,transcriber's ,note3 ,every effort has been made to replicate this text as faithfully as possible1 including inconsistencies in spelling and hyphenation4 ,no changes have been made to the printed text4 ,,of ,,the ,,just ,,shaping ,,of ,,letters ,,by ,,albrecht ,d,ue,r,e,r ,,translated ,,by ,r4 ,t4 ,,nichol ,,from ,,the ,,latin ,,text ,,of ,,the ,,edition ,,of ,,mdxxxv ,,of ,,the ,,just ,,shaping ,,of ,,letters ,,from ,,the ,,applied ,,geometry ,,of ,,albrecht ,d,ue,r,e,r ,,book ,,jjj ,,dover ,,publications1 ,,inc4 ,,new ,,york ,,albrecht ,d,ue,r,e,r ,,to ,,wilibald ,,pirckheimer ,,his ,,patron ,,and ,,very ,,good ,,friend ,,greeting3 ,illustration ,in our ,germany1 most excellent ,wilibald1 are to be found at the present day many young men of a happy talent for the ,art ,pictorial1 who without any artistic training whatever1 but taught only by their daily exercise of it1 have run riot like an unpruned tree1 so that unhesitatingly and without compunction they turn out their works1 purely according to their own judgment4 ,but when great and ingenious artists behold their so inept performances1 not undeservedly do they ridicule the blindness of such men2 since sane judgment abhors nothing so much as a picture perpetrated with no technical knowledge1 although with plenty of care and diligence4 ,now the sole reason why painters of this sort are not aware of their own error is that they have not learnt ,geometry1 without which no one can either be or become an absolute artist2 but the blame for this should be laid upon their masters1 who themselves are ignorant of this art4 ,since this is in very truth the foundation of the whole graphic art1 it seems to me a good thing to set down for studious beginners a few rudiments1 in which ,i might1 as it were1 furnish them with a handle for using the compass and the rule1 and thence1 by seeing ,truth itself before their eyes1 they might become not only zealous of the arts1 but even arrive at a great and true understanding of them4 ,now1 although in our own time1 and amongst ourselves1 the ,art ,pictorial is in ill repute with some1 as being held to minister incitement to idolatry1 yet a ,christian man is no more enticed to superstition by pictures or images1 than is an honest man girt with a sword to highway robbery4 ,certes he would be a witless creature who would willingly adore either pictures or images of wood or stone4 ,on the contrary1 a picture is the rather edifying and agreeable to ,christian religion and duty1 if only it be fairly1 artificially1 and correctly painted4 ,in what honour and dignity this art was anciently held amongst the ,greeks and ,romans1 the old authors sufficiently testify2 though afterwards all but lost1 while it lay hid for more than a thousand years4 ,it has now at length1 only within the last two hundred years1 by some ,italians been brought again to light4 ,for it is the easiest thing in the world for the ,arts to be lost and perish2 but only with difficulty1 and after long time pains are they resuscitated4 ,wherefore ,i hope that no wise man will defame this laborious task of mine1 since with good intent in behoof of all who love the ,liberal ,arts have ,i undertaken it3 nor for painters alone1 but for goldsmiths too1 for sculptors1 and stonecutters1 and woodcarvers1 and for all1 in short1 who use compass1 and rule1 and measuring line--that it may serve to their utility4 ,nor is anyone compelled whether or no to spend gainful hours on these exercises of mine2 although ,i am not ignorant that whoever is well exercised in them will thence acquire not only the principles of his own art1 but by daily practice1 an exactitude of judgment1 with which he will proceed to higher investigations discover many more things than ,i have here pointed out4 ,but since1 illustrious ,sir1 it is clearer than light that you are yourself1 so to speak1 an asylum of all the noble ,arts1 it has been my pleasure1 out of a singular love ,i bear towards you1 to dedicate to you this book2 not because ,i desire to appear therein as rendering you any great service1 but because thereby you may understand how engaged my mind is to you2 and since by my work ,i can confer on you but little favour1 at least by the exhibition of a ready mind ,i may repay the benefits you shower upon me4 ,farewell4 ,illustration ,,of ,,the ,,just ,,shaping ,,of ,,letters ,,from ,,the ,,applied ,,geometry ,,of ,,albrecht ,d,ue,r,e,r ,,book ,,iii4 ,now1 since architects1 painters others at times are wont to set an inscription on lofty walls1 it will make for the merit of the work that they form the letters correctly4 ,accordingly ,i am minded here to treat briefly of this4 ,and first ,i will give rules for a ,latin ,alphabet1 and then for one of our common ,text3 since it is of these two sorts of letters we customarily make use in such work2 and first1 for the ,roman letters3 ,draw for each a square of uniform size1 in which the letter is to be contained4 ,but when you draw in it the heavier limb of the letter1 make this of the width of a tenth part of the square1 and the lighter a third as wide as the heavier3 and follow this rule for all letters of the ,alphabet4 ,first1 make an ,a after this fashion3 ,indicate the angles of the square by the letters a4 b4 c4 d4 7and so do for all the rest of the letters73 then divide the square by two lines bisecting one another at right angles--the vertical e4 f4 the horizontal g4 h43 then1 in the lower line1 take two points1 i4 and k41 distant respectively one-tenth of the space c4 d4 from the points c4 and d43 then1 from the point i4 draw upwards to the top of the square the lighter limb2 thence downwards the heavier limb1 so that the outer edges of both may touch1 respectively1 the points i4 and k43 then let a triangle be left between the limbs1 and a point e4 be fixed at top in the middle of the letter1 and next join both limbs beneath the horizontal line1 and let this limb be a third as broad as the heavier limb4 ,now let the arc of a circle1 applied to the top of the outside edge of the heavier limb1 project beyond the square4 ,then cut off the top of the letter with a serpentine or curving line1 so that the concavity decline towards the lighter limb1 and prolong acutely either limb of the letter at the bottom to either side1 so as to meet the angles of the square at c4 and d43 this you shall make with the arc of a circle1 whose semi-diameter is one-seventh of the side of the square2 but the two lower curves1 mutually opposite1 permit to extend so that each is a third of the heavier limb1 and this you shall obtain by the arc of a circle whose diameter is equal to the breadth of the heavier limb4 ,moreover1 this same letter ,a you may cut off at top with the side of the square1 and