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  [ Transcriber's Note:

    Every effort has been made to replicate this text as faithfully
    as possible, including inconsistencies in spelling and hyphenation.
    No changes have been made to the printed text.
  ]




  OF THE JUST SHAPING OF LETTERS

  BY ALBRECHT DÜRER

  TRANSLATED
  BY
  R. T. NICHOL

  FROM THE LATIN TEXT OF
  THE EDITION OF
  MDXXXV




  OF THE JUST
  SHAPING OF
  LETTERS

  FROM THE APPLIED
  GEOMETRY OF
  ALBRECHT
  DÜRER
  BOOK
  JJJ

  DOVER PUBLICATIONS, INC. NEW YORK




ALBRECHT DÜRER TO WILIBALD PIRCKHEIMER HIS PATRON AND VERY GOOD FRIEND
GREETING:


[Illustration]

In our Germany, most excellent Wilibald, are to be found at the present
day many young men of a happy talent for the Art Pictorial, who without
any artistic training whatever, but taught only by their daily exercise
of it, have run riot like an unpruned tree, so that unhesitatingly and
without compunction they turn out their works, purely according to their
own judgment. But when great and ingenious artists behold their so inept
performances, not undeservedly do they ridicule the blindness of such
men; since sane judgment abhors nothing so much as a picture perpetrated
with no technical knowledge, although with plenty of care and diligence.
Now the sole reason why painters of this sort are not aware of their own
error is that they have not learnt Geometry, without which no one can
either be or become an absolute artist; but the blame for this should be
laid upon their masters, who themselves are ignorant of this art. Since
this is in very truth the foundation of the whole graphic art, it seems
to me a good thing to set down for studious beginners a few rudiments,
in which I might, as it were, furnish them with a handle for using the
compass and the rule, and thence, by seeing Truth itself before their
eyes, they might become not only zealous of the arts, but even arrive at
a great and true understanding of them.

Now, although in our own time, and amongst ourselves, the Art Pictorial
is in ill repute with some, as being held to minister incitement to
idolatry, yet a Christian man is no more enticed to superstition by
pictures or images, than is an honest man girt with a sword to highway
robbery. Certes he would be a witless creature who would willingly adore
either pictures or images of wood or stone. On the contrary, a picture
is the rather edifying and agreeable to Christian religion and duty, if
only it be fairly, artificially, and correctly painted.

In what honour and dignity this art was anciently held amongst the
Greeks and Romans, the old authors sufficiently testify; though
afterwards all but lost, while it lay hid for more than a thousand
years. It has now at length, only within the last two hundred years, by
some Italians been brought again to light. For it is the easiest thing
in the world for the Arts to be lost and perish; but only with
difficulty, and after long time & pains are they resuscitated. Wherefore
I hope that no wise man will defame this laborious task of mine, since
with good intent & in behoof of all who love the Liberal Arts have I
undertaken it: nor for painters alone, but for goldsmiths too, & for
sculptors, and stonecutters, and woodcarvers, and for all, in short, who
use compass, and rule, and measuring line--that it may serve to their
utility.

Nor is anyone compelled whether or no to spend gainful hours on these
exercises of mine; although I am not ignorant that whoever is well
exercised in them will thence acquire not only the principles of his own
art, but by daily practice, an exactitude of judgment, with which he
will proceed to higher investigations & discover many more things than I
have here pointed out.

But since, illustrious Sir, it is clearer than light that you are
yourself, so to speak, an asylum of all the noble Arts, it has been my
pleasure, out of a singular love I bear towards you, to dedicate to you
this book; not because I desire to appear therein as rendering you any
great service, but because thereby you may understand how engaged my
mind is to you; and since by my work I can confer on you but little
favour, at least by the exhibition of a ready mind I may repay the
benefits you shower upon me.

Farewell.

[Illustration]




OF THE JUST SHAPING OF LETTERS




FROM THE APPLIED GEOMETRY OF ALBRECHT DÜRER

BOOK III.


Now, since architects, painters & others at times are wont to set an
inscription on lofty walls, it will make for the merit of the work that
they form the letters correctly. Accordingly I am minded here to treat
briefly of this. And first I will give rules for a Latin Alphabet, and
then for one of our common Text: since it is of these two sorts of
letters we customarily make use in such work; and first, for the Roman
letters: Draw for each a square of uniform size, in which the letter is
to be contained. But when you draw in it the heavier limb of the letter,
make this of the width of a tenth part of the square, and the lighter a
third as wide as the heavier: and follow this rule for all letters of
the Alphabet.

First, make an A after this fashion: Indicate the angles of the square
by the letters a. b. c. d. (and so do for all the rest of the letters):
then divide the square by two lines bisecting one another at right
angles--the vertical e. f. the horizontal g. h.: then, in the lower
line, take two points, i. and k., distant respectively one-tenth of the
space c. d. from the points c. and d.: then, from the point i. draw
upwards to the top of the square the lighter limb; & thence downwards
the heavier limb, so that the outer edges of both may touch,
respectively, the points i. and k.: then let a triangle be left between
the limbs, and a point e. be fixed at top in the middle of the letter,
and next join both limbs beneath the horizontal line, and let this limb
be a third as broad as the heavier limb.

Now let the arc of a circle, applied to the top of the outside edge of
the heavier limb, project beyond the square. Then cut off the top of the
letter with a serpentine or curving line, so that the concavity decline
towards the lighter limb, and prolong acutely either limb of the letter
at the bottom to either side, so as to meet the angles of the square at
c. and d.: this you shall make with the arc of a circle, whose
semi-diameter is one-seventh of the side of the square; but the two
lower curves, mutually opposite, permit to extend so that each is a
third of the heavier limb, and this you shall obtain by the arc of a
circle whose diameter is equal to the breadth of the heavier limb.

