Editing ancient technical and mathematical figures: Tools and
Contents
1. Figure 16 The reconstruction of DRaFT s diagram with Inkscape Our junctions are not all perfect and there are other defects but these could probably have been improved in places We have decided to leave the problems as they are for instance at point j where two lines overlap It is important to use round caps for the end of lines and this can be selected in Inkscape It will improve the appearance of oblique junctions 20 but it will be easier to set them right away By comparison in DRaFT points are defined and then curves are de fined going through the points This approach somewhat brings us closer to a programmatic one but in fact it was above all simpler to make it that way in DRaFT It is possible that future versions of DRaFT will al low a more interactive construction of a curve but this has advantages and drawbacks When curves are defined in Inkscape especially when tracing over an image it is best to highlight the curves by giving them a good color and thickness An important problem when tracing curves is that of distinct curves meeting at some points If no special care is taken the curves will not meet well like in some of the examples shown earlier If the endpoints of two curves should coincide then it is possible to extend one curve by the other and to merge the two points within Inkscape However if the points to be merged are not both endpoints then the best way to merge the points is to2. 3 1 3 Labels We also need to be able to add complex labels perhaps involving ancient alphabets 3 1 4 Overlays We may want to write over existing figures that is not only to base paths on existing figures but also to retain the underlying figures figure 13 14 North pole N F le Sud South pole Fig 48 Figure 13 An example of overlay Taking as a base the first bitmap im age we have added several labels as well as a red dot All these additions are vectorial as zooming in will show 15 3 2 Choices 3 2 1 Intentions In order to reconstruct a figure one has in fact to get a certain understand ing of the figure and also of the intentions of its author For instance drawing a diagram goes by steps points or starting points of lines are chosen then lines are drawn There are intentions A person may want to draw a straight line or may want a line to be understood as straight but either does not exactly manage to draw such a line or does not care to draw the line that he she could draw In the case of figures 5 and 6 and any other reconstruction we have to distinguish the intention of the original figure and author and the intention of the reconstructed figure and author These may not be the same 3 2 2 Tracing or abstracting away The edition of ancient figures represents an interpretation of the figures The most faithful representation is that of a facsimile In certain cases facsi
3. TEX 2015 CTAN macros latex contrib polyglossia 3 Gregg De Young Diagrams in ancient Egyptian geometry Survey and assessment Historia Mathematica 36 321 373 2009 4 Gregg De Young Mathematical diagrams from manuscript to print examples from the Arabic Euclidean transmission Synthese 186 21 54 2012 5 Gregg De Young Takanori Suzuki Ken Saito and Ken ichi Takahashi Diagrams in Greek mathematical texts 2011 http www greekmath org 6 Marie Th r se Debarnot Kitab Magalid ilm al Hay a la trigonom trie sph rique chez les Arabes de l Est la fin du X si cle Damas Institut fran ais de Damas 1985 7 Johan B C Engelen How to include an SVG image in FTX 2013 CTAN info svg inkscape 8 Joran Friberg Amazing traces of a Babylonian origin in Greek mathematics Singapore World Scientific Publishing Co 2007 9 Michel Goossens The X TEX Companion TEX meets OpenType and Unicode 2010 10 Michel Goossens Frank Mittelbach Sebastian Rahtz Denis Roegel and Herbert Vof The ETEX Graphics Companion Boston Addison Wesley 2007 11 Ebenezer Henderson Life of James Ferguson F R S etc Edinburgh A Fullarton amp Co 1867 facsimile reprint by Cambridge University Press in 2010 12 John D Hobby METAPOST a user s manual 2013 www tug org docs metapost mpman pdf 48 13 Guy Le Meur Le r le des diagrammes dans quelques trait s de la Petite ast
4. This is what we did but it is rather cumbersome A much easier way to draw curves is to use the hobby ex tension of TikZ named after John Hobby the author of METAPOST s path drawing algorithm which makes it possible to define curves the way they are defined in METAPOST This extension requires a rather recent version of ATEX since it uses advanced LTEX3 features and we haven t used it here Another possibility in order to facilitate the positioning of points is to use the TikZEdt editor http www tikzedt org Like we did with METAPOST we have used TikZ to trace the figure used in DRaFT s tutorial figures 23 25 We have in particular used this example to illustrate label switching that is the possibility to go from one set of labels for instance Arabic to another for instance Latin In order to achieve this functionality all labels are defined with the lab command which takes two parameters A boolean is then used to select which language we wish to display This could also have been done with METAPOST We have also used the onimage package which provides the simple tikzonimage environment One of the arguments of this environment is the base figure An option to this environment will add help lines And once the trac ing is done the original figure can be hidden by setting an appropriate boolean to false Points could have been drawn using various styles such as the one popularized by Neugebauer see fig
5. and the way some of its features are represented may be objectionable but this is another story and we will come back to that matter in a moment Figure 2 Zooming in on the first figure highlights the fact that it is a bitmap 1 2 Bitmaps The main defect of the figure then is to our modern eyes the fact that it is a bitmap figure Or more precisely the figure which is included in this document is a bitmap If you try to zoom on it it will look terrible figure 2 especially when put next to vectorial fonts or figures Other examples of bitmap drawings are given in figure 9 and are unfortunately common in today s publications This problem should be distinguished from problems of the original figure which may have to do with the print ing process or with some logical features of the drawing For instance the dashes between X and gt are at times irregular but apart from such minor details there are no noticeable micro problems 3 Figure 3 An original figure from 1867 left and its reprint in 2010 by Cam bridge University Press right A number of details have entirely been lost in the reprint The figures have been scanned at 600dpi from the 1867 original and from the reprint 11 Interestingly the text of the reprint not the text of the figures is well reproduced see the caption The Crank Or rery and the problem is not merely one of scanning resolution It seems that the text was
