first.
Ex. 89. 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)
34 FREE-HAND GEOMETRICAL DRAWING.
letters on a large scale, 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. 90. The varieties of curves being innumerable, a few are
here annexed by way of suggestion. The student can devise
many others.
The group of four parallel curves affords an excellent example for practice, each curve being nearly straight in the middle, and sharply curved at the ends, while its left-hand half
is convex upwards, and its right-hand half equally so downwards; 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.
CURVES AND CURVED OBJECTS IN GENERAL. 35
The use of tangents in sketching curves is also illustrated in
these examples.
2)
Ex. 91. An exercise of peculiar utility, is found in sketching
easy curves through several given scattered points. This operation frequently occurs in geometrical drawing, when other than
circular curves are to be described. The essential things to be
observed in these cases, are,ftrst, to avoid all sudden, irregular,
and unnecessary variations in the rate or degree of curvature,
and, second, especially to avoid making an angle at any point
BC
in the intended curve. These important requirements can be
met by keeping at least three successive points in view at once.
36 FREE-HAND GEOMETRICAL DRAWING.
Thus, while joining A and B in the figure, keep C in view, and
operate likewise in making all the figures.
i'7
The student should practice extensively on this example, jcrst
/f~ -N\~ ~taking the points, in many different relative positions, and then
s D _ running easily flowing curves
through them.
-^ m~ ^ ^'Ex. 92. In several of the preceding examples, curves have
been drawn tangent to straight
lines previously drawn. We here add an example of drawing
tangent 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. 93. Finally, the examples of this chapter close with practice in the very nice operation of drawing symmetrical figures
with variously curved outlines. By symmetrical figures, are
meant those which are divided by a centre line into two exactly 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
CURVES AND CURVED OBJECTS IN GENERAL. 37
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.
1. 2. 3. 4.
5. 6.'7.
38 FREE-HAND GEOMETRICAL DRAWING.
8. 9.
10. 11.
Ex. 94. Representing a few elementary corner pieces, illustrates some of
the foregoing principles. 10 is inferior to 9 because its main spur seems
~12. ~ 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 corners at each side of it.
Thus the skeleton of every corner should embody a good idea,
for no richness of detail in ornament can redeem bad governing
outlines.
Attention to such simple principles as these will guide in the
design or selection of borders, and prevent the necessity of presenting 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 ornamental devices.
Another principle, disregard of which through disproportion
CLUVES AND CURVED OBJECTS IN GENERAL. 39
ate interest in some trivial thing, has spoiled many a drawing,
is this. Ornamental devices on drawings of solid worth, should
never represent any thing essentially mean, or rudely comic, or
even a reminiscence of anything, which however pleasant, is of
transient interest. Such things as the latter, when preserved at
all, should be in a separate form. In fact, sense of humor is
best delighted, and the happiest laughter excited, by the simple
sight of a beautiful border, or other work, some simple quality
of which, such as its compact neatness, or clean firmness, is
highly suggestive of analogous attributes in its maker.
CHAPTER VIII.
LETTERING.
General Principles.
LETTERING, though not strictly a part of a drawing, is a necessary 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: first, uniformity of size and proportions,
and, second, beauty and regularity of form in each letter. Illshaped 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 proportional distances and angles, should enable him to secure uniformity 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 pages.
LETTERING. 41
ROMAN CAPITALS.
Before entering upon a general discussion of all the varieties
of letters, we will make a special study of the common Roman
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 designing fanciful letters.
PLATE I. The 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 embraces 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 letters which are largely made up of curved lines.
Letters, as large as those of this plate, may be made by instruments, 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 P1. I. 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 P1. I. will now be sufficient. I, the simplest of all
the letters, consists of a vertical column, whose width may properly 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 proportions may be observed in the wide parts and caps of all the
letters.
We thus have for an I the following complete proportions:
Divide its height into sixteen equal parts. Then its height -
16, its total width s-, width of column 4, projection of cap 1-,
and thickness of cap A-. These dimensions are to be preserved
in the vertical columns of all the letters. Also all wide columns
are to be of 4-4perpendicular 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
42 FREE-HAND GEOMETRICAL DRAWING.
Each column of the H is like an I. The extreme width of
this letter allowing -1 between the caps is equal to K — of its total
height.