Moreover, this same letter A you may cut off at top with the side of the
square, and then produce to a fine point in either direction, as you did
the feet below, yet so that the longer production shall be to the
fore-side (namely, the left); but in this case it will be necessary to
draw in the limb k. a little closer.

Likewise the same A you may draw in yet another manner--that is, pointed
at top. In that case let the limbs slope towards one another yet more
closely; then lower the transverse a little and double its width. You
may also cut off the limb at top bluntly, or sharpen it on the
fore-side. You ought to make yourself familiar with these three forms,
or whichever of them pleases you best.

And note likewise that in exactly the same fashion in which this letter
is acutely prolonged at top & bottom, are the other letters to be so
prolonged which are drawn with oblique lines, as V, X, Y, although a few
changes may be necessary, as you shall hear below.

I have here subjoined an engraving of this letter.

[Illustration: A]




OF THE LETTER B.


And now you shall draw B in its square thus: First divide the square
horizontally by the line e. f.; then bisect the lines a. e. and b. f. by
the line g. h. Next, you must first set properly the broad vertical limb
of the letter, distant its own breadth from the side a. c. of the square
a. b. c. d. Then erect the line i. k. on the inner side of the limb
already drawn, and distant from it one-tenth of a side of the square,
and let it cut the line g. h. in the point l.

Next, draw strips narrower and horizontal (to be produced hereafter into
the convex limbs) from the vertical band to meet the vertical line
i. k.--namely, at top, below the line a. b.; next, above the line e. f.;
and at bottom, above the line c. d.

Now set a leg of the compass on the point l. and describe a semicircle
to the right of the transverse strips, so that the extremities of the
circumference, in the vertical line i. k., below the side a. b., and
above the line e. f. may coincide with those short transverse lines.
Then bisect the narrow transverse strip which is above the line e. f. in
the line i. k. by the point m.; and indicate the breadth of the letter,
to the right of the semicircle, by the point n. in the line g. h.; and
afterwards draw from the point m. above the line e. f. in the direction
of f. a short horizontal line as great as need be: then describe a
semicircle which shall include this line, and the point n., and, at the
top, the side a. b.; and through n. let pass a vertical line. These all
combine to form, below, the concave of the curved limb, and above, its
convex.

Next, produce the transverse strip above c. d., in the direction of d.,
as far as required, and mark this q. Then bisect the line m. q. by the
line o. p., cutting the line n. in the point r.; and next describe a
semicircle touching the horizontal line e. f., the point r., and the
position q. Then indicate the breadth of this limb of the letter by the
point s. to the right of the point r. in the line o. p. and describe a
semicircle, touching the line m., the point s., and the side of the
square c. d. There will then remain in the letter three right angles to
be eliminated: the interior and lower one may be shaped into a curve by
a circle whose semi-diameter is two-thirds of the breadth of the broad
limb of the letter, and the exterior ones you shall fine to a point by
circular lines whose semi-diameter is equal to the breadth of that limb.


Another method.

Or you may make your B in this fashion: Let the side a. c. of the square
be divided into nine equal parts, and cut off the four superior parts by
the horizontal line e. f. Then erect your vertical limb as described
above; and the superior curved limb you shall make between a. b. and
e. f.; the inferior between e. f. and c. d.

Now divide a. b. into nine equal parts, and cut off four parts towards
b. in the point g.; then divide c. d. into five equal parts, and the
last, towards d. mark off in the point h. and join g. and h. by the line
g. h. which should touch on their exterior edges the superior and
inferior limbs of the letter. Now these limbs must be drawn of a
particular form; and the compass, in drawing the circular lines, must be
moved up and down their diagonals: and these two diagonals you shall
determine in this wise.

Divide a. e. into four parts; the lowest, above e., call i. e.; the
lowest of the five remaining, above c., call c. k. Then join the points
i. and b. and k. and f. respectively, by the lines i. b. and k. f. Upon
these lines move and turn your compass, & in this way you shall describe
both curved limbs: and they must both be broader towards the top than
towards the bottom, as follows naturally with the stroke of a pen, and,
moreover, while approximately round, they are not to be circular;
therefore you will have to move your compass at need along the
diagonals, and withal to assist it also with the hand, as I have done in
the picture on the following page.

[Illustration: B]




OF THE LETTER C.


Next you shall make the letter C in its own square thus: Bisect the
square a. b. c. d. by the horizontal line e. f. and in it let i. be the
middle point. From this point as the centre, & i. f. or i. e. as the
radius, describe a circle touching interiorly all four sides of the
square. Now move the leg of the compass, but without varying its span,
to a point k. a little to the right of i. in the line e. f., letting the
space i. k. denote the greatest breadth of the letter you desire; & from
the centre k. describe another circle which shall cut twice the line
b. d., and whose circumference to the left will mark the required
breadth of the letter. Next, draw the vertical line g. h., parallel to
b. d., distant from b. a tenth part of the line a. b. This will cut off
for you at top and bottom the letter C as the ancients were accustomed
to use it. But I would have you cut off the lower limb in the middle
point between g. h. and b. d.: then make the limbs somewhat finer and
rounder on the inside towards top and bottom from the point where the
circles intersect; and for its greater perfection round out the letter,
above and below, to touch the sides of the square a. b. and c. d. Next,
low down, where the letter with one foot crosses the line g. h., there,
under the circular line make the form a little more incurved, yet so
that with the tip of its end it shall again touch the circular line.
Similarly, but higher up, make the foot more hollow on the inside than
the circle left it: and thus two circular lines will give you very
nearly the whole form of the letter.


Another method.