6. A node below sloped K c B node above sloped K a C node above sloped K b A end tikzpicture end document Figure 31 Chinese labels in a TikZ figure with XJATEX 43 eo Nf 3 lt A B C db C C e Y amp v My 7 A B a cN B RS c C C v V Y oN o d a 2 a S 2 e s CA B Ae c A IIHHA C B C C amp os 0 CS KN ee c chiu d i c Figure 32 TikZ figures using various languages and fonts with XHATEX 44 5 4 METAPOST code examples verbatimtex A amp latex documentclass article usepackage CJKutf8 begin document newcommand korean 1 begin CJK UTF8 mj 1 end CJK etex numeric u u 1cm beginfig 1 pair A B C A origin B A 5u 0 C fullcircle scaled 8u shifted A intersectionpoint fullcircle scaled 6u shifted B draw A B C cycle label 11ft btex korean 1 etex A label 1rt btex korean 11 etex B label top btex korean t etex C label bot btex korean 0 sqrt 3 2 472 5 etex rotated angle B A 5 A B label btex korean Z 0 3 etex rotated angle B C 5 B C 2 5labeloffset unitvector B C rotated 90 label btex korean 2Z0 4 etex rotated angle C A 5 A C 2 5labeloffset unitvector C A rotated 90 endfig end dore Figure 33 Korean labels in a METAPOST figure with pdfTEX 45 A 5 Jab B A IJHHa 5 B Figure 34 META
7. EPS file within a METAPOST figure Figure 20 shows the original figure after it was traced together with the grid used for setting the points as well as the traced figure alone In order to obtain this capa bility we had to make a modified version of the ext eps extension since the available version was meant for a permanent figure inclusion Figures 21 and 22 show the METAPOST source code for the first figure with the included EPS The core of the code is made of the definition of various points z0 3u 28u etc of various curves using epsdraw 22 and of labels using epslabel1 Setting the boolean grid to true displays a grid which is useful for finding the coordinates of the points expressed as a fraction of the whole width When tracing is done the original figure can be hidden by setting showeps to false G D L E Figure 17 A METAPOST figure for a proof of Pythagoras theorem Euclid Elements Book I Proposition 47 Figure 18 shows the source code The pink areas are equal and so are the blue ones In the first case the area of the square BAGF is twice that of the triangle FBC And this triangle is equal to the triangle ABD which is itself equal to half of the rectangle BDLK The second case is similar 23 beginfig 1 numeric u u lcm pair A BC DEP H I Koby B orioiine C B Eu 0 A fiullcircle scared Gu shifted C 1LitersectionpoilLnte frulleirels scaled Su shiftec 5 s F B G A AR ro
8. course an abstraction of some concept or object When we reproduce a figure when we recreate it we create a new abstraction we add a layer to the interpretation of the concept or object If Bouasse s figure is corrected we may be satisfied of having improved his figure but at the same time we conceal the original work and we are perhaps betraying the author s intentions Whenever there is a doubt it is probably best to show both the original drawing and its recreation Finally an exact reconstruction sometimes helps to locate errors or omissions in an original figure As an example we can consider the rather complex drawing of gears by Olivier figure 4 from 22 Several prob lems were found during the reconstruction of that figure 3For a possible solution to these problems see our work on the drawing of great circles and parallels on a sphere 24 Although our work is not totally general and cumbersome in places it does highlight a real problem in that kind of drawings 6 Figure 4 An example of a partial reconstruction of a complex drawing with METAPOST 22 Top original drawing Bottom reconstruction 2 The quality of reconstructions Depending on the purpose of a reconstruction some tools may be more adapted than others If the accuracy of the geometry is important and if some points on separate lines should coincide such as when two circles are tangent then exact posit
9. few hours on computer and using the above mentioned basic components Such drawings will be called diagrams f le Nord ON Fole Sud Fig 48 Figure 1 A figure in a book by Bouasse published in 1925 1 1 Issues 1 1 A first example An example of a rather elaborate diagram that we might consider is given in figure 1 and is taken from one of the many books written by Henri Bouasse A number of issues are raised by that figure The first is that of its use Since we have a good scan of the original figure we can use it in another publication if needed The figure could then be supplemented by a description which would mention point y and give the meaning of other features So far so good Admittedly the scan could be better and we have not removed the numerous stains originating either in the original paper or in the photocopy which was used But we may also want to do more or at least something else If for instance we plan to publish a new edition of Bouasse s original book it might be desirable to produce a faithful reproduction of that figure but a reproduction which is perfect and without defects This suggests that the original figure has some defects Of course if we look at the figure from the 1925 vantage point the figure is nearly perfect It has the quality which might be expected from figures in scientific books of that period The content of the figure or more precisely the way it was drawn
10. from Inkscape the mp output was slightly edited removing the second coordinates and introducing a variable 244 457001 719 875000 371 734009 JIS 867004 p 215 164001 681 492004 s N N N of of o ol Me Il 225 55 7 604980 343 089996 226 AGS 7TLOVDY 3622811006 Z21 474 773010 483 500000 this was forgotten and we recompute it here zZ28 whatever z0 zl whatever z2 z3 Z29 z726 0 20 scaling everything down Lor 1 0 wore 29s x i eee fy SR le yf ale cles Sy endfor Figure 27 METAPOST code for the reconstruction of DRaFT s example Only the coordinates of the points were set with Inkscape begin 35 draw 271 278 79 710 711 zl2 z13 z14 craw 26 74 71 70 228 72 73 74 55 Craw z10 727 CHAN ZilQ Z15 5 draw z0 z4 Gheewy ZI SIG 21706218 Clicam Zzi1 229 222 draw 4 226554255 A2 A Craw z20 724 crea 2215542355424 54 225 Ciaw 211 219 6 42205 clan 221 55420 654125 5429554225 Craw 2i 2zZ22 5 chaw 2Z1L8 Z2 is labels pickup pencircle scaled 2pt row 1 0 uote 29 draw zlil Sl loel ctoo clecimal i Zi wicacolor rec endfor pickup pencircle scaled 5 label top btex selectlanguage arabic etex z15 endfig end Figure 28 METAPOST code for the reconstruction of DRaFT s example Only the coordinates of the points were set with Inksca
