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Teaching According to Robin Hartshorne

n the fall semester of 1988, I taught an un- 1482 up to about 1900. Billingsley, in his preface dergraduate course on Euclidean and non- to the first English of the Elements . I had previously taught (1570) [1], writes, “Without the diligent studie of courses in and algebraic Euclides Elements, it is impossible to attaine unto Igeometry, but this was my first time teach- the perfecte knowledge of Geometrie, and conse- ing Euclidean geometry and my first exposure to quently of any of the other Mathematical .” non-Euclidean geometry. I used the delightful book Bonnycastle, in the preface to his edition of the by Greenberg [8], which I believe my students en- Elements [4], says, “Of all the works of antiquity joyed as much as I did. As I taught similar courses in subsequent years, which have been transmitted to the present time, I began to be curious about the origins of geome- none are more universally and deservedly esteemed try and started reading Euclid’s Elements [12]. Now than the Elements of Geometry which go under I require my students to read at least Books I–IV the name of Euclid. In many other branches of sci- of the Elements. This essay contains some ence the moderns have far surpassed their masters; reflections and questions arising from my but, after a lapse of more than two thousand years, encounters with the text of Euclid. this performance still retains its original preemi- nence, and has even acquired additional celebrity Euclid’s Elements for the fruitless attempts which have been made A called the Elements was written approxi- to establish a different .” Todhunter, in the mately 2,300 years ago by a man named Euclid, of preface to his edition [18], says simply, “In England whose life we know nothing. The Elements is divided the text-book of Geometry consists of the Elements into thirteen books: Books I–VI deal with geom- of Euclid.” And Heath, in the preface to his defin- etry and correspond roughly to the material taught in high school geometry courses in the United States itive English translation [12], says, “Euclid’s work today. Books VII–X deal with and in- will live long after all the text-books of the present clude the Euclidean , √the infinitude of day are superseded and forgotten. It is one of the primes, and the irrationality of 2. Books XI–XIII noblest monuments of antiquity; no deal with , culminating in the con- worthy of the name can afford not to know Euclid, struction of the five regular, or platonic, solids. the real Euclid as distinct from any revised or Throughout most of its history, Euclid’s Elements rewritten versions which will serve for schoolboys has been the principal manual of geometry and or engineers.” indeed the required introduction to any of the These opinions may seem out-of-date today, when sciences. Riccardi [15] records more than one thou- most modern mathematical have a history sand editions, from the first printed edition of of less than one hundred years and the latest logi-

Robin Hartshorne is professor of at the Uni- cal restructuring of a subject is often the most prized, versity of California at Berkeley. His e-mail address is but they should at least engender some curiosity [email protected] about what Euclid did to have such a lasting impact.

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How Geometry Is Taught Today A typical high school geometry course contains results about congruent trian- gles, , lines, the Pythagorean , similar , and of rectilinear plane figures that are familiar to most of us from our own school days. The material is taught mainly as a collection of truths about geometry, with little attention to and proofs. However, one does find in most texts the “ruler ”, which says that the points of a can be put into one-to-one correspondence with the real in such a way that the between two points is the difference of the corresponding real numbers. This is presumably due to the influence of Birkhoff’s article [3], which advocated the teaching of geometry based on mea- surement of and angles by the real numbers. It seems to me that this use of the real numbers in the is analysis, not geometry. Is there a way to the study of geom- etry on purely geometrical concepts? A college course in geometry, as far as I can tell from the textbooks currently available, provides a potluck of different topics. There may be some “modern Euclidean geometry” containing fancy about triangles, , and their special points, not found in Euclid and mostly discovered during a period of intense activity in the mid-nineteenth century. Then there may be an intro- duction to the problem of parallels, with the discovery of non-Euclidean geome- Figure 1. The , in Simson’s translation [17]. The try; perhaps some projective geometry; proof shows that the ABD is congruent to the triangle FBC. Then and something about the role of trans- the BL, being twice the first triangle, is equal to the GB, formation groups. All of this is valuable which is twice the second triangle. Similarly, the rectangle CL equals the material, but I am disappointed to find square HC. Thus the on the sides of the triangle ABC, taken to- that most textbooks have somewhere a gether, are equal to the square on the base. hypothesis about the real numbers equiv- alent to Birkhoff’s ruler axiom. nitudes of the same kind could be compared as to This use of the real numbers obscures one of size: less, equal, or greater, and they could be the most interesting aspects of the development added or subtracted (the lesser from the greater). of geometry: namely, how the concept of continuity, They could not be multiplied, except that the op- which belonged originally to geometry only, came eration of forming a rectangle from two line seg- gradually by analogy to be applied to numbers, ments, or a volume from a and an leading eventually to Dedekind’s of , could be considered a form of multiplication the field of real numbers. of magnitudes, whose result was a magnitude of a different kind. Number versus Magnitude in Greek In Euclid’s Elements there is an con- Geometry cept of (what we call ) for line seg- In classical Greek geometry the numbers were ments, which could be tested by placing one seg- 2,3,4,… and the unity 1. What we call negative ment on the other to see whether they coincide numbers and zero were not yet accepted. Geo- exactly. In this way the equality or inequality of line metrical such as line segments, angles, segments is perceived directly from the geometry areas, and were called magnitudes. Mag- without the assistance of real numbers to