Or, secondly, you may make the letter C thus: Draw in the square a
diagonal c. b.; set the leg of your compass on its middle point i. and
with the other leg describe the exterior circle as before, terminating
it above at the diagonal c. b.; but below, make your circle pass a
little beyond the former sweep. Then set the leg of your compass, but
without changing its gauge, as far above i. in the diagonal as the
letter's greatest width, and describe your inner circle; and, as though
made with a pen, let the descending stroke be heavier than the
ascending. The rest you may elaborate with your hand; & let the trimming
of the ends of the letter, above, slope upwards, & below, downwards,
exactly as I have here drawn the shapes.

[Illustration: C]




THE LETTER D.


The letter D you shall make thus: Divide its square by the perpendicular
or vertical line g. h. and by the horizontal line e. f. into four small
squares, and call their point of intersection i.: then draw the broader
limb of the letter from the side a. b. downwards, to meet the side c. d.
and at the distance of its own width from a. c.; and produce the limb at
top and bottom to a sharp point at the angles a. and c. as was shown
above in B; using the same method in all straight limbs in the remaining
letters. Next you are to produce from this limb two narrower tracts
horizontally, and from these are to be described the circular arcs of
the letter between the line a. b. at top and the line c. d. at bottom,
and extending as far as the perpendicular g. h.; next, with your compass
join g. f. h. Then, in the line e. f. lay off a portion equal in breadth
to the widest limb of the letter, at the point k.; next, set one foot of
your compass on k. and let the other cut the said line e. f. in l.; let
this be the immovable leg of your compass, and with the other, beginning
from k., describe internally, to the narrower transverse limbs, an arc
which shall touch both, completing your acute angle above, but rounding
out the lower one by a circular arc of the same diameter as the one by
which you sharpened your exterior subtending angle.


Another method.

You may make the round limb of the same D in another fashion; namely, as
a pen naturally would, broader above than below. For this, draw the
diagonal c. b. and describe your exterior arc as before; but to describe
the interior, in the line c. b. take a point m. lower down than i. and
distant from it the width of the broader limb, and without altering your
compass describe an interior line; but where the limb must needs be
narrower, there you are to accommodate it with your hand, both below and
above, as in the following cut.

[Illustration: D]




THE LETTER E.


The letter E you shall form in its square thus: Draw a transverse line
e. f. bisecting a. b. and c. d. in e. and f.; then draw the great
vertical limb of the letter, to the left, as you did for D. Next draw
also an upper transverse limb of narrower dimensions, parallel to a. b.
and in length six-tenths minus one-third of one-tenth of the length of
a. b.; and the end of this bend downwards one-tenth of the length of
a. b. and use this as the diameter of the circle with which you round
out the inner angle of this extremity: then draw your narrow middle limb
parallel to the median line e. f. and above it, so that it may be
shorter than the upper limb by one-tenth of the length of a. b., but at
its terminus double as wide; & you are to round it out (in either
direction) by the arc of a circle whose diameter is one-sixth the length
of e. f. Now construct your lowest limb upon the line c. d., so that at
its ultimate angle it may exceed in length the upper limb by one-tenth
of c. d.; the cusp, however, you are to prolong beyond this part by
two-thirds of one-tenth part, and erect above it to one-sixth of the
length of c. d., and round out the same by a circle whose semi-diameter
is also a sixth of c. d. In like manner the final angle of the letter
you shall round out by an arc of the same circle by which you rounded
out the middle transverse limb: the other angles you are to leave acute,
as in the following cut.

[Illustration: E]




THE LETTER F.


The letter F you are to form in the same manner as E; except that you
shall omit the lower limb altogether, and, in its place, round out the
letter on both sides below, as you did E on one side only, as I have
shown you below.

[Illustration: F]




THE LETTER G.


Likewise the letter G you are to make as you did C, before described;
this, however, excepted: that in front (that is, to the left) of the
line g. h. is to be erected the broad limb of the letter, upwards from
the curve to the line e. f., and above it is to be rounded to a point,
on either side, as before was said; but below, both angles are to
remain.

Or, you shall form G in the following fashion in the said square,
divided as before: Draw the diagonal c. b. and set your compass with one
leg on the point i. and with the other describe an arc from e. to the
middle point c. d. and mark this point l.; in like manner also, describe
an arc upwards to the line a. b. so as to meet the perpendicular line
g. h. & mark that point z. Then, in the line g. h., take a point m. so
that the part m. h. shall be one-tenth of the line g. h.; then, with a
sweep of your hand join l. & m. with the curved line l. m. Next, you are
to draw from z. a line upwards, as broad as the standard of the letter,
but oblique and in direction midway between your circular line & the
perpendicular g. h. and from the extremity of this line you must draw a
curved line to meet a. b. at the point where your circular line touches
it. Next, cut off from the bottom of g. h. a part one-third of its
length, & indicate this by the point n., & to this height, from the
level of m. upwards, produce the broad limb of the letter, and let its
extremities above be finished in either direction, of the same size.
After this set the leg of your compass on the diagonal c. b. the breadth
of the standard of the letter above i. & at the distance e. i. describe
an arc, which above shall touch the exterior boundary a. b. but below
shall stop short above l.; & from this point you must with your hand
draw a line to the vertical limb at the height of m.

And the same you shall do above in drawing the narrower limb of the
letter, as seen in the following diagram.

[Illustration: G]




THE LETTER H.


The letter H is to be formed of two broad, great, & vertical limbs of
the height of the square, in such fashion that their extremities, being
produced exteriorly, shall touch the four angles of the square, a. c.
and b. d. respectively. Now in what fashion the projections of the
broader limbs of letters are to be rounded out at top and bottom and on
either side, you have been instructed; for in any letter you please, any
broad and vertical limb is to be depicted at top and bottom thrice as
broad as at its middle: provided always it is not joined to a narrower
limb. So when this has been accomplished, then draw your narrower
transverse limb upon the line e. f. as is shown below.