11. several graphical tools and several ones based on programming 4 1 Tools with mostly mouse interaction In this category of tools a figure is created using the mouse and by select ing various options in an interface or in menus Parts of a drawing can be selected and modified Here we will only discuss two tools one very specialized the other much more general 4 1 1 DRaFT DRaFT is a small Java program created by Ken Saito for the purpose of editing figures from ancient mathematical manuscripts 27 29 Its main features are the ability to trace lines or curves over an original figure given by a JPG file DRaFT also allows for the inclusion of labels Labels can be scaled or rotated Lines can be colored or dashed When using DRaFT the user defines points and these points are listed in a special window Then the user joins some of these points using several types of curves DRaFT produces an EPS file which can then be postprocessed and in cluded in a document It has been used in several recent articles in partic ular those by Saito De Young Sidoli Malpangotto and Le Meur 26 31 32 3 15 5 4 28 30 13 33 An example of a figure traced with DRaFT is given in figure 15 based on an original manuscript figure 14 4 1 2 Inkscape Inkscape is a general open source program for vector graphics Itis not targetted towards the edition of ancient diagrams but it can easily be used SAT is free open source and av
12. 89 5 810 2ll 2Z12 5 SESMSLOMI 25 648s HIS owls vaivacolor Greene epsdraw Zils censiomil 2 21l4 censionl 25215 oo tens iLeMml 2 24 wiitineolor blues eosciraw 24 216 eicgimt zly zilS mithcoler 1 5 7 5 epsdran ZIZ Zi3 waltclace lloe blue epodan Wiki 72170 25621 wivtacolor 23 5 3 2 eoscraw v4 5 42193 220 219 waicieolor 3 8 3 Gjoscleawy zo 221 wWalielueollow die Goscraw 23 722 walclncollorw DLNS Gosciraw Zz23 2ZZ4 22 5 22Z6 22 I 24 29 22 9 Z30 withcolor red drawoptions withpen pencircle scaled 2u epsdraw z25 z29 withcolor green dashed evenly iciLCke drawoptions withpen pencircle scaled 2u epsdraw z1 0 2u zl 0 Zia s epsdraw z2 0 2u 223 0 Zw 3 drawoptions withcolor red epslabel top btex ar etex z0 Sos llalosil ulke locexx euet so Giese wil p endeps endfig end Figure 22 The METAPOST code with exteps 2 2 28 4 2 2 TikZ Figures can also be traced with TikZ 34 a generic graphical TEX environ ment The approach is somewhat similar to that of METAPOST With TikZ points are put in place and lines are drawn Curved lines are a bit more tricky because the curves provided by default in TikZ make it hard to go through a sequence of points Bezier curves can be drawn but control points have to be carefully selected
13. Editing ancient technical and mathematical figures Tools and traps Denis Roegel 21 June 2015 Abstract This study examines a number of problems and issues in the edi tion of ancient technical and mathematical figures and review a num ber of solutions and possible extensions to better suit the needs of historians Introduction Scientific articles books and manuscripts may contain technical drawings Contrary to more artistic and free drawings technical drawings aim at expressing some abstract properties or try to give a more or less accurate representation of an object an instrument a machine and so on Although it is quite possible to imagine border cases the features which usually define a technical drawing are its basic components which are points segments curves colored surfaces and various labels Here we will focus on drawings which could be produced by hand albeit perhaps not with the same perfection We therefore deliberately ignore those draw ings which can only be generated by a computer for instance drawings containing a very large number of elementary features such as segments The simulation of a tree or other natural objects are examples of drawings that will intentionally be excluded from our study A good rule of thumb might be that most of the drawings of interest to us are those that can be Denis Roegel LORIA BP 239 54506 Vandceuvre l s Nancy cedex roegel loria fr drawn in at most a
14. H Y T G D draw H E Z G W draw dashed green E G NOWwawe N sey draw X X1 draw OX controls R2a ame RZD RL two ticks draw OM 0 007 0 02 s S M 0 010 0 02 Ss Ndraw OR 0 007 OM OL NS NES Ki 0 0107 0 02 Ss node color blue above at A Large lab A node color blue at M 0 05 0 02 S Large lab M end tikzonimage end document Figure 25 The source code of the reconstruction using TikZ 2 2 32 4 2 3 Inkscape METAPOST Another solution which lies between purely graphical tools and program ming solutions is to use a tool of the first kind in order to produce a sketch of the tracing and to modify it using another tool Inkscape for instance generates a SVG file and that file could be edited directly It could also be postprocessed An even better solution is to convert the SVG file to METAPOST using the following commands cSwe Comyerc rile Swe i joe iiile joclic psto dit q Tiles pdf i npost Lr1il me However the METAPOST code will be rather low level and labels will not be converted Nevertheless labels and other features can be added within METAPOST In order to make this useable it is best to set only points in Inkscape But since points are actually circles they produce complex curves in META POST Even simpler is to draw small ticks in Inkscape and this w
15. POST figures using various languages and fonts with Pdf TEX 46 Conclusion The examination of a number of figures drawn in the current literature has shown the existence of a number of problems and issues Modern figures produced using graphical interfaces often contain subtle problems which were not intended by their authors but crept in because the interfaces make it difficult to get the drawings right Certain drawings produced by tracing figures such as ancient mathematical diagrams may intentionally contain imperfect junctions but these drawings nevertheless also have a more perfect side for instance smooth curves or consistency of strokes and so on There therefore appears to exist a real need to achieve both a certain perfection in tracing usually by some hand tracing and at the same time to be able to control other features at a higher level and this can often best be obtained by programming Several solutions have been examined and we have argued for the extension of some tools to even better suit the needs of the user We hope that our study will help histo rians to produce even better critical figures especially with the advent of UTF8 OpenType compliant versions of TEX 47 References 1 Henri Bouasse Propagation de la lumi re Th orie de la r flexion vitreuse et m tallique Paris Delagrave 1925 2 Francois Charette and Arthur Reutenauer Polyglossia An alternative to Babel for Xq4TRxX and Lua