then produce to a fine point in either direction1 as you did the feet below1 yet so that the longer production shall be to the fore-side 7namely1 the left72 but in this case it will be necessary to draw in the limb k4 a little closer4 ,likewise the same ,a you may draw in yet another manner--that is1 pointed at top4 ,in that case let the limbs slope towards one another yet more closely2 then lower the transverse a little and double its width4 ,you may also cut off the limb at top bluntly1 or sharpen it on the fore-side4 ,you ought to make yourself familiar with these three forms1 or whichever of them pleases you best4 ,and note likewise that in exactly the same fashion in which this letter is acutely prolonged at top bottom1 are the other letters to be so prolonged which are drawn with oblique lines1 as ,v1 ,x1 ,y1 although a few changes may be necessary1 as you shall hear below4 ,i have here subjoined an engraving of this letter4 ,illustration3 ,a ,,of ,,the ,,letter ,b4 ,and now you shall draw ,b in its square thus3 ,first divide the square horizontally by the line e4 f42 then bisect the lines a4 e4 and b4 f4 by the line g4 h4 ,next1 you must first set properly the broad vertical limb of the letter1 distant its own breadth from the side a4 c4 of the square a4 b4 c4 d4 ,then erect the line i4 k4 on the inner side of the limb already drawn1 and distant from it one-tenth of a side of the square1 and let it cut the line g4 h4 in the point l4 ,next1 draw strips narrower and horizontal 7to be produced hereafter into the convex limbs7 from the vertical band to meet the vertical line i4 k4--namely1 at top1 below the line a4 b42 next1 above the line e4 f42 and at bottom1 above the line c4 d4 ,now set a leg of the compass on the point l4 and describe a semicircle to the right of the transverse strips1 so that the extremities of the circumference1 in the vertical line i4 k41 below the side a4 b41 and above the line e4 f4 may coincide with those short transverse lines4 ,then bisect the narrow transverse strip which is above the line e4 f4 in the line i4 k4 by the point m42 and indicate the breadth of the letter1 to the right of the semicircle1 by the point n4 in the line g4 h42 and afterwards draw from the point m4 above the line e4 f4 in the direction of f4 a short horizontal line as great as need be3 then describe a semicircle which shall include this line1 and the point n41 and1 at the top1 the side a4 b42 and through n4 let pass a vertical line4 ,these all combine to form1 below1 the concave of the curved limb1 and above1 its convex4 ,next1 produce the transverse strip above c4 d41 in the direction of d41 as far as required1 and mark this q4 ,then bisect the line m4 q4 by the line o4 p41 cutting the line n4 in the point r42 and next describe a semicircle touching the horizontal line e4 f41 the point r41 and the position q4 ,then indicate the breadth of this limb of the letter by the point s4 to the right of the point r4 in the line o4 p4 and describe a semicircle1 touching the line m41 the point s41 and the side of the square c4 d4 ,there will then remain in the letter three right angles to be eliminated3 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 letter1 and the exterior ones you shall fine to a point by circular lines whose semi-diameter is equal to the breadth of that limb4 ,another method4 ,or you may make your ,b in this fashion3 ,let the side a4 c4 of the square be divided into nine equal parts1 and cut off the four superior parts by the horizontal line e4 f4 ,then erect your vertical limb as described above2 and the superior curved limb you shall make between a4 b4 and e4 f42 the inferior between e4 f4 and c4 d4 ,now divide a4 b4 into nine equal parts1 and cut off four parts towards b4 in the point g42 then divide c4 d4 into five equal parts1 and the last1 towards d4 mark off in the point h4 and join g4 and h4 by the line g4 h4 which should touch on their exterior edges the superior and inferior limbs of the letter4 ,now these limbs must be drawn of a particular form2 and the compass1 in drawing the circular lines1 must be moved up and down their diagonals3 and these two diagonals you shall determine in this wise4 ,divide a4 e4 into four parts2 the lowest1 above e41 call i4 e42 the lowest of the five remaining1 above c41 call c4 k4 ,then join the points i4 and b4 and k4 and f4 respectively1 by the lines i4 b4 and k4 f4 ,upon these lines move and turn your compass1 in this way you shall describe both curved limbs3 and they must both be broader towards the top than towards the bottom1 as follows naturally with the stroke of a pen1 and1 moreover1 while approximately round1 they are not to be circular2 therefore you will have to move your compass at need along the diagonals1 and withal to assist it also with the hand1 as ,i have done in the picture on the following page4 ,illustration3 ,b ,,of ,,the ,,letter ,c4 ,next you shall make the letter ,c in its own square thus3 ,bisect the square a4 b4 c4 d4 by the horizontal line e4 f4 and in it let i4 be the middle point4 ,from this point as the centre1 i4 f4 or i4 e4 as the radius1 describe a circle touching interiorly all four sides of the square4 ,now move the leg of the compass1 but without varying its span1 to a point k4 a little to the right of i4 in the line e4 f41 letting the space i4 k4 denote the greatest breadth of the letter you desire2 from the centre k4 describe another circle which shall cut twice the line b4 d41 and whose circumference to the left will mark the required breadth of the letter4 ,next1 draw the vertical line g4 h41 parallel to b4 d41 distant from b4 a tenth part of the line a4 b4 ,this will cut off for you at top and bottom the letter ,c as the ancients were accustomed to use it4 ,but ,i would have you cut off the lower limb in the middle point between g4 h4 and b4 d43 then make the limbs somewhat finer and rounder on the inside towards top and bottom from the point where the circles intersect2 and for its greater perfection round out the letter1 above and below1 to touch the sides of the square a4 b4 and c4 d4 ,next1 low down1 where the letter with one foot crosses the line g4 h41 there1 under the circular line make the form a little more incurved1 yet so that with the tip of its end it shall again touch the circular line4 ,similarly1 but higher up1 make the foot more hollow on the inside than the circle left it3 and thus two circular lines will give you very nearly the whole form of the letter4 ,another method4 ,or1 secondly1 you may make the letter ,c thus3 ,draw in the square a diagonal c4 b42 set the leg of your compass on its middle point i4 and with the other leg describe the exterior circle as before1 terminating it above at the diagonal c4 b42 