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I am reading through the Dover publication of Gauss' work on curves and curved surfaces called "General Investigations of Curved Surfaces". This Dover publication presents two major papers authored by Gauss in 1827 and 1825 (the 1825 paper was actually published later than the 1827 paper).

The subject matter is not much different than taught in a typical Advanced Calculus class that presents the beginnings of Differential Geometry. However, the book is still interesting reading although a tough travel due to the more crude notation and the total lack of any diagrams.

This whole topic was made for the use of graphic diagrams to help with the presentation and yet nothing of that sort appears in the entire book. The only diagrams that appear are in the end notes that briefly introduce the reader to Spherical Geometry. These end notes are not part of Gauss' original work.

I submit that this whole paper could be condensed by half or maybe to a third of its current size by the use of well appointed diagrams to help illustrate the mathematical development. Indeed, reading through this is like reading a thousand words instead of the one well placed "picture". And, the very first section (aka paragraph) could be replaced with a diagram of the Unit Sphere with radial vectors displayed to denote points (x,y,z) on the surface of the sphere.

Which brings me to my question. Obviously these writers did not have LaTeX and TikZ at their disposal for writing and the idea of printing papers with diagrams was likely difficult, costly, or maybe impossible (without hiring an artist) in those earlier years. At what point did diagrams become more prevalent in published papers. I am referring of course to papers where a diagram is well suited to illustrating the concepts being described. For the papers contained in this Dover book, I would guess almost every section could be augmented by a diagram.

I am guessing not before 1900 and maybe even later did diagrams become more common. I am probably older than most of the readers of this forum but my textbooks (physics & math) were loaded with diagrams (mid-1960 but many textbooks published in mid to late 1950s) and this was before Knuth, TeX, and of course, TikZ.

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    $\begingroup$ I may be wrong, but I seem to recall reading that it was a stated objective of the analytic geometers of the time to avoid the use of diagrams whenever possible. $\endgroup$
    – nwr
    Jun 28, 2016 at 23:15
  • $\begingroup$ I remember in my Classical Mechanics class, the professor was drawing out the Euler Angles on the blackboard (yes, black with just white chalk). As he was busy creating the detailed diagram he commented: "If you can't draw diagrams, you will never make it in Physics". $\endgroup$
    – K7PEH
    Jun 28, 2016 at 23:31
  • $\begingroup$ Just because I was curious, I checked two texts that I have. My Advanced Calculus textbook, copyright 1952 with printing in 1959 of the edition I owned, had 18 diagrams in 67 pages covering Differential & Vector Calculus in 3-space -- average of one diagram every 4 pages. A more modern text, copyright 2012, chapters covering the same topics in about 100 pages has multiple diagrams on almost every single page and they are in color! I much prefer the newer text although I think it is dumbed down a little bit from my 1952/1959 textbook. $\endgroup$
    – K7PEH
    Jun 28, 2016 at 23:58

3 Answers 3

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Two points:

  1. @Nick R: I don't think geometers doing pure or analytic geometry wanted to avoid diagrams. They were educated with Euclides, reasoning on figures/diagrams. This trend (doing geometry almost without figures) has flowed like a tsunami especially in the 1970s...

  2. Something that has not been underlined enough is the technical limitations of the time. From early 18th century, and in particular at the time of Gauss, copper engraving was dominant. This process is time-consuming, costly, and last but not least, necessitates a very able "artist" (figures carved on wood can be found in mathematical texts till the early 18th, but this technique hadn't the precision that can be obtained with copper). For this reason, when figures were present in a book, they were generally gathered at the end of the book. Many new imaging processes were invented, mainly in the second half of the 19th century, under the influence of the huge development of massive (newspaper) printing techniques, lithography, photoengraving, offset printing etc... providing mathematical treatises or textbooks with very good overall quality (printing, binding, ...) and especially good figures inserted in the text at affordable prices everywhere in the 1900s.

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  • $\begingroup$ when figures were present in a book, they were generally gathered at the end of the book --- This is also the case for most journals during this time, which I've seen from my extensive browsing of 18th century journals (checked out from a nearby university library, volume-by-volume for nearly a decade), such as Nouvelles Annales de Mathematiques and Mathesis Recueil Mathematique and Archiv der Mathematik und Physik and many others. (continued) $\endgroup$
    – Dave L Renfro
    May 23, 2018 at 7:31
  • $\begingroup$ Incidentally, google digitalizations of such books and journals often mess up the back diagram pages. For example, compare the google and the University of Michigan scans of C. R. M. Talbot's 1860 English translation and extensive commentary of Newton's Enumeration Linearum Tertii Ordinis. $\endgroup$
    – Dave L Renfro
    May 23, 2018 at 7:34
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Probably not what you are really asking. But it seems interesting anyway.

This page of Euclid is from the Ninth Century. There is a diagram. It has other modern innovations, such as spaces between words.

Euclid

Source

The papyrus fragment below (also Euclid) is from the First Century, one of the earliest known instances of a geometric diagram.

EUclid

Source

A parchment from the Ninth Century with a diagram representing three dimensions.

Euclid

Source

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  • $\begingroup$ I like this answer but it does seem off-topic. The important part in OP question being "more prevalent". $\endgroup$
    – VicAche
    Jun 30, 2016 at 22:48
  • $\begingroup$ When did they introduce space between words? I can imagine it being a nightmare during that time for anyone who wished to learn to write. $\endgroup$
    – bjd2385
    Jul 10, 2016 at 15:10
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I'm not a specialist in history of science and math but I have recently read something about your question accidentely in the book by MacLane "Categories for the Working Mathematicians" in the Notes to the Chapter I.

enter image description here

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  • $\begingroup$ That comment seems specific to the use of an arrow to show functional maps from domain to codomain which to me classifies this as notation. But, functional map diagrams are also the result but still I am assuming that the 1940 date is unique to functional maps and not relevant to the general idea of including diagrams in mathematical papers and texts. $\endgroup$
    – K7PEH
    Jun 30, 2016 at 15:06
  • $\begingroup$ Of course, this answer was not thought for being exhaustive. It only provides a somewhat definite date for the introduction of some type of diagram. But commutative diagrams are very general and, now, the main graphical objects which do not reduce to a mere picture (in my opinion, of course), so I thought that it would be been interesting to fix this date. $\endgroup$
    – user91126
    Jun 30, 2016 at 15:26
  • $\begingroup$ Yes, it is interesting on the date and I do not discount that aspect. I was merely saying that my interest was really about the advent of diagrams as being a key part of mathematical publication. This resulted from reading Gauss' work on Curves & Surfaces and the total absence of any diagrams versus today where an area of math where such topic publications use a plethora of graphs and diagrams. $\endgroup$
    – K7PEH
    Jun 30, 2016 at 15:34

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