The height of the arm of the L is 7 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 4- of the height. The
ends of the arms must be 1y thick. F is like an L turned upside 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 extreme right-hand line of the letter. E differs from F only in
having another arm. Some designers make this letter a little
wider (14) 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 1- of its total height. Notice, on all these
arms, that their curved sides are nearly quarter circles, 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 elements. V having its average width only equal to half its extreme 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 k between
their caps. This width will then be 1-_8 of the whole height
Let the perpendicular width of all narrow columns be 6, and
the horizontal width of V and A at their points I.
Observe, that the left hand column is the wide one, and
that in all letters having slanting columns, except Z, the heavy
column slants downward 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 vertical 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 1-\7 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
LETTERING. 43
to V and A. The length of full caps to oblique wide columns
being Ad, and to vertical narrow ones -Ad, the total width at top,
allowing I between caps, if there were a full cap at the left
upper corner, will be 16y. There is no cap at the lower righthand 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
- of its total height. The point of the V-shaped part is on the
bottom line, and midway between the inner lines of the adjacent vertical columns. W, the widest letter of the alphabet,
is of an extreme width equal to 2y76 of its extreme height. Its
oblique lines are parallel to the corresponding lines of V. The
extreme width of Z is equal to \-14 of the height. Its arms are
lengthened, as there are no caps opposite to them. The lower
one is 1 0 long and 8- high, the upper 9- 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 1-5 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 1 7 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 1 of its
height. Its larger curve, convex downward, has for a chord a horizontal line, at a height above the bottom equal to 5g of the height.
The extreme width of U is 14 of D', of P, and B 1 5 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 1- beyond the upper one. R is 1 6 wide at bottom.
It differs 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 threefourths of the same width below the body of the letter. In
either case the extreme width equals 14 of the height. The
extreme width of C equals 1 5. The highest and lowest points of
its outer curve are in the middle of the extreme width; and the
44 FREE-HAND GEOMETRICAL DRAWING.
corresponding points of the inner curve are half way between
the inner point of the lower curved arm and the vertical tangent
to the inner curve. In a letter as large as this, it is well to let
the upper 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 necessary 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 P1. 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 modification of the design, ending in a large circle.
The proportions here given are not absolute, but only relative.
Thus an ordinary letter, as an H, 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, expanded and condensed letters.
Directions, much more minute than the preceding, are sometimes 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 in
groups.
LETTERING. 45
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 letters may be included in two grand divisions.
I.-Geometrical letters are all those which have a definite
geometrical outline, which could be made with drawing instruments; and
II.-Free-hand letters, or those of so irregularly varied outline that they must be made by hand only, guided mainly by
the fancy of the designer.
Since the letters 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., will have
materials for a valuable scrap-book of letters, which may be
arranged according to the classifications presently to be given,
and which will be useful for reference, and will contain numerous practical illustrations of the explanations which follow.
By examining such a collection, it will be seen that in all ordinary letters three things may be distinguished(a) the essential elements.
(b) the complementary additions.
(c) the decorations.
By the essential elements of letters, are meant those portions
which are necessary, and sufficient, to enable one to recognize
those letters. The first half of each of the first two rows of P1.
II. are letters formed of essential elements only.
By complementary additions, are meant the caps, and the
hanging parts of the arms, etc. The letters of the first four
rows of P1. II. 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 section. 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.
Straight, including polygonal letters, as 0
Nature - - - Ca CD
Curved. rtC
e Rectangular, as - H. CD
Simple - - - - - ~' Oblique, as — *-V.
I * ^CD
Curved, as - -S.
Combination- CD
r Rectangular and -.
Mixed, embracing Oblique, as.