[Illustration: H]




THE LETTER I.


The letter I you are to make of a single broad vertical tract in the
midst of its square, touching the latter top and bottom; and of this, at
both ends, and on either side, you are to round out the productions or
projections as below is shown.

[Illustration: I]




THE LETTER K.


Now for K: You are to make the first tract vertical, in the same manner
as you formerly did for H; then draw another narrower limb from the
broader and erect one, so that it may, at its lower end, impinge
obliquely on the transverse line e. f. and above may ascend to the right
till it meet the line a. b., taking care to make it parallel to the
diagonal c. b.; and this, at the top, you are to produce in both
directions so that each production may represent a tenth part of the
line a. b. The hitherward projection you are to round out with a circle
of which the diameter must not exceed the breadth of the lesser limb;
but of the other arc, by means of which you round out the farther
projection, you shall make the diameter double as great as the diameter
of the arcs by which you have customarily hollowed out the preceding
extensions of the broad and vertical limbs. Next, from the narrow limb
so constructed draw in a downward direction another broad limb, so that
it too may be parallel to a diagonal of the square; & of this the
beginning is to be taken from the acute angle which the narrower limb
makes with the broad vertical limb, and let it be drawn with its
projection to the angle d., yet in this fashion: take two points this
side of d. after this manner, so that the first point may be distant
from d. the tenth part of the line c. d. & the second as far again from
the first; then let the said tract be drawn within the space which is
between the two points, but in blind and invisible lines. Afterwards you
shall add the extension, which you shall make this way: take before f.
in the line e. f., a point g. no farther distant from f. than the
breadth of the narrower limb; on this point set one leg of your compass,
& let the other be extended to the angle d., from which let it be guided
back along the broad but invisible blind limb: thence will result the
lower convexity of the tail you seek; but its upper concavity look for
in this way: divide f. d. in its middle point h.; on this set one leg of
your compass, and with the other describe an arc passing through d. to
meet the broad limb.

Or you may make K in this manner: First, let your broader vertical limb,
and your upper narrow one remain as they have been described, except
that the interior angle which the narrower limb forms with a. b. shall
remain acute, but the exterior one shall be rounded out, as has been
said. Then let there be drawn the lower broad limb, obliquely from the
angle which is included between e. f. and the vertical limb, and let it
descend to meet the side c. d. so that between d. and the limb the width
of the limb be left vacant; and the hither angle is to be left, but the
farther, towards d., shall be rounded out a little, as shown below.

[Illustration: K]




THE LETTER L.


As for the letter L, you shall make it by a combination of parts of two
of the preceding letters: namely, you shall make the first vertical
broad limb, as you did a while back in I; and to this join a foot as you
did at the bottom of E, when you made it. Such is L depicted below.

[Illustration: L]




THE LETTER M.


The letter M you shall form in two ways within its square. In the first,
draw the narrower limb of the letter vertical, to the right of a. c.,
distant from a. one-tenth of the distance a. b.: draw the other, &
broader limb, on the near side of b. d., also a tenth part of the whole
distant from b. & in such fashion that both limbs touch the square at
top and bottom; then, between the two, bisect the line c. d. in the
point e. and draw a broad limb from the inner angle of the narrow limb,
downwards to the point e., & next a narrow one upwards from e. to the
inner angle of the broader vertical limb; and the inner angles at top
you must not round out, but leave acute; the exterior angles, however,
at the top, and both exterior and interior at bottom of both vertical
limbs, you are to adorn with the customary projections, as you have done
in the preceding letters. You are to know, too, that when these letters
are drawn with a pen, they are to be described with a single stroke. But
for your guidance is this letter, in the manner in which I have
instructed you, depicted below.


Another method.

Another way is thus: Divide the side a. b. of the square into six equal
parts & mark off the two extreme parts, one at either end, by the points
f. and g.; then draw the inner and broader limb, with its point at e. as
above; and to this, in an upward direction, a narrower one, so that
between f. g. be left a vacant space, and so more readily the letter
slope forward. Then you are to draw the two lateral and vertical
limbs--the near and slender, and the farther broad one--at the top,
indeed, as in the first sketch, but at the bottom produce them to the
two angles c. and d. and finally add projecting cusps, as you were
instructed in the first M; but the projection below will pass beyond the
square at the points c. and d. Or you shall make M at top with acute
angles, in which case the lateral limbs will slope the more; or shear
them off obtusely, and in this fashion (whichever pleases you best) make
them as you see them depicted in the following diagrams.

[Illustration: M]




THE LETTER N.


Likewise the letter N you shall make in its square thus: First you are
to draw two standards vertical and slender, so that at top & bottom they
may touch the square, & that being produced, the nigh one at the bottom,
and the farther at the top, they may touch the angles at c. and b. Now
join these two by a broad oblique limb, running from the angle a. to the
point e., by which is denoted the remote side of the farther limb, where
you shall allow the acute angle to remain; but at the top, this limb,
produced beyond the angle a., you are to round out to a fifth part of
the length of a. b. This prolongation should incurve below, a fifteenth
part of the distance a. b. projected on two arcs, the upper one the
greater, the lower the less. For the lesser arc, therefore, you shall
take as diameter of its circle, a line the fifth part of the distance
a. b. and its centre is to be taken outside the square, so that the foot
of the compass may touch the tip of the extension and the angle a.; then
extend a little the feet of the compass, and shift its centre until the
arc touch both the tip of the part produced, & the broad oblique limb,
in the middle point between the side a. c. & the nearer of the two
slender vertical limbs.

