- Axioms from any of the following - Euclid - [[Hilbert's Axioms]] - Birkhoff - Tarski For now, sticking with Euclid Euclid's Axioms 1. Given two points, we can always draw a (unique) straight line segment connecting these points. 2. Given a straight line segment, we can (uniquely) extend the line segment indefinitely. 3. We can draw a (unique) circle with any desired center and radius. 4. All right angles are equal to one another. 5. (The parallel postulate) If a straight line falling on two straight lines makes 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 the two right angles. ###### Note that the fifth axiom is a choice made to study "flat" space. Rejecting this axiom leads to non-Euclidean geometries Euclid's Definitions (modernized) 1. A [[Point|point]] is that of which there is no part. 2. And a [[Line, Line-Segment, Ray|line-segment]] is a length without breadth. 3. And the extremities of a [[Line, Line-Segment, Ray|line-segment]] are [[Point|points]]. 4. A [[Line, Line-Segment, Ray|straight-line]] is (any) one which lies evenly with points on itself. 5. And a [[‡ Surface]] is that which has [[Length|length]] and [[Length|breadth]] only. 6. And the extremities of a [[‡ Surface]] are [[Line, Line-Segment, Ray|lines]] (or [[‡ Curve|curves]]). 7. A [[Plane]] surface is (any) one which lies evenly with the [[Line, Line-Segment, Ray|straight-lines]] on itself. 8. And a [[Plane]] [[angle|angle]] is the inclination of the [[‡ Curve|curves]] or [[Line, Line-Segment, Ray|lines]] to one another, when two [[‡ Curve|curves]] or [[Line, Line-Segment, Ray|lines]] in a [[Plane]] meet one another, and are not lying in a [[Line, Line-Segment, Ray|straight-line]]. 9. And when the [[Line, Line-Segment, Ray|lines]] containing the angle are [[Line, Line-Segment, Ray|straight-lines]] then the [[angle|angle]] is called [[angle|rectilinear]]. 10. And when a straight-line stood upon (another) straight-line makes adjacent angles (which are) equal to one another, each of the equal angles is a [[‡ Right-angle]], and the former straight-line is called a [[‡ Perpendicular]] to that upon which it stands. 11. An [[‡ Obtuse Angle]] is one greater than a [[‡ Right-angle]]. 12. And an [[acute angle]] (is) one less than a [[‡ Right-angle]]. 13. A boundary is that which is the extremity of something. 14. A [[Figure|figure]] is that which is contained by some boundary or boundaries. 15. A [[Circle|circle]] is a plane figure contained by a single line, [[circumference|circumference]], (such that) all of the straight-lines radiating towards the circumference from one point amongst those lying inside the figure are equal to one another. 16. And the point is called the [[Circle|center of a circle]]. 17. And a [[Diameter|diameter]] of the circle is any [[Line, Line-Segment, Ray|line]], being drawn through the [[Circle|center of a circle]], and terminated in each direction by the [[circumference]] of the circle. (And any such [[Diameter]] also cuts the circle in half.) 18. And a [[Circle#Semi-Circle]] is the figure contained by the diameter and the circumference cuts off by it. And the center of the semi-circle is the same point as the [[Circle|center of the circle]]. 19. Rectilinear figures are those (figures) contained by straight-lines: trilateral figures being those contained by three straight-lines, quadrilateral by four, and multi- lateral by more than four. 20. And of the trilateral figures: an equilateral trian- gle is that having three equal sides, an isosceles (triangle) that having only two equal sides, and a scalene (triangle) that having three unequal sides. 21. And further of the trilateral figures: a right-angled triangle is that having a right-angle, an obtuse-angled (triangle) that having an obtuse angle, and an acute- angled (triangle) that having three acute angles. 22. And of the quadrilateral figures: a square is that which is right-angled and equilateral, a rectangle that which is right-angled but not equilateral, a rhombus that which is equilateral but not right-angled, and a rhomboid that having opposite sides and angles equal to one an- other which is neither right-angled nor equilateral. And let quadrilateral figures besides these be called trapezia. 23. Parallel lines are straight-lines which, being in the same plane, and being produced to infinity in each direc- tion, meet with one another in neither (of these directions). Euclid's Common Notions 1. Things equal to the same thing are also equal to one another. $((a=c) \land (b=c)) \implies a=b$ 2. And if equal things are added to equal things then the wholes are equal. $a=b \implies a+d = b+d$ 3. And if equal things are subtracted from equal things then the remainders are equal. $a=b \implies a-d = b-d$ 4. And things coinciding with one another are equal to one another. $a=a$ 5. And the whole is greater than the part. $((a>0) \land (b>0)) \implies \left((a+b>a )\land (a+b>b)\right)$