In 2005, I was a newly-minted algebraic geometry PhD who had earned a job at a small liberal arts school with two senior algebraists already on staff. Not surprisingly, I wasn’t assigned any algebra classes. Instead, I inherited the upper calculus and analysis classes. Unfortunately, I had (and have) no talent or background in these areas. While I was able to teach the content, I lacked (and still lack) the depth of understanding and experience with the material to create an engaging classroom. I needed to improve my very traditional, very by-the-textbook, very boring lectures. And I needed to do it quickly, as my teaching evaluations weren’t magically improving on their own.

Figure 2. Advertisement for “Study the Masters.” Ohio Focus 8(6)(Spring 2008): 4.

So, in the summer of 2008 I attended an Ohio MAA Short Course entitled, “Study the Masters: Using Primary, Historical Sources in Teaching and Research.” The course was run by Danny Otero from Xavier University and David Pengelley from New Mexico State University.^[1] The many benefits to using primary sources were central to the short course and were later summarized in an AMS blog post by members of the TRIUMPHS grant team:^[2] 

Teaching from primary sources has long been common practice in the humanities and social sciences. Reading texts in which individuals first communicated their thinking offers an effective means of becoming mathematically educated in the broad sense of understanding both traditional and modern methods of the discipline. The use of original sources in the classroom promotes an enriched understanding of the subject and its genesis for instructors as well as students.

In contrast to many textbook expositions, which often present mathematical ideas in a distilled form far removed from the questions that motivated their development, original sources place these ideas in the context of the problem the author wished to solve and the setting in which the work occurred. Problems and the motivations for solving them are more apparent and natural in primary sources, and the works of these thinkers are more compelling than traditional textbook expositions. Exposing the original motivations behind the development of “esoteric” mathematical concepts may be especially critical for placing the subject “within the larger mathematical world,” thereby making it more accessible to students. Further, primary texts seldom contain specialized vocabulary (which comes with later formalism), thereby promoting access to the ideas by students with varied backgrounds [Barnett et al. 2015].

This was exactly what I needed: a technique to enrich my classroom and improve learning that didn’t depend on years of experience with the material. As added bonuses, this method was one that I personally found interesting and it was general enough to be applied across several classes.

Since that short course, my evaluations have improved and my professional activity has expanded. I've been lucky enough to write six Primary Source Projects (PSPs)^[3] under the TRIUMPHS grant; multiple publications in Convergence,^[4] The College Mathematics Journal,^[5] and Mathematics Magazine;^[6] and uncountably many^[7] anecdotes, stories, and other primary source teaching materials. While I started by constructing projects for real analysis, I’ve now pivoted into differential equations, with multivariable calculus in my crosshairs.

I always hope that creating a lesson using primary sources will be smooth, but it never is. Challenges inevitably arise, and new projects seem to bring new dilemmas. The process often feels like Cantor’s diagonalization argument—as soon as you think you've listed all the issues out . . . you find another that wasn’t on the list. So, please don’t think of the following pages as exhaustive, but rather just as a personal collection of pitfalls and potential solutions that I share to hopefully save you time and headaches. The issues are presented roughly in the order that you may encounter them in the creation of a project, and so there may be a higher probability of experiencing those that appear earlier in this article. Luckily, those are also accompanied by more and better words of advice.

 

So you’ve decided on a topic. You now need to find
primary sources, but where to look first?

Example

This one you just can’t escape. Unless you’re a historian of mathematics, extremely well-versed in a certain topic, or just very lucky, you probably don’t have a file of primary sources appropriate to use in a classroom. Every project, PSP, anecdote, or story I use has required some outside research, since undergraduate texts treat math content like it is in the public domain. After all, most of our classes involve mathematics that is well over 100 years old. So you must look elsewhere. But where?

It may help to know that . . .

• Mathematical sourcebooks collect and translate (and often annotate) important mathematical works. While some are dedicated to specific topics—I’ve made extensive use of George Birkhoff's A Source Book in Classical Analysis [1973]—cultures, and eras, I have found John Fauvel and Jeremy Gray’s The History of Mathematics: A Reader^[8] [1987] and Ron Calinger’s Classics of Mathematics [1994] to be the most general and useful.
• History of mathematics textbooks often have very robust bibliographies and are a great place to start. I have found Morris Kline’s Mathematical Thought from Ancient to Modern Times [1990] and Victor Katz’s History of Mathematics [1993] to be the most helpful.
• While modern popular textbooks may not provide a good starting point, older textbooks are temporally much closer to the discovery (or creation) of the mathematics and hence are more likely to give credit for specific results.
• Conversely, sometimes later editions of a book or reprints in a collection of works contain additional useful information.
• Biographies of the mathematician in question are another great spot to look for sources and memorable stories.
• While not the norm, there are also modern textbooks (often with titles of the form ________ from a Historical Perspective), which have robust citations to primary sources. They exist in many areas of math, but since I’ve been concentrating on analysis and differential equations, I’ve found Analysis by its History [Hairer and Wanner 2008] and Ordinary Differential Equations [Ince 1944] to be the most helpful as I developed materials for my classes.
• Secondary sources in journals offer another good place to look. Harvard’s library states, “Secondary sources were created by someone who did not experience first-hand or participate in the events or conditions you’re researching. For a historical research project, secondary sources are generally scholarly books and articles. A secondary source interprets and analyzes primary sources” [Harvard Library 2023]. The most useful secondary sources that I’ve found have been in journals such as Historia Mathematica, Bulletin of the AMS, Mathematics Magazine, Archive for History of Exact Sciences, American Mathematical Monthly, The Mathematical Gazette, The British Journal for the History of Mathematics, The British Journal for the Philosophy of Science, and The British Journal for the History of Science. Even the non-historical articles in these journals often include robust background and bibliographies. And, of course, be sure to check out Convergence.