Or you may make the letter N in such fashion that its upper nigh
extension shall remain within the square; or you may make from it an
acute angle as shown overleaf.

[Illustration: N]




THE LETTER O.


Now O you shall make this way in its square. Set in the square the
diameter c. b. and bisect it in the point e., so that e. may form a
middle point between the two points f. and g. which are to be your two
centres; and from each let a circle be described touching two sides of
the square; & where the circles cut one another, there with your hand
you must shape the slender outline of the letter to a juster proportion,
as below is shown.

[Illustration: O]




THE LETTER P.


P you shall make in its square in this wise. Divide the square
a. b. c. d. by the median horizontal line e. f.; then divide a. e. &
b. f. equally by the line g. h. Next draw, first the broad vertical limb
for this letter P, as you did a short while ago for K, and afterwards
erect the line i. k. the distance of its own breadth to the right of
your vertical limb; (here you must ever observe that in a lettered
square we speak of the angle a. as the "hither" angle, that is, to the
left; & the angle b. as the "farther" angle, that is, to the right).
Then where the line i. k. cuts g. h. call the point l., and next draw
two slender horizontal limbs, the upper below a. b., the lower above
e. f., from the broad vertical limb as far as the line i. k. Set one leg
of the compass on the point l., extending the other to the lower side of
the lower horizontal limb near k.; then describe an arc through the line
g. h. as far as the other slender horizontal limb of this same P, &
where it cuts the line g. h. set the point m. Next, on the far side of
m. measure the width of the large limb of the letter, along the line
g. h. to the point n. and let your compass be stretched so that with one
foot it may touch the line a. b. and with the other the point n.; then
set one foot of the compass on n. & the other on the line g. h. to the
right, in the point o., in which this foot is to be left standing
immovable, and with the other is to be described an arc, passing through
the point n. and touching the lines a. b. and e. f.

Or you may form the loop of this letter in the following manner. Set a
leg of the compass under the transverse g. h. in the line i. k., in a
place median between the line e. f. & the lower part of the upper
transverse of the slender limb, in the point p. and describe an arc as
before, passing through m. so that the loop will be acute at the bottom,
and its tip will end in the middle space between the line i. k. and the
broad vertical limb of the letter.

Or make this same P with a circular sweep, by shifting the compass upon
the diameter, so that that sweep may be broader at the top (as though
made with a pen) as will be shown in the diagram on the following page.

[Illustration: P]




THE LETTER Q.


Make your Q in its square in the self-same manner as was prescribed for
O; but add to it its tail thus: Draw a diameter of the square, the line
a. d., about which, starting from the curved outline, begin to draw a
long tail, producing it through the angle d. in such fashion that d. may
be in the middle of the thickest part of the tail; but where the tail
begins let it be a little narrower than in the angle d., where it should
attain its real thickness. Then let it be drawn out, beyond the angle d.
to the length of the entire diameter, and in a downward direction, yet
so that it curves while it slopes, & that its tip shall not fall lower
than a third of the side below the lowest side of the square, and shall
tend, as it nears the point, to grow sharper little by little, and at
length end in a very fine point indeed.

Or you shall give Q a shorter tail in this fashion, to wit: set your
compasses to the length of the side c. d. and draw a tail from the bulge
of the same letter, describing through the point d. its inner arc of the
same length as c. d., taking care that the tail bend upwards until it
again reach c. d. produced, in the point h.; then shift your compasses,
& with the other leg again describe from the bulge of the letter an arc
below d. & continue it until again it reach h., but in such fashion that
the tail shall find its greatest thickness at the start, as in the
following figure is doubly depicted.

[Illustration: Q]




THE LETTER R.


Moreover R you must make in its square just as was directed for P; but
then erect a right line q. r. through the middle point of the square, &
let it cut the exterior arc of the rounded limb in s., from which point,
downwards towards the angle d., let there be drawn a broad tract, almost
equal to that which you made above for the letter K., but this is to be
somewhat bent in, and so shaped by your hand that its tip, well formed,
may arrive directly on the angle d.

Or make R in such fashion that its rounded sweep, as though made with a
pen, shall be above broader, & narrower below. To accomplish this, you
must shift your compasses on the diameter q. e. & not allow the rounded
limb to touch the vertical one, as was described in P. Besides, the
oblique limb is to be deduced from the rounded one with a little more of
a curve; just as I have drawn overleaf.

[Illustration: R]




THE LETTER S.


Next, the letter S you shall make as follows in its square, a. b. c. d.
First draw the horizontal line e. f. and the median & vertical one g. h.
and let them bisect one another in the point m. Then choose the main
thickness of the letter, and set it in the line g. h. so that the point
m. may divide it, having one-third of the thickness below it; next, set
the lesser thickness, at the top beneath g., indicating it by the point
i., and at bottom, above h. in the point k.; and the thickness of the
letter indicate above by n. and below by l.

Next, set a leg of your compasses on the line g. h. in the mid-point
between i. and n., and with the other describe a circle passing through
i. and n.; in like manner, upon the line g. h. set your compasses upon
the mid-point of g. l. and describe a circle passing through g. & l.
Then once more set your compass on the same line g. h. in the mid-point
of n. h. and describe a circle through n. & h.; and lastly, in the
mid-point of l. k. you must set one leg, & with the other is a circle to
be described through these same points l. & k.; afterwards cut off by
vertical section the upper portion of this letter, so that the part thus
cut off may contain in its extremity the maximum thickness of the letter
and a third part besides, & also that its tip may project downwards so
far as to stand midway between the centre of the circle i. n. and the
side b. d.; in other words, let the tip be distant on the right, from
the circle i. n. the first third of the interval between the greater and
lesser circles.