Rectangular and lp,
GEOMETRICAL,
GEOMETRICAL, Essential, as in A. Curved, as- - - -..
whose elements CD
are in their Importance - Complementary, A
Ias in -A *'' In single lines, as ECD C
% E qu alIn double lines, E
as - -CD
mQ Las —- - 19as0' Narrow, elements in Co 2.
~W q ~ ~ Relative size- Esni I single lines, as - o -
E- Essential elements r.
E-4X^~~~~ widest, as in - A. - o
z^~~~~~~4 Cmlmnay ele- I Narrow, elements in
Unequal...- -- CGomplementary ele-l double lines as - \
FREE, as Ger- ments widest, as lm ~
man Text, Rus- Plain. (Italian type.)
tic, etc., Letters Finish an type.)
decorated with Decorated
L free ornaments. (geometrically.) I
LETTERING. 47
It follows from this, that there cannot be very many radically different forms of letters; therefore, before proceeding to
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 given.
1~. By altering the proportions of height and width, forming
ea2panded or condensed letters.
20. By retaining or omitting the complementary additions.
3~. By making the wide columns of the letter massive or
slender.
4~. By making the letters as if they were fat 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 outlines, 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.
(a.) The thickness of the caps is the same as that of the narrow essential elements.
(b.) In pencilling letters, never pencil the ornaments, unless
the letters are of extraordinary size, but pencil the outlines
only, in very fine lines.
(c.) It is better to do all the pencilling by hand, since instruments would perpetually be hiding portions of the letters, and
so preventing the eye from judging readily of their proper proportions.
(d.) 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.
(e.) The fifth row of P1. II. shows a simple free-hand or
" rustic" letter, in two sizes and styles. These are barJc letters.
Log letters are often seen in handbills, etc.
f.) The sixth row embraces " skeleton" and "full faced"
" small " Roman letters and italics. A common error consists
in making the stems of the b's, p's, etc., too long. The total
48 FREE-HAND GEOMETRICAL DRAWING.
height of such letters need not be more than one and a half
times the height of their bodies.
(g.) 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.
(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 appreciable 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.) P1. II. illustrates most of the distinctions of form mentioned in the preceding table, except the inelegant and unused
Italian type. This plate, or one of similar nature, should be
constructed by the student.
(1.) 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.
PL. I.
l^YXLLXI~~r _\JCXC A)L I
IFTIIP^~~~~~~~~~~~~I)~
A^ U ^J L~~~~~ll~~li~O)
PL. II.
ABCDE FGH IJKLM NOPQRSTUVWXYZ &
GfWSHVY (
Ibacefghlj L ii- op c ^t-Ivwxyz a/ io / Jjk II i
CHAPTER IX.
GEOMETRIC SYMBOLISM.
Definitions.
A symbol is anything apparent to sense, which yet, of itself,
naturally expresses, represents, or suggests to the mind some
truth of life.
In this, a symbol is quite different from an emblem, or
a type, as may be sufficiently seen by reflection on the
universal use of the words. Thus every one says " the national
enblem," 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 may be equally
strong, whatever the thing chosen may be; but it does not
depend on any inherent relation of the thing to the thought,
being established in the act of choosing the emblem.
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.
To illustrate: The mingled verdure and bloom of spring, are
symbols of the freshness, modesty and promise of unperverted
youth. The tints and fruits of autumn, or a sunset in crimson
and gold, 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 symbol of a broken life. The American flag is an
emblem of the nation's life. Its rivers are the symbol of the scale
of its life, its ideas, and its actions. Its best, and its worst, treatment of the Indians, are types of its highest and of its lowest
humanity seen in all other relations of life.
Other illustrations. Water, by its properties, is a type of
fluids generally. The ocean is of itself a symbol of eternity. A
ring placed upon the finger becomes, by the manner of the act,
an emblem of whatever put it there.
4
50 FREE-HAND GEOMETRICAL DRAWING.
The oak, with its mighty and horizontal arms, is a symbol of
all sufficient rugged strength. The elm is a symbol of united
strength and grace, of cultured rather than native qualities.
Hence the avenues of cities are lined with elms, rather than
with oaks.
With the idea of symbols thus awakened, the following exam.
ples of geometric symbols will suffice to lead the mind into
action upon the subject.