 
Figure 3. Gottfried Wilhelm Leibniz, Jacob Bernoulli, and Johann Bernoulli. Convergence Portrait Gallery.

• You may find contradictory information. When trying to determine who solved the “Bernoulli differential equation” I went to the sources I described above [Ince 1944; Hairer and Wanner 2008; Katz 1993] and unfortunately found different information in each place! What follows is from the resulting publication [Parker 2013].

Here's a mystery to ponder: who first solved the Bernoulli differential equation

\[\frac{dy}{dx} +P(x) y = Q(x) y^n ? \]

Was it Gottfried Wilhelm Leibniz (1646–1716)—the German mathematician, philosopher, and developer of the calculus? According to Ince [1944, p. 22], “The method of solution was discovered by Leibniz, Acta Erud. 1696, p.145.”

Or was it Jacob (James, Jacques) Bernoulli (1655–1705)—the Swiss mathematician best known for his work in probability theory? Whiteside [1981, p. 97], in his notes to Newton's papers, states “The ‘generalized de Beaune’ equation \(dy/dx = py+q y^n\) was given its complete solution in 1695 by Jakob Bernoulli.”

Or was it Johann (Jean, John) Bernoulli (1667–1748)—Jacob’s acerbic and brilliant younger brother? Pierre Varignon (1654–1722) [Hairer and Wanner 2008, p. 140] wrote to Johann Bernoulli in 1697 that “In truth, there is nothing more ingenious than the solution that you give for your brother’s equation; and this solution is so simple that one is surprised at how difficult the problem appeared to be: this is indeed what one calls an elegant solution.’

Was it all three? Kline [1990, p. 474] says, “Leibniz in 1696 showed it can be reduced to a linear equation by the change of variable \(z=y^{1-n}\). John Bernoulli gave another method. In the Acta of 1696 James solved it essentially by separation of variables.”

I, personally, love it when this happens, as there is now an opportunity to add clarity to the literature. If you’re curious, Jacob proposed the problem. A solution was published by Leibniz, though all the pertinent details required to actually carry out the technique were omitted. Later Johann gave those details, along with a second solution using variation of parameters.

• Don’t forget that Google (or your own favorite search engine) is a thing. I’m sure you’re not surprised that I usually start there. I’ve also found that the librarians at my institution have great ideas for tracking down sources.
• You are not alone. You have friends you didn’t even know about. The Primary Source “group” of mathematicians is extremely welcoming and excited to help others see the benefits of using primary sources in their classrooms. They are members of the above-mentioned TRIUMPHS Society, the International Study Group on the Relations between the History and Pedagogy of Mathematics (HPM), or the History of Mathematics Special Interest Group of the MAA (HOM SIGMAA). You can reach out to the authors and editors of papers you find interesting, or contact the organizers of and speakers in conference sessions who reference primary sources. I have found all to be very generous with time and advice when I get stuck. I won’t speak for everyone else, but you can always reach out to me.^[9]

 

You have identified the primary source with the seed of the topic,
but when you go to find it . . . it isn’t there.

Example

I was working with Janet Heine Barnett on polishing my PSP on higher order linear differential equations with constant coefficients [Parker 2021a], when she noticed that I had cited a certain letter between Leonhard Euler (1707–1783) and Johann Bernoulli as:^[10]

Der Briefwechsel zwischen Leonhard Euler und Johann I. Bernoulli – III. 1739–1746, Bibl. Math. (3) 63 (1905) 16–87.

while in an earlier paper [Parker 2016], I had cited the same article as:

Der Briefwechsel zwischen Leonhard Euler und Johann I Bernoulli [Teil] III, 1739–1746 (Seventeen Letters from Euler to Johann I Bernoulli Part III, 1739–1746). Biblioth. Mathem., 53: 16–87, 1905.

She understandably wondered which volume was correct. The Euler Archive lists the citation of E863 as:

Originally published by G. Eneström in the three articles: Der Briefwechsel zwischen Leonhard Euler und Johann I. Bernoulli. I – III; Bibl. math. 43, 1903, p. 344–388; 53, 1904, p. 248–291; 63, 1905, p. 16–87 [Eneström 2018].

It also links to scans of the three articles, from which I used three letters contained in the third of the three articles. Unfortunately, the scans don’t include the front-page information and so, to answer Janet's question, I decided to go directly to Bibliotheca Mathematica to see which volume number was correct. But when I looked, I couldn’t find any Volume 43, 53, or 63! They just weren’t there. So, the short answer to the question is that neither citation is correct!

Now, should I have realized this earlier? Absolutely. But I didn’t. It still needed to be fixed.

It may help to know that . . .

