The fourth book of Euclid’s Elements, a 2,300-year-old geometry text, includes directions for constructing a 15-sided polygon inside a circle. The first step is familiar to geometry students: Draw an equilateral triangle and a regular pentagon so their vertices touch the circle and the two shapes share one vertex. In addition to text directions, the Elements included diagrams that illustrated the method.

Line segments have been drawn and erased in the oldest complete copy of the Elements, a ninth-century manuscript housed in the Vatican Library. Image courtesy of the Library of Congress Online Catalog, Prints and Photographs Division.

It's impossible to know what Euclid's original diagrams looked like, but the surviving manuscripts reveal surprising variations in the representation of geometric shapes such as the pentadecagon. To modern eyes, these variations look like errors: In some medieval versions of the text, line segments measure the wrong length. In the oldest copy of the Elements, a ninth-century manuscript housed in the Vatican library, segments have been drawn and erased. Another ninth-century text at Oxford University displays the sides of the pentadecagon within the circle as erratic and curved, not straight. A 12th-century copy in Paris also uses curves, but they're slightly less curvy than in the older version at Oxford. In Vienna, one can find a text from the 11th or 12th century in which the original lines were the right length and straight, but someone later added curved segments (1).

The Elements gets a lot of attention, but it's not the only historical scientific text with these diagrammatic issues. They turn up in copies of works by Ibn al-Haytham, Archimedes, Aristotle, and Ptolemy. Variations include parallel lines that aren't, mislabeled shapes, equal lines or angles drawn unequally, or unequal angles that might be shown as congruent. A 10th-century manuscript of the Archimedes Palimpsest uses an isosceles triangle to represent a parabola, for example. These may seem like little more than quirks of history, but some researchers see intriguing clues amongst drawings, hints at how math has evolved over millennia.

Picture This Researchers are starting to study those variations to explore how mathematical ideas spread and to gain insights into how different peoples approached the subject. Traditionally, math historians who study ancient Greek texts have focused on the words and numbers, dismissing the diagrams as mere illustrations of the text. That focus omits part of the story and the history, asserts math and science historian Nathan Sidoli at Waseda University in Tokyo who, along with his collaborator Ken Saito at Osaka Prefecture University, observed in a 2012 essay the diagrammatic changes in the pentadecagon and other proofs (1). Math is rich with abstraction, and over time people have found various ways to visualize those abstractions. “From a young age, we're trained to understand generalities in certain visual ways,” Sidoli says. “By looking at this stuff we can remind ourselves that these are not universal ways of seeing.” Diagrams have been a part of math for thousands of years of human history. Babylonians calculated square roots and knew the principle of the Pythagorean Theorem more than one thousand years before the time of Pythagoras or Euclid. Evidence comes in the form of a clay tablet from the 17th century BCE, which shows a geometric diagram of a square and its diagonals together with corresponding numbers. Data-visualization pioneer Edward Tufte, a professor emeritus in political science, computer science, and statistics at Yale, calls the tablet a “graphic witness” to the knowledge of the Babylonians. Some researchers argue that diagrams can form an integral part of the mathematics and are messengers across time—warts and all. If an error that originates in one copy propagates through future versions, it may show that the copyists didn't understand the math or didn't value accuracy. On the other hand, some scholars used diagrams to add to the knowledge in the Elements. Where Euclid may have originally described properties of an acute angle, for example, a later scribe might have added similar properties for obtuse and right angles, as well. This fragment of the Elements was part of the Oxyrhynchus Papyri, a group of manuscripts discovered in 1897 at an ancient garbage dump near Oxyrhynchus, Egypt. Approximately 2,000 years old, the text alludes to the fifth proposition of Book 2 of the Elements. Image courtesy of Bill Casselman (University of British Columbia, Vancouver).

