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== The geometric principles ==
== The geometric principles ==
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A point is a mark without quantity. A line is length lacking width. Also of lines, some are straight, others curved. A surface is that which has length and width. Of surfaces, some are planar, others curved. An angle is the indirect coming together of lines. Of angles, some are rectilinear, others in other ways. A straight line is perpendicular to another straight line, when either makes equal angles, they are called right angles. An angle greater than right is called obtuse; less is called acute. A terminal is a limit or a border. A figure is that which is closed by a terminal or terminals. A circle is a figure whose center is equally distant from its perimeter: a diameter led through the center divides it into semicircles: a straight line beside the center divides it into unequal portions. Of the rectilinear figures certain ones are trilateral, certain ones are quadrilateral, others are multilateral. Now of the triangles, certain are equilateral: certain are Isosceles, which has only two sides equal: certain are Scalene, which has three unequal sides. Likewise, some are orthogonal, where one of the angles is right: others are amblygonal, where one is obtuse: others are oxygonal, which has all acute. Always, however, the largest side is opposite the largest angle. And the three angles joined together make two right angles. There are five kinds of quadrilateral figures: the square, rhombus, rectangle, rhomboid, and trapezium. Only the first and second of these have equal sides, and the first and third have right angles. The second and fourth have opposite angles equal. The third and fourth have opposite sides equal. The last is neither equilateral nor equiangular. Now parallel, or equidistant, straight lines are those having been drawn in the same plane and do not contact no matter how long or in what way they are extended. A parallelogram is that of which opposite sides are equidistant. The first four types of equilaterals are of such a sort. The limits and intersections of lines are points. Of whatever rectilinear figures are chosen, all their angles joined make up in total as many pairs of right angles as triangles they are divided into. From where the four angles of a quadrilateral figure forge four right angles, because it is resolved into two triangles. The angles of a pentagonal figure forge six right angles, because it is cut into three triangles, and so on and so forth.
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'''{{sc|
A point
}}'''
is a mark without quantity. A line is length lacking width. Also of lines, some are straight, others curved. A surface is that which has length and width. Of surfaces, some are planar, others curved. An angle is the indirect coming together of lines. Of angles, some are rectilinear, others in other ways. A straight line is perpendicular to another straight line, when either makes equal angles, they are called right angles. An angle greater than right is called obtuse; less is called acute. A terminal is a limit or a border. A figure is that which is closed by a terminal or terminals. A circle is a figure whose center is equally distant from its perimeter: a diameter led through the center divides it into semicircles: a straight line beside the center divides it into unequal portions. Of the rectilinear figures certain ones are trilateral, certain ones are quadrilateral, others are multilateral. Now of the triangles, certain are equilateral: certain are Isosceles, which has only two sides equal: certain are Scalene, which has three unequal sides. Likewise, some are orthogonal, where one of the angles is right: others are amblygonal, where one is obtuse: others are oxygonal, which has all acute. Always, however, the largest side is opposite the largest angle. And the three angles joined together make two right angles. There are five kinds of quadrilateral figures: the square, rhombus, rectangle, rhomboid, and trapezium. Only the first and second of these have equal sides, and the first and third have right angles. The second and fourth have opposite angles equal. The third and fourth have opposite sides equal. The last is neither equilateral nor equiangular. Now parallel, or equidistant, straight lines are those having been drawn in the same plane and do not contact no matter how long or in what way they are extended. A parallelogram is that of which opposite sides are equidistant. The first four types of equilaterals are of such a sort. The limits and intersections of lines are points. Of whatever rectilinear figures are chosen, all their angles joined make up in total as many pairs of right angles as triangles they are divided into. From where the four angles of a quadrilateral figure forge four right angles, because it is resolved into two triangles. The angles of a pentagonal figure forge six right angles, because it is cut into three triangles, and so on and so forth.