but below1 make your circle pass a little beyond the former sweep4 ,then set the leg of your compass1 but without changing its gauge1 as far above i4 in the diagonal as the letter's greatest width1 and describe your inner circle2 and1 as though made with a pen1 let the descending stroke be heavier than the ascending4 ,the rest you may elaborate with your hand2 let the trimming of the ends of the letter1 above1 slope upwards1 below1 downwards1 exactly as ,i have here drawn the shapes4 ,illustration3 ,c ,,the ,,letter ,d4 ,the letter ,d you shall make thus3 ,divide its square by the perpendicular or vertical line g4 h4 and by the horizontal line e4 f4 into four small squares1 and call their point of intersection i43 then draw the broader limb of the letter from the side a4 b4 downwards1 to meet the side c4 d4 and at the distance of its own width from a4 c42 and produce the limb at top and bottom to a sharp point at the angles a4 and c4 as was shown above in ,b2 using the same method in all straight limbs in the remaining letters4 ,next you are to produce from this limb two narrower tracts horizontally1 and from these are to be described the circular arcs of the letter between the line a4 b4 at top and the line c4 d4 at bottom1 and extending as far as the perpendicular g4 h42 next1 with your compass join g4 f4 h4 ,then1 in the line e4 f4 lay off a portion equal in breadth to the widest limb of the letter1 at the point k42 next1 set one foot of your compass on k4 and let the other cut the said line e4 f4 in l42 let this be the immovable leg of your compass1 and with the other1 beginning from k41 describe internally1 to the narrower transverse limbs1 an arc which shall touch both1 completing your acute angle above1 but rounding out the lower one by a circular arc of the same diameter as the one by which you sharpened your exterior subtending angle4 ,another method4 ,you may make the round limb of the same ,d in another fashion2 namely1 as a pen naturally would1 broader above than below4 ,for this1 draw the diagonal c4 b4 and describe your exterior arc as before2 but to describe the interior1 in the line c4 b4 take a point m4 lower down than i4 and distant from it the width of the broader limb1 and without altering your compass describe an interior line2 but where the limb must needs be narrower1 there you are to accommodate it with your hand1 both below and above1 as in the following cut4 ,illustration3 ,d ,,the ,,letter ,e4 ,the letter ,e you shall form in its square thus3 ,draw a transverse line e4 f4 bisecting a4 b4 and c4 d4 in e4 and f42 then draw the great vertical limb of the letter1 to the left1 as you did for ,d4 ,next draw also an upper transverse limb of narrower dimensions1 parallel to a4 b4 and in length six-tenths minus one-third of one-tenth of the length of a4 b42 and the end of this bend downwards one-tenth of the length of a4 b4 and use this as the diameter of the circle with which you round out the inner angle of this extremity3 then draw your narrow middle limb parallel to the median line e4 f4 and above it1 so that it may be shorter than the upper limb by one-tenth of the length of a4 b41 but at its terminus double as wide2 you are to round it out 7in either direction7 by the arc of a circle whose diameter is one-sixth the length of e4 f4 ,now construct your lowest limb upon the line c4 d41 so that at its ultimate angle it may exceed in length the upper limb by one-tenth of c4 d42 the cusp1 however1 you are to prolong beyond this part by two-thirds of one-tenth part1 and erect above it to one-sixth of the length of c4 d41 and round out the same by a circle whose semi-diameter is also a sixth of c4 d4 ,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 limb3 the other angles you are to leave acute1 as in the following cut4 ,illustration3 ,e ,,the ,,letter ,f4 ,the letter ,f you are to form in the same manner as ,e2 except that you shall omit the lower limb altogether1 and1 in its place1 round out the letter on both sides below1 as you did ,e on one side only1 as ,i have shown you below4 ,illustration3 ,f ,,the ,,letter ,g4 ,likewise the letter ,g you are to make as you did ,c1 before described2 this1 however1 excepted3 that in front 7that is1 to the left7 of the line g4 h4 is to be erected the broad limb of the letter1 upwards from the curve to the line e4 f41 and above it is to be rounded to a point1 on either side1 as before was said2 but below1 both angles are to remain4 ,or1 you shall form ,g in the following fashion in the said square1 divided as before3 ,draw the diagonal c4 b4 and set your compass with one leg on the point i4 and with the other describe an arc from e4 to the middle point c4 d4 and mark this point l42 in like manner also1 describe an arc upwards to the line a4 b4 so as to meet the perpendicular line g4 h4 mark that point z4 ,then1 in the line g4 h41 take a point m4 so that the part m4 h4 shall be one-tenth of the line g4 h42 then1 with a sweep of your hand join l4 m4 with the curved line l4 m4 ,next1 you are to draw from z4 a line upwards1 as broad as the standard of the letter1 but oblique and in direction midway between your circular line the perpendicular g4 h4 and from the extremity of this line you must draw a curved line to meet a4 b4 at the point where your circular line touches it4 ,next1 cut off from the bottom of g4 h4 a part one-third of its length1 indicate this by the point n41 to this height1 from the level of m4 upwards1 produce the broad limb of the letter1 and let its extremities above be finished in either direction1 of the same size4 ,after this set the leg of your compass on the diagonal c4 b4 the breadth of the standard of the letter above i4 at the distance e4 i4 describe an arc1 which above shall touch the exterior boundary a4 b4 but below shall stop short above l42 from this point you must with your hand draw a line to the vertical limb at the height of m4 ,and the same you shall do above in drawing the narrower limb of the letter1 as seen in the following diagram4 ,illustration3 ,g ,,the ,,letter ,h4 ,the letter ,h is to be formed of two broad1 great1 vertical limbs of the height of the square1 in such fashion that their extremities1 being produced exteriorly1 shall touch the four angles of the square1 a4 c4 and b4 d4 respectively4 ,now in what fashion the projections of the broader limbs of letters are to be rounded out at top and bottom and on either side1 you have been instructed2 for in any letter you please1 any broad and vertical limb is to be depicted at top and bottom thrice as broad as at its middle3 provided always it is not joined to a narrower limb4 ,so when this has been accomplished1 then draw your narrower transverse limb upon the line e4 f4 as is shown