B P PROP. XXXIV. THE OR, YRAMIDS that have equal bases and altitudes Let ABCD, EFGH be two pyramids that have equal bases BCD, FGH, and also equal altitudes, viz. the perpendiculars drawn from the vertices A and E upon the planes BCD, FGH: The pyramid ABCD is equal to the pyramid EFGH. If they are not equal, let the pyramid EFGH exceed the E Book VII. pyramid ABCD by the folid Z. Then, a feries of prifms of the fame altitude may be defcribed about the pyramid ABCD that fhall exceed it, by a folid less than Za; let these be a 33. 7. the prisms that have for their bases the triangles BCD, NQL, ORI, PSM. Divide EH into the fame number of equal parts into which AD is divided, viz. HT, TU, UV, VE, and through the points T, U and V, let the fections TZW, UEX, VOY be made parallel to the base FGH. The fection NQL is equal to the section WZTb; as alfo ORI to XEU, and PSM to YOV; b 32.70 and therefore, also the prifms that stand upon the equal fections R 3 are C I. Cor. 28. 7. Book VII are equal c, that is, the prism which stands on the base BCD, and which is between the planes BCD and NQL is equal to the prifm which stands on the bafe FGH, and which is between the planes FGH and WZT; and fo of the reft, because they have the fame altitude; wherefore, the fum of all the prifms described about the pyramid ABCD is equal to the fum of all those described about the pyramid EFGH. But the excefs of the prisms described about the pyramid ABCD above F A the pyramid ABCD is lefs than Z; and therefore, the excess of the prifms defcribed about the pyramid EFGH above the pyramid ABCD is alfo lefs than Z. But the excess of the pyramid EFGH above the pyramid ABCD is equal to Z, by hypothefis; therefore, the pyramid EFGH exceeds the pyramid ABCD, more than the prifms defcribed about EFGH exceed the fame pyramid ABCD. The pyramid EFGH is therefore greater than the fum of the prifms described about it, which is impoffible. The pyramids ABCD, EFGH, therefore, are not unequal, that is, they are equal to one another. Therefore, pyramids, &c. Q. E. D. 3 PROP. Book VII. E VERY prism having a triangular base may be divided into three pyramids that have triangular bases, and that are equal to one another. Let there be a prifm of which the base is the triangle ABC, and let DEF be the triangle oppofite to the bafe: The prifm ABCDEF may be divided into three equal pyramids having triangular bafes. D F b 34. 7. Join AE, EC, CD; and because ABED is a parallelogram, of which AE is the diameter, the triangle ADE is equal a to the triangle ABE: therefore the pyramid of which a 34. 1. the bafe is the triangle ADE, and vertex the point C, is equal to the pyramid, of which the bafe is the triangle ABE, and vertex the point C. But the pyramid of which the base is the triangle ABE, and vertex the point C, that is, the pyramid ABCE is equal to the pyramid DEFC f, for they have equal bafes, viz. the triangles ABC, DFE, and the fame altitude, viz. the altitude of the prifm ABCDEF. Therefore, the three pyramids ADEC, ABEC, DFEC are equal to one another. But the pyramids ADEC, ABEC, DFEC make up the whole prifm ABCDEF; therefore, the B prifm ABCDEF is divided into three equal pyramids. Wherefore, &c. Q. E. D. COR. I. From this it is manifeft, that every pyramid is the third part of a prifm which has the fame base, and the fame altitude with it; for if the base of the prifm be any other figure than a triangle, it may be divided into prifms having triangular bases. COR. 2. Pyramids of equal altitudes are to one another as their bafes; because the prifms upon the fame bases, and of the same altitude, are c to one another as their bases. C I. cor.28. ELEMENT S OF GEOMETRY. AN N arch of a circle is any part of the circumference, II. The perimeter of any figure is the length of the line or lines by which it is bounded. AXIOM. The leaft line that can be drawn between two points, is a ftraight line; and if two figures have the fame ftraight line for their bafe, that which is contained within the other, if its bounding line or lines be not any where convex toward the bafe, has the least perimeter. COR. 1. Hence, the perimeter of any polygon inferibed in a circle is lefs than the circumference of the circle. COR. 2. If from a point two ftraight lines be drawn touching a circle, these two lines are together greater than the arch Book VIII. |