GEOMETRIC ILLUSTRATIONS.
A straight line is the symbol of repose, monotony, and death.
It is so by reason of its monotony of form, 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 inaction 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 proportioned
forms. Massive 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
their immense stones, all illustrate this.
Also, in foundations generally, where permanence is most
desirable, the main lines are mostly straight and horizontal.
But in a church, the multitudinous flowing lines should only
express the endlessly varied, yet only beautiful, individual and
concerted life, that should, visibly, centre in, and flow from the
stirring exercises and activities within it.
The circle is a symbol of monotonous routine, and hence, as
a symbol of eternity, represents only a dormant, unprogressive
one. It is thus, by reason of its single centre and uniform distance from that to the circumference, and its consequent uniform rate of variation of direction at all points, and its perpetual return to the same point of beginning.
Hence it was peculiarly appropriate that the Egyptians,
whose earthly life was so largely expressed by the stiff, dead
straightness of a right line, should have adopted the circle as
their symbol of eternity, an eternity of endless dull repetitions
GEOMETRIC SYMBOLISM. 51
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.
Again, life is either sensual or spiritual; and, in a given
amount of it, as the one prevails, the other is wanting. Now
monotony of life indicates absence of thought-activity, and hence,
secondarily, the circle as the symbol of monotonous routine,
unenlivened by varied thought, is also a symbol of sensuous,
more than of intellectual existence. Hence the Romans, who
were a grosser, and more materialistic people than the Greeks,
made great use of the circle in their architecture, while the
Greeks rejected it.
Thus the coarseness of the compound circular moulding is
apparent in contrast with that of the freely varied curve of the
second figure,* whose infancy quickly turns at maturity into a
prolonged career of elegance, gracefully and quickly brought
to a close, when its work is done.
The ellipse being only the general form, of which the circle
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 governing 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 are less
pleasing than semi-circular topped ones.
Quite otherwise from the foregoing is it with the yperbola,
* "Greek lines," Atlantic Monthly, June and July, 1861.
52 FREE-HAND GEOMETRICAL DRAWING.
which is sufficiently defined for present purposes by saying that
it consists of two equal, opposite, and infinite branches, AF and
BG, to which a pair of lines, M and N, crossing at the centre C,
are tangent only at an infinite distance. Such lines are called
asymptotes. The fixed points H and K are called itsfoci, each
one, afocUs.
C R
The complete symbolism of this line is remarkable for its
ready and striking truthfulness.
The general idea of the infinite approach of a curve to a
straight tangent, as a symbol of an infinite progress towards
perfection, or the absolute ideal, never actually attained, has
long been familiar; but is realized in the case of any of the
many very different curves which have asymptotes. But the
complete symbolism of the hyperbola has perhaps never been
defined.
First, and for this world only, as proper to be mentioned here,
there is material civilization, and there is moral or spiritual
civilization. Also there is material barbarism and there is
moral barbarism. There is the material civilization of Paris,
and Berlin and London, and there is the moral civilization of
bhe Quaker, and the philanthropist of New England, and of the
little mountain democracies of Switzerland.
There is the material barbarism of the savage, and the hideous, appalling and exasperating moral barbarism of the hordes
who are only all the more savage in nature, as they are more
acquainted with arts and opportunities, which they pervert to
savage uses, and who, as such, have blotted the annals of the
world; as in the early chapters of the history of European enterprises, particularly in South America, Africa, and Asia.
GEOMETRIC SYMBOLISM. 53
Then, in the hyperbola, asymptote M may represent material
civilization or perfection, or material barbarism either individual or social, and in opposite directions, as right and left, respectively, from C. Asymptote N may likewise represent the
ideal of moral civilization, or of moral barbarism. Then, as
each branch of the curve is divided symmetrically by the line or
axis ED, if DA and DF, tangent to N and M at infinite distance from D, represent infinite progress towards the perfect
T-oral and material ideals, respectively, then EB and EG, tangent in like manner to the same lines, may represent unlimited
progress in material and moral evil, degradation and the bad.
