课程与习作 | Linda Myers：欧几里得在博雅——“欧几里得与罗巴切夫斯基”课程设计，及优秀课程作业
几何原本
按语：
本次为大家介绍的是St. John's College的Lynda Myers老师于2015学年春季学期在博雅学院开设的《欧几里得与罗巴切夫斯基》的课程情况。课程采用St. John's College传统的seminar形式，即老师在授课过程中主要负责引导学生讨论，而同学们则以课前细读文本、课上分组进行课堂展示、小班讨论的方式进行学习。
本篇推送包括两方面的内容：首先是课程简介，含课程大纲、阅读书目以及课程要求；第二部分将展示潘昌昀、盛汀苑、王祁涛和吴狄四位2015级本科生的习作。
课程感言
Myers老师以欧几里得的《几何原本》为出发点，却不以几何学为唯一讲授内容，而是结合近现代非欧几何中的一些理论，带领同学们深入认识了其中精细的内在逻辑，体验了经典中非同一般的智慧。其在领域上扩展的程度之广，体现了博雅教育的真正精髓，让我们收获无穷。
课程采用小班讨论课（seminar）的授课形式。讨论课便于同学们积极参与到课程当中，而严格控制的班级人数让老师能够照顾到每一个学生的情况。
Myers老师平易近人，讲课耐心细致，批改论文一丝不苟、认真负责，同学们在课程中收获良多。
——2015级本科 潘昌昀
结课合影
课程大纲
Euclid/Lobachevski Course Syllabus
Dr. Lynda L. Myers
Reading List:
Euclid, The Elements, Book I (Thomas Heath translation). Recommended edition: Green Lion Press.
Proclus, Selections from Commentary on the First Book of Euclid’s Elements, Glenn Morrow translation. To be handed out.
Girolamo Sacchieri, Selections from Euclides Vindicatus (Euclid Freed of Every Fleck), George Halstead translation. To be handed out.
Nicholas Lobachevski, Geometrical Researches on the Theory of Parallels, George Halstead translation. To be handed out.
Stephen F. Barker, “Geometry,” from The Encyclopedia of Philosophy, Paul Edwards, Editor in Chief (New York: 1967), Vol. III, pp. 285-290. To be handed out.
1482年出版印刷的《几何原本》书影
Schedule:
Week 1
Session
A. Euclid, Elements, Book 1, Definitions 1-22 (overview)
B. Def. 1-7
C. Postulates 1-3; Common Notions; Proposition 1 (introduction)
Week 2
Session
A. Propositions 1-3; Def. 15
B. Structure of a Euclidean proposition; problems v. theorems
C. Proposition 4; Common Notion 4
Week 3
Session
A. Prop. 5-6; logic of immediate inference; reductio ad absurdum
B. Prop. 7-8
C. Prop. 9-12 (First essay due)
Week 4
Session
A. Prop. 13-15; Postulate 4
B. Prop. 16-17
C. Postulate 5
Week 5
Session
A. Prop. 18-21
B. Prop. 22-23; (read 24-25); Prop. 26
C. Review theory of congruence
Week 6
Session
A. Prop. 27-28; Definition 23
B. Prop. 29; Postulate 5
C. Prop. 30 (cf. Playfair’s axiom); Prop. 31; Prop. 32 (cf. Props. 16-17)
Week 7
Session
A. Prop. 33-35 (Read 36-41)
B. Prop. 42-43
C. Prop. 44-45
埃及出土的《几何原本》残页
Week 8
Session
A. Prop. 46-8
B. Overview of Euclid, Elements, Book I
C. Overview continued (Second essay due)
Week 9
Session
A. Proclus, Commentary on the First Book of Euclid’s Elements, paragraph 175-177; 191-193
B. Ibid., para. 361-368
C. Ibid., para. 369-373.
Week 10
Session
A. Saccheri, Euclid Freed of Every Fleck, Preface, Book I, Propositions 1-2
B. Ibid., Prop. 3
C. Ibid., Corollaries 1-3
Week 11
Session
A. Lobachevski, Theory of Parallels, Author’s preface, Theorems 1-10
B. Idem
C. Idem
Week 12
Session
A. Theorem 16
B. Idem
C. Th. 17
Week 13
Session
A. Th. 18; Euclid X.1, Lob. Th. 19
B. Th. 20
C. Idem (Third essay due)
Week 14
Session
A. Th. 21-22
B. Th. 23
C. Idem
Week 15
Session
A. Th. 24
B. Original proofs
C. Idem
Week 16
Session
A. “Geometry”
B. Idem
C. Idem; Final essay due
Week 17
Session
A. Review and Conclusion
B. Idem
C. Meetings with individual students to discuss final essay
Week 18
Session
A. Meetings with individual students to discuss final essay
B. Continued
C. Continued
1607年，利玛窦和徐光启共同翻译出版了中文版《几何原本》。
Evaluation of Students’ Work:
This class will be a discussion (seminar-style) class, not a lecture class. Therefore regular attendance, preparation and class participation are essential. When called upon, students will be expected to present at the blackboard demonstrations of the propositions that they have studied. These demonstrations will be the basis for our conversations about important details of our authors’ mathematical arguments. In our conversations, students will be expected to raise questions about the text and to be willing to suggest and explore possible answers to those questions. Our more general, philosophical discussions will arise out of these explorations. Thus, the evaluation of students’ work will be based on several factors: regular class attendance, ability to demonstrate propositions at the board carefully and accurately, willingness to raise and explore thoughtful questions about the details of the texts, and willingness to speculate about more general philosophical issues raised by the texts. Students will be required to write four short (2-3 page) papers on topics assigned by Mrs. Myers. There will be no quizzes or final exam.
《几何原本》中的证明
优秀作业
Final Essay
Pan Changyun
From the preface of Lobachevski’s Theory of Parallels we know that the aim of Lobachevski’s writing this book was mostly to revise the imperfections and to bridge the so-called ‘gaps’ in Euclid’s Elements, and I think he did so. We can see the strong attempt to do that merely from the first few lines of his Theorems.
Theorem 1 in the Theory of Parallels, compared to Definition 4 in Euclid’s book, focused its attention on another aspect of characteristic of a straight line. While Euclid described how every part of a straight line positioned itself to make up a straight line, Lobachevski explained what feature a straight line maintains in all of its possible positions. If Euclid somehow provided a ‘microscope’ to take the most detailed picture of a straight line, Lobachevski simply showed us a macroscopic view of what it is capable of. This way, the Russian went one step further in exploring the features of the basic elements in the world of geometry, which explains how he bridged the gaps.
However, merely from the Theorems we have read and discussed during the last few classes, Theorem 1 in particular does not quite show its practical use in solving geometrical problems that have been hard to deal with in Euclid’s book. I am still confused about why the author specially highlighted the unmoving of the line during the revolution of surface. Perhaps it will be playing its role in the 3-dimensioned problems.
Theorem 2 and 3, on the other hand, was another story to tell. Not only can we see the adjustment and revision Lobachevski made to Euclid’s definitions from those, we are also able to find how mighty the Theorems can be in real, practical problems. The features of straight lines mentioned in the Theorems seem to be obvious, but once we are faced with problems that need them to be solved, it turns to be not that obvious to see. In short, what mentioned in the theorems was beyond our common notion, but far from needing to be proved. They are right what Lobachevski calls ‘gaps’ in Euclid’s book. Take Theorem 17 as an example of their practical use. When the triangle has been constructed, the author introduced Theorem 2 and 3 in succession smoothly, in order that the straight line goes into a triangle through one of its three sides, and gets out of it through another. There is no doubt that a straight line will always go like this, but take Theorem 16 into consideration, what seems absurd may still be logically true. So the guarantee given by the Theorems is necessary and does a lot of help to the construction of the system of the non-Euclidian geometry.
In a nutshell, different aspects of the same item can lead us to quite different world. Though being called non-Euclidian geometry, Lobachevski’s geometry is not a break and a re-building to Euclid, but more like a complement to make the system more complete.
Postulate V
Sheng Tingyuan
My version of postulate V is: That, if a straight line falling on two infinite straight lines make the interior angles on the same side unequal to two right angles, the three lines will contain a triangle on the side which sustains the angles less than the two right angles. Compared to Euclid’s version, mine, firstly, indicates the infinity of lines at the very beginning. (I wondered why Euclid didn’t do so, but added “if produced indefinitely” afterwards.) Secondly, I bring up the formation of the triangle to replace Euclid’s idea of the meeting of the two lines, namely replace the idea of a 1D meeting by the idea of a 2D shape. Does it help us to gain a better understanding of postulate V? I will answer the question at the end of my essay, along with the conclusion of my inquiry into postulate V.
Before I move on to make some general observations about Euclid’s use of postulate 5, I’d like to raise my questions first. It is because the answer of the one I’m most interested in, the second one, lies in the observation that follows:
Why did Euclid wait until 1.32 to prove the equality between the exterior angle and the interior and opposite angles when he could prove it using only postulate V and the proposition before 1.16?