16. TAPOST is used through LuaBTEX then OpenType fonts can be used An example of the use of Chinese OpenType labels in META POST through LualTgx is given in figure 29 but it requires a fairly recent version of Lual TEX As an illustration we give below a number of examples of figure la bels in various languages both with standard LaTeX and METAPOST and with XHTEX and TikZ In Lual4Tpx it is now possible to use TikZ with or without OpenType fonts METAPOST as an external program with standard fonts and METAPOST included with luamplib package with or without OpenType fonts Note that you should have a fairly recent and complete installation of TeX and its derivatives to get all this working A basic installation of TEX or one which is five years old will not do 40 documentclass article usepackage fontspec usepackage luamplib newfontfamily cjkfont Script CJK AR PL UKai CN begin document begin mplibcode numeric u u 1cm loxexejalinuic Ley 1L 9 pair ZA 1B CF A origin B A 5u 0 C fullcircle scaled 8u shifted A intersectionpoint GEullileireele s ealecl Gu Simaiziceicl 8 5 draw A B C cyels lebel bor bres NORTON NS NS Gt 3 24 2 55 GTER rotated angle B A 5 A B endfig end mplibcode end document Figure 29 Example of inclusion of METAPOST in a Lual Tpx file 41 5 3 XBIEX code examples documentclass article usepackage tikz usetikzli
17. a perfect drawing from scratch We will first review some of the basic components and operations in volved in the creation of figures and then consider what are the choices that we are offered and how we should decide on what type of reconstruc tion is needed 3 1 Features 3 11 Setting points In order to work on ancient diagrams there are a number of important requirements First of all we need to set points by obtaining them from another figure There must therefore be some way to relate an original drawing to a new one This however does not entail that we have to use an interface or even to include the original diagram as an overlay It is quite possible to take measurements on a point without scanning a figure 3 1 2 Tracing and cropping Tracing is the operation of creating paths based on existing images The most common paths are straight lines segments and curved lines or arcs We should be able to draw lines of various thicknesses colors and styles such as dashed lines There should also be different styles of points for instance the possibility to draw Neugebauer style points figure 12 Some figures may be cropped using closed paths We will not enter into the details of cropping but some tools make it possible to crop a figure according to a user defined path 13 fig 1 Figure 12 An example of a figure using the Neugebauer style points 6 This style is for instance also used in figures 6 11 and 17
18. abic using the ASCII encoding and still obtain a genuine Arabic output However things are much simpler when one uses an encoding capable of encoding all major scripts and at the same time fonts using this same encoding This is the case with UTF 8 5 1 Interfaces With Inkscape labels can be entered in UTF8 either directly or by copy pasting from another source such as an editor or a web page Various fonts are available through Inkscape Inkscape can also include TEX for mule although we haven t used this capability DRaFT can also use fonts in the same way but cannot use MITEX math ematical expressions 5 2 LTEX With LTEX there are a number of possibilities depending on how text is encoded and which version of TEX and which graphical environment one is using For instance some fonts such as OpenType fonts are only avail able with XqlTEX or Lua TEX There are many available OpenType fonts and a good way to find out which ones are installed on your computer is to check the fonts available in LibreOffice 5 2 1 Versions of T X The multiple versions of TEX may be confusing for newcomers and even for longtime users who have clung to their old habits TEX is the funda mental program and FTRxX is a large superset of TEX TEX uses TEX pdfTEX 39 is an extension of TEX outputting PDF files XJI X 20 21 9 is a more re cent extension of TEX in which OpenType fonts can be used and which is natively based on
19. ailable on all usual operating systems 7We will not consider other tools such as automatic tracers for instance pot race which trace a figure without manual intervention and produce vector graphics as these tools still leave the burden of simplifying the original figure 17 Figure 14 The original diagram used in DRaFT s tutorial 29 source file Tbn_Sahl_manuscript jpg on Wikimedia for that purpose Inkscape outputs SVG As an example we have tried to obtain more or less the same diagram as the one given in DRaFT s tutorial The two figures the one by DRaFT figure 15 and the one by Inkscape figure 16 are not identical because we did not try to make them totally identical Instead in both cases the original image was loaded in the program the image was manually traced by the introduction of points and adequate lines and arabic labels have been added scaled and rotated Since Inkscape has many capabilities it may prove more difficult to use especially for beginners because we have to wander through many buttons and menus that we do not need In that sense a simple program such as DRaFT may be more practical for focused beginners But the fact is that Inkscape is quite usable for that task In our experience in order to trace a figure as quickly as possible one should set the straight or curved lines with Bezier curves placing points in several positions in particular wherever several lines will meet or
20. brary intersections usepackage fontspec setmainfont Linux Libertine 0 setsansfont Arial newfontfamily greekfont Script Greek Linux Libertine 0 newfontfamily greekfontsf Script Greek Linux Libertine 0 usepackage polyglossia setdefault Language greek begin document pagestyle empty def firstcircle 0 0 circle 4 cm def secondcircle 5cm 0 circle 3 cm begin tikzpicture coordinate label left A A at 0 0 coordinate label right B B at 5cm 0 path name path c1 firstcircle path name path c2 secondcircle path name intersections of cl and c2 coordinate label above C C at intersection 1 draw A node below sloped u koc c B node above sloped u koc a C node above sloped unkoc b A end tikzpicture end document HiKos C Figure 30 Greek labels in a TikZ figure with Xq TEX 42 documentclass article usepackage tikz usetikzlibrary intersections usepackage xeC JK setCJKmainfont AR PL UKai CN begin document pagestyle empty def firstcircle 0 0 circle 4 cm def secondcircle 5cm 0 circle 3 cm begin tikzpicture coordinate label left A A at 0 0 coordinate label right B B at 5cm 0 path name path ci firstcircle path name path c2 secondcircle path name intersections of c1 and c2 coordinate label above C C at intersection 1 draw
21. cross For then these points can serve as anchors for future points If this is not done points can be added later 8Of course a good drawing takes time and we are not suggesting that work should be messy but we are looking for efficiency and some approaches are more efficient than others 18 new file linestyle_edit xml file edit options background shapes styles operations info Nlln POS ES Eur 8 290 0 342 0 9 283 0 149 0 10 279 0 225 0 add 11 254 0 211 0 12 258 0 392 0 13 255 0 244 0 14 143 0 391 0 15 77 0 390 0 16 222 0 147 0 17 104 0 148 0 18 20 0 147 0 19 411 0 225 0 rhoint data identifier subpoint of label se visible 7 shapes groups uw ey pom tart p 12 point 11 add add brol start p 8 point 9 R broken ime stari p 6 point 7 broken line start p 1 point 18 broken line start p 5 point 4 broken line start p 3 point 19 gt aae brol Figure 15 DRaFT s interface above from 29 and the diagram produced by DRaFT as given in the tutorial 29 Notice by zooming that the lines apparently meeting at point amp actually do not meet although they were probably meant to There are also other anomalies 19 it View Layer Object Path Text Filters Extensions he OBE ve O root No objects selected Click Shift clic nr z 71