Next cut the lower limb of the letter to the left, by a vertical line
through the mid-point between the two circles, & in such fashion that
the part so cut off may be a fourth part wider & higher than the upper,
and that its tip may rise to the height of the centre of the circle
n. h.


Another method.

Yet another way may you make the letter S. In the square a. b. c. d.
bisect the horizontal line e. f. in the point m.; then set one leg of
your compass upon the mid-point between g. and m. & with the other
describe a segment of a circle in the direction of a. e. passing through
the points m. and g.; next, set your compass upon the mid-point between
m. and h. and describe a segment of a circle through m. and h. in the
direction of f. d. The two arcs will touch above, in front, and below,
in the rear, the exterior curvatures of this same letter S.

Next, draw through m. the diameter c. b. and at its middle indicate the
maximum thickness of the letter by the two points p. and q. from which
let there be drawn two right lines, one up, & one down, to those two
arcs; & next, from the two points p. & q. draw two curved parallels to
the same arcs, regulating the distance between them, their elevation &
depression from the centres of the same circles. Next, indicate below g.
and above h. the minimum thickness of the letter; and from these points
you will with your hand fashion the inner shape of the letter, both
above and below, & produce the limb of S, above towards b. Cut it off so
that its lower tip may touch the segment, & that the part cut off
upwards may contain a tenth part of a. b. and that the segment may still
exceed the part cut off. Then construct a vertical line r. s. to the
right of e. c. and distant from it a fifth part of c. d.; let it cut the
diagonal c. b. in t. and to the angle just formed produce the extremity
of the letter, making the part so cut off a third broader than the upper
portion. Lastly, you will have to produce the tip ever so little beyond
t.; as I have briefly indicated.

[Illustration: S]




THE LETTER T.


Set the broad limb of T in the midst of its square erect, produced &
drawn to a point on either side below, just as you did before in the
letter I; then take two points e. and f., distant respectively one-tenth
of the whole space from a. and b., and let the transverse limb of the
letter be drawn below e. f. and of an equal length with it; but the
projecting extremities of this line are to be cut obliquely, and the
tips of these projections shall so far extend above the line a. b. to
the right as below they depend to the left. The oblique lines of these
projections are to be each a fifth part of the length of a. b.; & the
angles of these projections you shall round out by means of circles of
diverse radius--namely, for the lesser angle you are to use a diameter
only two-thirds of the width of the broader limb; but for the greater
angle you shall take a diameter equal to the side of a square contained
between the broad and vertical limb and the intercepted portion of the
line a. b.


Another method.

Or you make T thus in its square: Take your point e. as before, to the
right of a., and cut your transverse limb diagonally, as before, yet so
that the projection be dimidiated to the right, and at top the angle
remain as it falls; and so at the other extremity, only the point f.
must be moved as near again to b., the cutting line to be a little more
erect, & the projection formed a trifle broader than at the hither end;
otherwise shall everything remain as before; as I have delineated for
you on the opposite page.

[Illustration: T]




THE LETTER V.


V you shall thus make in its square: Bisect c. d. in the point e.; then
set the point f. one-tenth of the whole line a. b. beyond a., and in
like fashion g. to the hither side of b. Then draw the broad limb of
your letter downwards from f. to e. and sharpen it; & thence draw
upwards your slender limb to g.; and at the top produce it in either
direction, as you did before at the bottom of A; just as you see it
shown below.

[Illustration: V]




THE LETTER X.


X you shall form thus: Draw two vertical lines e. f. and g. h. distant
respectively one-tenth part of the line a. b. from the sides a. c. and
b. d. Then draw the two limbs intersecting one another in the form of a
cross--the broad one so that at top, & with its hither side it shall
touch e., & at the bottom, and with its farther side h.; but the narrow
limb so that at top, and with its farther side it may touch g., & at
bottom, with its hither side f. Then add its projections, touching, at
top and bottom, the four angles a. b. c. d., & choose a semi-diameter of
the larger circle of the length of a fifth part of a. b.; & with that
you shall round out the four greater angles; but for the lesser circle
you shall take a diameter as long as two-thirds the width of the broader
limb.

Or you may vary X thus: Let everything be left as before except the
narrower limb, which at top you shall make more erect by one-half the
breadth of the wider limb; and so the upper part of the letter shall be
less and narrower than the lower, and shall have a different aspect, as
is shown below.

[Illustration: X]




THE LETTER Y.


Y you shall achieve in the midst of its square, as far as its lower half
is concerned, after the instructions before given for I; but its upper
part you shall divide so that its hither limb shall contain two-thirds,
and its farther one-third of the broad standard; and let them slope to
either side so that produced they may touch the two angles a. and b.;
and the greater circles, by which you are to round out their obtuse or
greater angles, make of a diameter as great as a containing side of the
square enclosed between the standard and the sides of the great square,
as in T was shown; but the diameters of the circles which you apply to
the lesser angles, make double the width of the broad standard, as
below.

[Illustration: Y]




THE LETTER Z.


Z you shall make thus: Set upon either side, both beneath & beyond the
angle a., two points e. & f., each at a distance of the tenth part of
a. b.; so also, set two other points g. and h. both before and above the
angle d. and with right lines join e. f. and g. h.; then draw your
narrower transverse limb, beneath a. b. backwards as far as the angle
b.; from thence draw your broad limb diagonally to c.; and then again a
narrower one from c. to g.; and with your hand round out the two tips e.
and h.

Or make Z thus: Divide the square a. b. c. d. by the vertical line e. f.
and in this reduced space construct the letter as before; but so that
the two transverse limbs be cut short, above on the nigh side, and below
on the far, by the vertical lines a. c. and e. f. respectively as below.

