...Euc. n. 12, 13, instead of Euc. i. 47, the truth of the theorem may be proved. 26. This is to shew that the square of the diagonal of a rectangular parallelopiped is equal to the sum of the squares of its three edges. 27. Let a rectangular parallelogram ABCD be formed by four squares, each...

...i. 47, and note page 82, the truth of the property is shewn. 45. This is to shew that the square on the diagonal of a rectangular parallelopiped is equal to the sum of the squares on its three edges. 46. This mav be effected in several ways, the most 'simple is by drawing...

...equal. Therefore these diagonals are all equal to each other. QED THEOREM VIII. 692. The square of each diagonal of a rectangular parallelopiped is equal to the sum of the squares of the three edges which meet at any vertex. Hypothesis. Same as in Theorem VII. Conclusion....

...The edges of a regular tetraedron are each 2e. Show that the surface is 4« 2 j/3. 6. The square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of three edges meeting at a common vertex. 7. Two prisms, or pyramids, having equivalent bases,...

...and ABC give by aid of §§ 338 and 528, A'C2 = AA* + AC2 = AA!*- + A& + AD\ That is, the square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of the three edges meeting at any vertex. PROPOSITION VIII. THEOREM. 534. The sum of the squares...

...597. COR. 1. The diagonals of a rectangular parallelopiped are equal. 598. COR. 2. The square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of the three edges meeting at any vertex. For, if AG is a rectangular parallelopiped, the rt....

...to the cone. Ex. 511. The diagonals of a parallelopiped bisect each other. Ex. 512. The square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of its three dimensions. PROPOSITION XXXVI. THEOREM. 668. Every section of a circular cone...

...of their bases. 766. The diagonals of a rectangular parallelopiped are equal. 767. The square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of the three diagonals meeting in any vertex. 768. The volume of a triangular prism is equal...

...by projecting the edges of a cube upon a plane perpendicular to a diagonal? 433. The square of any diagonal of a rectangular parallelopiped is equal to the sum of the squares of the three edges meeting at any vertex. 434. The sum of the squares of the four diagonals...

...EXERCISES. 1. Show that the diagonals of a parallelopiped bisect each other. 2. Show that the square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of the three edges meeting at any vertex. PROPOSITION VI. THEOREM. 427. The volume of any parallelopiped...