The Reader Intervention The Elements, which contains 13 volumes, has appeared in at least hundreds of editions, and until the last century it was the second-best-selling book in the world. (The Bible was first.) But not everything in the Elements came from Euclid. The volumes represent a collection of mathematics knowledge known to the Greeks at the time. Physicist Stephen Hawking described Euclid as “the greatest mathematical encyclopedist of all time,” likening him to Noah Webster, who assembled the first English language dictionary (2). The Elements was translated from Greek, Arabic, Latin, Hebrew, and other languages. The treatise evolved as it grew and migrated—and so did the diagrams. Readers made notes in the margins and inserted changes. Later readers and translators saw both the manuscript and the additions and made revisions that seemed appropriate for their time. Those interactions are captured in transcriptions of the proofs and diagrams in the Elements, and the act of copying became an act of transformation, says Eunsoo Lee, a PhD student at Stanford University studying the evolution of diagrams over time in the Elements. “We may easily forget about the role of readers in the making of diagrams,” says Lee, noting that they could intervene or intermingle by marking on the manuscript. Later, scribes took those notes into consideration. “If they determined that the marginal diagrams [were] superior to the main diagrams,” explains Lee, “the marginal diagrams were adopted as the main diagrams for later generations.” These visual changes conveyed mathematical ideas in ways that couldn't be transmitted through text. It’s too simplistic to call these changes errors. Some of the changes may have been intended as improvements; others arose from cultural practices. Arabic reads right-to-left, for example, so in early Arabic versions of the Elements the orientations of its diagrams were often flipped—angles that opened to the left in ancient Greek manuscripts opened to the right in the Arabic versions. However, when those Arabic versions were translated into Latin, some scribes didn’t flip the diagrams back. Mathematician Robin Hartshorne, retired from the University of California, Berkeley, further argues that it’s not necessarily fair to see changing the diagrams as a corrective process. Even with curves and erasures, those pentadecagon diagrams got the point across. Printing the Elements with accurate diagrams reflects the values of a time, he says, but it's a practice disloyal to earlier versions. “I would call it redrawing the diagram to the taste of modern mathematicians who like to see metrical exactness,” says Hartshorne. “These are hand-drawn diagrams of things that are not necessarily easy to represent,” adds science historian Courtney Roby, who studies ancient scientific texts at Cornell University, in Ithaca, New York. “Diagrams are the creations of individual authors and scribes, and their creativity and experimentation and change.”

An Elemental Evolution Lee focuses on manuscripts from the ninth century through the first printed version of the Elements, which appeared in 1482 with the advent of the printing press. From that time on, says Lee, the Elements became a standard textbook in many European universities, and its diagrams became teaching tools. As a result, “you find totally different shapes of diagrams in the age of printing culture,” says Lee, who is digitizing a collection that includes at least 5 papyri, 32 Geek manuscripts, 92 translated manuscripts, and 32 printed editions of the Elements. Until the 19th century, Euclid’s treatise was held up as the model of rigorous, structured mathematical arguments. Those arguments needed drawings to make sense. “They don’t work without the diagrams,” says philosopher John Mumma at the California State University, San Bernardino, who has argued that diagrams in the Elements are more than just teachable visuals and that they serve an important role within the argument of the proof itself (3) In the late 19th and early 20th centuries, mathematicians challenged the supremacy of the Elements partly because of Euclid's reliance on diagrams. German mathematician David Hilbert in particular called for a more formal approach to mathematics that used logic alone and didn’t require diagrams in its proofs and truths, which he regarded as a sort of mathematical crutch. “Euclid's Elements was dismissed as not being rigorous,” says Mumma. “He was thought to use diagrams in a kind of intuitive, open-ended way.” For example, where a diagram in the Elements might show a point on a line between two other points, Hilbert wanted an analytical description of what he called “betweenness,” without relying on a sketch. British philosopher and logician Bertrand Russell also criticized Euclid’s approach, noting that many of the Greek’s proofs were weak because they drew their reasoning power from diagrams and not from logic alone. “A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid’s earlier proofs fail before this test,” Russell wrote in 1902 (4). (The first proof in the Elements shows how to make an equilateral triangle by using two intersecting circles. However, it relies on the diagram to justify the intersection point—rather than proving its existence rigorously.) Many modern historians of math, however, now regard Euclid’s approach as another way of seeing math—one that's not necessarily weaker just because it uses diagrams. These scholars argue that the diagram makes the proof, and there is no universal way of understanding math. “You can really make clear, precisely what information the diagram is holding for the argument,” says Mumma. “It’s not just an illustration.” Contemporary research focused on diagrams began in large part in the 1990s, when Reviel Netz at Stanford University and Kenneth Manders at the University of Pittsburgh, in Pennsylvania, began arguing that ancient mathematical diagrams deserved another look. Netz says the field focused on two aspects: The visuals themselves and how people used them (5, 6). He says Lee’s work at Stanford University, by comparing diagrams across centuries, brings those two areas together to advance the field. “He is the first person to do both at the same time,” says Netz, who is also Lee’s advisor. Netz says Lee’s work will help historians understand how “science pivoted away from the theoretical geometry of the Greeks to … more applied and physical uses of geometry for the real world.” After the Elements, Lee wants to analyze diagrams in Euclid’s Optics—an early treatise on the physics of light—and then focus on works by Ptolemy and Archimedes. He says he hopes his research incites interest among historians, philosophers, and mathematicians in analyzing how people have used diagrams—and still do—to probe deep ideas in mathematics. “We tend to think of them as things that can be removed,” he says. “But some ideas cannot be conveyed by text. They have to be conveyed by diagrams.”