When Euclid was used as a textbook, what exercises did students do?

Question

Until fairly recently, it was common for students in school to learn Euclidean geometry from a translation of Euclid. I get the impression that ca. 1700 this would have been in college and only for a tiny fraction of the population, whereas maybe ca. 1900 in the US it would have been at a younger age and perhaps in some cases the entire population.

I'm curious how this actually worked. In particular, what did these students do to practice their skills? Would students have mainly proved theorems, or would there have been more of mix? Were there editions of Euclid that had a set of theorems created for students to practice by proving them? Or maybe separate anthologies of problems? Can such materials still be found online?

Were people using a straight translation of Euclid, like Heath in the English-speaking world, or modernizations? Were there any particularly good or widely used modernizations? The coolest one I'm aware of is Byrne.

Answer 1

For a history of how Geometry textbooks (and the way they were used) in the United States evolved from the mid-19th century into the 20th century, see:

Herbst, P. (2002) Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics 49(3), 283-312.

Herbst distinguishes among three eras in Geometry instruction in US high schools:

1. The Era of Text, beginning in the 1840s
2. The Era of Originals, beginning with the texts of Greenleaf (1858) and Chauvenet (1870)
3. The Era of Exercise, resulting from the recommendations of the Committee of Ten

Regarding the Era of Text, Herbst writes:

American high schools started to offer geometry courses in the 1840’s as universities started to make it a requisite for admission (Quast, 1968, p. 36). During this period, which I call the Era of Text, the study of geometry entailed mastering the Euclidean body of knowledge as developed by a text. Texts that served that purpose included Robert Simson’s (1756) Elements of Euclid, John Playfair’s (1795/1860) Elements of Geometry, and Adrien-Marie Legendre’s Elements of Geometry (translated and edited by John Farrar in 1819). There are notable differences in how the three authors – Simson, Playfair, and Legendre – develop the Euclidean body of knowledge (Jones, 1944), but as far as what proving could have meant for students who studied those texts, the three texts are similar...

The purpose of studying geometry in school was to grasp the necessary character of the relationships between geometrical objects. It was essential to that acquisition to study the proofs of those relationships. The word demonstration was used in the text, but it was neither used as the name of an object to be studied nor used as the name of a singular skill to be trained on...

The study of geometry was done through reading and reproducing a text; such work would train the reasoning faculties of students. But, the texts do not hint at the existence of official mechanisms to verify or steer the evolution of students’ reasoning. To know geometry and to be able to prove the theorems of geometry were indistinguishable. And the difference between knowing geometry and remembering the text was immaterial in school (see Quast, 1968, p. 40). As the geometry course became more common in high schools, geometry instruction began to move away from simply reproducing a geometry text. (Emphasis added)

As has been remarked by others, during this era geometry texts (including Euclid's) did not contain problems for students to solve, and it was not expected that students write any proofs other than the ones that were already in the text, which they memorized and had to reproduce. The first textbooks to include "originals" (that is, original problems for students to solve) were those of Greenleaf (1858) and Chauvenet (1870), regarding which Herbst writes:

The texts by Greenleaf (1858) and later by William Chauvenet (1870) differed from those of the Era of Text in that they included exercises at the end. As these exercises were spoken of as opportunities for students to do original work, I call this transitional period in the evolution of proving the Era of Originals. These originals were corollaries of propositions proved in the main text and additional theorems that might not have deserved a place in the main text. Thus, they afforded opportunities for students to use their reasoning to further their geometric knowledge. In his preface to a later edition of Chauvenet’s (1887) text, William Byerly said the purpose of originals was to “compel the student to think and to reason for himself” beyond just learning “to understand and demonstrate a few set propositions” (p. 5). Changes in the way later texts would give original exercises seem to respond to that purpose of training students’ “power of grasping and proving a simple geometrical truth,” a power that “can never be gained by memorizing demonstrations” (Chauvenet, 1887, p. 5). In that spirit, George Wentworth (1878) would give long lists of original exercises at the end of each chapter, and Wentworth (1888) would also interject some of those original exercises in between the propositions of a given chapter. The presence of originals presumed that students would learn to reason by reasoning. (Emphasis added)

There is much more in Herbst's paper than I have mentioned in this brief summary; in fact as the title suggests most of the paper is devoted to the evolution of the "two-column proof" as both an instructional form and as a norm specific to the genre of high school Geometry textbooks. But the background material on how things were done before the Era of Exercise is very relevant for this question.

Answer 2

Euclid’s Elements was used as a Geometry textbook in essentially the same way that Virgil’s Æneid was used as a Latin textbook. Neither contained exercises anything else pedagogical, and both were used in the same way, in about the same time period.

Their heyday was in the days before schooling as we know it existed. They would be used between a student and his Latin or math tutor, a person or people who worked one-on-one with the student, often at the well-to-do student’s house.

