HYPOTHESES. If the same straight line is If two straight lines be cut If two parallel planes be cut CONSEQUENCES. They are parallel to one an- They shall be cut in the same The plane which passes This straight line shall fall Their common sections with A triangle may be made of Z. Of Solid Figures. HYPOTHESES. If three straight lines be pro- If four straight lines be pro- If four straight lines be con- If solid parallelopipeds are If solid parallelopipeds have CONSEQUENCES. The solid parallelopiped de- The similar solid parallel- As the first is to the fourth, They are to one another in the They are to one another as They are equal to one another. And are upon triangular bases Idem. If there be two triangular {Their bases and altitudes are The prisms shall be equal to The opposite planes are simi- ograms. They are equal and similar CONSEQUENCES. They have to one another the ratio which is the same with the ratio compounded of the ratios of their sides. They are equal to one another. It divides the whole into two solids, the base of one of which shall be to the base of the other, as the one solid is to the other. It shall be cut into two equal parts. The common section of the planes passing through the points of division, and the diameter of the solid parallelopiped, cut each other into two equal parts. (It may be divided into three pyramids that have triangular bases, and are equal to one another. (It may be divided into two equal and similar pyramids having triangular bases, and which are similar to the whole pyramid; and into two equal prisms which together are greater than half of the whole pyramid. They are to one another in the triplicate ratio of that of their homologous sides. They are to one another as their bases. Their bases and altitudes are reciprocally proportional. They are equal to one another. It is the third part of a prism which has the same base and altitude, PROBLEMS. A. Relating to Straight Lines. To draw a straight line perpendicular to a plane, from a given point above it. To draw a straight line at right angles to a given plane, from a point given in that plane. From a given finite straight line to cut off any required part. To divide a given straight line similarly to a given divided To cut a given straight line in extreme and mean ratio. To find a mean proportional between two given straight To find a third proportional to two given straight lines. B. Relating to Rectilineal Angles. To divide a given right angle into five equal parts. C. Relating to Triangles. To construct an isosceles triangle, in which each of the angles at the base shall be double of the angle opposite to the same. F. Relating to Inscribed Figures: In a given circle to inscribe a straight line equal to a given straight line, which is not greater than the diameter of the circle. To inscribe a circle in a given triangle. To circumscribe a circle about a given triangle. In a given circle to inscribe a triangle equiangular to a given triangle. |