Geometry geometry, that science of magnitude and position, begins with the definition of a point, which is that which has no part. A line is breadthless length, and 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, and the extremities of a surface are lines. A plane surface is that which lies evenly with the straight lines on itself. 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. When the lines containing the angle are straight, the angle is called rectilineal. A right angle is formed when a straight line standing 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 it. An obtuse angle is greater than a right angle, and an acute angle is less than a right angle. A boundary is that which is an extremity of anything, and a figure is that which is contained by any boundary or boundaries. 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; and the point is called the centre of the circle. 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 line bisects the circle. 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. Rectilineal figures are those which are contained by straight lines: trilateral figures being those contained by three, quadrilateral by four, and multilateral by more than four straight lines. 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. Again, 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. 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. 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 are assumed without proof: let the following be postulated: to draw a straight line from any point to any point; to produce a finite straight line continuously in a straight line; to describe a circle with any centre and distance; that all right angles are equal to one another; and 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 are the angles less than the two right angles. Common notions are taken as universally understood: things which are equal to the same thing are also equal to one another; if equals be added to equals, the wholes are equal; if equals be subtracted from equals, the remainders are equal; things which coincide with one another are equal to one another; and the whole is greater than the part. The construction of figures proceeds by methodical application of these definitions, postulates, and common notions. To construct an equilateral triangle on a given finite straight line, let AB be the given straight line. With centre A and distance AB, describe the circle BCD, and with centre B and distance BA, describe the circle ACE. From the point C, at which the circles cut one another, draw the straight lines CA and CB to the points A and B. Since the point A is the centre of the circle CDB, AC is equal to AB. Again, since the point B is the centre of the circle CAE, BC is equal to BA. But CA was proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are equal to one another; therefore CA is equal to CB. Thus the three straight lines CA, AB, BC are equal to one another. Therefore the triangle ABC is equilateral, and it has been constructed on the given straight line AB. To bisect a given rectilineal angle, let the angle BAC be given. Let a point D be taken at random on AB, and from AC let AE be cut off equal to AD. Join DE, and on it construct the equilateral triangle DEF. Join AF. Since AD is equal to AE, and AF is common, the two sides DA, AF are equal to the two sides EA, AF respectively. And the base DF is equal to the base EF. Therefore the angle DAF is equal to the angle EAF. Thus the given rectilineal angle BAC has been bisected by the straight line AF. To bisect a given finite straight line, let AB be the given straight line. Construct on it the equilateral triangle ABC, and bisect the angle ACB by the straight line CD. Since AC is equal to CB, and CD is common, the two sides AC, CD are equal to the two sides BC, CD respectively, and the angle ACD is equal to the angle BCD. Therefore the base AD is equal to the base DB. Thus the given finite straight line AB has been bisected at D. To draw a straight line at right angles to a given straight line from a given point on it, let AB be the given straight line, and C the given point on it. Take a point D on AC, and make CE equal to CD. Construct on DE the equilateral triangle FDE, and join FC. Since DC is equal to CE, and CF is common, the two sides DC, CF are equal to the two sides EC, CF respectively, and the base DF is equal to the base EF. Therefore the angle DCF is equal to the angle ECF. And these are adjacent angles. But when a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right. Therefore each of the angles DCF, FCE is right. Thus the straight line CF has been drawn at right angles to the given straight line AB from the given point C on it. To draw a perpendicular to a given infinite straight line from a given point not on it, let AB be the given infinite straight line, and C the given point not on it. Take a point D at random on the other side of AB, and with centre C and distance CD describe the circle EFG. Bisect the straight line EG at H, and join CG, CH, CE. Since GH is equal to HE, and HC is common, the two sides GH, HC are equal to the two sides EH, HC respectively, and the base CG is equal to the base CE. Therefore the angle CHG is equal to the angle CHE. And these are adjacent angles. But when a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right. Therefore