# How to study the history of mathematics from the drawings in the "Principles" of Euclid

In the fourth book, The Beginnings, by Euclid, a text on the geometry of the age of 2,300 years, is indicated on the construction of a 15-sided polygon inside a circle. The first step is well known to those who study geometry: building an equilateral triangle and a regular pentagon so that their vertices lie on a circle and both figures have one common vertex. In addition to textual instructions, the "Beginnings" contained drawings illustrating the method.

In the oldest complete copy of the Beginnings , a ninth-century manuscript stored in the Vatican Library, straight line segments were erased and erased. Image from Library of Congress Online Catalog, Prints and Photographs Division.

It is impossible to know what the original diagrams of Euclid himself looked like, but in the surviving manuscripts there are amazing variations in the display of such geometric figures as the fifteen-square. To the modern observer such variations seem to be mistakes: in some medieval versions of the text, the straight line segments have the wrong length. In the ninth-century manuscript, the oldest copy of The Beginnings , stored in the Vatican Library, the segments were drawn and erased. In another ninth-century text stored at Oxford University, the sides of the fifteen-square inside the circle are curved and erratic, not straight. In the Paris copy of the twelfth century, curves are also used, but they are slightly less tortuous than in the old Oxford version. The text of the eleventh or twelfth century is stored in Vienna, in which the original lines were of the correct length and straight, but later someone added curved segments to them (1).

The “beginnings” are of great interest, but this is not the only historical scientific text with problems in the drawings. It turns out that they are found in copies of the works of Ibn al-Haytham, Archimedes, Aristotle and Ptolemy. Among the variations, there are parallel lines that are not actually parallel, incorrectly marked shapes, equal segments or angles, drawn unequal, or unequal angles that may look equal. For example, in the tenth century Archimedes palimpsest manuscript, an isosceles triangle is used to denote parabota. This may seem like a simple historical oddity, but some researchers find intriguing hints among the drawings about how mathematics evolved over the millennium.

## Visualization

Researchers are beginning to study these variations in order to learn how mathematical ideas spread and to understand how different people approached this topic. Traditionally, historians of mathematics who study ancient Greek texts focus on words and numbers and skip drawings, as simple illustrations to the text. Science historian Nathan Sidoli from Waseda University in Tokyo and his colleague Ken Saito from Osaka Prefecture University, who noted the schematic changes in the 15-sided square and other evidence, due to this focus on the text, we miss part of the story (1).

Mathematics is rich in abstraction, and over time people have discovered many ways to visualize these abstractions. “From our very youth, we have been learning to understand general concepts in certain visual ways,” says Sidoli. “Looking at these works, we can remind ourselves that this is not a universal way of seeing.”

Drawings and diagrams were part of the mathematics of thousands of years of human history. The Babylonians calculated square roots and knew the principle of the Pythagorean theorem for more than a thousand years before Pythagoras or Euclid. Evidence can serve as a clay tablet, dated seventeenth century BC, on which is drawn a drawing of a square and its diagonals with the corresponding numbers. Edward Taft, a pioneer of data visualization, a professor of political science, computer science, and statistics from Yale, calls the sign a "graphic witness" of the Babylonian knowledge.

Some researchers believe that the drawings themselves can be an integral part of mathematics and a carrier of information between the centuries, despite all its flaws. If the error that appeared in one copy spread to subsequent versions, then it shows that the scribes did not understand mathematics or did not appreciate accuracy. On the other hand, some pundits used the blueprints to supplement the knowledge outlined in the Beginnings . For example, where Euclid described the properties of an acute angle only, later scribes could add similar properties for obtuse and right angles.

This piece of "The Beginning" was part of the Oxyrhynus papyrus, a group of manuscripts, discovered in 1897 at an ancient garbage dump near the city of Oxyhirnh in Egypt. Text about 2,000 years old refers to the fifth theorem of the second volume, “Beginning” . Image by Bill Casselman (University of British Columbia, Vancouver).

The Beginnings , consisting of thirteen volumes, were published in at least hundreds of publications, and until the last century it was the second largest book in sales in the world. (The first is the Bible.) But not everything in the Beginnings was derived by Euclid. The volumes presented a collection of mathematical knowledge, known to the ancient Greeks of the time. Physicist Stephen Hawking called Euclid “the greatest mathematics encyclopaedist of all time,” and compared him with Noah Webster, who compiled the first English dictionary (2).

The "beginnings" were translated from Ancient Greek, Arabic, Latin, Hebrew and other languages. The treatise in the process of growth and migration evolved, as did the drawings in it. Readers left notes in the margin and made edits. Subsequent readers and translators saw the manuscript and the additions, and edited the work in accordance with what was appropriate for their time. Such interactions are recorded in the translation of evidence and the drawings of the Beginnings , and the act of copying itself became, according to the student of the evolution of the drawings of the Beginnings, a graduate student at Stanford University, Jensu Lee, an act of transformation.

“We can easily overlook the role of readers in creating blueprints,” says Lee, stressing that they could interfere and contribute by making notes in the manuscript. Later, the scribes took these notes into account. “If they thought that the field drawings were more important than the main drawings,” explains Lee, “the drawings on the fields by subsequent generations turned into basic ones.” These visual changes conveyed mathematical ideas in ways that are impossible to convey with text.

