The King of Infinite Space
Page 11
Vulnerant omnes, ultima necant. Every hour wounds, the last one kills.
IF EUCLIDEAN GEOMETRY, narrowly understood, is exhausted, and if the way of life that Euclid offered is itself open to revision—the first step inevitably to rejection—wherein the peculiar and powerful claim that Euclid makes? Western art has always been a servant in a great Euclidean house. In an interview with Émile Bernard, Paul Cézanne remarked that there is “an invisible scaffolding of spheres, cones and cylinders in nature.” Critics have been less successful in observing as much in the paintings of Claude Monet or J. M. W. Turner. Turner’s early works and especially his drawings reflect an architectural appreciation of Euclidean forms. His mature paintings do not. About the great oils, Kenneth Clark says what is obvious: “They have no ready assonance in geometry.” Looking at one and the same thing—nature, after all—each man came away with markedly different ideas about that invisible scaffolding. No very good sense can be given to the idea that the elements of Euclidean geometry may be found in nature because either everything is found in nature or nothing is. Euclidean geometry is a theory, and the elements of a theory may be interpreted only in terms demanded by the theory itself. Euclid’s axioms are satisfied in the Euclidean plane.
Nature has nothing to do with it.
EUCLID HAS ACHIEVED a permanent hold on the human imagination for reasons that go beyond his manner, his method, the details of his proofs, or even the many ideas he has offered the mathematical community. Beyond any other book, it is the Elements that has offered an uncompromising appreciation of the world of shapes—one that it created. The Elements is an exaltation of geometry. Euclid made a conscientious but unsuccessful effort to incorporate into his thoughts the numbers and their properties, but it is to geometry that his heart owes its allegiance. Because this is so, he was able to offer mathematicians what mathematicians so rarely offer, and that is a vision.
The vision offered, mathematicians could ask the question that only a vision could make possible: what form of unity lies beneath the numbers and the shapes—le coeur dans le coeur, the deepest structure, the heart of the profound identity between shapes and numbers.
They reckon ill who leave me out
When me they fly, I am the wings
I am the doubter and the doubt,
And I the hymn the Brahmin sings.3
Of this form of unity, we know more than Euclid could have known. The quest for unity will continue, and, of course, it will always fail. And this, too, we know. Whatever the form of unity mathematicians acquire, the world’s diversity will in time overwhelm them, as it overwhelms us all.
Euclid remains what sensitive men and women always thought he was, a great partisan, an unequivocal voice, a part of the drama in which opposites are forever resolved and then as often dissolved.
I am writing about Euclid of Alexandria; the Euclid of the Elements; the Euclid of geometry, dust boards and diagrams, procedures and proofs, points and planes.
I am writing of Euclid the Great.
1. René Thom, Semio Physics (Reading, MA: Addison-Wesley, 1990), 32.
2. “Mathematics: The Loss of Certainty,” ScienceBlog, 2005, review of Brian Davies, “Whither Mathematics?” Notices of the American Mathematical Society 52 (December 2005): 1350–1356.
3. From Ralph Waldo Emerson, “Brahma.”
Teacher’s Note
In his twenty-ninth proposition, Euclid says . . .
To write about Euclid is to imagine oneself linked as a companion in art to men and women long dead but still shuffling toward the Euclidean blackboard. The low murmur of their vows may be heard in Greek, Latin, or Arabic; it may be heard in all of the languages in which books are made and memories preserved.
Everyone teaching from or writing about Euclid’s Elements does so from his own perspective, of course, but Euclidean plane geometry is not a subject that encourages pedagogical innovation. The elements are always the same: Euclid’s common notions, his definitions, his axioms, and then his theorems and their proofs. There is a sense, at times subdued and at times ebullient, that this very old system merits a form of devotion, teachers and their students participating in a ritual whose full meaning is not easily grasped and never grasped at once.
The Euclidean tradition stretches from the ancient world to our own, but its value is not in the end the propositions that it makes possible. These we know and we have known them for a long time. “What can be shown,” Wittgenstein remarked in the Tractatus, “cannot be said” (Was gezeigt werden kann, kann nichts gesagt werden).
As much is true of Euclid’s Elements.
The book demands both effort and concentration. The proofs do not come easy. A way of life is requested, and if it is not forthcoming, it is demanded. It is a way of life and a form of dedication that has a striking moral value. It is noble. This the Elements does not say, but everyone coming to the book understands that this is what it shows.
The Euclidean academy is remarkably stable. It has lasted a long time. Teachers and writers and their students enter the academy and are then lost on the sands of time. It does not matter. The academy confers a form of immortality on its academicians. It is the immortality that arises from having participated in one of the arts of civilization. This is the only form of immortality that any of us can share.