• Lots of journals have very similar names, even if they have nothing to do with each other. Do you know the difference between the currently published journals The Mathematica Journal, Mathematica, and Mathematika? I don’t.
• The converse is also true, as the same publication may have many different names. Journals aren’t static. They start and end. They take breaks and reappear. They split and combine with other journals. The Euler Archive notes that the flagship publication of the Academy of Sciences in St. Petersburg (in which Euler published much of his work) has gone through several name changes.
  • 1726–1746 Commentarii Academiae Scientiarum Imperialis Petropolitanae
  • 1750–1776 Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae
  • 1777–1782 Acta Academiae Scientiarum Imperialis Petropolitanae
  • 1783–1802 Nova Acta Academiae Scientiarum Imperialis Petropolitanae
  • 1803–1916? Mémoires de l’Académie Impériale des Sciences de St. Pétersbourg, avec l’Histoire de l’Académie
    • Series V: 1803–1830?
    • Series VI: 1830?–1859?
    • Series VII: 1859?–1897?
    • Series VIII: 1897?–1916?

All of these St. Petersburg publications should not be confused with the St. Petersburg Mathematical Journal, which is an AMS publication and is the English translation of the Russian journal Algebra i Analiz.

• The format of information contained in citations is not consistent. As you can see above, the same article was cited in three different formats in Mathematics Magazine, the PSP, and the Euler Archive. Citations were even less standardized in the past.
• Abbreviations for journal titles only recently became standardized in the USA by Mathematical Reviews. Prior to this, sources were named in many different ways and formats. For example, a citation of “Acta” would require context to determine which of the dozens of Acta journals in the classification it is referring to, or perhaps one of the older non-cataloged journals such as the Acta Eruditorum.
• Many authors have published articles or books with extremely similar, or identical, names. While researching classroom materials on Wronskians, I found that Giusippe Peano (1858–1932) wrote two short articles, “Sur le Déterminant Wronskien” and “Sur les Wronskiens.” The similar titles became more confusing when I learned that the articles appeared in the same issue of the same journal, [Peano 1889a] and [Peano 1889b]!
• The organization of texts was sometimes more elaborate than it is now, which can lead to misleading citations. One that often confuses me is Euler’s “Institutionum calculi integralis.” The first Volume consists of an Introduction with 39 Problems, Solutions, Theorems, Demonstrations, Corollaries, Applications, or Examples [Euler 1768]. (For simplicity lets call these PSTDCAEs.) The Introduction is followed by three Sections (with 356 PSTDCAEs spread over 9 chapters, 270 PSTDCAEs spread over 7 chapters, and 37 PSTDCAEs spread over 0 chapters, respectively). After all that, Euler published two additional Volumes, all in the same work.
• References sometimes don’t refer to the original article, but to a republication in the collected works of the mathematician. For example, the above letters to and from Bernoulli to Euler do appear in three different volumes of Bibliotheca Mathematica. However, the Euler Archive notes that they can also be found in the collected works of Euler (called the Opera Omnia) in Series 1, Volume 23, pp. 450–455.^[11]

Because of all of this, things get mixed up. In an attempt to clear up confusion for my PSP, I researched the journal in question: Bibliotheca Mathematica. Joe Dauben wrote that Gustav Eneström (1852–1923) started it as an addendum to Gösta Mittag-Leffler’s (1846–1927) Acta Mathematica in order to catalog mathematical literature. Today, this is considered the first series of the journal. In 1887 Eneström split it from the Acta and focused it on math history, which began its second series of 13 volumes. The third series consisted of 14 volumes between 1900 and 1914 [Dauben 1999]. Hence, the volumes in question can’t possibly be 43, 53, and 63. So, what should they be?

I went back to the first page of the first article and there Eneström cited earlier work in Biblio. Math. but included no volume numbers. However, on the first pages of the second and third articles, the references to previous work now included volume numbers—just in an odd format. See Figure 4.

Figure 4. Citations from the first and third of Eneström's articles. Screenshots provided by author.

Putting this all together, I learned I wasn't referring to volumes \(43\), \(53\), and \(63\), but rather \(4_3\), \(5_3\) and \(6_3\). Somewhere along the line, the subscripts got lost and everything became hard to follow. In today's notation, the citations probably should have read “Series 3, Volume 6” instead of “63”.

 

What happens when you can't read the language that the source is written in?

Example

Danny Otero explained this particular challenge well:

The problem of getting these readings into the hands of my students has been most formidable. I have been able to make do with standard source books in the history of mathematics, but no single source for all of these readings currently exists. I have used Calinger’s Classics of Mathematics as a textbook with some success; it has the added benefit of including some well-written summaries of the history of mathematics that make for fine supplemental reading. By my count, 10 of the 21 readings that form the core of the course can be found in some form in Calinger’s book. One or two other texts can be found in Struik’s Source Book and Fauvel & Gray’s Reader. Still, some of the readings cannot be found in any of these and must be hunted down through the literature and made available to the students through the reserve desk of the university library. Even worse, four of the readings (by Fibonacci, St. Vincent, Briggs, and Cauchy) do not exist in print in English at all. . . . I have been forced to prepare my own translations of these texts for distribution to the students [Otero 1999, p. 68]. 

Anything other than English is problematic for me. I can get the gist of French because I’m old enough that I passed a language exam for my PhD, and I chose French. I can sometimes recall my high school Spanish. I can guess at Latin. But I’m practically fluent in these compared to my understanding of things written with Greek, Cyrillic, Asian or Indian alphabets. How is it that my most popular projects are from Carl Runge (1856–1927) and Martin Kutta (1867–1944) in German and Giuseppe Peano (1858–1932) in Italian? Here are some strategies.

It may help to know that . . .