Calling such changes an error would be too trite. Some of the changes were supposed to be improvements; others came from cultural practices. For example, the Arabic text is read from right to left, so in the early Arabic versions of “Beginnings” the orientation of the drawings was often mirrored — the angles that were revealed to the left in ancient Greek manuscripts, in the Arabic versions were revealed by the right. However, when these Arabic versions were translated into Latin, some scribes did not turn the drawings back.

The mathematician Robin Hartshorn, who previously worked at the University of California at Berkeley, even argues that it is not always fair to see the change of drawings as an editing process. Even with all these curves and curves, the drawings of the fifteen-squares conveyed the necessary meaning. The “Start” print with accurate drawings reflects the values ​​of time, he says, but this practice is disloyal to previous versions. “I would call it a redrawing of drawings to the tastes of modern mathematicians who are seeking to see metric accuracy,” says Hartshorn.

“These were hand-drawn drawings of concepts that are not always easy to submit in writing,” adds science historian Courtney Roby, who studies ancient scientific texts at Cornell University. “Drawings are the creations of specific authors and copyists, their creativity, experiments, and changes.”

## Evolution began

Lee worked on manuscripts from the ninth century to the first printed version of The Beginning , which appeared in 1482 after the invention of the printing press. Since Lee says, "The Beginning" has become the standard textbook in many European universities, and their drawings have become a teaching tool. As a result, “in the era of print culture, we see completely different kinds of drawings,” says Li, who digitizes a collection of at least five papyrus, 32 ancient Greek manuscripts, 92 translated manuscripts, and 32 printed editions, “Started .

Until the nineteenth century, Euclid’s treatise was considered a model of rigorous and structured mathematical evidence. To make sense, this evidence requires drawings. “They are useless without blueprints,” explains philosopher John Mumma of the University of California, who claims that the “Beginnings” drawings are not just a visual teaching tool, they are also important for proving the statements themselves (3)

At the end of the 19th and the beginning of the 20th centuries, mathematicians questioned the superiority of the "Beginnings" and partly due to the dependence of Euclid on the drawings. In particular, the German mathematician David Hilbert called for a more formal approach to mathematics that uses only logic and does not require drawings for proof, which he considered a kind of "crutches" of mathematics.

“Euclid’s“ Beginnings ”was refused, because they didn’t seem very strict,” says John Mumma. "It was believed that he used the drawings intuitively and too freely."

For example, in the "Beginnings" was a drawing showing a point on a straight line between two other points. Hilbert needed an analytical description of what he called "intermediateness," without the use of pictures. The British philosopher and logician Bertrand Russell also criticized Euclid’s approach: he noted that many ancient Greek evidence is weak because they take the power of their reasoning from the blueprints, and not exclusively from logic. “Genuine evidence must remain valid even in the absence of painted figures, but very much Euclid’s evidence does not pass this test,” wrote Russell in 1902 (4). (The first proof in the Principles shows how to construct an isosceles triangle with the help of two intersecting circles. However, the intersection point is justified from the drawing, its existence is not proved strictly.)

However, many modern historians of mathematics perceive the Euclidean approach as another way of seeing mathematics - and it is not necessarily weak simply because it uses drawings. These scientists argue that drawing is evidence, and that there is no universal way to understand mathematics. “We can actually understand everything by using the information contained in the drawing as a proof,” Mumma says. "This is not just an illustration."

Modern studies have focused on drawings for the most part since the 1990s, when Revil Netz of Stanford University and Kenneth Menders of the University of Pittsburgh stated that ancient mathematical drawings deserve to be viewed from a different angle. Netz says the field of study focuses on two aspects: the graphical representation itself and how people use drawings (5, 6). He argues that the work of Lee from Stanford University in comparing the drawings of different centuries combines these two aspects, allowing you to expand the field of study.

Netz says that Lee's work will help historians understand how "science has moved from the theoretical geometry of the ancient Greeks to ... a more applied and physical use of geometry for the real world."

After “Beginning,” Lee wants to analyze the drawings in Euclid ’s Optics , an early work on the physics of light, and then focus on the works of Ptolemy and Archimedes. He hopes that his research will attract the interest of historians, philosophers and mathematicians to the analysis of how people used (and continue to use) blueprints to study deep mathematical ideas. “We tend to get rid of the blueprints,” he says. “But some ideas are impossible to convey in the text. They need to be transferred graphically. "

## Reference materials

1. Saito K, Sidoli N (2012) Diagram of Comparisons and Mathematics. The Chemla K ed. (Cambridge Univ Press, Cambridge, UK), pp 135-162. Google scholar
2. Hawking S, ed (2002) On the Shoulders of Giants (Running Press, Philadelphia). Google scholar
3. Mumma J (2010) Proofs, pictures, and Euclid. Synthese 175: 255–287. CrossRef Web of Science Google Scholar
4. Russell B (1902) The teaching of Euclid. Math Gaz 2: 165–167. Google scholar
5. Netz R (1998) Greek mathematical diagrams: Their use and their meaning. Learn Math 18: 33–39. Google scholar
6. Manders K (1995) Diagram-based geometric practice. The Philosophy of Mathematical Practice, ed Mancosu P (Oxford Univ Press, Oxford), pp 65–79. Google scholar

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