22. digitized at one resolution and that the figures were dig itized at another It is also possible that the publisher understood that the figures required special care but that merging figures and text caused the figures somehow to lose most of their interest It is however surprising that this went down to the printing and was not caught earlier On the other hand more and more established publishers for instance Hachette with the French National Library are giving in to low quality print on demand reproductions The fact that the bitmap nature of the included figure is apparent sug gests that the original figure was not digitized at an adequate resolution This is indeed often the case in many of today s reprints of ancient edi tions including by well established publishing houses figure 3 But the truth is that the original printing had defects and that no matter which resolution is taken either the ruggedness of the original drawing or the insufficient scanning resolution will show up when one has a closer look at such a drawing This makes such a drawing awkward for modern com puter usages but it may even be a problem in printed editions featuring reproductions Of course most of these problems were irrelevant in the original editions because nobody would have read a book with a micro scope Now people do 13 Digitizing a drawing Although the main point of this article is to detail tools for recreating or improving t
23. echnical or geometrical figures we should give some guide lines about the inclusion of facsimiles Original drawings if reproduced at the same scale should be digitized at at least 300dpi and a larger resolution will often be better The thresh hold of 300dpi is related to the usual reading conditions and to the resolu tion of our own eyes If a drawing is shown in closeup a larger resolution will hide the fact that the reproduction is a reproduction Otherwise if the resolution is insufficient it will show up even though it may appear seamless when printed at the original scale Manuscripts and also older prints should be digitized in color with at least 24 bits per pixel 1 4 Correctness Another problem with the original drawing of figure 1 is that it is techni cally incorrect The drawing shows a representation of the celestial sphere with three diameters two meridians and half of the equator and ecliptic There are two important errors First if the vantage point does not lie in the plane of the equator the two poles should not lie on the boundary of The necessary scanning resolution actually does not depend only on how much a drawing or photograph is magnified but also on how much useful information it con tains The physical constraints in particular the optical constraints of the production of an original such as a digital photograph put a limit to that information and hence to the useful resolution For a deeper a
24. es which are part of web pages and such figures should when possible use a vector format such as SVG e second complex drawings should be parameterized so that a figure can easily be scaled and the features of its lines curves points or the colors etc easily changed In addition the user should have a fine control over labels and be able to compute accurately the positions of some points if there is such a need Of course any repetitive work should not be done with a graphical interface For instance if we want to change the thickness of a number of lines or change globally the size of labels or their colors etc a program will do much better 38 5 Languages Technical or mathematical drawings usually contain labels The figures shown earlier have already illustrated the use of Latin and Arabic labels but there may be a need to enter labels in other languages or to add more complex annotations such as mathematical expressions How this can be done and even if this can be done depends on the tool used to produce a drawing Whatever tool or environment is used the characters are typed using a certain encoding This can be ASCII ISO Latin 1 UTF 8 or one of many other encodings Such encodings are called input encodings and in certain cases they may be changed within a same text Fonts on the other hand may use the same or a different encod ing as the input encoding It is for instance possible to type Ar
25. h we can obtain a good curve going through a number of points and in fact there have been efforts to import this capability in TikZ Although this is useable and we use it an even better approach would be a way to position a number of points graphically possibly in a rela tive way to name them to connect them and then work out the remain ing part in a purely textual way using the names given to the points in order to have all the benefit of automation of consistency and so on None of the tools described in this survey provide all of these functionali ties Many if not all of these features could be made available through a purely graphical interface but it would entail creeping featurism which eventually would deter new users On the other hand creeping featurism in a programming language does not deter new users for the very reason that new users do not see the new features You only see what you use In any case whatever the tools that one uses there are two important priorities to have in mind when tracing a figure or producing an entirely new one e first it is important to stick to vector graphics as much as possi 37 ble and to keep lines smooth a figure produced by a tool such as Inkscape DRaFT METAPOST or another one should never be scanned but directly included as vector graphics for instance as a PDF file but PDF files can also contain bitmaps so that one should be careful this also applies to figur
