# 《幾何原本（英文版）》歐幾里得.pdf

Table of Contents Prematter Introduction Using the Geometry Applet About the text Euclid A quick trip through the Elements References to Euclid s Elements on the Web Subject index Book I. The fundamentals of geometry: theories of triangles, parallels, and area. Definitions (23) Postulates (5) Common Notions (5) Propositions (48) Book II. Geometric algebra. Definitions (2) Propositions (13) Book III. Theory of circles. Book VII. Fundamentals of number theory. Definitions (22) Propositions (39) Book VIII. Continued proportions in number theory. Propositions (27) Book IX. Number theory. Propositions (36) Book X. Classification of incommensurables. Definitions (11) Propositions (37) Book IV. Constructions for inscribed and circumscribed figures. Definitions (7) Propositions (16) Book V. Theory of abstract proportions. Definitions (18) Propositions (25) Book VI. Similar figures and proportions in geometry. Definitions (11) Propositions (37) Definitions I (4) Propositions 1-47 Definitions II (6) Propositions 48-84 Definitions III (6) Propositions 85-115 Book XI. Solid geometry. Definitions (28) Propositions (39) Book XII. Measurement of figures. Propositions (18) Book XIII. Regular solids. Propositions (18) Table of contents a71 Propositions (18) Propositions Proposition 1. If a straight line is cut in extreme and mean ratio, then the square on the greater segment added to the half of the whole is five times the square on the half. Proposition 2. If the square on a straight line is five times the square on a segment on it, then, when the double of the said segment is cut in extreme and mean ratio, the greater segment is the remaining part of the original straight line. Lemma for XIII.2. Proposition 3. If a straight line is cut in extreme and mean ratio, then the square on the sum of the lesser segment and the half of the greater segment is five times the square on the half of the greater segment. Proposition 4. If a straight line is cut in extreme and mean ratio, then the sum of the squares on the whole and on the lesser segment is triple the square on the greater segment. Proposition 5. If a straight line is cut in extreme and mean ratio, and a straight line equal to the greater segment is added to it, then the whole straight line has been cut in extreme and mean ratio, and the original straight line is the greater segment. Proposition 6. If a rational straight line is cut in extreme and mean ratio, then each of the segments is the irrational straight line called apotome. Proposition 7. If three angles of an equilateral pentagon, taken either in order or not in order, are equal, then the pentagon is equiangular. Proposition 8. If in an equilateral and equiangular pentagon straight lines subtend two angles are taken in order, then they cut one another in extreme and mean ratio, and their greater segments equal the side of the pentagon. Proposition 9. If the side of the hexagon and that of the decagon inscribed in the same circle are added together, then the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon. Proposition 10. If an equilateral pentagon is inscribed ina circle, then the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon inscribed in the same circle. Proposition 11. If an equilateral pentagon is inscribed in a circle which has its diameter rational, then the side of the pentagon is the irrational straight line called minor. Proposition 12. If an equilateral triangle is inscribed in a circle, then the square on the side of the triangle is triple the square on the radius of the circle. Proposition 13. To construct a pyramid, to comprehend it in a given sphere; and to prove that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid. Lemma for XIII.13. Proposition 14. To construct an octahedron and comprehend it in a sphere, as in the preceding case; and to prove that the square on the diameter of the sphere is double the square on the side of the octahedron. Proposition 15. To construct a cube and comprehend it in a sphere, like the pyramid; and to prove that the square on the diameter of the sphere is triple the square on the side of the cube. Proposition 16. To construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the icosahedron is the irrational straight line called minor. Corollary. The square on the diameter of the sphere is five times the square on the radius of the circle from which the icosahedron has been described, and the diameter of the sphere is composed of the side of the hexagon and two of the sides of the decagon inscribed in the same circle. Proposition 17. To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the dodecahedron is the irrational straight line called apotome. Corollary. When the side of the cube is cut in extreme and mean ratio, the greater segment is the side of the dodecahedron. Proposition 18. To set out the sides of the five figures and compare them with one another. Remark. No other figure, besides the said five figures, can be constructed by equilateral and equiangular figures equal to one another. Lemma. The angle of the equilateral and equiangular pentagon is a right angle and a fifth. Elements Introduction - Book XII. Table of contents a71 Propositions (18) Propositions Proposition 1. Similar polygons inscribed in circles are to one another as the squares on their diameters. Proposition 2. Circles are to one another as the squares on their diameters. Lemma for XII.2. Proposition 3. Any pyramid with a triangular base is divided into two pyramids equal and similar to one another, similar to the whole, and having triangular bases, and into two equal prisms, and the two prisms are greater than half of the whole pyramid. Proposition 4. If there are two pyramids of the same height with triangular bases, and each of them is divided into two pyramids equal and similar to one another and similar to the whole, and into two equal prisms, then the base of the one pyramid is to the base of the other pyramid as all the prisms in the one pyramid are to all the prisms, being equal in multitude, in the other pyramid. Lemma for XII.4. Proposition 5. Pyramids of the same height with triangular bases are to one another as their bases. Proposition 6. Pyramids of the same height with polygonal bases are to one another as their bases. Proposition 7. Any prism with a triangular base is divided into three pyramids equal to one another with triangular bases. Corollary. Any pyramid is a third part of the prism with the same base and equal height. Proposition 8. Similar pyramids with triangular bases are in triplicate ratio of their corresponding sides. Corollary. Similar pyramids with polygonal bases are also to one another in triplicate ratio of their corresponding sides. Proposition 9. In equal pyramids with triangular