[Illustration: Z]

So likewise, in other fashion, can we make all the letters already
drawn, on a scale of ninths, just as we have now drawn them on a scale
of tenths; in just the same manner, according to the due proportion of
each, in its own square, a. b. c. d., dividing them into nine, as just
now into ten parts; & that this may be the better understood, I have
chosen to append here letters of such fashion. Also these letters are to
be made five parts high when written small & rapidly, by hand. In such
writing the versals are made of the same proportion and form, but
one-third larger than the ordinary letters of the writing.

  [Illustration:
  A A A A
  B B B B
  C C C C
  D D D D]

  [Illustration:
  E E F F
  G G G H
  H I I K
  K K L L]

  [Illustration:
  M M M
  M N N
  N N O O
  P P P P]

  [Illustration:
  Q Q Q
  R R R
  S S S T
  T T V V]

  [Illustration:
  X X X
  Y Y Z
  Z Z]




DIRECTIONS FOR THE CONSTRUCTION OF THE TEXT OR QUADRATE LETTERS


The letters which are usually called "text," or quadrate, it was
formerly customary so to write, although they are now imitated by the
new art, as presently I shall show below. Although the alphabet begins
with the writing of A, yet shall I (not needlessly) in the first place
undertake to draw an I; because almost all the other letters are formed
after this letter, although always something has to be added to it or
taken away.

First make your I of equal squares, of which three are properly set one
over the other; and the top of the top one, and the bottom of the bottom
one, divide in two points, that is to say, into three equal parts: then
set a square equal to the others in an oblique manner, so that its
diagonal be vertical, and its angle on the first point of the top
square. In this way, this oblique square shall extend with its angles
more to the left than the right. Then produce upwards on either side,
after the width of the superposed squares, right lines to meet the sides
of the oblique set square. Next do below precisely as you did above,
except that you must set the angle of the oblique square on the second
point, that is, the one farthest to the right in the bottom of the
lowest square; and let fall your lines on either side upon the
transposed square: so will I be perfect; only above it draw with a fine
pen a tiny in-crescent.

So shall you make N from two standards of this same I, set so that their
angles at top and bottom touch; and in this manner the space between the
two shall be narrower than the breadth of either: also, you shall no
longer put little crescents above them; you must make of the same length
all the short letters throughout the alphabet.

In like manner make M of three standards, just as you made N of two.

R make as I, except only that at top you must set an equal square
diagonally, to the right, so that angle touch angle. R you may make also
in this fashion: below leave its foot as before, but above add two
diagonal squares, which shall touch each other with their angles in the
middle point of the vertical limb, and then produce upwards both sides
of the latter to meet the diagonal square.

V is made in three ways. First let it be made simply, as N; only that in
the farther limb, you shall omit at the top the diagonal square; and
instead shall draw an oblique line, so that it may make two angles in
this limb (produced) of which the farther shall be of the same height as
that of the topmost angle of the diagonal square, & the higher angle of
the same height as the angle nearest to it in the said square.

The second V, which we use at the beginning of a sentence, make thus:
Draw the first limb as before for I; only, at the bottom, push the
diagonal square a little further to the right, so that its hither angle
does not project beyond the side of the vertical limb, but falls in the
line of its descent. Then set the second limb to the right of this, and
cut it off below by an oblique line, drawn from the lowest angle upwards
to the right, as far as the middle point of the lowest of the three
superposed squares.

Next make W (i. e., double V) just as you made V-simplex; only you shall
set before it the standard limb of I.

B make as the second V-simplex; but in the first upright omit the
diagonal square at the top, and set upon the three original squares
three others similar, but the seventh one you shall cut off diagonally
from its lower hither angle.

Likewise when your B so made is turned upside down, then it will be a Q.

X you shall construct from I. Append from top angle to the right a
diagonally set square, as you did before in R; and at the bottom draw an
acute tail to the left from the diagonal, and at the middle of the
vertical limb describe a transverse, in such way that the former is cut
before and aft by the latter's diagonal; let the hither and lower angle
be terminated as far in front of the upright as would measure one-half
of the cutting diagonal, which at top shall just touch the upright; but
to the right let the transverse at top project to a point just below the
angle of the oblique square; from thence downwards let it be cut off by
an oblique line parallel to the anterior diagonal.

C you shall construct from I after this fashion: Remove the top diagonal
square, & let lines be produced on either side to the proper height of
the letter, and cut off the hither angle by a diagonal; then draw at top
a broad transverse, projecting beyond the vertical to the right the
width of the latter, and cut this off by a diagonal in such a way that
it project below only half as far as above.

The vertical standard of E you shall make as for C; but from above let
there descend to the right a broad limb from the diagonal bisecting the
right angles of one square, and one-third again as long as broad; and
let there be drawn from its lower angle a small diagonal line to the
vertical limb.

T shall be made like C, except that at top something is added to its
diagonal, so that its tip converges to a fine point, and the like to the
left on the hither side of the broad standard, just as at the top: and
because of this is T at top more elegant than C, and has not the same
incurved appearance.

L you are to make below like I; only six squares are to be set on end;
then cut off the hither side of the seventh by a diagonal, and so the
apex of the letter shall remain to the right.

The letter S you shall make as L; except that at top to the right must
be drawn a broad limb of the length of the diagonal, which afterwards
you are to cut off by a line parallel to the diagonal.

F you shall make as S, just adding to it a transverse limb at the height
of the shorter letters and double as long as broad, so that the point on
the hither side & below shall project as far as half the limb's breadth,
so that the two diagonal abscissions may be equidistant from one
another.

