Punnett Squares and Hybrid Crosses: How Mendelians Learned Their Trade by the Book

In 1901, Erich von Tschermak (1871–1962) produced a critical edition of Gregor Mendel’s (1822–1884) paper on “Versuche über Pflanzen­hybriden”; and in the same year, William Bateson (1861–1926) submitted an English translation entitled “Experiments in Plant Hybridization” to the readers of the Journal of the Royal Horticultural Society.[1] Tschermak’s edition appeared as volume 121 of the renowned series Ostwalds Klassiker der exakten Natur­wissen­schaften (Ostwald’s Classic Texts in the Exact Sciences). Historians have rarely noted the paradox that lies in the fact that a paper, which scientists like von Tschermak and Bateson had lifted from obscurity just a year earlier, was almost instantaneously included in the Pantheon of classical contributions to the “exact” sciences. The discipline that Mendel supposedly founded, namely genetics, did not yet exist in 1901, and his alleged “discovery” of laws of inheritance would remain highly contested for at least another decade, even involving accusations of scientific misconduct.[2]


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To resolve this paradox, it is important to note that Mendel’s paper did not only present a new theoretical perspective on the problem of inheritance. It also introduced a unique and highly counter-intuitive combination of mathematical and biological methods—in essence, methods of combinatorial analysis and artificial hybridization.[3] This raises the possibility that Mendel’s paper was not so much taken up by early Mendelians for its theoretical content but in order to acquire and rehearse the practical skills needed to carry out experiments. If we follow this historical interpretation, Mendel’s paper was read as a treatise in applied mathematics and experimental design, and its rapid reception among biologists would be better explained as the diffusion of a computational and experimental protocol.

In order to see if this interpretation was viable, we looked at Mendel’s paper itself in a fresh way and followed up its reception in some of the early textbooks and manuals that taught how the new science of genetics was supposed to be done and not necessarily how biological phenomena were supposed to be understood. How did these early genetics manuals present the subject? What kinds of difficulties did their readers encounter? How did graphs and diagrams contribute to understanding experimental and computational protocols? How engaged were the readers of these publications?

To answer these questions, we visited a variety of libraries and examined copies of introductions to Mendelian genetics published in the first quarter of the twentieth century. We found readers had annotated these books by underlining text sections; leaving various other reading marks such as lines connecting different parts of the text; inserting question or explanation marks; and adding comments, explanations, or criticisms to the margin of the pages. We examined copies of Tschermak’s edition of Mendel’s paper, R. C. Punnett’s Mendelism, W. Johannsen’s Elemente der exakten Erblichkeitslehre (Elements of an Exact Science of Heredity), W. Bateson’s Mendel’s Principles of Heredity, E. Baur’s Einführung in die experimentelle Vererbungslehre (Introduction to the Experimental Science of Heredity), W. E. Castle’s Genetics and Eugenics, and R. A. Fisher’s Statistical Methods for Research Workers. All these textbooks went through several editions and remained in use for decades, well into the second half of the twentieth century. Generations of researchers and students read and annotated them as they learned the trade of genetics by repeating the experiments and calculations printed in those books. Some of the copies we examined in Uppsala, Svalöf, Copenhagen, Berlin, and Woods Hole had distinguished owners. We traced books belonging to renowned geneticists such as Wilhelm Johannsen (1857–1927) and Herman Nilsson-Ehle (1873–1949). In other cases, however, we were unable to identify the readers who left marginal notes and other markings in these books. This was the case, for example, with the copies preserved at the Berlin State Library, where only the accession records, not the lending registers, are still available.

Placed one next to the other, the genetics manuals we examined appear at first glance as an uneven set. The slim second edition of Reginald Punnett’s (1875–1967) Mendelism, printed in octavo with less than one hundred pages makes a sharp contrast with the encyclopaedic Elemente, by Johannsen, a book of over five hundred pages with a trim size double the octavo. Punnett’s book was conceived as a short, accessible essay, as the author explained in the introduction. It could only give a taste of Mendelian genetics, whereas Johannsen’s was a complex treatise with detailed discussions of experimental and computational protocols related both to Mendel’s laws and to Johannsen’s pure line theory. In some cases, the same author produced more than one manual for different readerships. This was the case, for instance, with Erwin Baur (1875–1933), whose 1911 Einführung was followed ten years later by a new textbook prepared for farmers, gardeners, and forestry workers only half the size of the previous manual.

Despite the undeniable differences in their agendas and intended readerships, these manuals all contributed to the formation of a homogeneous visual language for genetics. Arrows, straight lines, and tree-like structures were used to depict experimental crossings on the page. The square diagram introduced by and now named after Punnett became a visual instrument for carrying out the combinatorial calculations that had been embedded in Mendel’s paper in textual and mathematical form.[4] Other authors of genetics manuals, for instance Bateson, quickly integrated this tool in their own textbooks because it effectively explained complex crossings to readers less at ease with computations. The analysis of variance method developed by the statistician and geneticist Ronald Fisher (1890–1962) was also a tool designed to facilitate statistical calculations: the experimental results were displayed in a table, and subdivided by classes according to their cause of variation. This tabular arrangement made the structure of the experiment clearer and the arithmetic simpler.[5]

Photographic plates and colour drawings were widely used in many early genetics manuals. Browsing the pages of Bateson’s, Baur’s and Castle’s manuals, for instance, even the modern reader appreciates the detail and elegance of their iconography. Plants and animals (peas, primulas, snapdragons, maize, butterflies, fowls, mice, rabbits, and more) together with their inheritance laws for color and other features are revealed to the reader. These lavish illustrations made it into foreign-language editions too. In the 1910 German translation of Punnett’s manual, for instance, the images and drawings are maintained, but with translated captions.

