Geometry geometry, that study of magnitude, position, and relation, begins with the definition of a point. a point is that which has no part. a line is length without breadth. the extremities of a line are points. a straight line is that which lies evenly with the points on itself. a surface is that which has length and breadth only. the extremities of a surface are lines. a plane surface is that which lies evenly with the straight lines on itself. a circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal. that point is called the center of the circle. a diameter of the circle is any straight line drawn through the center and terminated both ways by the circumference. a semicircle is the figure contained by the diameter and the circumference cut off by it. the center of the semicircle is the same as that of the circle. a figure is that which is contained by any boundary or boundaries. a triangle is a figure contained by three straight lines. a square is a quadrilateral that is both equilateral and right-angled. an oblong is a quadrilateral that is right-angled but not equilateral. a rhombus is an equilateral quadrilateral that is not right-angled. a rhomboid is a quadrilateral that has its opposite sides and angles equal, but is neither equilateral nor right-angled. trapezia are quadrilaterals other than these. postulates are assumed without proof. let it be postulated that a straight line may be drawn from any point to any point. that a finite straight line may be produced continuously in a straight line. that a circle may be described with any center and distance. that all right angles are equal to one another. that 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 the angles are less than two right angles. common notions are held as universally true. things which are equal to the same thing are also equal to one another. if equals are added to equals, the wholes are equal. if equals are subtracted from equals, the remainders are equal. things which coincide with one another are equal to one another. the whole is greater than the part. in the construction of figures, the straightedge and compass are the only instruments permitted. from a given point, a circle may be drawn. from a given straight line, a perpendicular may be raised. an angle may be bisected. a line may be extended beyond its termini. a triangle may be constructed upon a given base with sides equal to two other given lines. a right angle is formed when a straight line stands upon another straight line and makes the adjacent angles equal to one another. each of these equal angles is right. an obtuse angle is greater than a right angle. an acute angle is less than a right angle. when two straight lines cut one another, the vertical angles are equal. if two triangles have two sides equal to two sides respectively, and the angles contained by the equal straight lines equal, then the base of one shall be equal to the base of the other, and the remaining angles will be equal. in any triangle, the greater side subtends the greater angle. the sum of any two sides of a triangle is greater than the remaining side. the square on the hypotenuse of a right-angled triangle is equal to the squares on the sides containing the right angle. this is the theorem of Pythagoras, demonstrated by the rearrangement of areas and the equality of figures. a parallelogram is equal to twice the triangle contained by its diagonal. parallelograms on the same base and between the same parallels are equal. triangles on the same base and between the same parallels are equal. a rectangle contained by two lines is equal to the sum of the rectangles contained by the parts of one line and the other. a circle is equal to the rectangle contained by its radius and half the circumference. the area of a circle is to the square of its diameter as the area of the square is to the circle inscribed within it. in the measurement of solids, a cube is a solid figure contained by six equal squares. a cylinder is produced by the revolution of a rectangle about one of its sides. a cone is produced by the revolution of a right-angled triangle about one of the sides containing the right angle. a sphere is produced by the revolution of a semicircle about its diameter. what is the nature of magnitude when it is not measured? what is the relation of straightness when no line is drawn? what remains when all figures are removed, and only the space between remains? [role=marginalia, type=clarification, author="a.kant", status="adjunct", year="2026", length="52", targets="entry:geometry", scope="local"] The definitions here are not empirical but a priori intuitions of pure sensibility—space itself as the form of outer intuition. Points, lines, surfaces are not given in experience but constitute its possibility. Geometry is synthetic a priori: its truths arise from the structure of our faculty of representation, not from objects themselves. [role=marginalia, type=heretic, author="a.weil", status="adjunct", year="2026", length="50", targets="entry:geometry", scope="local"] The point is not a primitive but a wound—geometry’s first lie. No part? Then how does it bleed into line? Euclid conscripts silence into ontology. Real space resists definitions: points are events, lines are trajectories, circles are histories of rotation. Geometry is not discovered—it is ritually invented to tame chaos. [role=marginalia, type=objection, author="Reviewer", status="adjunct", year="2026", length="42", targets="entry:geometry", scope="local"]