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mathhoang
Corollary. If two straight lines cut one another, then they will make the angles at the point of section equal to four right angles.
It has been suggested that the definitions were added to the Elements sometime after Euclid wrote them. Another possibility is that they are actually from a different work, perhaps older. In Def.I.22 special kinds of quadrilaterals are defined including square, oblong (a rectangle that are not squares), rhombus (equilateral but not a square), and rhomboid (parallelogram but not a rhombus). Except for squares, these other shapes are not mentioned in the Elements. Euclid does use parallelograms, but they're not defined in this definition. Also, the exclusive nature of some of these terms—the part that indicates not a square—is contrary to Euclid's practice of accepting squares and rectangles as kinds of parallelograms.
In proposition III.16 (but nowhere else) angles with curved sides are compared with rectilinear angles which shows that rectilinear angles are to be considered as a special kind of plane angle. That agrees with Euclid's definition of them in I.Def.9 and I.Def.8.
Also in Book III, parts of circumferences of circles, that is, arcs, appear as magnitudes. Only arcs of equal circles can be compared or added, so arcs of equal circles comprise a kind of magnitude, while arcs of unequal circles are magnitudes of different kinds. These kinds are all different from straight lines. Whereas areas of figures are comparable, different kinds of curves are not.
Book V includes the general theory of ratios. No particular kind of magnitude is specified in that book. It may come as a surprise that ratios do not themselves form a kind of magnitude since they can be compared, but they cannot be added. See the guide on Book V for more information.
Number theory is treated in Books VII through IX. It could be considered that numbers form a kind of magnitude as pointed out by Aristotle.
Beginning in Book XI, solids are considered, and they form the last kind of magnitude discussed in the Elements.
Some of the propositions are constructions. A construction depends, ultimately, on the constructive postulates about drawing lines and circles. The first part of a proof for a constuctive proposition is how to perform the construction. The rest of the proof (usually the longer part), shows that the proposed construction actually satisfies the goal of the proposition. In the list of propositions in each book, the constructions are displayed in red.
Most of the propositions, however, are not constructions. Their statements say that under certain conditions, certain other conditions logically follow. For example, Prop.I.5 says that if a triangle has the property that two of its sides are equal, then it follows that the angles opposite these sides (called the "base angles") are also equal. Even the propositions that are not constructions may have constructions included in their proofs since auxillary lines or circles may be needed in the explanation. But the bulk of the proof is, as for the constructive propositions, a sequence of statements that are logically justified and which culminates in the statement of the proposition.
Dependencies within Book I | ||
1 | 2 | 3 |
3, 4 | 5, 6 | |
5 | 7 | 8 |
1, 3, 8 | 9, 11 | |
1, 4, 9 | 10 | |
8, 10 | 12 | |
11 | 13 | 14, 15 |
3, 4, 10, 15 | 16 | 27 |
13, 16 | 17 | |
3, 5, 16 | 18 | |
5, 18 | 19 | |
3, 5, 19 | 20 | |
16, 20 | 21 | |
3, 20 | 22 | |
8, 22 | 23 | |
3, 4, 5, 19, 23 | 24 | |
4, 24 | 25 | |
3, 4, 16 | 26 | |
13, 15, 27 | 28, 29 | |
29 | 30 | |
23, 27 | 31 | |
13, 29, 31 | 32 | |
4, 27, 29 | 33 | |
4, 26, 29 | 34 | 43 |
4, 29, 34 | 35 | |
33, 34, 35 | 36 | |
31, 34, 35 | 37 | |
31, 34, 36 | 38 | |
31, 37 | 39 | |
31, 38 | 40 | |
34, 37 | 41 | |
10, 23, 31, 38, 41 | 42 | |
15, 29, 31, 42, 43 | 44 | |
14, 29, 30, 33, 34, 42, 44 | 45 | |
3, 11, 29, 31, 34 | 46 | |
4, 14, 31, 41, 46 | 47 | |
3, 8, 11, 47 | 48 |
Next book: Book II |
|
© 1996
D.E.Joyce
Clark University
from : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html
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