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A solid is a body contained under the triple dimension, that is, what is length, breadth, and depth. A perpendicular line into a plane is that which makes right angles with straight lines having been led on the plane. Parallel planes are those which never meet no matter how long and no matter in what direction they are extended. Parallelopiped solids are those whose opposite bases are parallel. Types of solids are pyramids, columns, prisms, and polyhedral figures. A solid angle is made from the meeting of three or more plane angles, it is necessary that four of these are less than right angles. Of the polyhedral figures there are five such solids which are called regular, since they are each bounded under equilateral, equiangular, and equal bases within themselves. The pyramid has four triangles. The octahedron, eight. The cube has six squares. The icosahedron has twenty triangles. The dodecahedron has twelve pentagons. A cone is a round pyramid above a circular base. A cylinder is a round column having as bases equal and parallel circles. In these, an axis is led through the peak and centers of the bases. When the axis is perpendicular to the base, it is called a right cone and cylinder. Now, cut, it is scatenus. As two straight lines cutting each other mutually, so too does every rectilinear triangle lie in one plane. A sphere is a solid enclosed by one surface from which the center in the middle is equally distant. Its diameter or axis goes through the center, as [[w:Theodosius_of_Bithynia|Theodosius]] says. Or as Euclid says, a sphere is a solid which is described by a semicircle revolved around a fixed diameter. A ratio, or proportion, is a comparison of quantities of the same kind. Similar, same, or equal ratios are those which are either of the same name, or is simultaneously bigger by whatever named ratio, or simultaneously smaller. Now the ratio is named by numbers. The quantities of the same ratios are called proportional.
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'''{{sc|
A solid
}}'''
is a body contained under the triple dimension, that is, what is length, breadth, and depth. A perpendicular line into a plane is that which makes right angles with straight lines having been led on the plane. Parallel planes are those which never meet no matter how long and no matter in what direction they are extended. Parallelopiped solids are those whose opposite bases are parallel. Types of solids are pyramids, columns, prisms, and polyhedral figures. A solid angle is made from the meeting of three or more plane angles, it is necessary that four of these are less than right angles. Of the polyhedral figures there are five such solids which are called regular, since they are each bounded under equilateral, equiangular, and equal bases within themselves. The pyramid has four triangles. The octahedron, eight. The cube has six squares. The icosahedron has twenty triangles. The dodecahedron has twelve pentagons. A cone is a round pyramid above a circular base. A cylinder is a round column having as bases equal and parallel circles. In these, an axis is led through the peak and centers of the bases. When the axis is perpendicular to the base, it is called a right cone and cylinder. Now, cut, it is scatenus. As two straight lines cutting each other mutually, so too does every rectilinear triangle lie in one plane. A sphere is a solid enclosed by one surface from which the center in the middle is equally distant. Its diameter or axis goes through the center, as [[w:Theodosius_of_Bithynia|Theodosius]] says. Or as Euclid says, a sphere is a solid which is described by a semicircle revolved around a fixed diameter. A ratio, or proportion, is a comparison of quantities of the same kind. Similar, same, or equal ratios are those which are either of the same name, or is simultaneously bigger by whatever named ratio, or simultaneously smaller. Now the ratio is named by numbers. The quantities of the same ratios are called proportional.
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'''{{sc|Similar}}''' planar and similarly positioned figures are those whose each and every angle is equal and all the same. And, each and every side is proportional and equidistant. Similar and similarly located solids are those contained under similar, the same number of, and parallel bases. And it is sometimes possible to make it so that two sides of figures in similar position of planes, or pairs, simultaneously are congruent. And in similar location of solids, two bases or a pair or a triple may unite on one plane, with the rest equidistant. Correlative sides or correlative bases, each and every correlative to be collected. Likewise, similar cones or similar cylinders are those whose axes are proportional with respect to the diameter of the bases, and are inclined rightly or equally. Now every two circles and every two spheres are mutually similar, since they always have proportional diameters of their perimeters. Likewise, in circles, proportional chords of the diameter cut off similar portions, which receive equal angles positioned either to the center or to the periphery. Also in spheres, similar circles (the diameters of which are proportional to the diameter of the sphere) cut off spherical portions. Now, just as a parallelogram is double its triangle, a tetragonal column is double its Serratile. Likewise, just as a column is triple its pyramid, a cylinder is triple its cone. Likewise, two triangles, two parallelograms, two columns, two pyramids, or cones constituted on equal bases are proportional in their peak. But if they are of the same height, they are proportional in their bases. Likewise, the angles in circles, whether it terminates at the center or at the periphery, are proportional in their received peripheries.
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'''{{sc|Now, similar}}''' planar figures are in double ratio of their responding sides. Thus also two circles are in double ratio of their diameters. And similar solids are in triple ratio of their diameters. Now in other figures, whether triangles on planes or parallelograms you may bring together, or in solid pyramids or parallelopipeds or columnns you may bring together, the ratio of the figures brought together is always constructed from the ratios of the bases and heights. From there, if the bases are reciprocal with respect to the heights, the figures must be equal, and vice-versa.