Since postulate V is a crucial part of parallel constructions and proofs, why does it appear directly only in 1.29 and 1.44? (According to my observation, “fundamental significance”, “little direct appearance”, “much indirect appearance” are the universal features of the five postulations. Common notions, on the other hand, only appears directly.)
Can we still prove or construct the following propositions when we replace postulate V with Playfair’s axiom?
According to my middle school teachers, any quadrilateral figure with equal and parallel extremities is a parallelogram. Why didn’t Euclid state so in 1.33? He was getting really close to that conclusion!
In order to get a better idea of postulate V, I’d like to begin with my observation of 1.27, where we first approach the problem of parallels (without using postulate V, though):
1.27, 1.28 offers us the three main features of parallels, with which we can easily recognize most of the parallels. 1.29 is the converse of 1.27 and 1.28, and it enables us to actually USE the property of parallels to solve other problems. 1.30 expands the parallel relationship between two lines to more than two lines, and since parallels are crucial components of parallelograms, we are therefore able to build relationships (or positional relationships, to be specific) between parallelograms (see in 1.45). Then comes 1.31, which enables us to CONSTRUCT parallels. By far, we have obtained the three fundamental set of tools to solve parallel problems: 1.27-28 (to recognize), 1.29 (to use), and 1.31 (to build). 1.32, a left-over of 1.16, seems isolate, for it is never used by other proposition in book 1.
After his thorough discussion of parallels, Euclid moved on to parallelograms——figures defined only by parallels. 1.33-34 showed the property of parallelograms, and 1.29 appears in each of them. It is then explicit that the properties of parallelograms are closely related to those of parallels. 1.35-42 is an elegant set of propositions, building spatial relationships between parallelograms and triangles in the same parallels as well as giving us new insight into the meaning of “equal”. However, instead of appearing in each of the proposition, 1.29 only appeared in the first one of the whole set of propositions, which is 1.35. Is it still of the same significance? Yes! Though there isn’t any logical sequence of appearance in 1.33 and 1.34, a very strict one can be seen in 1.35-42, which indicates that if 1.29 appears in 1.35, the fundamental one, it will be as important to 1.36-42 as it is to 1.33-35.
Based on the proved triangle-parallelogram relationship, 1.42-45 take a further step. By bringing up the very frequently used figure in 1.43, they enable us to compare the space of multilateral figures by applying them to rectangles with equal bases. The other crucial function of 1.29 first appears directly in 1.44, although the idea of sending off two parallels that meet at one point, thus creating a parallelogram on the basis of a given rectilineal angle instead of making postulate I constructions or crossing parallels has been formerly indicated in 1.33. 1.46-48 are brilliant insight into the relationship between lines and spaces, and the elegancy of the use of triangle-parallelogram reaches its peak in 1.47. Since it is based on 1.41, 1.41 on 1.34 and 1.34 on 1.29, postulate V also has an important yet indirect influence on it.
Now that I’ve observed and discussed the use of 1.29 in Book 1, I’ve gained some understanding of it. To get to know the function of postulate V, one should examine its function in 1.29 and 1.44. For the former, I think that postulate V is the key that opens the boundless possibilities of parallels. Without postulate V, we would stop at knowing what parallels are and how to construct them, but never are we able to solve problems, such as converting lengths, spaces and positions; build relationship between sides of triangles or parallelograms and triangles. For the latter, I think it is postulate V that help us gain much more control of parallelograms——not only are we able to construct them in any angle and base, we are also able to make comparison of their spaces, and make them into mediators.
Last but not least, let’s return to the first question raised in the beginning. I think that Euclid’s expression is better, for it gives us a point instead of a figure, and while a figure is yet to be explored, a point is much more basic and handier to use. Moreover, the idea of “meeting” instead of “forming” a figure correspond with the phrase “do not meet” in the definition of parallels, thus it can be regarded as a complement of definition 23.
The Exegesis of Postulate 1, Theorem 1 and Theorem 2
Wang Qitao
I’d like to discuss the relationship between postulate 1, theorem 1 and theorem 2.
The vivid distinction between postulate 1 and theorem 2 is that the former focuses on ‘two points’ and ‘line’ which the points construct while the latter concentrates on ‘two lines’ and ‘common point’ which they have.
To accomplish my essay, I’d like to translate postulate 1 and theorem 2 into other words:
Postulate 1: Two points must determine a straight line. Two points can determine only one straight line.