26. ill result in small strokes and not dots in METAPOST The second coordinates can then be manually or automatically discarded This is what we did in figures 26 28 With this hybrid technique we could for instance also add irregular hand traced dashed lines and still benefit from all of METAPOST s fea tures Incidentally Inkscape also has the capability to output both a SVG and a BT X file so that the figure included in a document will have fonts matching those of the document 7 4 2 4 Other ATEX environments There are several other TEX environments or packages which could be used for our task The simplest of these environments is certainly the stan dard picture environment It will however be far from convenient as tracing curves is not that easy Bezier curves can be drawn but the ease of drawing curves with METAPOST will be seriously missed Another possibility is the pst ricks package with which tracing a fig ure should be very similar to the way it was done with TikZ 33 Figure 26 The reconstruction of DRaFT s example by setting a number of points in Inkscape but by doing all further work in METAPOST Here we made use of only one arabic label 34 verbatimtex S amp latex documentclass article usepackage LAE LFE T1 fontenc usepackage ut 8 inputenc usepackage arabic english babel begin document etex numeric u u lcm beginfig l coordinates obtained
27. ions or intersections should be used This is not always what is desired For instance in points A and G of figure 5 traced from a manuscript one may wonder if the lines were intended to overlap I believe they were meant to This does however raise the issue that if something is meant to overlap then the overlap should be visible without zooming If we want to stress some feature of a drawing we should magnify it not conceal it as this then leaves some doubt about the intent of the author Therefore if the lines reaching A and G had been meant to overlap we think that they should have overlapped much more However the same author then gives a new construction which only keeps some of the features of the original and which is not traced from the original One might therefore expect an exact reconstruction or at least good junctions and this is not the case figure 6 Figure 7 is another simple example with an unfinished arc F and in adequate junctions at B It would surprise us if such features had been intended Another example is given by Friberg in figure 8 The figure is rather complex and it is obvious that Friberg must have gone to a great length to get it as right as possible with whatever software was used But when you look closely at Q and many other points it will look rather messy Yet another subtle example is shown in figure 11 where point E was obviously placed by hand without computing the actual in
28. miles can be made more readable by enhancing faded lines but such enhancements should be made explicit that is they should not be con cealed If facsimiles are not used then we usually only want to show the rele vant features of a drawing We have to decide which features are impor tant to us and which ones are not Sometimes we want to highlight that an original drawing had lines that did not meet But we can achieve this without having to trace the original drawing Once this is understood reproducing a drawing can be made much simpler If we want to show the differences between the figures of various manu scripts a good idea might be to give facsimiles but also approximate re constructions Tracing the original figures is perhaps not always necessary Of course some features are most naturally drawn by hand for in stance the edges of a broken tablet Figure 19 shows an example of a figure drawn entirely in METAPOST without any tracing and aimed at being an approximation of the earlier figure 11 The dimensions and angles could be fine tuned but in many cases an exact reproduction is not sought 16 4 Tools In this section we are going to survey a number of tools and use them in order to illustrate the essential features of technical drawings Our survey will not be comprehensive though We will restrict ourselves to the pro duction of drawings for ATEX And we will only consider representative cases We will examine
29. n of the figure given earlier figure 11 but with the circled points correctly positioned This figure was not traced from the earlier one only constructed by visually guessing the dimensions and deducing the position of B from those of D E A and Z 25 80 y 70 60 50 CG 40 30 y 2 20 10 Ce 0 0 OT 07 0 0F 09 09 04 08 06 Figure 20 The reconstruction of DRaFT s example with METAPOST and the exteps extension 26 verbatimtex S amp latex documentclass article usepackage LAE LFE T1 fontenc usepackage ut f8 inputenc usepackage arabic english babel begin document newcommand ar 1 selectlanguage arabic 1 eTe input exteps roegel modified version of exteps showeps true Sshoweps false prologues 2 beginfig 1 pegineps TSarto iig eps base s 0 0 scale g 0 n 0 A7 overall scaling GE Frgure clipping true grid true u pct labeloffset 1 5u distance of labels defaultscale 2 scaling of labels zZ0 3u 28u z1 20u 28u z2 42u 28u 5 z3 54u 28u z21 22u 69u z28 28u 64u z29 48 5u 47 5u Z30 53 50 43 Figure 21 The METAPOST code with exteps 1 2 27 drawoptions withpen pencircle scaled 5u epsdraw ZO0 gil Z2 23 74 745 26 27 Wwallncoloir iceclp Sosciraw 24 4
30. n tracing a drawing are to be able to hide the original drawing after the tracing is completed and to have help lines for positioning additional points Also useful is the ability to switch between various sets of labels latin arabic etc 4 2 1 METAPOST METAPOST 12 is a companion program to TEX and is particularly well suited for technical drawings A METAPOST file is a text file which is processed by the METAPOST program and can produce either PostScript EPS or SVG We will not enter into the details of the various formats but formats can be converted and METAPOST outputs can be included in various documents not only in TEX documents such as the present one Figures 17 and 18 show an example of a figure constructed with META POST with one point A obtained as an intersection of two circles not shown and other points determined using rotations This example il lustrates a proof of Pythagoras theorem taken from Euclid s Elements Book I Proposition 47 Figure 4 shown earlier is a much more elaborate reconstruction In that example the METAPOST figure depends only on a few parameters and aims at producing an exact construction in descriptive geometry Such an exact construction may help and in fact did help to identify omissions or errors in the original drawing We have used METAPOST to trace the figure used in DRaFT s tutorial This was achieved using the exteps extension which allows for the in clusion of an
31. nalysis of the problems of digitization in libraries and archives see our recent study on sustainable digitization 25 the sphere but more inside Second the great circles should be tangent to the outer circles which they are not especially for one of the meridi ans and for the ecliptic These representation choices were common in the past and the figures were not always meant to be accurate We therefore have to consider whether these features should be kept or not In some cases the drawings might be improved 1 5 Reproducing or improving The problems of Bouasse s figure leads us to ponder the questions of re production or improvement By reproduction we mean a faithful recre ation of a figure without logical improvements Usually only the lines or curves become perfect but the points are those of the original figure An improvement instead will introduce deeper changes When reproducing a figure the first question to ask is why it is repro duced Do we merely want to show the original drawing Or do we want to edit it as part of a critical edition or a translation Do we want to trans late the labels and make them readable Is the original figure so faded that it needs to be edited If we recreate or reconstruct a figure we have to ask ourselves whether we want to improve it whether we want to show it as it was intended or whether we only want to retain some of its features and which ones A figure is of