below4 ,illustration3 ,h ,,the ,,letter ,i4 ,the letter ,i you are to make of a single broad vertical tract in the midst of its square1 touching the latter top and bottom2 and of this1 at both ends1 and on either side1 you are to round out the productions or projections as below is shown4 ,illustration3 ,i ,,the ,,letter ,k4 ,now for ,k3 ,you are to make the first tract vertical1 in the same manner as you formerly did for ,h2 then draw another narrower limb from the broader and erect one1 so that it may1 at its lower end1 impinge obliquely on the transverse line e4 f4 and above may ascend to the right till it meet the line a4 b41 taking care to make it parallel to the diagonal c4 b42 and this1 at the top1 you are to produce in both directions so that each production may represent a tenth part of the line a4 b4 ,the hitherward projection you are to round out with a circle of which the diameter must not exceed the breadth of the lesser limb2 but of the other arc1 by means of which you round out the farther projection1 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 limbs4 ,next1 from the narrow limb so constructed draw in a downward direction another broad limb1 so that it too may be parallel to a diagonal of the square2 of this the beginning is to be taken from the acute angle which the narrower limb makes with the broad vertical limb1 and let it be drawn with its projection to the angle d41 yet in this fashion3 take two points this side of d4 after this manner1 so that the first point may be distant from d4 the tenth part of the line c4 d4 the second as far again from the first2 then let the said tract be drawn within the space which is between the two points1 but in blind and invisible lines4 ,afterwards you shall add the extension1 which you shall make this way3 take before f4 in the line e4 f41 a point g4 no farther distant from f4 than the breadth of the narrower limb2 on this point set one leg of your compass1 let the other be extended to the angle d41 from which let it be guided back along the broad but invisible blind limb3 thence will result the lower convexity of the tail you seek2 but its upper concavity look for in this way3 divide f4 d4 in its middle point h42 on this set one leg of your compass1 and with the other describe an arc passing through d4 to meet the broad limb4 ,or you may make ,k in this manner3 ,first1 let your broader vertical limb1 and your upper narrow one remain as they have been described1 except that the interior angle which the narrower limb forms with a4 b4 shall remain acute1 but the exterior one shall be rounded out1 as has been said4 ,then let there be drawn the lower broad limb1 obliquely from the angle which is included between e4 f4 and the vertical limb1 and let it descend to meet the side c4 d4 so that between d4 and the limb the width of the limb be left vacant2 and the hither angle is to be left1 but the farther1 towards d41 shall be rounded out a little1 as shown below4 ,illustration3 ,k ,,the ,,letter ,l4 ,as for the letter ,l1 you shall make it by a combination of parts of two of the preceding letters3 namely1 you shall make the first vertical broad limb1 as you did a while back in ,i2 and to this join a foot as you did at the bottom of ,e1 when you made it4 ,such is ,l depicted below4 ,illustration3 ,l ,,the ,,letter ,m4 ,the letter ,m you shall form in two ways within its square4 ,in the first1 draw the narrower limb of the letter vertical1 to the right of a4 c41 distant from a4 one-tenth of the distance a4 b43 draw the other1 broader limb1 on the near side of b4 d41 also a tenth part of the whole distant from b4 in such fashion that both limbs touch the square at top and bottom2 then1 between the two1 bisect the line c4 d4 in the point e4 and draw a broad limb from the inner angle of the narrow limb1 downwards to the point e41 next a narrow one upwards from e4 to the inner angle of the broader vertical limb2 and the inner angles at top you must not round out1 but leave acute2 the exterior angles1 however1 at the top1 and both exterior and interior at bottom of both vertical limbs1 you are to adorn with the customary projections1 as you have done in the preceding letters4 ,you are to know1 too1 that when these letters are drawn with a pen1 they are to be described with a single stroke4 ,but for your guidance is this letter1 in the manner in which ,i have instructed you1 depicted below4 ,another method4 ,another way is thus3 ,divide the side a4 b4 of the square into six equal parts mark off the two extreme parts1 one at either end1 by the points f4 and g42 then draw the inner and broader limb1 with its point at e4 as above2 and to this1 in an upward direction1 a narrower one1 so that between f4 g4 be left a vacant space1 and so more readily the letter slope forward4 ,then you are to draw the two lateral and vertical limbs--the near and slender1 and the farther broad one--at the top1 indeed1 as in the first sketch1 but at the bottom produce them to the two angles c4 and d4 and finally add projecting cusps1 as you were instructed in the first ,m2 but the projection below will pass beyond the square at the points c4 and d4 ,or you shall make ,m at top with acute angles1 in which case the lateral limbs will slope the more2 or shear them off obtusely1 and in this fashion 7whichever pleases you best7 make them as you see them depicted in the following diagrams4 ,illustration3 ,m ,,the ,,letter ,n4 ,likewise the letter ,n you shall make in its square thus3 ,first you are to draw two standards vertical and slender1 so that at top bottom they may touch the square1 that being produced1 the nigh one at the bottom1 and the farther at the top1 they may touch the angles at c4 and b4 ,now join these two by a broad oblique limb1 running from the angle a4 to the point e41 by which is denoted the remote side of the farther limb1 where you shall allow the acute angle to remain2 but at the top1 this limb1 produced beyond the angle a41 you are to round out to a fifth part of the length of a4 b4 ,this prolongation should incurve below1 a fifteenth part of the distance a4 b4 projected on two arcs1 the upper one the greater1 the lower the less4 ,for the lesser arc1 therefore1 you shall take as diameter of its circle1 a line the fifth part of the distance a4 b4 and its centre is to be taken outside the square1 so that the foot of the compass may touch the tip of the extension and the angle a42 then extend a little the feet of the compass1 and shift its centre until the arc touch both the tip of the part produced1 the broad oblique limb1 in the middle point between the side a4 c4 the nearer of the two slender vertical limbs4 ,or you may make the letter ,n in such fashion that its upper nigh extension shall remain