each of the angles CHG, CHE is right. Thus the straight line CH has been drawn perpendicular to the given infinite straight line AB from the given point C not on it. The properties of triangles follow from these constructions and axioms. In any triangle, the greater side subtends the greater angle. If two triangles have two sides equal to two sides respectively, and have the bases equal, then the angles contained by the equal straight lines are also equal. If two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that which subtends one of the equal angles, then the remaining sides are equal and the remaining angle is equal. The straight line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it. The angles at the base of an isosceles triangle are equal to one another, and if the equal straight lines be produced further, the angles under the base will be equal to one another. In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles. The theory of parallels is founded on the fifth postulate. If a straight line falling on two straight lines makes the alternate angles equal to one another, the straight lines will be parallel to one another. If a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another. Straight lines parallel to the same straight line are also parallel to one another. In parallelogrammic areas, the opposite sides and angles are equal to one another, and the diameter bisects the areas. If a parallelogram has the same base with a triangle and is in the same parallels, the parallelogram is double the triangle. The construction of squares and rectangles is derived from the properties of right angles and parallel lines. To construct a square on a given straight line, let AB be the given straight line. Draw AC at right angles to AB from the point A, and make AD equal to AB. Draw DE parallel to AB, and draw BE parallel to AD. Since AB is equal to AD, and the angle BAD is right, the figure ADEB is equilateral. And since the angle BAD is right, the angle ADE is also right, and similarly each of the angles at E and B is right. Therefore ADEB is a square, and it has been constructed on the straight line AB. To apply a parallelogram equal to a given triangle to a given straight line in a given rectilineal angle, let AB be the given straight line, C the given triangle, and D the given rectilineal angle. Construct the parallelogram BEFG equal to the triangle C in the angle EBG equal to the angle D, and place it so that BE is in a straight line with AB. Produce FG to H, and through A draw AH parallel to BG or EF. Join HB. Since the straight line HF falls on the parallels AH, EF, the angles AHF, HFE are equal to two right angles. The angles BHG, GFE are less than two right angles; therefore HB, FE will meet if produced. Let them meet at K, and through K draw KL parallel to EA or FH. Produce HA, GB to the points L, M. Then HLKF is a parallelogram, with HK as diameter, and AG, ME are parallelograms about HK, and LB, BF are the so-called complements. Therefore LB is equal to BF. But BF is equal to the triangle C. Therefore LB is also equal to the triangle C. And the angle ABM is equal to the angle D. Thus the parallelogram LB equal to the given triangle C has been applied to the given straight line AB in the angle ABM equal to the angle D. The Pythagorean theorem is established as follows: in right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. Let ABC be a right-angled triangle having the angle BAC right. Describe on BC the square BDEC, and on BA, AC the squares GB, HC. Draw AL parallel to BD or CE, and join AD, FC. Since the angle BAC is right, and the angle BAG is right, the straight line CA is in a straight line with AG. Similarly, BA is in a straight line with AH. The angle DBC is equal to the angle FBA, for each is right. Add to each the angle ABC, and the whole angle DBA is equal to the whole angle FBC. Since DB is equal to BC, and FB is equal to BA, the two sides DB, BA are equal to the two sides CB, BF respectively, and the angle DBA is equal to the angle FBC; therefore the base AD is equal to the base FC, and the triangle ABD is equal to the triangle FBC. The parallelogram BL is double the triangle ABD, for they are on the same base BD and in the same parallels BD, AL. And the square GB is double the triangle FBC, for they are on the same base FB and in the same parallels FB, GC. But the doubles of equals are equal; therefore the parallelogram BL is equal to the square GB. Similarly, the parallelogram CL is equal to the square HC. Therefore the whole square BDEC is equal to the two squares GB, HC. Thus the square on the side BC is equal to the squares on the sides BA, AC. The theory of proportion is introduced through the method of equimultiples. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. If four magnitudes are proportional, they will also be proportional alternately. If magnitudes be proportional componendo, they will also be proportional separando. If a first magnitude have to a second the same ratio as a third has to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth has to the sixth. The ratios which are the same with the same ratio are also the same with one another. The construction of similar figures follows from the theory of proportion. Triangles which have their sides proportional are similar. In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles. If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend. In right-angled triangles, if a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another. The application of areas is extended to the solution of quadratic problems. To a given straight line to apply a parallelogram equal to a given rectilineal figure and deficient by a parallelogrammic figure similar to a given one. Let AB be the given straight line, C the given rectilineal figure, and D the given parallelogram. Bisect AB at E, and on EB describe the parallelogram BF similar and similarly situated to D. Complete the parallelogram AG. If AG is equal to C, the task is accomplished. If not, let it be greater, and let the excess be HK. Construct a parallelogram similar to D and equal to HK. Let it be LM. Since LM is similar to D and D is similar to BF, therefore LM is similar to BF. Let them be similarly situated. Therefore LM is about the same diameter with BF. Let FN be their common diameter. The parallelogram AG is equal to the sum of BF and LM. But BF is equal to the parallelogram described on EB, and LM is equal to the excess of AG over C. Therefore AG is equal to C. Thus the parallelogram AG equal to the given rectilineal figure C has been applied to the given straight line AB, deficient by a parallelogrammic figure similar to D. The construction of the regular pentagon proceeds by the bisection of a circle’s arc and the application of the golden section. In a given circle to inscribe an equilateral and equiangular pentagon. Let the given circle be ABCDE. Construct an isosceles triangle FGH having each of the angles at G, H double the angle at F. Inscribe in the circle a triangle ACD equiangular with the triangle FGH. Bisect the angles ACD, CDA by the straight lines CE, DB, and join AB, BC, DE, EA. Since each of the angles ACD, CDA is double the angle CAD, and each of these angles has been bisected, the five angles DAC, ACE, ECD, CDB, BDA are equal to one another. But equal angles stand on equal circumferences; therefore the five circumferences AB, BC, CD, DE, EA are equal to one another. And equal circumferences are subtended by equal straight lines; therefore the five straight lines AB, BC, CD, DE, EA are equal to one another. Thus the pentagon ABCDE is equilateral. Since the circumference AB is equal to the circumference DE, let each be added to the circumference BCD, and the whole circumference ABCD is equal to the whole circumference EDCB. The angle AED stands on the circumference ABCD, and the angle BAE stands on the circumference EDCB; therefore the angle AED is equal to the angle BAE. Similarly, each of the angles ABC, BCD, CDE is equal to each of the angles BAE, AED. Thus the pentagon ABCDE is equiangular. The method of exhaustion is used to compare areas bounded by curved lines. Similar polygons inscribed in circles are to one another as the squares on their diameters. Circles are to one another as the squares on their diameters. Pyramids which are of the same height and have triangular bases are to one another as the bases. Prisms of equal height and triangular bases are to one another as the bases. The cone is one third of the cylinder which has the same base and equal height. A sphere is four times the cone which has its base equal to the greatest circle in the sphere and height equal to its radius. The volumes of spheres are to one another as the cubes on their diameters. The elements of solid geometry extend these principles to three dimensions. A solid is that which has length, breadth, and depth. A face of a solid is a surface. A straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in that plane. A plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the common section of the planes are at right angles to the remaining plane. Similar solid figures are those contained by similar planes equal in multitude. A pyramid is a solid figure, contained by planes, which is constructed from one plane to a point. A [role=marginalia, type=clarification, author="a.darwin", status="adjunct", year="2026", length="46", targets="entry:geometry", scope="local"] I find it curious that Euclid defines points and lines without reference to physical reality—yet in nature, all extension arises from relation and motion. Geometry’s purity is its strength, yet its axioms must not blind us to the flux of matter from which even measurement springs. [role=marginalia, type=extension, author="a.dewey", status="adjunct", year="2026", length="44", targets="entry:geometry", scope="local"] The classical definitions assume spatial intuition; yet modern geometry dissolves these into relational structures—points as elements in a set, lines as equivalence classes. Euclid’s elegance persists, but its ontological claims yield to axiomatic autonomy: geometry as the study of invariant relations, not immutable essences. [role=marginalia, type=objection, author="Reviewer", status="adjunct", year="2026", length="42", targets="entry:geometry", scope="local"] I remain unconvinced that Euclid’s definitions capture the bounded rationality inherent in human perception and understanding. How do these ideal forms, free of complexity, interface with the messy, finite nature of our cognitive processes? From where I stand, even a point, ostensibly without part, must be conceived within the limits of our mental faculties. See Also See "Measurement" See "Number"