• While not common, sometimes mathematicians were deliberately vague about their discoveries in order to prevent plagiarism and maintain priority. So don’t be discouraged if sometimes you hit a dead end—that might have been the intent of the author. This occurred in many ways, but perhaps the most interesting was the use of anagrams to encode discoveries.^[12]

The history of anagrammatism is a part of the history of superstition, but it sometimes concerns the historian of science. Not only in a general way, as truth is not always sharply separated from error, nor reason from unreason, but also because anagrams were occasionally used by men of science who desired to announce a discovery and establish their priority without running the risk of being plagiarized by unscrupulous rivals.

. . .

Another famous scientific anagram was introduced twenty years later than Huygens’ by no less a person than Newton.^[13] In the latter’s very long letter of Oct. 24, 1676 to the Secretary of the Royal Society, Henry Oldenburg, he concealed anagrammatically his discovery of the infinitesimal calculus [Sarton 1936, pp. 136, 138]. 

Figure 5. The first page of [Sarton 1936].

The referenced letter by Newton was intended for Leibniz and came to be known as the “Epistola Posterior.” It actually contains two different anagrams:

The foundation of these operations is evident enough, in fact; but because I cannot proceed with the explanation of it now, I have preferred to conceal it thus: 6a cc d æ 13e ff 7i 3l 9n 4o 4q rr 4s 8t 12v x.^[14]

and

At present I have thought fit to register them both by transposed letters, lest, through others obtaining the same result, I should be compelled to change the plan in some respects. 5accdæ 10effh 12i 4l 3m 10n 6oqqr 7s 11t 10v 3x: 11ab 3cdd 10eaeg 10ill 4m 7n 6o 3p 3q 6r 5s 11t 7vx 3acæ 4egh 6i 4l 4m 5n 8oq 4r 3s 6t 4v, aaddaeeeeeeeiiimmnnooprrrssssttuu.^[15]

You could hardly be blamed for not seeing this as a very early statement of calculus.

• Much mathematical symbolism in undergraduate mathematics has been relatively unchanged over the past 200 years. So even if you can’t read the language, you may be able to understand things from mathematical context. Otero points out limits to this, though: “Mathematicians who lived before Euler used mathematical notation only idiosyncratically, and before Descartes’ day rarely, if at all. The mathematical manuscripts of the late Renaissance (for example, Cavalieri’s work on computing areas, the source of his famous Principle) are largely opaque to the modern reader, and require considerable exegesis to extract their meaning, even in English translation” [Otero 1999, p. 60].
• Google Translate, Babelfish, and other online translation sites can give you an idea of what your text is about. If you have a document with Optical Character Recognition (OCR), it can simply be copied into some of the translation websites. You may want to remove the mathematical symbolism and any LaTeX commands.
• You might be more literate than you think. In general, mathematicians wanted their work to be seen by others^[16] and that could be done in two ways. First, they would write in their native language but use simple structure so non-native speakers could understand. Or, they would use a more popular language (such as Latin) that the author wasn’t native in, which would necessitate simple structure. In either case, the texts tended to be easier to read than standard writings in that language. As Barnett noted above, primary sources rarely use “specialized vocabulary” [Barnett et al. 2015].
• While a source may not be translated into English, it may be translated into another language (often French or German) which would be easier for you to read or translate.
• Your colleagues in language departments can be assets. Just remember that their expertise extends well beyond simple translation, so asking them to read long passages for you can be interpreted as insulting. But I’ve found them very collegial for specific questions.
• Your students can also be assets since some will have majors, minors, or interests in foreign languages and can help you out. This has by far been my most productive path. I have had students translate and analyze passages in French, German, and Latin, and that work became beneficial for all involved.^[17]

 

You have the source and the language isn't a problem. It's the mathematics.

Example

I was working on a project to use primary sources to teach existence / uniqueness theorems for first-order differential equations. After some research, I was led to Peano's 1890 paper, “Démonstration de l’intégrabilité des équations différentielles ordinaires” [Peano 1890], which supposedly contained the correct proof of his existence theorem (he published the result with an incorrect proof earlier, in 1886). However, when I started looking through the paper, it was impossible (for me) to figure out what was happening, and it wasn’t because I couldn’t read French. The mathematics was entirely symbolic logic and was indecipherable to me. As near as I can tell, Figure 6 displays the statement of the theorem I was trying to teach.^[18]

Figure 6. Peano’s existence theorem . . . I think [Peano 1890, p. 204].

Its proof is even more opaque, and I had no idea how to approach this primary source. I admit I was relieved to see that others had similar reactions to this passage. Fulvia Skof in Giuseppe Peano between Mathematics and Logic^[19] wrote

It should be remarked that the strong defense of symbolic writing which characterizes his work as a whole contributed to win him the esteem he enjoys today in the mathematical community, and especially among logical mathematicians. But it was also an obstacle to the timely achievement of the success promised by Peano’s brilliant results in Analysis in the early years and by his innovative ideas; for the presentation of the results in ideographic form, distancing the majority of readers, in the end made them little known. This is what happened, for instance, to the important theorem on the existence of a solution to Cauchy’s problem for first-order differential equations and the subsequent extensions to systems [Skof 2008, p. 4] .

There is no way that this is an appropriate primary source for my class. But, can something be salvaged from it?

It may help to know that . . .