within the square2 or you may make from it an acute angle as shown overleaf4 ,illustration3 ,n ,,the ,,letter ,o4 ,now ,o you shall make this way in its square4 ,set in the square the diameter c4 b4 and bisect it in the point e41 so that e4 may form a middle point between the two points f4 and g4 which are to be your two centres2 and from each let a circle be described touching two sides of the square2 where the circles cut one another1 there with your hand you must shape the slender outline of the letter to a juster proportion1 as below is shown4 ,illustration3 ,o ,,the ,,letter ,p4 ,p you shall make in its square in this wise4 ,divide the square a4 b4 c4 d4 by the median horizontal line e4 f42 then divide a4 e4 b4 f4 equally by the line g4 h4 ,next draw1 first the broad vertical limb for this letter ,p1 as you did a short while ago for ,k1 and afterwards erect the line i4 k4 the distance of its own breadth to the right of your vertical limb2 7here you must ever observe that in a lettered square we speak of the angle a4 as the hither angle1 that is1 to the left2 the angle b4 as the farther angle1 that is1 to the right74 ,then where the line i4 k4 cuts g4 h4 call the point l41 and next draw two slender horizontal limbs1 the upper below a4 b41 the lower above e4 f41 from the broad vertical limb as far as the line i4 k4 ,set one leg of the compass on the point l41 extending the other to the lower side of the lower horizontal limb near k42 then describe an arc through the line g4 h4 as far as the other slender horizontal limb of this same ,p1 where it cuts the line g4 h4 set the point m4 ,next1 on the far side of m4 measure the width of the large limb of the letter1 along the line g4 h4 to the point n4 and let your compass be stretched so that with one foot it may touch the line a4 b4 and with the other the point n42 then set one foot of the compass on n4 the other on the line g4 h4 to the right1 in the point o41 in which this foot is to be left standing immovable1 and with the other is to be described an arc1 passing through the point n4 and touching the lines a4 b4 and e4 f4 ,or you may form the loop of this letter in the following manner4 ,set a leg of the compass under the transverse g4 h4 in the line i4 k41 in a place median between the line e4 f4 the lower part of the upper transverse of the slender limb1 in the point p4 and describe an arc as before1 passing through m4 so that the loop will be acute at the bottom1 and its tip will end in the middle space between the line i4 k4 and the broad vertical limb of the letter4 ,or make this same ,p with a circular sweep1 by shifting the compass upon the diameter1 so that that sweep may be broader at the top 7as though made with a pen7 as will be shown in the diagram on the following page4 ,illustration3 ,p ,,the ,,letter ,q4 ,make your ,q in its square in the self-same manner as was prescribed for ,o2 but add to it its tail thus3 ,draw a diameter of the square1 the line a4 d41 about which1 starting from the curved outline1 begin to draw a long tail1 producing it through the angle d4 in such fashion that d4 may be in the middle of the thickest part of the tail2 but where the tail begins let it be a little narrower than in the angle d41 where it should attain its real thickness4 ,then let it be drawn out1 beyond the angle d4 to the length of the entire diameter1 and in a downward direction1 yet so that it curves while it slopes1 that its tip shall not fall lower than a third of the side below the lowest side of the square1 and shall tend1 as it nears the point1 to grow sharper little by little1 and at length end in a very fine point indeed4 ,or you shall give ,q a shorter tail in this fashion1 to wit3 set your compasses to the length of the side c4 d4 and draw a tail from the bulge of the same letter1 describing through the point d4 its inner arc of the same length as c4 d41 taking care that the tail bend upwards until it again reach c4 d4 produced1 in the point h42 then shift your compasses1 with the other leg again describe from the bulge of the letter an arc below d4 continue it until again it reach h41 but in such fashion that the tail shall find its greatest thickness at the start1 as in the following figure is doubly depicted4 ,illustration3 ,q ,,the ,,letter ,r4 ,moreover ,r you must make in its square just as was directed for ,p2 but then erect a right line q4 r4 through the middle point of the square1 let it cut the exterior arc of the rounded limb in s41 from which point1 downwards towards the angle d41 let there be drawn a broad tract1 almost equal to that which you made above for the letter ,k41 but this is to be somewhat bent in1 and so shaped by your hand that its tip1 well formed1 may arrive directly on the angle d4 ,or make ,r in such fashion that its rounded sweep1 as though made with a pen1 shall be above broader1 narrower below4 ,to accomplish this1 you must shift your compasses on the diameter q4 e4 not allow the rounded limb to touch the vertical one1 as was described in ,p4 ,besides1 the oblique limb is to be deduced from the rounded one with a little more of a curve2 just as ,i have drawn overleaf4 ,illustration3 ,r ,,the ,,letter ,s4 ,next1 the letter ,s you shall make as follows in its square1 a4 b4 c4 d4 ,first draw the horizontal line e4 f4 and the median vertical one g4 h4 and let them bisect one another in the point m4 ,then choose the main thickness of the letter1 and set it in the line g4 h4 so that the point m4 may divide it1 having one-third of the thickness below it2 next1 set the lesser thickness1 at the top beneath g41 indicating it by the point i41 and at bottom1 above h4 in the point k42 and the thickness of the letter indicate above by n4 and below by l4 ,next1 set a leg of your compasses on the line g4 h4 in the mid-point between i4 and n41 and with the other describe a circle passing through i4 and n42 in like manner1 upon the line g4 h4 set your compasses upon the mid-point of g4 l4 and describe a circle passing through g4 l4 ,then once more set your compass on the same line g4 h4 in the mid-point of n4 h4 and describe a circle through n4 h42 and lastly1 in the mid-point of l4 k4 you must set one leg1 with the other is a circle to be described through these same points l4 k42 afterwards cut off by vertical section the upper portion of this letter1 so that the part thus cut off may contain in its extremity the maximum thickness of the letter and a third part besides1 also that its tip may project downwards so far as to stand midway between the centre of the circle i4 n4 and the side b4 d42 in other words1 let the tip be distant on the right1 from the circle i4 n4 the first third of the interval between the greater and lesser circles4 ,next cut the