Thus we have the crowning antithesis of the good and the bad,
both material and moral, with unlimited possible progress towards each destiny, and in each form; all, expressed by the
hyperbola, without obscurity, or confusing admixture with
other considerations.
The parabolac, each of whose points is equidistant from a
fixed point, F, the focus, and a fixed
line, D, the directrix, has but one
branch, and no asymptotes. Whatever symbolism it may have, inde-
pendent of the hyperbola, may not D _
be yet apparent. Compared with
that, its single branch may signify a
career only good, and both material
and spiritual as before, since it has
bilateral symmetry about the axis DF.
The absence of the asymptote is expressive as showing that,
being perfect in itself, it needs no outer standard to which to
approximate; or conform itself, but is only perfect, though
free.
As having a governing point within itself, while also conditioned by the line D, which is isnfinite, it is a symbol of the
normal life of any perfect finite rational creatures; conditioned
in their action by their own wills, and yet, in the final results
of their action, by the will of the infinite.
This attractive image, from which the hateful thought of the
bad is excluded, makes the gothic arch, formed of two opposite
semli-parabolas, a beautifully appropriate entrance to any place
where the ideal is to be sought, enjoyed, and if possible so
54 FREE-HAND GEOMETRICAL DRAWING.
realized; at least in thought and purpose, as to determine the
progressive improvement of the life.
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.
If, however, aesthetic thought in this department should ever
lead to the general adoption of an educational symbolism, then
buildings for the successive departments of any large institution might be arranged on a spiral, from the preparatory school
to that in which resident graduates should remain to pursue
special studies.
The conchoid, however, exceeds in ready and convenient expressiveness of grand fundamental ideas, any curve yet tried in
the field of educational symbolism.
The conchoid is a curve of two parts, or branches, all of whose
points are at the same fixed distance from a given line, meas
GEOMETRIC SYMBOLISM. 55
ured on lines drawn from a fixed point. When this point is
nearer the fixed line than the fixed distance, one branch of the
curve will be looped. Thus E E is the fixed line, and A the
DO \
P1\!
t-p
fixed point. Then d C d e; b a= b c, etc.
Now the primary ideas to be expressed in the organization
of a comprehensive institution, are: 10. A Central Course of
56 FREE-HAND GEOMETRICAL DRAWING.
Stud.y, from which various professional courses shall radiate;
20. A personal directing body; 3~. A series of buildings devoted to the professional courses; 4~. A series of inferior
buildings devoted to the lower purposes of organization; 5K.
A select group of structures devoted to the most refined
purposes of the institution. See now how perfectly the conchoid, when laid out on a grand scale on the ground, permits
the symbolical expression of these ideas, in the material orgalization of an institution, as its published curriculum exhibits
them in the printed expression of the logical organization.
1~. A grand building, surmounted with a dome, as the symbol 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 courses.
20. Professors, as the personal determining element in the
life and work of an institution, should have residences 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 professional schools, and reached from A by paths on the radial lines
as a b c, which determine the points where they stand.
4~. B B, being the inferior branch of the curve, should be
devoted to the gymnasium, janitor's lodge, bathing house,
society (open) rooms, etc.
50. The loop A e, as a separate and peculiar feature, should
be set apart for an elegant garden enclosure, with fountains.
etc., and faced by the observatory, chapel, and library buildings.
It may be added that, having regard to the beauty of a varied
over a monotonous curve, like the circle, the superior branch of
the conchoid would afford superior beauty as a reverse curve for
the caps of doors or windows.
The bilateral symmetry of the conchoid, that is, the equality
of the parts on each side of the line, C e admits another point of
significance.
In a university, for example, such as some' believe in, of all
embracing comprehensiveness, schools for the professions based
immediately upon the constitution of man, viz., Theology,
Law, Medicine, Politics, and higher Teaching, might be located
GEOMETRIC SYMBOLISM. 57
on one side of C e, while those of the professions based on
the constitution of nature, as adapted to subserve man, as
Engineering, Architecture, Mining, etc., might be on the other
side of C e.