• The source might still have value. Though I obviously don't ask my students to understand Peano's symbolic writing, I do mention the story and show his version when presenting the theorem. It takes 20 seconds and does them no harm.
• Your confusion might not be your fault. It might be due to a typo, or because the passage is simply not true. Historically, there has been little oversight in scientific publication. Generally, the acceptance of a paper was at the discretion of the editor, or perhaps a small group of advisors. There was no guarantee that anyone deciding its acceptance was an expert, much less a mathematician, much less a scientist. Alex Csiszar noted

It was only near the turn of the twentieth century that the idea began to take hold that editors and referees, taken as one large machinery of judgment, ought to ensure the integrity of the scientific literature as a whole. . . . Outside the Anglophone scientific world, referee systems remained rare. Albert Einstein, for example, was shocked when an American journal sent a paper of his to a referee in 1932. The idea that any legitimate scientific journal ought to implement a formal referee system began to take hold in the decades following the Second World War [Csiszar 2016, p. 308]. 

Little oversight combined with tedious printing technology led to many mistakes and much confusion . . . not just yours.

• It might still be worth the trouble. And you should give your students some credit. Otero noted

But as I contemplated the audience I was planning to work with, I realized that these students are already quite accustomed to reading equally difficult original texts in their other humanities courses. After all, in philosophy courses they will often read Plato, Aristotle, Thomas Aquinas, Kant, Kierkegaard, Wittgenstein—in translation, of course, but otherwise unadulterated. In their literature courses they grapple with the works of Aristophanes, Dante, Shakespeare, Voltaire, Tolstoy, Joyce. What’s wrong with asking them to study some of the classics in mathematics, too? [Otero 1999, p. 60]

• You can look elsewhere. Others may have found the passages challenging and worked to clarify them, especially editors of collected works. Skof continued

Beppo Levi recalls, in his commemoration of his Master, that they were virtually unknown until the German mathematician Gustav Mie, in his “Beweis der Integrierbarkeit gewoehnlicher Differentialgleichungssysteme nach Peano,” based on Peano’s 1890 paper—with addition of a study of the uniqueness of the solution—supplied, as Levi says: . . . free re-exposition of Peano's Memoir . . . , freeing it from the new hindrance to reading caused by the use of logical ideography, which Peano had introduced in his own exposition, and from that prolixity that resulted from an excessive preciseness in the statement of introductory observations. It was only after Mie’s work that Peano’s result and his procedure to proof could be universally appreciated at its true value, and could prompt further studies by De La Valle-Poussin, Arzel, and Osgood [Skof 2008, p. 5].

• It’s possible that the mathematics may not be as difficult as it first appears and may just be notationally inconvenient or algebraically intensive. This is how my RK-4 PSP came to be [Parker 2021b]. The original article from Runge skipped a lot of steps, which made using it in the classroom very difficult. So, I had an undergraduate student translate and fill in all the gaps as an honors thesis.
• It might be best to pivot. Skof referenced the existence result of William Fogg Osgood (1864–1943). It, and the uniqueness result of Émile Picard (1856–1941), are both understandable and usable in the classroom. And, luckily, both are translated in [Calinger 1994]. See Figure 3.

Figure 7. Osgood’s existence and Picard’s uniqueness theorems. Screenshots provided by author.

• It might be best to skip it altogether. Remember that the goal is to use primary sources that aid in the teaching and learning of mathematics. If a text doesn’t help with that, then it should not be used.

 

The history is interesting and pedagogically valuable,
but you can't justify the time required.

Example

A barrier (real or perceived) to utilizing primary sources in the classroom is the amount of time required to teach in this non-conventional way. This is mitigated as we must consider that students aren’t understanding the material as well with our standard lectures. On the other hand, we don’t have unlimited class time to cover the syllabus. And so we have to do a cost / benefit analysis. I’ll illustrate that with two projects that fell on opposite sides of that decision.

After teaching the ODE course multiple times, I noticed that my students repeatedly messed up the connection between the linear dependence of functions and their Wronskian.^[20] It isn’t the most clear statement as the converse of the relevant theorem is false. But it can be made true by adding an easy condition that is almost always satisfied by functions we study. Upon researching this theorem, I discovered an interesting story. For years very respected mathematicians, including Charles Hermite (1822–1901), Camille Jordan (1838–1922), and Pierre Laurent (1813–1854), made the exact same mistake that my students make, and Paul Mansion (1844–1919) gave an incorrect additional condition to validate the converse. Giuseppe Peano was the first to notice the mistake in 1889 (luckily this time with with no symbolic logic) [Peano 1889a; 1889b].

Contrast this with my research on the method of reduction of order,^[21] with the hope of using it in the classroom. This also involved an interesting story, with two different methods published in two papers (“Solution de différens problémes de calcul intégral” and “Extract de différentes lettres de M. d’Alembert à M. De La Grange écrites pendant les années 1764 & 1765”) by two famous mathematicians (Joseph-Louis Lagrange (1736–1813) and Jean le Rond d’Alembert (1717–1783)), one right after each other (pp. 179–380 and pp. 381–396) in the same volume (3) of the same journal (Miscellanea Taurinensia) [Lagrange 1766; d’Alembert 1766]! The standard method in use today is d’Alembert’s.

For both of these topics, covering the material using primary sources takes longer than teaching the concepts traditionally, but there are other benefits that may justify the time spent. In the first case, in addition to gaining a better understanding of the Wronskian, the benefit was developing a better comprehension of the underlying logic, which is obviously necessary for success across the mathematics curriculum. As such, it was worth creating a PSP for use in the classroom. In the second case, in addition to a better understanding of reduction of order, the benefit was gaining a comprehension of linear and adjoint operators. These concepts aren’t as transferrable to other parts of the standard curriculum, so I judged that it was not worth creating a PSP based on these texts.