lower limb of the letter to the left1 by a vertical line through the mid-point between the two circles1 in such fashion that the part so cut off may be a fourth part wider higher than the upper1 and that its tip may rise to the height of the centre of the circle n4 h4 ,another method4 ,yet another way may you make the letter ,s4 ,in the square a4 b4 c4 d4 bisect the horizontal line e4 f4 in the point m42 then set one leg of your compass upon the mid-point between g4 and m4 with the other describe a segment of a circle in the direction of a4 e4 passing through the points m4 and g42 next1 set your compass upon the mid-point between m4 and h4 and describe a segment of a circle through m4 and h4 in the direction of f4 d4 ,the two arcs will touch above1 in front1 and below1 in the rear1 the exterior curvatures of this same letter ,s4 ,next1 draw through m4 the diameter c4 b4 and at its middle indicate the maximum thickness of the letter by the two points p4 and q4 from which let there be drawn two right lines1 one up1 one down1 to those two arcs2 next1 from the two points p4 q4 draw two curved parallels to the same arcs1 regulating the distance between them1 their elevation depression from the centres of the same circles4 ,next1 indicate below g4 and above h4 the minimum thickness of the letter2 and from these points you will with your hand fashion the inner shape of the letter1 both above and below1 produce the limb of ,s1 above towards b4 ,cut it off so that its lower tip may touch the segment1 that the part cut off upwards may contain a tenth part of a4 b4 and that the segment may still exceed the part cut off4 ,then construct a vertical line r4 s4 to the right of e4 c4 and distant from it a fifth part of c4 d42 let it cut the diagonal c4 b4 in t4 and to the angle just formed produce the extremity of the letter1 making the part so cut off a third broader than the upper portion4 ,lastly1 you will have to produce the tip ever so little beyond t42 as ,i have briefly indicated4 ,illustration3 ,s ,,the ,,letter ,t4 ,set the broad limb of ,t in the midst of its square erect1 produced drawn to a point on either side below1 just as you did before in the letter ,i2 then take two points e4 and f41 distant respectively one-tenth of the whole space from a4 and b41 and let the transverse limb of the letter be drawn below e4 f4 and of an equal length with it2 but the projecting extremities of this line are to be cut obliquely1 and the tips of these projections shall so far extend above the line a4 b4 to the right as below they depend to the left4 ,the oblique lines of these projections are to be each a fifth part of the length of a4 b42 the angles of these projections you shall round out by means of circles of diverse radius--namely1 for the lesser angle you are to use a diameter only two-thirds of the width of the broader limb2 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 a4 b4 ,another method4 ,or you make ,t thus in its square3 ,take your point e4 as before1 to the right of a41 and cut your transverse limb diagonally1 as before1 yet so that the projection be dimidiated to the right1 and at top the angle remain as it falls2 and so at the other extremity1 only the point f4 must be moved as near again to b41 the cutting line to be a little more erect1 the projection formed a trifle broader than at the hither end2 otherwise shall everything remain as before2 as ,i have delineated for you on the opposite page4 ,illustration3 ,t ,,the ,,letter ,v4 ,v you shall thus make in its square3 ,bisect c4 d4 in the point e42 then set the point f4 one-tenth of the whole line a4 b4 beyond a41 and in like fashion g4 to the hither side of b4 ,then draw the broad limb of your letter downwards from f4 to e4 and sharpen it2 thence draw upwards your slender limb to g42 and at the top produce it in either direction1 as you did before at the bottom of ,a2 just as you see it shown below4 ,illustration3 ,v ,,the ,,letter ,x4 ,x you shall form thus3 ,draw two vertical lines e4 f4 and g4 h4 distant respectively one-tenth part of the line a4 b4 from the sides a4 c4 and b4 d4 ,then draw the two limbs intersecting one another in the form of a cross--the broad one so that at top1 with its hither side it shall touch e41 at the bottom1 and with its farther side h42 but the narrow limb so that at top1 and with its farther side it may touch g41 at bottom1 with its hither side f4 ,then add its projections1 touching1 at top and bottom1 the four angles a4 b4 c4 d41 choose a semi-diameter of the larger circle of the length of a fifth part of a4 b42 with that you shall round out the four greater angles2 but for the lesser circle you shall take a diameter as long as two-thirds the width of the broader limb4 ,or you may vary ,x thus3 ,let everything be left as before except the narrower limb1 which at top you shall make more erect by one-half the breadth of the wider limb2 and so the upper part of the letter shall be less and narrower than the lower1 and shall have a different aspect1 as is shown below4 ,illustration3 ,x ,,the ,,letter ,y4 ,y you shall achieve in the midst of its square1 as far as its lower half is concerned1 after the instructions before given for ,i2 but its upper part you shall divide so that its hither limb shall contain two-thirds1 and its farther one-third of the broad standard2 and let them slope to either side so that produced they may touch the two angles a4 and b42 and the greater circles1 by which you are to round out their obtuse or greater angles1 make of a diameter as great as a containing side of the square enclosed between the standard and the sides of the great square1 as in ,t was shown2 but the diameters of the circles which you apply to the lesser angles1 make double the width of the broad standard1 as below4 ,illustration3 ,y ,,the ,,letter ,z4 ,z you shall make thus3 ,set upon either side1 both beneath beyond the angle a41 two points e4 f41 each at a distance of the tenth part of a4 b42 so also1 set two other points g4 and h4 both before and above the angle d4 and with right lines join e4 f4 and g4 h42 then draw your narrower transverse limb1 beneath a4 b4 backwards as far as the angle b42 from thence draw your broad limb diagonally to c42 and then again a narrower one from c4 to g42 and with your hand round out the two tips e4 and h4 ,or make ,z thus3 ,divide the square a4 b4 c4 d4 by the vertical line e4 f4 and in this reduced space construct the letter as before2 but so that the two transverse limbs be cut short1 above on the nigh side1 and below on the far1 by the vertical lines a4 c4 and e4 f4 respectively as below4 ,illustration3 ,z ,so likewise1 in other fashion1 can we make all the letters already drawn1 on a scale of ninths1 just as we have