The rich iconography of these genetics manuals, however, could be a helpful learning tool only for the engaged reader who went from text to image, from image to computation, and from computation back to text to fully understand the crossings described. And we found clear indications of the readers’ engagement in the annotated copies we examined. For instance, readers carefully followed computations: they repeated them in the book margins and added values not explicitly calculated, but they also dutifully amended mistakes when they discovered them (Figure 1).

Figure 1  This is an excerpt from W. Johannsen’s Elemente (2nd edition, 1913, p. 51; copy available at the Staatsbibliothek Berlin, Call No. La1983(2)). Johannsen is providing the example of a method for checking the accuracy of computations. The reader, who has followed the process step by step, is ready to point out a mistake (14 × 16 = 224, not 225) in the final column of the table.

Some of the most striking and illuminating examples for the practical engagement of prominent readers with these texts come from Tschermak’s edition of Mendel’s paper. Until recently, the personal copy of Nilsson-Ehle was preserved in the library of the plant breeding station at Svalöf, Sweden, which now belongs to the private company Lantmännen. On it’s back flyleaf, Nilsson-Ehle tried in vain, and with clear signs of frustration, to derive the Mendelian ratios for a bihybrid cross, a problem any student of genetics would consider trivial today (Figure 2). As his research notes preserved in the library of the University of Lund show, he later achieved quite some mastery in deriving Mendelian ratios for complex crosses, making use of the Punnett square.[6] But even someone as mathematically adept as Johannsen needed to take recourse to auxiliary lines to follow the basic combinatorics involved in this problem (Figure 3). Auxiliary lines drawn between text and diagram in his personal copy of the third edition of Punnett’s Mendelism demonstrate that he also relied on this visual instrument to intuit complex crosses, in this case involving three genetic factors (Figure 4).

Figure 2  Verso side of back flyleaf of Tschermak’s edition of Mendel’s paper. Nilsson-Ehle is trying to work out the possible combinations between two pairs of alleles (A – a and B – b) and their numerical ratios, but fails, as is evident from deletions and abandoned attempts. Svalöv Weibull AB Research Library. Reproduced from a xerox copy that was made in 1998 by one of the authors (SMW). The company Savalöv Weibull AB was taken over by Lantmännen and the current whereabouts of the library are unknown.
Figure 3  Lines drawn by Wilhelm Johannsen to figure out the combinations of alleles that can result from a dihybrid cross. From his own copy of Tschermak’s edition of Mendel’s paper preserved in the Science Library of Copenhagen University (Call No. 80-33).
Figure 4  Two pages from Johannsen’s personal copy of the third edition of Punnett’s Mendelism. Johannsen’s connects symbolic representations of allelic combinations in the text with the respective fields in a Punnett square. Copy preserved in the Royal Library of Copenhagen (Call No. 8° N. hist. 20481).

At the same time, neither Johannsen nor Nilsson-Ehle showed any interest in the “Concluding Remarks” of Mendel’s paper, in which he speculated about the mechanisms by which traits are inherited. This lack of engagement suggests that, for early twenty-first-century researchers, the really intriguing aspect of Mendel’s paper was how it could be used to figure out the combinatorial nature of hybridization experiments. Visual tools like the Punnett square played a key role in this process because the mathematical arguments Mendel employed were far from intuitive. They needed to be rehearsed again and again to become what is second nature for geneticists today.

Staffan Müller-Wille is Associate Professor of History and Philosophy of the Life Sciences at the University of Exeter, and Editor-in-Chief of the journal _History and Philosophy of the Life Sciences_. Giuditta Parolini is Postdoctoral Researcher (_wissenschaftliche Mitarbeiterin_) at the Technical University of Berlin.


  1. Gregor Mendel, Versuche über Pflanzenhybriden: Zwei Abhandlungen, 1865 und 1869, ed. Erich Tschermak, Ostwald’s Klassiker der exakten Wissenschaften 121 (Leipzig: Wilhelm Engelmann, 1901), https://archive.org/details/versucheberpfla00tschgoog (https://archive.org/details/versucheberpfla00tschgoog); Gregor Mendel, “Experiments in Plant Hybridisation, with an introductory note by W. Bateson, M.A., F.R.S.,” Journal of the Royal Horticultural Society 26 (1901): 1–32, https://www.biodiversitylibrary.org/item/164047#page/8/mode/1up.  ↩
  2. Gregory Radick, “Beyond the ‘Mendel–Fisher Controversy,’” Science 350, no. 6257 (2015): 159–60, https://doi.org/10.1126/science.aab3846.  ↩
  3. Hans-Jörg Rheinberger and Staffan Müller-Wille, The Gene: From Genetics to Postgenomics (Chicago: University of Chicago Press, 2017), chap. 3.  ↩
  4. Anthony W. F. Edwards, “Punnett’s Square,” Studies in History and Philosophy of Biological and Biomedical Sciences 43 (2012): 219–24, https://doi.org/10.1016/j.shpsc.2011.11.011.  ↩
  5. Giuditta Parolini, “The Emergence of Modern Statistics in Agricultural Science: Analysis of Variance, Experimental Design and the Reshaping of Research at Rothamsted Experimental Station, 1919–1933,” Journal of the History of Biology 48 (2015): 301–35.  ↩
  6. Staffan Müller-Wille, “Early Mendelism and the Subversion of Taxonomy: Epistemological Obstacles as Institutions,” Studies in History and Philosophy of Biological and Biomedical Sciences 36 (2005): 465–87.  ↩
Suggested citation: Staffan Müller-Wille & Giuditta Parolini, “Punnett Squares and Hybrid Crosses: How Mendelians Learned Their Trade by the Book,” History of Knowledge, May 8, 2018, https://historyofknowledge.net/2018/05/08/punnett-squares-and-hybrid-crosses/.