Teachers and writers alike hope that having been taught, they will be able to teach others in turn. It is a hope.
But no one writing about Euclid is entitled to end a book in doubt.
Now in his thirtieth proposition, Euclid says . . .
A Note on Sources
All references to Euclid are from Euclid, The Elements: Books I–XII, complete and unabridged, edited and translated by Thomas L. Heath (New York: Barnes & Noble, 2006). The original edition of Heath’s text was published in 1906; his textual comments, although sometimes valuable, are, of course, out of date.
Appendix
Euclid’s Definitions
Definitions in boldface represent the core of Euclid’s scheme, the load-bearing structures.
1. A point is that which has no part.
2. A line is length without breadth.
3. The extremities of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.
6. The extremities of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.
8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
9. And when the lines containing the angle are straight, the angle is called rectilineal.
10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
11. An obtuse angle is an angle greater than a right angle.
12. An acute angle is an angle less than a right angle.
13. A boundary is that which is an extremity of anything.
14. A figure is that which is contained by any boundary or boundaries.
15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;
16. And the point is called the center of the circle.
17. A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.
19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained
by four, and multilateral those contained by more than four straight lines.
20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.
21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.
22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
Index
Addition, 21, 24, 103, 104, 110, 112, 142, 143
Alexandria, library in, 5
Algebra, 7, 70–71, 91, 142
geometrical algebra, 95, 96, 98, 111 (see also Geometry: analytic geometry)
straight line defined by equation, 98, 112–113
triumph of, 103, 105, 106
Analytical mechanics, 144
Angles, 41, 88, 119
acute, 160, 161
base angles of an isosceles triangle, 58
and Beltrami pseudosphere, 133
of curvilinear triangle, 139
as equal, 26, 50–51, 52, 58, 64, 67, 68, 73–74, 81, 89–90, 159
obtuse, 160, 161
rectilineal, 159
right angles, 7, 50–51, 135, 160, 161 (see also Pythagorean theorem)
in spherical geometry, 125
trisecting, 47–48, 147
Apollonius, 6
Appel, Kenneth, 151
Approximations, 4
Arabic archipelago, 70
Arab renaissance, 120
Archimedes, 6, 9
Architecture, 1, 4, 7, 64, 79, 128, 133
Arguments, 15–16, 17, 20
Aristippus, 1–2
Aristotle, 15, 16, 17, 23, 31, 45
Arithmetic, 69, 80, 92, 112
and completeness axiom, 109
consistency/inconsistency of, 107
fundamental theorem of, 100–101
geometrical axioms as arithmetical theorems, 114
as incomplete, 150
triumph of, 103
See also Geometry: unity of geometry and arithmetic; Mathematics; Numbers
Art, 7, 79, 80, 133, 152, 155. See also Paintings
Associative laws, 105
Associative operation, 142
Assumptions, 11, 12, 27, 45, 46, 55, 83, 119
of existence of points/lines/planes, 49, 107
hidden, 30–31
that the parallel postulate is false, 120, 121
Atiyah, Michael, 91
Atoms, 41–42, 43, 44
Axiomatic systems, 11, 12, 14, 149
and arguments, 17
new, 107
as way of life, 9, 106, 148, 152, 156
Axioms, 45–56, 80, 90, 152
Archimedean axiom, 108–109
Cantor-Dedekind axiom, 100
completeness axiom, 108, 109
of connection, order, congruence and continuity, 107
consistency/inconsistency of, 106–107, 131
for fields, 104–105, 112
fifth axiom, 53–56, 118 (see also Axioms: Playfair’s axiom; Parallel postulate)
first axiom, 113–114
first three axioms, 46, 48–49, 51, 61, 66, 86
fourth axiom, 50–51, 73
fourth proposition as axiom, 27
Hilbert’s axioms, 109, 111
interpreted in arithmetic, 113–114
made theorems, 46
of neutral geometry, 131
Playfair’s axiom, 54, 55, 91, 137, 138
relationship between axioms and theorems, 12, 14, 19, 149
as self-evident, 46
See also Axiomatic systems
Babylonians, 8, 69
Bacon, Francis, 77
Beltrami, Eugenio, 121, 132–133
Bolyai, János, 118, 122–123, 126, 127–128
Bolyai, Farkas, 127–128
Boole, George, 23
Boundaries, 160
Breadth, 33, 35, 36, 159
Bridge of Asses, 64, 65(fig.). See also Propositions: fifth proposition
Calculus, 39, 94
differential calculus, 41, 59
of segments, 110
Cantor, Georg, 93
Cantor-Dedekind axiom, 100, 101, 109
Cardioids, 99, 99(fig.)