The inverse-proposition of the latter is ‘two points can determine more than one straight line’. That is to say: two points can have two straight lines, which is one of the conditions in the inverse-proposition.
It is contradictory to theorem 2 which says two straight lines can’t intersect in two points. It is lucky that if postulate 1 is true, that sad event will not happen.
Therefore, if postulate 1 is true, theorem 2 will be true. Postulate 1 is a prerequisite of theorem 2.
Now, let’s translate theorem 2.Theorem 2: Two straight lines can intersect in one point or not meet with each other.
The inverse-proposition of it is that ‘two straight lines can intersect in more than one point.’ That is to say: two straight lines can have two common points.
It is contradictory to the latter of postulate 1 ‘two points can only determine one straight line’.
In the same way, if theorem 2 is true, the latter of postulate 1 will be safe.
In conclusion, theorem 2 is equal to the latter of postulate 1.
Then, let’s throw a light on postulate 1 and theorem 1.Theorem 1 is very long. It presents a property of the straight line. The property can be used to check whether a given line is straight. Lobachevski revolves the surface, observing the position of the line on it. The key word is ‘two unmoving points’. The points lay on both the line and the surface.
In this time, let’s analyze theorem 1.Theorem 1: The first sentence is the core. The second one is to explain it. Prerequisites: Two unmoving points lay on the straight line which belongs to the surface.Operation: Revolve the surface.
Conclusion: The unmoving line.
The only thing that doesn’t change its position during a revolution is the axis of rotation. It is clear that two unmoving points are part of the axis of rotation, so is the unmoving straight line.
Why two unmoving points? Neither one nor three points are needed. I think theorem 1 is based on postulate 1. If only one point is unmoving, only that point will not change its position during the revolution of the surface and no unmoving line will exist. When three or more points are fixed, including two unmoving points, the prerequisites are surplus. How do we know two points are enough? Postulate 1: Two points can determine one straight line.
Can theorem 1 replace postulate 1? No, it can’t. Because the former of postulate 1 can’t be reflected from in 1.
Straight Line in Euclidean and non-Euclidean Geometry
Wu Di
All of the chosen definitions from Euclid’s Elements and theorems from Theory of Parallels are about straight line. The definition of straight line points out the relationship between point and straight line, though it is very vague. The key word of this definition must be ‘evenly’, which seems to be a visualized concept coming from experience in daily life, and it is not strict. Lobachevski tries to perfect it. Unlike Euclid, he seems to recognize its weakness, and tries to clarify it with his second sentence. His theorem is a way to explain ‘evenly’, which means ‘symmetrically’, and the symmetry can be tested by turning the line over, and the line coincides with its original position. In other word, straight lines are the fixed-point-sets of spatial rotations. This theorem is easier to understand than the definition which Euclid has given, but it is still a visualized concept which is not strict enough. What’s more, Lobachevski has never proved that his theorem satisfies the definition of straight line.
What confuses me is that since Theorem 1 is not with perfect proof, and Lobachevski never uses Theorem 1 in the following complicated theorems, so Lobachevski must put it as the first theorem for some important reasons. The necessity of Theorem 1 is that it just shows one nature of straight line. In other word, the theorem has ruled out many situations which may not happen to straight line.
By the way, I have some thoughts about the definition of straight line. If we want to have a specific definition of straight line, we should hold a nature only straight line has, such as ‘between two fixed points, the shortest line is straight line’. However, another question has been raised, which is that how can we describe ‘shortest’? Now, we find one key point which called ‘length’, but in geometry world, we can only describe length based on straight line. Then I fall in a logic cycle.
Postulate 1 in Euclid’s Element and Theorem 2 in Theory of Parallels both focus on a certain theme called ‘has one and only’. This comes from our own understandings, since Euclid never says ‘only’, and Lobachevski does not say both. I want to say that this two are equivalent, because if one of them is true, we can conclude that the other one is true, too. At first, I think this is obvious, but I ignore an order implied in the postulate and theorem. From Postulate 1, we draw straight line based on two points with the support of definition 2, but we guarantee the point based on straight line in Theorem 2, which needs the support of the definition of straight line in Euclidean Geometry. We do not need to focus on ‘evenly’, but the key word ‘point’. This is a good example to show the importance of definitions in proving theorems or propositions.
Lobachevski begins his book with confused theorems, the first and the second one look like axioms, as they do not admit proof on the basis of Euclid’s axioms, but an interpretation of Euclid axioms.
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