vol 4: Glossary
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Euclid's elements
Definitions
1. a point is that which has no part.
2. a line is a breadthless length.
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.
8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie on 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 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 centre of the circle.
17. A diameter of the circle is any straight line drawn through the centre 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 centre 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 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-angle triangle that which has an obtuse angle and an acute-angled triangle that which has its three angles acute.
22. Of the 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 that 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.
Postulates
Let the following be postulated:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance.
4. That all right angles are equal to one another.
5. That if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.
Common notions
1. Things that are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be substracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part. Heath, I, 153-155.
David Hilbert has produced a more modern set of axioms for geometry. Hilbert.
Books
| Heath, Thomas L, Thirteen Books of Euclid's Elements (volume 1, I-II), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.' Amazon back |
| Hilbert, David, Foundations of Geometry (Grundlagen der Geometrie), Open Court 1999 Jacket: 'Along with the writings of Hilbert's friend and correspondent Frege, Hilbert's Grundlagen der Geometrie is the major prop that set the stage for Russell and Whitehead's Principa Mathematica. Hilbert presents a new axiomatization of geometry, the reduction of geometry to algebra, and introduces the distinction between mathematics and metamathematics, with a new theory of proof. This edition is translated from the tenth German edition, including all the improvements which Hilbert derived from his own reflections and the contributions of other writers. Amazon back |
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