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their . Similarly, angles form a kind of mag- perception occur? How and when were the real nitude that can be compared directly as to equality numbers introduced into geometry? Was Euclid or inequality without any numerical measure of size. already using something equivalent to the real Two magnitudes of the same kind are commen- numbers in disguised form? surable if there exists a third magnitude of the same kind such that the first two are (whole number) mul- Development of the Real Numbers tiples of the third. Otherwise they are incommen- In , as we saw, the only numbers surable. So Euclid does not say the of were (positive) integers. What we call a rational two (a number) is irrational (i.e., not a rational num- number was represented by a of integers. ber). Instead he says (and proves) that the Any other was represented as a geomet- of a square is incommensurable with its side. rical magnitude. This of view persisted even The difference between classical and modern to the time of Descartes. In Book III of La Géométrie language is especially striking in the case of area. [6], when discussing the roots of cubic and quar- In the Elements there is no measure tic , Descartes considers of the area of a plane figure. Instead, equality of with integer coefficients. If there is an integer root, plane figures (which I will call equal content) is ver- that gives a numerical solution to the problem. ified by cutting in pieces and adding and sub- But if there are no roots, the solutions tracting congruent triangles. Thus the Pythagorean must be constructed geometrically. A quadratic theorem (Book I, 47, “I.47” for short) gives rise to a plane problem whose so- says that the squares on the sides of a right triangle, lution can be constructed with ruler and . taken together, have the same content as the square Cubic and quartic equations are solid problems on the . This is proved as the culmi- that require the of conics for their so- nation of a of demonstrating lution. The root of the equation is a certain line seg- equal content for various figures (for example, tri- ment constructed geometrically, not a number. angles with congruent bases and congruent alti- Halley [10] improves the method of Descartes tudes have the same content). to find roots of equations of up to six, For the theory of similar triangles, a modern using intersections of cubic in the plane. But text will say two triangles are similar if their sides he also shows an interest (following Newton) in are proportional, meaning the of the lengths finding approximate decimal numerical solutions of corresponding sides are equal to a fixed real num- to an equation. He comments that the geometri- ber. Euclid instead uses the theory of proportion, cal method gives an exact theoretical solution but due to Eudoxus, that is developed in Book V of the that for practical purposes one can get a more ac- Elements. Two magnitudes a, b of the same kind are curate solution—“as near the truth as you please”— said to have a ratio a : b. This ratio is not a num- by an arithmetical . ber, nor is it a magnitude. Its main role is explained One hundred years later the acceptance of by the following fifth definition of Book V: Two approximate numerical solutions had progressed ratios a : b and c : d are equal (in which case we say so far that Legendre [14, p. 61], in discussing the that there is a proportion a is to b as c is to d, and theory of proportion, says write a : b :: c : d) if for every choice of whole If A, B, C, D are lines [line segments], numbers m, n, the multiple ma is less than, equal one can imagine that one of these lines, to, or greater than the multiple nb if and only if or a fifth, if one likes, serves as a com- mc is less than, equal to, or greater than nd, mon measure and is taken as unity. respectively. Then A, B, C, D represent each a certain No operations (, multiplica- number of unities, whole or fractional, tion) are defined for these ratios, but they can be commensurable or incommensurable, ordered by size. In Book V a number of rules of op- and the proportion among the lines eration on proportions are proved, using the above A, B, C, D becomes a proportion of definition—for example, one called alternando numbers. (V.16), which says if a : b :: c : d , then also a : c :: b : d. Legendre’s uncritical acceptance of numbers rep- The whole theory of similar triangles is devel- resenting geometrical magnitudes makes his proofs oped in Book VI based on the definition that two easier but at the expense of rigor, for he has not said triangles are similar if their corresponding sides what kind of numbers these are, nor has he proved are proportional in pairs. that they obey the usual rules of arithmetic. Thus Euclid develops his geometry without It was Dedekind [5] who provided a rigorous de- using numbers to measure line segments, angles, finition of the real numbers. He was dissatisfied or areas. His theorems have the same appearance with the appeal to geometric intuition for matters as the ones we learn in high school, yet their - of limits in the calculus and wanted ing is different when we look closely. So the to give a solid theory of continuity based on num- questions arise: How and when did this change of bers. He saw the property of continuity expressed

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in the property of a line: that if one divides its axes in the plane and choosing an in- points into two nonempty subsets A, B, with every terval to serve as unit, one can establish a one-to- point of A lying to the left of every point of B, then one correspondence between the points of the plane there exists exactly one point of the line that marks and ordered pairs of real numbers. This this division. This prompted him to define a real correspondence creates a dictionary between number as a partition of the of rational num- geometry and , so that geometrical problems bers into two nonempty subsets A, B, with a