now drawn them on a scale of tenths2 in just the same manner1 according to the due proportion of each1 in its own square1 a4 b4 c4 d41 dividing them into nine1 as just now into ten parts2 that this may be the better understood1 ,i have chosen to append here letters of such fashion4 ,also these letters are to be made five parts high when written small rapidly1 by hand4 ,in such writing the versals are made of the same proportion and form1 but one-third larger than the ordinary letters of the writing4 ,illustration3 ,a ,a ,a ,a ,b ,b ,b ,b ,c ,c ,c ,c ,d ,d ,d ,d ,illustration3 ,e ,e ,f ,f ,g ,g ,g ,h ,h ,i ,i ,k ,k ,k ,l ,l ,illustration3 ,m ,m ,m ,m ,n ,n ,n ,n ,o ,o ,p ,p ,p ,p ,illustration3 ,q ,q ,q ,r ,r ,r ,s ,s ,s ,t ,t ,t ,v ,v ,illustration3 ,x ,x ,x ,y ,y ,z ,z ,z ,,directions ,,for ,,the ,,construction ,,of ,,the ,,text ,,or ,,quadrate ,,letters ,the letters which are usually called text1 or quadrate1 it was formerly customary so to write1 although they are now imitated by the new art1 as presently ,i shall show below4 ,although the alphabet begins with the writing of ,a1 yet shall ,i 7not needlessly7 in the first place undertake to draw an ,i2 because almost all the other letters are formed after this letter1 although always something has to be added to it or taken away4 ,first make your ,i of equal squares1 of which three are properly set one over the other2 and the top of the top one1 and the bottom of the bottom one1 divide in two points1 that is to say1 into three equal parts3 then set a square equal to the others in an oblique manner1 so that its diagonal be vertical1 and its angle on the first point of the top square4 ,in this way1 this oblique square shall extend with its angles more to the left than the right4 ,then produce upwards on either side1 after the width of the superposed squares1 right lines to meet the sides of the oblique set square4 ,next do below precisely as you did above1 except that you must set the angle of the oblique square on the second point1 that is1 the one farthest to the right in the bottom of the lowest square2 and let fall your lines on either side upon the transposed square3 so will ,i be perfect2 only above it draw with a fine pen a tiny in-crescent4 ,so shall you make ,n from two standards of this same ,i1 set so that their angles at top and bottom touch2 and in this manner the space between the two shall be narrower than the breadth of either3 also1 you shall no longer put little crescents above them2 you must make of the same length all the short letters throughout the alphabet4 ,in like manner make ,m of three standards1 just as you made ,n of two4 ,r make as ,i1 except only that at top you must set an equal square diagonally1 to the right1 so that angle touch angle4 ,r you may make also in this fashion3 below leave its foot as before1 but above add two diagonal squares1 which shall touch each other with their angles in the middle point of the vertical limb1 and then produce upwards both sides of the latter to meet the diagonal square4 ,v is made in three ways4 ,first let it be made simply1 as ,n2 only that in the farther limb1 you shall omit at the top the diagonal square2 and instead shall draw an oblique line1 so that it may make two angles in this limb 7produced7 of which the farther shall be of the same height as that of the topmost angle of the diagonal square1 the higher angle of the same height as the angle nearest to it in the said square4 ,the second ,v1 which we use at the beginning of a sentence1 make thus3 ,draw the first limb as before for ,i2 only1 at the bottom1 push the diagonal square a little further to the right1 so that its hither angle does not project beyond the side of the vertical limb1 but falls in the line of its descent4 ,then set the second limb to the right of this1 and cut it off below by an oblique line1 drawn from the lowest angle upwards to the right1 as far as the middle point of the lowest of the three superposed squares4 ,next make ,w 7i4 e41 double ,v7 just as you made ,v-simplex2 only you shall set before it the standard limb of ,i4 ,b make as the second ,v-simplex2 but in the first upright omit the diagonal square at the top1 and set upon the three original squares three others similar1 but the seventh one you shall cut off diagonally from its lower hither angle4 ,likewise when your ,b so made is turned upside down1 then it will be a ,q4 ,x you shall construct from ,i4 ,append from top angle to the right a diagonally set square1 as you did before in ,r2 and at the bottom draw an acute tail to the left from the diagonal1 and at the middle of the vertical limb describe a transverse1 in such way that the former is cut before and aft by the latter's diagonal2 let the hither and lower angle be terminated as far in front of the upright as would measure one-half of the cutting diagonal1 which at top shall just touch the upright2 but to the right let the transverse at top project to a point just below the angle of the oblique square2 from thence downwards let it be cut off by an oblique line parallel to the anterior diagonal4 ,c you shall construct from ,i after this fashion3 ,remove the top diagonal square1 let lines be produced on either side to the proper height of the letter1 and cut off the hither angle by a diagonal2 then draw at top a broad transverse1 projecting beyond the vertical to the right the width of the latter1 and cut this off by a diagonal in such a way that it project below only half as far as above4 ,the vertical standard of ,e you shall make as for ,c2 but from above let there descend to the right a broad limb from the diagonal bisecting the right angles of one square1 and one-third again as long as broad2 and let there be drawn from its lower angle a small diagonal line to the vertical limb4 ,t shall be made like ,c1 except that at top something is added to its diagonal1 so that its tip converges to a fine point1 and the like to the left on the hither side of the broad standard1 just as at the top3 and because of this is ,t at top more elegant than ,c1 and has not the same incurved appearance4 ,l you are to make below like ,i2 only six squares are to be set on end2 then cut off the hither side of the seventh by a diagonal1 and so the apex of the letter shall remain to the right4 ,the letter ,s you shall make as ,l2 except that at top to the right must be drawn a broad limb of the length of the diagonal1 which afterwards you are to cut off by a line parallel to the diagonal4 ,f you shall make as ,s1 just adding to it a transverse limb at the height of the shorter letters and double as long as broad1 so that the point on the hither side below shall project as far as half the limb's breadth1 so that the two diagonal abscissions may