Cathedrals, 64
Causality, 13
Cézanne, Paul, 152
Change, 43, 44, 52
Chesterton, G. K., 11
China, 9
Cicero, 1
Circles, 7, 13, 25, 46, 79
center/circumference of, 98, 130, 135, 136, 160
diameter of, 160
and geodesics, 125
and proposition one, 61–62
radii of, 49, 62, 98, 130
semicircles, 160
See also Poincaré, Henri: Poincaré disk
Clark, Kenneth, 78, 152
Clay tablets, 8
Coincidence, 21, 23, 25–26, 27, 39, 41, 67
and concrete vs. abstract models of geometry, 28–29
Common beliefs/notions, 19–32, 90
fifth, 29–30
first, 24
fourth, 23
second/third, 24, 62, 74
Common sense, 36, 64, 91, 118, 124, 130, 139
Commutative laws, 105
Compass, 63. See also Straight-edge and compass
Complexity, 55, 107
Computers, 150
Congruence, 26, 39, 67, 73, 74, 75, 107, 130
Consistency/inconsistency, 106–107, 131
Contradictions, 17, 83, 87, 89, 100, 120, 121, 131. See also Reductio ad absurdum
Contrapositives, 83, 84(n), 86, 86(fig.)
Converse relationship, 69, 81(n), 82, 83
Coordinate Method, The (Gelfand, Glagoleva, and Kirillov), 99–100
Coordinate systems, 97, 97(fig.), 115
Critique of Pure Reason, The (Kant), 117
Cultures, 3, 4, 9
Curvature, 38, 99, 125, 139
extrinsic, 40, 41
negative, 133
and straight lines, 39
Das Kontinuum (Weyl), 44
Davies, Brian, 150, 151
De Architectura (Vitruvius Pollio), 1–2
Dedekind, Richard, 102. See also Cantor-Dedekind axiom
Deduction, 45, 149
Definitions, 20, 33–44, 51, 90, 159–161
eighth and ninth, 51–53
fifteenth, sixteenth, and seventeenth, 62
fifth, 35
first seven and twenty-third, 33–34
fourth, 38
of hyperbolic lines/distance, 134–135, 136, 139
nineteenth, 60, 84, 85–86
ninth through twenty-second, 34
and real ordered fields, 113
of rectilinear figures, 60
seventh, 38
of shape, 49
tenth, 73
third, 43
twentieth, 60
twenty-third, 37–38, 43, 84
Degrees of freedom, 37
Democritus, 41, 42, 44
De Morgan, August, 84(n)
Descartes, René, 45, 96
Dieudonné, Jean, 115
Dimensions, 35, 36, 37, 40, 70, 125, 141, 144
Distance, 23, 37, 39, 41, 56, 69–70, 87, 88, 125, 132, 144
hyperbolic distance, 135–137, 139
Distributive laws, 105
Division, 93, 95, 103, 104, 110, 112
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Egyptians, 11
Einstein, Albert, 118
Elementary Geometry from an Advanced Standpoint (Moise), 94
Elements (Euclid), 9, 44, 80, 90, 91, 123, 153
Book I, 6, 119
Book II, 6
Book V, 7, 93–94, 110
Books V through IX, 7
Book VII, 93, 110
Book X, 93–94, 100, 110
books in, 6–7
first four books, 7
as having limited symbolic reach, 71
as illustrated, 59, 64–65, 79–80, 87, 90
and mountain-climbing pastoral, 57–58
as textbook, 5–6, 7, 155–156
Eliot, George, 45
Ellipses, 13, 98–99
Empson, William, 57–58
Encyclopedia Britannica, 28
Equality, 21, 22–25, 26, 36, 62, 63, 148
definition of, 25
“less than or equal to,” 105
of right angles, 50–51
of squares, 75
transitivity of, 24
See also Angles: as equal
Equator, 125
Erlangen program, 140, 142
Ethics, 123
Euclid, 21–22, 43, 89, 140, 145, 152–153
and Aristotle, 15, 17
birth/death of, 5
double insight of, 12
Euclidean ideal, 150
Euclidean style, 148–149
Euclidean tradition, 155–156
and fifth axiom (parallel postulate), 54–55, 118–119, 139–140
as a mathematician, 6
modern versions of, 8
predecessors of, 6
as a teacher, 5–6, 26–27, 79
translations of, 8
and unity beneath diversity of experience, 11
Euclides ab omni naevo vindicatus (Saccheri), 121
Eudoxus, 6, 94, 108