The near limb of the letter H make like L, and to it join by its top, in
the proper place, the farther made like I; but below, for the diagonal
square, substitute a fourth square in line with the others, and the
fifth and lowest cut off on the far side by its diagonal.

Of K make the near limb like L; and to the right of it append a diagonal
square, from the lowest angle of which let a line be obliquely produced
to meet the said vertical limb; and next from this line let a broad limb
be obliquely drawn, and this, at the bottom, you are to cut off by a
diagonal, in such fashion that the space below, between the two tips
shall not be more than the diagonal of a single square.

D in its lower half make like B; but at top let the anterior limb ascend
upwards to the maximum height of the letters, and then cut off the
hither angle by its diagonal; next superpose to the same height half a
square upon the other three squares of the farther limb, & once more do
here as you did below, and let this broken limb rest on the angle of the
near limb, and let it extend beyond it as far as the end of the upright
near limb; and so will it all but contain three conjunct squares; for
when it meets the near vertical limb, that fraction is to be cut off at
right angles.

O you are to make below as D, and also the same at the top as the
bottom, only, as it were, turning it upside down.

The anterior limb of P make like L inverted; but the posterior like the
standard of I: at bottom, however, you are not to add an oblique square,
but amputate the limb diagonally, & draw at the bottom a broad
transverse limb, which likewise is to be cut off diagonally, so that the
lower point shall project to the left, a distance of half the breadth of
the limb.

Likewise A in the lower half you are to make like N; but of its anterior
vertical limb, you are to cut off the hither angle of the middle square
by its diagonal; of the posterior, however, allow three squares to
remain superposed, and incline the top part (the fourth square) rather
to the left, so that if at this side is joined to it the half of a
square, then it shall attain the height of the letter; and cut off the
square obliquely, yet so that the lower point shall project farther than
the upper; then describe to the left a circle, sweeping downwards, so
that its contents shall embrace the farthest limit of the anterior limb.

Z is made in threefold fashion. First set a diagonal square which shall
touch the height of the letter; then add another like it, on the right,
joining their sides, & let these form a quadrangle sloping downwards on
the right: next set a diagonal square in straight line under the top
square, and distant from the lower one the length of its diameter: then
draw a diagonal line between the near angles of these two squares, or
make a rounded limb to reach the lower square; but from the said lowest
square of all you shall draw downwards and to the right, by the aid of
divers circles, a round extension, whose bottom shall mark the length of
the letter; and let its tip, sharp and tenuous, verge to the left. Or
construct Z of three oblique limbs, one above the other, & to connect
them draw the diagonal, which shall slope upwards to the right.

Another Z you may make in this way: Let three diagonal squares be set
atop of one another; and let the lowest have a rounded extension, as in
the first Z.

The first limb of G make below like I, and add at the bottom another
diagonal square, joining the two by their angles; but at top produce the
farther tip of this limb upwards to the height of the letter, & from
this point draw a diagonal downward to the left, as far as the hither
angle of the first right square of the three set one on other. Next draw
the farther vertical standard entire, of the same length as the hither
standard, and at the bottom draw a diagonal from the angle of the lowest
oblique square to touch the tip of the angle of the farther limb, & on
the inner side produce downwards the side of the limb, to meet the tip
of the said diagonal; to this also, by one line, join the lowest of the
hither squares. Now draw at top a transverse limb of the customary
breadth, from the back of the nearer vertical limb, passing through the
farther one, and reaching as far beyond this as its breadth; & this
limb, finally, you shall cut off by an oblique line parallel to that of
the near limb.

Y you shall make as N, only at bottom must be omitted the farther
diagonal square, & in its place is to be set a right square under the
other three superposed squares; then split the fifth square by a
diagonal, so that the tip shall be in front; from which let there be
produced a diagonal line, equal in length to a single side of the
square.

Curved, or short S, you shall make on this wise. At the middle height of
the letter, let there be set, close to one another, their angles
touching, two oblique squares; from the near square draw a broad
vertical limb to the height of the letter; and in the same fashion, from
the farther square let one fall downwards--just as you constructed I top
& bottom. Next cut off both these limbs, one at top and one at bottom,
by diagonals, in such fashion that the sharp tips of both may be on the
side near the middle. Then let there be drawn two broad limbs--namely,
from the upper, to the right, and downwards; and in like manner, from
the lower, upwards, and to the left; of the breadth of the limb, above
and below, but let them be produced no further than the breadth of the
distance between the limbs: then draw a diagonal downwards, from right
to left, which shall cut off both oblique limbs. To it also you must
produce the sides of the squares set in the midst.

So, accordingly, have I set them down--in skeleton in rotation, and in
proper order in black. This (as I said above) is the antique form of the
letters; but in these days there is used a more elegant text, and a
diagonal square is substituted in the middle place for a right square,
so that the lines of the letters are not so much curved; and there are
made certain limbs adjoined and cleft; and there are set one on another
three squares & a half; and spaces are left between two limbs as great
as their width. Letters of this sort also have I set forth on the third
page following; as also capital letters, which are called "versals,"
because they are customarily set at the beginning of a verse; and these
ought to be made one-third higher than the remaining shorter letters in
writing.

  [Illustration:
  i n m r r u v
  w b q x c e t l
  s f h k d o p
  a z z z g y s]

  [Illustration:
  a b c d e f g h
  i k l m n o p
  q r r s s t u v
  w x y z z z]

  [Illustration:
  A B C D E F G H
  I K L M N O P R
  S T V X Y Z Q *
  a b c d e f g h
  i k l m n o p q
  r s s t u v w x
  y z]

Here ends this little Book.






End of Project Gutenberg's Of the Just Shaping of Letters, by Albrecht Dürer