be equidistant from one another4 ,the near limb of the letter ,h make like ,l1 and to it join by its top1 in the proper place1 the farther made like ,i2 but below1 for the diagonal square1 substitute a fourth square in line with the others1 and the fifth and lowest cut off on the far side by its diagonal4 ,of ,k make the near limb like ,l2 and to the right of it append a diagonal square1 from the lowest angle of which let a line be obliquely produced to meet the said vertical limb2 and next from this line let a broad limb be obliquely drawn1 and this1 at the bottom1 you are to cut off by a diagonal1 in such fashion that the space below1 between the two tips shall not be more than the diagonal of a single square4 ,d in its lower half make like ,b2 but at top let the anterior limb ascend upwards to the maximum height of the letters1 and then cut off the hither angle by its diagonal2 next superpose to the same height half a square upon the other three squares of the farther limb1 once more do here as you did below1 and let this broken limb rest on the angle of the near limb1 and let it extend beyond it as far as the end of the upright near limb2 and so will it all but contain three conjunct squares2 for when it meets the near vertical limb1 that fraction is to be cut off at right angles4 ,o you are to make below as ,d1 and also the same at the top as the bottom1 only1 as it were1 turning it upside down4 ,the anterior limb of ,p make like ,l inverted2 but the posterior like the standard of ,i3 at bottom1 however1 you are not to add an oblique square1 but amputate the limb diagonally1 draw at the bottom a broad transverse limb1 which likewise is to be cut off diagonally1 so that the lower point shall project to the left1 a distance of half the breadth of the limb4 ,likewise ,a in the lower half you are to make like ,n2 but of its anterior vertical limb1 you are to cut off the hither angle of the middle square by its diagonal2 of the posterior1 however1 allow three squares to remain superposed1 and incline the top part 7the fourth square7 rather to the left1 so that if at this side is joined to it the half of a square1 then it shall attain the height of the letter2 and cut off the square obliquely1 yet so that the lower point shall project farther than the upper2 then describe to the left a circle1 sweeping downwards1 so that its contents shall embrace the farthest limit of the anterior limb4 ,z is made in threefold fashion4 ,first set a diagonal square which shall touch the height of the letter2 then add another like it1 on the right1 joining their sides1 let these form a quadrangle sloping downwards on the right3 next set a diagonal square in straight line under the top square1 and distant from the lower one the length of its diameter3 then draw a diagonal line between the near angles of these two squares1 or make a rounded limb to reach the lower square2 but from the said lowest square of all you shall draw downwards and to the right1 by the aid of divers circles1 a round extension1 whose bottom shall mark the length of the letter2 and let its tip1 sharp and tenuous1 verge to the left4 ,or construct ,z of three oblique limbs1 one above the other1 to connect them draw the diagonal1 which shall slope upwards to the right4 ,another ,z you may make in this way3 ,let three diagonal squares be set atop of one another2 and let the lowest have a rounded extension1 as in the first ,z4 ,the first limb of ,g make below like ,i1 and add at the bottom another diagonal square1 joining the two by their angles2 but at top produce the farther tip of this limb upwards to the height of the letter1 from this point draw a diagonal downward to the left1 as far as the hither angle of the first right square of the three set one on other4 ,next draw the farther vertical standard entire1 of the same length as the hither standard1 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 limb1 on the inner side produce downwards the side of the limb1 to meet the tip of the said diagonal2 to this also1 by one line1 join the lowest of the hither squares4 ,now draw at top a transverse limb of the customary breadth1 from the back of the nearer vertical limb1 passing through the farther one1 and reaching as far beyond this as its breadth2 this limb1 finally1 you shall cut off by an oblique line parallel to that of the near limb4 ,y you shall make as ,n1 only at bottom must be omitted the farther diagonal square1 in its place is to be set a right square under the other three superposed squares2 then split the fifth square by a diagonal1 so that the tip shall be in front2 from which let there be produced a diagonal line1 equal in length to a single side of the square4 ,curved1 or short ,s1 you shall make on this wise4 ,at the middle height of the letter1 let there be set1 close to one another1 their angles touching1 two oblique squares2 from the near square draw a broad vertical limb to the height of the letter2 and in the same fashion1 from the farther square let one fall downwards--just as you constructed ,i top bottom4 ,next cut off both these limbs1 one at top and one at bottom1 by diagonals1 in such fashion that the sharp tips of both may be on the side near the middle4 ,then let there be drawn two broad limbs--namely1 from the upper1 to the right1 and downwards2 and in like manner1 from the lower1 upwards1 and to the left2 of the breadth of the limb1 above and below1 but let them be produced no further than the breadth of the distance between the limbs3 then draw a diagonal downwards1 from right to left1 which shall cut off both oblique limbs4 ,to it also you must produce the sides of the squares set in the midst4 ,so1 accordingly1 have ,i set them down--in skeleton in rotation1 and in proper order in black4 ,this 7as ,i said above7 is the antique form of the letters2 but in these days there is used a more elegant text1 and a diagonal square is substituted in the middle place for a right square1 so that the lines of the letters are not so much curved2 and there are made certain limbs adjoined and cleft2 and there are set one on another three squares a half2 and spaces are left between two limbs as great as their width4 ,letters of this sort also have ,i set forth on the third page following2 as also capital letters1 which are called versals1 because they are customarily set at the beginning of a verse2 and these ought to be made one-third higher than the remaining shorter letters in writing4 ,illustration3 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 ,illustration3 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 ,illustration3 ,a ,b ,c ,d ,e ,f ,g ,h ,i ,k ,l ,m ,n ,o ,p ,r ,s ,t ,v ,x ,y ,z ,q 99 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 ,book4