You “knew” the material once you plowed through it. Virgil (and Ovid and Horace) were simply translated by the student as a way of practicing Latin skills. The poems were translated when the tutor said they were. Similarly, the Elements were read and demonstrated to the tutor’s satisfaction. Exercises? Why would you need exercises? That wouldn’t be the objective of the course.

Heath’s versions were mainly written for mathematicians, and were certainly not used in schools.

Our current course called Geometry doesn’t have much in common with reading the Elements. For decades, Geometry was taught in two courses, Plane and Solid. Those were the days when math schooling only went to Algebra II or so, with Freshman Math at colleges being what we call Pre-calculus. My dad, high school Class of 1949 who entered Clemson Agricultural College that same year, went through mathematics like this.

The set of postulates isn’t the same: many of the modern sets of postulates are due to contemporary writers. For example, the current formulation of Euclid’s 5th (“Given a line and a point not on that line, there is one and only one line through the point that is parallel to the given line”) is due to John Playfair, and is often referred to as Playfair’s axiom. In the full-on day’s of the 1950’s and 1960’s “New” Math, Edwin Moise’s Geometry book turned the SAS triangle congruence theorem into a postulate: proving the others (ASA, SSS, etc.) given one of them is not too bad, but having to prove an initial one is pure hell.

Moise was later joined by Floyd Downs, and their Geometry book may still be in print. Alfred Posamentier and H.S.M. Coxeter soon came out with competing books, mostly sticking to the Moise postulate model with tweaks. Those tweaks included exercises, applications, word problems, career sections, computer-aided learning, and all sorts of applied things that would never pollute the pure reader of Euclid. Those that needed practical Geometry would learn it as an apprentice to a craftsman.

Answer 3

As others have said, Euclid was often used as a reference book rather than what we now call a "textbook". Students were often required to memorize and reproduce the contents. For example, at Oxford, until the mid-19th century, students had to memorize two books of Euclid (to the letter) and were not asked to solve any original problems.

Textbooks with practical exercises began to be introduced in the 18th century with Clairaut in France (1741) and Kästner in Germany (1758). These initial textbooks were not always taken seriously, but d'Alembert made a call for serious, practical textbooks and a new approach to geometry in 1784. The transition to a more pedagogical approach occurred slowly in the early 19th century. LeGendre wrote a geometry textbook in France which became very popular and a translated version eventually became the most popular geometry textbook in the U.S. People continued to use Euclid as a reference, but increasingly turned to other books for teaching and exercises.

Most of the preceding history can be found in the historical introduction to a 1912 report (Slaught, H. E., & others. (1912). Final report of the national committee of fifteen on geometry syllabus. The Mathematics Teacher, 5(2), 46-131.)

Answer 4

Nick C mentions in a comment that there are a handful of Great Books colleges that currently use Euclid in their course of studies. I went to one of these and we devoted our entire freshman year of math (8 semester-hours total) to the Elements. No specific edition was required but almost everyone used the Heath translation as found in either the multi-volume Dover paperback set, or the Brittanica Great Books collection.

The general procedure was this: the teacher assigned a handful of propositions at the end of each class period; we had to learn them before the next class; and then in class the teacher would call students at random to demonstrate each of the assigned propositions at the board and field questions from the rest of the class. Then there would be open discussion of the prop. Using this method we covered a great deal of the Elements; it's been almost 20 years so I don't remember in a lot of detail but glancing over a summary of the text I'd say we worked through a good 75% of it.

We also spent a fair amount of time discussing the front matter of each book (definitions, postulates, etc.) and we had to write a paper or two as well as essay-format final exams at the end of each semester; in all of these we had to demonstrate a theoretical understanding of the geometrical concepts we had been working with throughout the year.

Some of the answers above suggest that the 'demonstrate propositions' approach to geometry is basically just rote memorization, but my experience was that after the first few propositions of Book I, it's more or less impossible to do this sort of thing by rote; if you don't understand why each step of each proposition follows from the previous one, and if you don't have a bigger-picture grasp of the overall logic and structure of each book, you will probably fail this sort of class -- because eventually you just won't be able to get through a proposition at the board, and you certainly won't be able to speak or write intelligently about it even if you can reproduce it mechanically.

That said, I have no idea how this neo-approach compares to historical approaches.

Answer 5

Some people theorize that Euclid's Elements should be studied (and was studied) by doing problems based on the theorems provided in The Elements.

It seems that Euclid does not use a figure until he has proven that it is possible to construct it. So, for example, he does not use squares until he proves, "To construct a square" in Book 1, Proposition 46. He can then use squares to prove Proposition 47, which is his proof of the Pythagorean Theorem.

Yet, Euclid does not prove how to construct an isosceles or scalene triangle. He only proves the construction of an equilateral triangle in Book 1, Proposition 1. Constructions of the scalene and isosceles are possible to do in a manner very similar to Proposition 1, and it is though that this might be one of the exercises Euclid's students may have been instructed to do.