32. pe end 36 43 Summary We have mainly examined two approaches for tracing an original figure one using a mostly graphical approach in which the mouse is used to reconstruct paths and position labels and another one in which everything is programmed Both approaches have advantages and drawbacks We found it time consuming to use Inkscape to draw paths especially because we always only saw the end result and that the way to the result had vanished There is no visible trace of which button and which menu option was used and this can be confusing It also means that different versions of the interface can force us to revise our habits and of course that it is difficult to tell someone else how something was achieved In terms of speed using an interface is certainly the fastest way to trace a figure but it is not the most flexible one With an interface it is easy to forget to have some lines meet especially if one line end meets another one in the middle as in figure 5 when it is supposed to not go beyond it This explains why some figures produced with such tools have defects An interface does not encourage perfect drawings and may even make it difficult But we have also found it very time consuming to position points with TikZ and to link these points Most awkward was the tracing of curves because we had to position control points in the void and to adapt them A much more practical approach is that of METAPOST in whic
33. problems become ap parent Most of the problems are about junctions lines do not meet or they overlap 11 Figure 10 A recent figure by Panza 17 Left complete figure Right de tail with wiggled curves The figure is actually taken from the preprint but the actual published online version is even more bizarre The fig ures shown here are exactly those of the original article Figure 11 Left a recent reconstructed figure where the circled dot at point E is slightly offset from the actual intersection 33 This is almost unno ticeable without zooming Right detail at point E The figures shown here are exactly those of the original article 12 3 New figures vs traced figures requirements We can distinguish two types of figures First a new figure may be based on an existing one but without the purpose of reproducing all of its fea tures For instance when a manuscript shows a square which is not totally square the new figure may show an exact square Traced figures on the other hand will try to retain some of the imperfections of the original fig ures such as squares with non parallel sides or with lines that do not exactly meet In both cases a number of techniques are useful for the edition or the creation of a new drawing These techniques overlap to a large extent so that it would be pointless to describe first those techniques useful for tracing a figure and then those useful for creating
34. ronomie Revue d Histoire des Math matiques 18 2 157 221 2012 14 LuaTEX Reference Manual 2015 www luatex org 15 Michela Malpangotto Graphical choices and geometrical thought in the transmission of Theodosius Spherics from Antiquity to the Renaissance Archive for the History of Exact Sciences 64 75 112 2010 16 Paolo Mascellani Pier Daniele Napolitani and Veronica Gavagna M2 5 2 Minutes of the workshop on The problem of diagrams and drawings criticism in mathematical texts 2005 17 Marco Panza The twofold role of diagrams in Euclid s plane geometry Synthese 186 1 55 102 2012 18 Manuel P gouri Gonnard A guide to Lua TEX 2013 19 Roshdi Rashed editor Histoire des sciences arabes volume 2 Paris Editions du seuil 1997 20 Will Robertson and Khaled Hosny The XqIgX reference guide May 21 2013 21 Will Robertson and Khaled Hosny The font spec package Font selection for X TEX and Lual4TRx 2015 CTAN macros latex contrib fontspec 22 Denis Roegel A complex drawing in descriptive geometry TUGboat 28 2 218 228 2007 23 Denis Roegel An early 1844 key driven adding machine IEEE Annals of the History of Computing 30 1 59 65 2008 24 Denis Roegel Spheres great circles and parallels TUGboat 30 1 80 87 2009 25 Denis Roegel Sustainable digitization 2012 http locomat loria fr 26 Ken Saito A preliminary study in the critical asses
35. select the snapping option so that one point falls on the other when they are sufficiently close But even if this is not done one always has the possibility to zoom on the drawing and to manually move the points so that they are sufficiently close This will not be perfect but it will be better than nothing Labels are easy to include and can either be written directly in one of the available fonts or copy pasted from another program In our recon struction of DRaFT s example we have used the Scheherazade Open Type font which is installed on our computer and which is freely avail able Inkscape produced a SVG file and this file was converted to PDF using the program rsvg convert It was then included in our document Incidentally we have departed from DRaFT s construction in a few ways either with the choice of labels with the two ticks on the horizon tal line or with a couple points that we chose to split in order to be more faithful to the original diagram 21 4 2 Tools with mostly programming Tracing a figure with an interface is not the only way to do it In fact we have to admit that we found it very cumbersome to trace a drawing with Inkscape having constantly to figure out on which button to click which mode we were in selecting the font moving some labels which were sometimes very small or very thin so that we tend to prefer the pro grammatic approach With such tools important requirements whe
36. sment of diagrams in Greek mathematical works SCIAMVS 7 81 144 2006 27 Ken Saito DRaFT reference manual for DRaFT release 1 1b25 2012 http www greekmath org 49 28 Ken Saito Traditions of the diagram tradition of the text A case study Synthese 186 7 20 2012 29 Ken Saito and Gregg De Young DRaFT tutorial 2013 http www greekmath org 30 Ken Saito and Nathan Sidoli Diagrams and arguments in ancient Greek mathematics lessons drawn from comparisons of the manuscript diagrams with those in modern critical editions In Karine Chemla editor The history of mathematical proof in ancient traditions pages 135 162 Cambridge Cambridge University Press 2012 31 Nathan Sidoli What we can learn from a diagram The case of Aristarchus s On the sizes and distances of the Sun and Moon Annals of Science 64 4 525 545 2007 32 Nathan Sidoli and Takanori Kusuba Nasir al Din al T sr s revision of Theodosius s Spherics Suhayl 8 9 46 2008 33 Nathan Sidoli and Takanori Kusuba Al HarawT s version of Menelaus Spherics Suhayl 13 149 212 2014 34 Till Tantau The TikZ and PGF Packages Manual for version 3 0 0 cvs 2014 http pgf sourceforge net 50