Figure 8. Josef-Maria Hoëné-Wronski (1776–1853) and Guiseppe Peano (1858–1932). Convergence Portrait Gallery.

It may help to know that . . .

• There is no general answer to this question. A project that is time-appropriate for use in one classroom might not be appropriate in another because of necessary prerequisites, learning outcomes, or type of students. And that same project may be inappropriate for other sections of the same class at the same school.
• You can think outside the box. In addition to differential equations, the Wronskian PSP also made a good project in a Linear Algebra course and an Introduction to Proofs class, as it really does concentrate on logic—the relationships between a statement and its converse, contrapositive, and inverse. As such it has been used by other instructors, and it ended up being my most popular PSP [Parker 2021c].
• A project that is borderline-appropriate for your classroom use might still be very useful to others and should be shared. It can make for an interesting math club talk or publication in a journal. Both my Wronskian and reduction of order projects appeared in Convergence / Loci [Engdahl & Parker 2011; Cummings & Parker 2015].^[22] In both cases my co-author was an undergraduate who undertook the translations.

 

 

It is possible that the early sources contain a method no longer used.

Example

Years ago I felt that I had a great idea for a PSP concerning systems of differential equations with constant coefficients. The first person to solve such a system was Jean le Rond d’Alembert (1717–1783) in 1743, though he gave a better, clearer, solution in his 1750 “Suite des recherches sur les calcul integral” [d’Alembert 1750].

Figure 9. D’Alembert’s system solution. Screenshots supplied by author.

This passage is short, easy to translate, and easy to understand. It uses no college-level mathematics besides knowing that the solution to \(dy/dx =y\) is \(y=e^{x}\), while the modern solution uses matrices, determinants, eigenvalues, and eigenvectors. I suggested creating a PSP around d’Alembert’s elementary solution, but others correctly pointed out that the passage doesn’t really help students learn the expected content. D’Alembert’s method is not useful when the coefficient matrix has repeated or complex eigenvalues, nor when the system is non-homogenous. There is a reason that matrices are the preferred language to describe the method of solution and that d’Alembert’s method didn't “take”.

It may help to know that . . .

• This is a common issue. It is often, or even typically, the case that the first appearance of a technique only solves an “easy” case of a more general problem. Sometimes, when the general solution was found, it expanded upon the simplified version; this would justify using that original source for teaching. Other times (as in the d’Alembert story above), the general solution doesn’t build on what appeared in the original source, and so that source isn't as useful.
• As with other challenges, it is still possible that while the passage isn’t appropriate for your classroom, it can be utilized elsewhere and by others. For example, I plan on giving d’Alembert’s passage to a student, then ask them to figure out how it is related to the matrix method. Perhaps his method can yet be expanded to include cases that d’Alembert hadn’t considered.

 

 

Sometimes the history actually hinders the presentation of the material.

Example

I’ve not experienced this in my differential equations course, but I do use primary sources periodically in other classes. And sometimes students latch onto the “wrong” stuff. Some prefer to learn more about the dispute between Newton and Leibniz than about actual calculus concepts. Some are fascinated by Georg Cantor’s mental and professional struggles, but they ignore his brilliant contributions to understanding infinity and set theory. Some are more interested in Bernard Bolzano’s isolation, politics, and religious views than the Intermediate Value Theorem.

While I haven’t used any of these mathematicians' writings in my teaching, I am very confident my students would be more interested in Paul Erdös’s peculiarities; Andre Weil’s near execution during World War II;^[23], Alexander Grothendieck’s reclusion; Évariste Galois’s death in a duel; Kurt Gödel’s, Niels Henrik Abel’s, and Srinivasa Ramanujan’s fatal malnourishment; or G. H. Hardy’s hatred of photographs and mirrors . . . than in any of their mathematical results.^[24] Mathematicians are often just interesting people.

Figure 10. Mathematics has a colorful history, which can both add interest and
create distractions in a mathematics classroom. Image created by the editors.

It may help to know that . . .

• At the end of the day you get to choose how much of the primary sources and history you include in your classroom. Remember that our colleagues often include none of the background when they are teaching mathematics, so anything you provide may be a benefit to the student.
• If certain material is distracting in class but you still wish to use it, consider assigning it as reading, homework, or extra credit, or simply making it a topic for direct instruction (e.g., a lecture or instructor-provided written explanation).
• Because it is a math classroom, you don’t need to tell all of the historical story. Concentrate on the truth, and nothing but the truth; but allow yourself to skip the whole truth.
• Otero again comes to our rescue. He notes, “In fact, I think being an expert historian of mathematics would make the course hard to teach, as it would be more difficult for the instructor to avoid teaching history and stay focused on the central issue, which is teaching calculus" [Otero 1999, p. 69]. I think of this often as I teeter on the edge of historical rabbit-holes.

 

There is rarely a geodesic from the genesis of an idea to its conclusion.