bases the bases are reciprocally proportional to the heights; and those pyramids are equal in which the bases are reciprocally proportional to the heights. Proposition 10. Any cone is a third part of the cylinder with the same base and equal height. Proposition 11. Cones and cylinders of the same height are to one another as their bases. Proposition 12. Similar cones and cylinders are to one another in triplicate ratio of the diameters of their bases. Proposition 13. If a cylinder is cut by a plane parallel to its opposite planes, then the cylinder is to the cylinder as the axis is to the axis. Proposition 14. Cones and cylinders on equal bases are to one another as their heights. Proposition 15. In equal cones and cylinders the bases are reciprocally proportional to the heights; and those cones and cylinders in which the bases are reciprocally proportional to the heights are equal. Proposition 16. Given two circles about the same center, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle. Proposition 17. Given two spheres about the same center, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface. Corollary to XII.17. Proposition 18. Spheres are to one another in triplicate ratio of their respective diameters. Next book: Book XIII Previous: Book XI Elements Introduction Table of contents a71 Definitions (28) a71 Propositions (39) Definitions Definition 1. A solid is that which has length, breadth, and depth. Definition 2. A face of a solid is a surface. Definition 3. 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 the plane. Definition 4. A plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane. Definition 5. The inclination of a straight line to a plane is, assuming a perpendicular drawn from the end of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the end of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up. Definition 6. The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the intersection at the same point, one in each of the planes. Definition 7. A plane is said to be similarly inclined to a plane as another is to another when the said angles of the inclinations equal one another. Definition 8. Parallel planes are those which do not meet. Definition 9. Similar solid figures are those contained by similar planes equal in multitude. Definition 10. Equal and similar solid figures are those contained by similar planes equal in multitude and magnitude. Definition 11. A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines, that is, a solid angle is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point. Definition 12. A pyramid is a solid figure contained by planes which is constructed from one plane to one point. Definition 13. A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar, and parallel, while the rest are parallelograms. Definition 14. When a semicircle with fixed diameter is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere. Definition 15. The axis of the sphere is the straight line which remains fixed and about which the semicircle is turned. Definition 16. The center of the sphere is the same as that of the semicircle. Definition 17. A diameter of the sphere is any straight line drawn through the center and terminated in both directions by the surface of the sphere. Definition 18. When a right triangle with one side of those about the right angle remains fixed is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone. And, if the straight line which remains fixed equals the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuse- angled; and if greater, acute-angled. Definition 19. The axis of the cone is the straight line which remains fixed and about which the triangle is turned. Definition 20. And the base is the circle described by the straight in which is carried round. Definition 21. When a rectangular parallelogram with one side of those about the right angle remains fixed is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cylinder. Definition 22. The axis of the cylinder is the straight line which remains fixed and about which the parallelogram is turned. Definition 23. And the bases are the circles described by the two sides opposite to one another which are carried round. Definition 24. Similar cones and cylinders are those in which the axes and the diameters of the bases are proportional. Definition 25. A cube is a solid figure contained by six equal squares. Definition 26. An octahedron is a solid figure contained by eight equal and equilateral triangles. Definition 27. An icosahedron is a solid figure contained by twenty equal and equilateral triangles. Definition 28. A dodecahedron is a solid figure contained by twelve equal, equilateral and equiangular pentagons. Propositions Proposition 1. A part of a straight line cannot be in the plane of reference and a part in plane more elevated. Proposition 2. If two straight lines cut one another, then they lie in one plane; and every triangle lies in one plane. Proposition 3. If two planes cut one another, then their intersection is a straight line. Proposition 4. If a straight line is set up at right angles to two straight lines which cut one another at their common point of section, then it is also at right angles to the plane passing through them. Proposition 5. If a straight line is set up at right angles to three straight lines which meet one another at their common point of section, then the three straight lines lie in one plane. Proposition 6. If two straight lines are at right angles to the same plane, then the straight lines are parallel. Proposition 7. If two straight lines are parallel and points are taken at random on each of them, then the straight line joining the points is in the same plane with the parallel straight lines. Proposition 8. If two straight lines are parallel, and one of them is at right angles to any plane, then the remaining one is also at right angles to the same plane. Proposition 9 Straight lines which are parallel to the same straight line but do not lie in the same plane with it are also parallel to each other. Proposition 10. If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then they contain equal angles. Proposition 11. To draw a straight line perpendicular to a given plane from a given elevated point. Proposition 12. To set up a straight line at right angles to a give plane from a given point in it. Proposition 13. From the same point two straight lines cannot be set up at right angles to the same plane on the same side. Proposition 14. Planes to which the same straight line is at right angles are parallel. Proposition 15. If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then the planes through them are parallel. Proposition 16. If two parallel planes are cut by any plane, then their intersections are p