37. t lt dv Er A amp iL jan 1S N So IR KT Another curious example is given in an article by Panza 17 There the curves of one of the figures strangely became wiggled figure 10 This was certainly not intended and perhaps even went unnoticed Some authors use software which is acceptable but somewhere on the line the figures get mangled Examples are for instance given by Malpan gotto and Le Meur figure 9 whose figures have all become bitmaps al though they were certainly vector graphics at the beginning 15 13 The loss of the vectorial nature of the drawing may have occurred with the publisher perhaps when EPS graphics were transformed in JPEG for in clusion in the articles A number of years ago the same happened with one of our articles 23 We had prepared a vector graphics and provided it as a PDF file Alas it was included as a bitmap This is very noticeable in the online version but almost invisible in print These examples show that the tools alone are not the solution but what also matters is how the tools are used Figure 9 Recent figures by Malpangotto 15 and Le Meur 13 These figures appear as bitmaps in the PDF and are shown here exactly as they appear in the original articles Now in most cases the imperfections are barely noticeable especially in print The authors often go to a great length to produce the best draw ings and it is only by scrutinizing the figures that
38. tated 90 H A I C C A rotated 90 B D C E K L C B rotated 90 A K whatever B D K whatever B C rill B A G F Cyele waliclacolowr 1L 7 21 o son hne cares orhe areas pickup pencircle scaled 2pt Chea A B Cey elep a wieweinclle dean A G Peb A Dee eSa Cle draw K L pickup pencircle scaled 5 dravi Cohen A Dpcheany B T Aran ARS draw A K dashed evenly pickup pencircle scaled 4pt Tor esky Br C ao bik CHAR Kens netipoime S s amp tor cirelec pointe SIAClIE Ore F pickup pencircle scaled 5 label togo ocex SES tex C label lft eorex SES etex F endfig Figure 18 The source code for figure 17 some parts omitted 24 numeric u u lcm circled points def cpoint text c forsuffixes c LEK pencirele scale 2 5077 draw p picikup pencirele scaled 1 55t draw withcolor white endfor enddef IOS Limie il il e pair Ap 2r C Di De A origin A G Suxdir45 7 G 5 4uxdir20 Depoitne 95 OE G A si poine 7 Oi E Z p B whatever D E whatever A 2Z draw A G Z draw A B D Ceci 4 2 G 1D 18 18 8 label top btex A etex A label llft btex G etex G label Lie forex B e tex B label vre oreks 2 CCTS A label ulft btex D etex D label top btex E etex E ENCHE LG end G B Figure 19 A better reconstructio
39. tersection We hope that our remarks are not taken as personal criticism We are particularly aware of how difficult it is to produce some drawings and how sometimes we become slaves of the tools and also how sometimes some figures are messed up by third par ties One might also argue that the faithfulness to the original diagram implies that one cannot be sure that the lines will actually meet On the other hand given that we use smooth curves which already depart from the original drawing why should one want to avoid exact junctions in the very case where an approximate junction can only be noticed when zooming This seems counterintuitive 8 Figure 5 An excerpt of a recent reconstructed figure 32 where some junctions overlap The overlap was probably intentional or rather the author intentionally did not try to obtain perfect junctions Left complete figure Right detail The figure shown here is exactly that of the original article Figure 6 A new figure based on figure 5 32 The intent of the author is now to show a perfect construction but that result is not achieved as every circled point is mispositioned and as the end curves are usually not tangent see for instance in E by zooming and especially on the right side of the figure The figure shown here is exactly that of the original article cx 1 i x lt o oO F d J S gt TE 010 z vA gt Ex l
40. the UTF 8 encoding And LuaTgx 14 18 is an even more recent version of TEX using Lua as a scripting language XqETEX is the TEX version of X IEX LualTgx is the TEX version of LuaTEX For end users X IRX and LuaTEX offer similar functionalities XIEX was the first TEX extension to provide support for OpenType fonts How ever most of its features are included in LuaTpx so that the latter should be considered as the real successor of pdfTEX and XATKX is probably bound to die This is somewhat ironic as only a very small fraction of TEX users have ever used XJI X Although standard TEX cannot use OpenType fonts directly if Xql4TEX is used to produce a figure using OpenType fonts that figure can be ex ported as a PDF and then included in a standard TEX file so that in a way OpenType fonts are not totally forbidden in I4Tpx For an introduction to multilingual writing with X TFX and Lual Tpx see the documentation of the polyglossia package 2 5 2 2 Graphical environments and labels Since TikZ is a TEX package it can use OpenType fonts if Xq TEX or LualTpX are used METAPOST is more problematic since it is currently tied to PostScript At the moment standalone METAPOST that is METAPOST not called from TeX can not include OpenType fonts but in many cases labels can be pro duced with other fonts It is possible that future extensions of standalone METAPOST will have the ability to use OpenType fonts However if ME
41. ure 11 29 Pf a ur Figure 23 The reconstruction of DRaFT s example with TikZ Care was taken to use rounded caps for the lines 30 documentclass article usepackage onimage usetikzlibrary calc usepackage LFE LAE T1 fontenc usepackage ut f8 inputenc usepackage arabic english babel newif ifshowimage showimaget rue makeatletter we add a boolean to this macro defined in onimage sty def tikzonimage opt 1 2 begin tikzpicture ifshowimage node inner sep 0 image Nine udecrannies itt else node inner sep 0 image phantom includegraphics 1 2 al begin tikzonnode image makeatother newif ifarabiclabels arabiclabelstrue arabiclabelsfalse newcommand lab 2 ifarabiclabels selectlanguage arabic 1 else 2 fi pagestyle empty begin document tikzset every path style line cap round Figure 24 The source code of the reconstruction using TikZ 1 2 31 newcommand yA 0 345 begin tikzonimage width textwidth saito fig jpg tsx show help lines a at 0 037 VA My at 0 27 yA K aw 00425 AVA ar 0 785 0 58 coordinate coordinate coordinate coordinate ccorcimare at 0 227 0 84 draw CA iL draw B C draw X 11 draw red
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