Example

I am currently struggling to solve this issue. I want to figure out a way to use history to help motivate Laplace Transforms. Teaching this concept has been a struggle for decades. In 1963, S. T. R. Hancock wrote of this challenge in a very contemporary way:

In technical colleges one is often called upon to introduce advanced mathematical techniques to students whose background is not very extensive. Such a method is the use of the Laplace Transform for the solution of linear differential equations with constant coefficients. Textbooks which deal with this topic, even those specifically written for engineers, derive the transform from the Fourier Integral, or from Heaviside’s Operational Calculus, or just brusquely define the process [Hancock 1963, p. 3].

Figure 11. Statement of Laplace Transform. Wikipedia.

Laplace Transforms are obviously named after Pierre-Simon Laplace (1749–1827). But that is the only simple thing about their evolution; for instance:

• Laplace didn’t actually create or use the modern Laplace Transform.
• What Laplace did had previously been done by Euler.
• The modern Laplace Transform wasn’t used until 1910, by Harry Bateman (1882–1946).
• Lots and lots and lots involving the development of Laplace Transforms happened between Euler's time and Bateman's.
• Laplace transforms can be motivated in many different ways. They are a formalization of Oliver Heaviside’s (1850–1925) Operational Calculus; they can be viewed as a way of measuring how similar two functions are; they can be understood as a specific example of a functional transform; or they can be considered as a generalization of power series and generating functions.

For years, I’ve struggled to find the right way to use the history of this topic to help with the learning. And, I’ve not figured it out. Yet. But, what is good for the goose is good for the struggling mathematician. I’ve . . .

• Googled the topic repeatedly.
• Checked older textbooks such as [Ince 1944].
• Referenced modern textbooks with a historical bent, including Differential Equations with Applications and Historical Notes [Simmons 2017].
• Found dozens of secondary sources from journals, such as Archive for History of Exact Sciences, American Mathematical Monthly, Mathematical Gazette, CODEE Journal, and more.
• Read Laplace’s biography by Gillispie et al. [1997].
• Reached out to friends^[25] for help, including Danny Otero, Dick Pulskamp (and his extensive website [Pulskamp n.d.]), Ken Monks, and others.
• Begun to disseminate the work in other ways, such as talks at MAA conferences.
• And so on.

Each of these steps have clarified my path—a bit. But as I’ve not yet cracked the problem, I clearly don’t have all the answers. Nonetheless, hopefully you now know a few common pitfalls to avoid along with some possibly novel potential solutions to help you as you fight through them. And, knowing is half the battle. Good luck with the other half!

 

Andre, Nicole R., Susannah Engdahl, and Adam E. Parker. 2012, July. An Analysis of the First Proofs of the Heine-Borel Theorem. Loci: Convergence 9.

Barnett, Janet Heine, Kathleen M. Clark, Dominic Klyve, Jerry Lodder, Daniel E. Otero, Nicholas A Scoville, and Diana White. 2015, 20 January. Using Primary Source Projects to Teach Mathematics. AMS Blog: On Teaching and Learning Mathematics.

Barrow-Green, June, Jeremy Gray, and Robin Wilson. 2022. The History of Mathematics: A Source-based Approach. Vol. 2. Providence, RI: MAA Press Imprint of the American Mathematical Society.

Birkhoff, George, ed. 1973. A Source Book in Classical Analysis. Cambridge: Harvard University Press.

Calinger, Ronald. 1994. Classics of Mathematics. Englewood Cliffs, NJ: Pearson. (Orig. pub. 1982.)

Csiszar, Alex. 2016, 21 April. Peer Review: Troubled From the Start. Nature 532:306–308.

Cummings, Sarah and Adam E. Parker. 2015, September. D’Alembert, Lagrange, and Reduction of Order. Convergence 12.

d’Alembert, Jean le Rond. 1750. Suite des recherches sur le calcul intégral, troisiémie partie. Histoire de l’Académie Royale des Sciences et des Belles-Lettres de Berlin 4:249–291.

d’Alembert, Jean le Rond. 1766. Extract de différentes lettres de M. d’Alembert à M. de la Grange écrites pendant les années 1764 & 1765. Miscellanea Taurinensia 3:381–396.

Dauben, Joseph W. 1999. Historia Mathematica: 25 Years/Context and Content. Historia Mathematica 26:1–28.

Eneström, Gustaf. 2018. Seventeen letters from Euler to Johann I Bernoulli, 1727–1740. Euler Archive. (Orig. pub. 1740.)

Engdahl, Susannah, and Adam E. Parker. 2011, March. Peano on Wronskians: A Translation. Loci: Convergence 8.

Euler, Leonhard. 1768. Institutionum calculi integralis. Vol. 1. St. Petersburg: Imperial Academy of Sciences.

Fauvel, John, and Jeremy Gray, eds. 1987. The History of Mathematics: A Reader. London, MacMillan Press. 

Gillispie, Charles C., Robert Fox, and Ivor Grattan-Guinness. 1997. Pierre-Simon Laplace, 1749–1827: A Life in Exact Science. Princeton, NJ: Princeton University Press.

Hairer, Ernst, and Gerhard Wanner. 2008. Analysis by its History. New York: Springer.

Hancock, S. T. R. 1963. The Laplace Transform. The Mathematical Gazette 47(361):215–219.

Harvard Library. 2023. Library Research Guide for the History of Science: Introduction.

HOM SIGMAA. 2023. History of Mathematics Special Interest Group of the MAA.

HPM. n.d. International Study Group on the Relations Between the History and Pedagogy of Mathematics.

Huygens, Christian. 1656. De Saturni luna observatio nova. The Hague, Netherlands: Adriann Vlacq.