Likewise the hyperbola, is a ready servant of educational symbolism, and perhaps even more perfectly, for the same purpose.
For, besides placing the first, or " humnanistic" class of professional schools on one branch of a hyperbolic avenue, laid out
upon a suitable tract of ground, with its foundation general school
at the focus, H, of that branch, and the second, or technological class of professional schools on the other branch, with its
appropriate college of preliminary general culture at' the focus
K, a further subdivision is provided for. Each class of professions has its industrial, and its artistic subdivision; and the
symmetry of each branch with respect to the axis, K H, permits
the tangible expression of this subdivision. Thus, on the humanistic side, Schools of Theology, Law, Medicine, etc., could
be on D A, and schools of Poetry, Oratory, and Vocal Music
could be on D F.
Likewise, Schools of Industrial Technology, as of Engineering,
Building, Mining, and Technical Chemistry, could be on E B,
and schools of Architecture, Decorative Design, Instrumental
Music, etc., could be on E G.
Finally, those disposed to contend for the equal rank and
dignity of the two main classes, humanistic and technological,
of professional pursuits, would adopt the hyperbola, to determine the arrangement of the assemblage of buildings composing the general zaterial organization of the university. Those
who should claim superior rank for the humanistic c:lass, would
employ the conchoid, and place the schools of that class on its
superior branch; while those who would avoid all such rivalries, as well as the bewildering unwieldliness of so colossal an
organization, would be likely to employ the conchoid as first
explained, and for one class only of professions.
The positions of lines have a significance, as well as their
forms. Thus a prevalence of vertical lines symbolizes aspiration, upward-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.
9 8 FREE-HAND GEOMETRICAL DRAWING.
IHence the betrayal of offensive vain consciousness, or of obtuseness, either in the maker or beholder, in adding an up-pointing
hand to the tip 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 meaning to
what it sees.
A prevalence of horizontal lines, is expressive of a clinging
to the earth, as in the 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 predominence of the horizontal in the Greek temples, with their flat
roofs and horizontal mouldings, and flat door and window tops.
Once more, and in a derivative manner, horizontal lines express firrness, decision, stability, and hence are the proper
characteristic lines of foundations and supports. The repose,
or death, which they primarily signify, leads to the secondary
meanings, unchangeableness, and thence decision, or stability, as
stated. Hence the curved outlines of mouldings on supporting
parts best flow into the horizontal top and bottom surfaces of
such parts.
Thus, the first figure and the one below, show a better rela
GEOMETRIC SYMBOLISM. 59
tion of the curved contour, as tangent to the bases, than the
second figure does.
Carvings.-Work becomes so costly as soon as straight outlines are abandoned, and especially as carved work begins to
be employed, that its consequent difficulty of attainment makes
it symbolical of the grace and beauty that can only come from
above, or, of man's best aspirations; while the plain lines of
ordinary work represent, by comparison, humbler human industries. Hence a bit of choice carving to crown, or tip, or face a
piece of otherwise plain work, happily symbolizes the cheerful
co-operation of earth and heaven, the descent of celestial beauty
to welcome and encourage the efforts of man.
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.
An entirely different principle, however, governs the employment of ornamental castings from really rich and beautiful designs. I-ere, the thought is the nobly generous one of bringing to every humble home, by means of a beneficent art of
multiplication, beauties of decoration which could not otherwise
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 purchasecan own and enjoy them.
60 FREE-HAND GEOMETRICAL DRAWING.
The "ginger-bread" products of the scroll saw, from inch
boards, are mean in origin, material, and execution, and are
therefore to be discarded for their inherent demerits; but good
castings, from beautiful designs, inherit and partake of the
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 containing a carved bust or other form of life, may be mentioned
as symbolizing the escape of the spirit from the hindrances and
imprisonment of the body.
Without further illustration, it may now be enough to add
that the foregoing may serve to set the thoughts in motion upon
the line indicated, so that the student will freely give to all his
works an attractive and elevating meaning, at the same time
that they fulfil the bare physical conditions required of them.