Huygens, Christian. 1659. Systema Saturnium. The Hague, Netherlands: Adriaan Vlacq.

Ince, Edward L. 1944. Ordinary Differential Equations. New York: Dover. (Orig. pub. 1927.)

Katz, Victor J. 1993. A History of Mathematics: An Introduction. New York, Harper Collins.

Kline, Morris. 1990. Mathematical Thought from Ancient to Modern Times. Vol. 3. Oxford: Oxford University Press. (Orig. pub. 1972.)

Lagrange, Joseph-Louis. 1766. Solution de différens problêmes de calcul intégral. Miscellanea Taurinensia 3:179–380.

Leanhardt, Aaron E., and Adam E. Parker. 2017. Fontaine’s Forgotten Method for Inexact Differential Equations. Mathematics Magazine 90(3):208–219.

MathSciNet. 2023. Abbreviations of Names of Serials. Mathematical Reviews.

Otero, Daniel E. 1999. Calculus from an Historical Perspective: A Course for Humanities Students. PRIMUS 9(1):56–72.

Parker, Adam E. 2013. Who Solved the Bernoulli Differential Equation, and How Did They Do It? College Mathematics Journal 44(2):89–97. .

Parker, Adam E. 2016. What’s in a Name: Why Cauchy and Euler Share the Cauchy-Euler Equation. College Mathematics Journal 47(3):191–198.

Parker, Adam E. 2020a. Solving First-Order Linear Differential Equations: Bernoulli’s (almost) Variation of Parameters Method. TRIUMPHS. Differential Equations, 3. https://digitalcommons.ursinus.edu/triumphs_differ/3/.

Parker, Adam E. 2020b. Solving First-Order Linear Differential Equations: Gottfried Leibniz’ “Intuition and Check” Method. TRIUMPHS. Differential Equations, 1. https://digitalcommons.ursinus.edu/ triumphs_differ/1/.

Parker, Adam E. 2020c. Solving First-Order Linear Differential Equations: Leonard Euler’s Integrating Factor Method. TRIUMPHS. Differential Equations, 4. https://digitalcommons.ursinus.edu/triumphs_differ/4/.

Parker, Adam E. 2020d. Wronskians and Linear Independence: A Theorem Misunderstood by Many. TRIUMPHS. Differential Equations, 2. https://digitalcommons.ursinus.edu/triumphs_differ/2/.

Parker, Adam E. 2021a. Leonhard Euler and Johann Bernoulli Solving Homogenous Higher Order Linear Differential Equations With Constant Coefficients. TRIUMPHS. Differential Equations, 6. https://digitalcommons.ursinus.edu/triumphs_differ/6/.

Parker, Adam E. 2021b. Runge-Kutta 4 (and Other Numerical Methods for ODEs). TRIUMPHS. Differential Equations, 7. https://digitalcommons.ursinus.edu/triumphs_differ/7/.

Parker, Adam E. 2021c, January. Wronskians and Linear Independence: A Theorem Misunderstood by Many – A Mini-Primary Source Project for Students of Differential Equations, Linear Algebra and Others. Convergence 18.

Parker, Adam E. 2022, July. Solving Linear Higher Order Differential Equations with Euler and Johann Bernoulli: A Mini-Primary Source Project for Differential Equations Students. Convergence 19.

Peano, Giuseppe. 1889a. Sur le Déterminant Wronskien. Mathesis 9:75–76.

Peano, Giuseppe. 1889b. Sur les Wronskiens. Mathesis 9:110–112.

Peano, Giuseppe. 1890. Démonstration de l’intégrabilité des équations différentielles ordinaries. Mathematische Annalen 37:182–228.

Pulskamp, Richard. n.d. Pierre Simon Laplace on Probability and Statistics. http://www.probabilityandfinance.com/pulskamp/Laplace/index.html.

Sarton, George. 1936. Notes on the History of Anagrammatism. Isis 26(1):132–138.

Simmons, George F. 2017. Differential Equations with Applications and Historical Notes. 3rd ed. Boca Raton, FL: CRC Press.

Skof, Fulvia, ed. 2008. Giuseppe Peano between Mathematics and Logic. Milan: Springer.

The Euler Archive. n.d. St. Petersburg Academy Publications.

TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources. 2023. Welcome to TRIUMPHS!

TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources. n.d. The TRIUMPHS Society.

Whiteside, Derek Thompson, ed. 1981. The Mathematical Papers of Isaac Newton. Vol. 3. Cambridge: Cambridge University Press. (Orig. pub. 1969.)

 

 

Acknowledgment

So many people have encouraged and helped me on the journey of including historical sources in my classroom. Danny Otero, Janet Barnett, Ken Monks, and Dominic Klyve, are just a few of the many that should be thanked. I also appreciate the countless^[25] students that gave feedback, along with the anonymous referees and editors that have improved everything I've written, certainly including this paper. I hope I can pass forward some of their kindness and expertise.

About the Author

Adam Parker is professor of mathematics at Wittenberg University in Springfield, Ohio. He earned degrees in mathematics and psychology at the University of Michigan and received his PhD in algebraic geometry from the University of Texas at Austin. To improve learning in his classes, Dr. Parker uses primary sources when teaching. Developing such material can be difficult, and this article is an attempt to provide advice to instructors who encounter problems when developing historical modules. In his spare time, he enjoys cooking, eating, trying not to kill houseplants, and spending time with his dog, Rosie.