PROPOSITION XXXVI. THEOREM. If three Right Lines be proportional, the Solid Parallelepipedon made of them, is equal to the Solid Parallelepipedon made of the Middle Line, if it be an Equilateral one, and Equiangular to the aforesaid Parallelepipedon. L ET three Right Lines A, B, C, be proportional, viz. Let A beto B, as B is to C. I fay, the Solid made of A, B, C, is equal to the equilateral Solid made of B, equiangular to that made on A, B, C. Let E be a folid Angle contained under the three plane Angles DEG, GEF, FED; and make DE, GE, EF, each equal to B, and compleat the solid Parallelepipedon EK. Again, put LM equal to A, *26 of this. and at the Point L, at the Right Line LM, make * a folid Angle contained under the Plane Angles NLX, XLM, MLN, equal to the folid Angle E; and make LN equal to B, and LX to C. Then because A is to B, as B is to C, and A is equal to LM, and B to LN, EF, EG, or ED, and C to LX; it shall be as LM is to EF, so is GE to LX. Andso the Sides about the equal Angles MLX, GEF, are reciprocally proportional. Wherefore the Parallelogram † 14.6. MX is f equal to the Parallelogram GF. And fince the two plane Angles GEF, XLM, are equal, and the Right Lines LN, ED, being equal are erected at the angular Points containing equal Angles with the Lines first given, each to each; the Perpendiculars Cor. 35 drawn † from the Points N, D, to the Planes drawn of this. thro' XLM, GEF, are equal one to another. Therefore the Solids LH, EK, have the same Altitude; but folid Parallelepipedons that have equal Bases, and the *31 of this. fame Altitude, are equal to each other. Therefore the Solid HL is equal to the Solid EK. But the Solid HL is that made of the three Right Lines A, B, C, and the Solid EK that made of the Right Line B. Therefore, if three Right Lines be proportional, the solid Parallelepipedon made of them, is equal to the solid Parallelepipedon made of the Middle Line, if it be an equilateral one, and equiangular to the aforesaid Parallelepipedon; which was to be demonftrated. PROPOSITION XXXVI. THEOREM. If four Right Lines be proportional, the folid Parallelepipedons fimilar, and in like manner described from them, shall be proportional. And if the folid Parallelepipedons, being fimilar, and alike described, be proportional, then the Right Lines they are described from, shall be proportional. LET the four Right Lines AB, CD, EF, GH, be proportional, viz. let AB be to CD, as EF is to GH, and let the fimilar and alike fituate Parallelepipedons KA, LC, ME, NG, be described from them. I say, KA is to LC, as ME is to NG. For because the solid Parallelepipedon KA is fimilar to LC, therefore KA to LC shall have * a Pro- *33 of this. portion triplicate of that which AB has to CD. For the fame Reason, the Solid ME to NG will have a triplicate Proportion of that which EF has to GH. But AB is to CD, as EF is to GH. Therefore AK is to LC, as ME is to NG. And if the Solid AK be to the Solid LC, as the Solid ME is to the Solid NG. I say, as the Right Line AB is to the Right Line CD, fo is the Right Line EF to the Right Line GH. For because AK to LC has † a Proportion triplicate of +33 of this. that which AB has to CD, and ME to NG has a Proportion triplicate of that which EF has to GH, and fince AK is to LC, as ME is to NG; it shall be as AB is to CD, so is EF to GH. Therefore, if four Right Lines be proportional, the Solid Parallelepipedons fimilar, and in like manner described from them, shall be proportional. And if the solid Parallelepipedons, being fimilar and alike described, be proportional, then the Right Lines they are described from, shall be proportional; which was to be demonftrated. PRO PROPOSITION XXXVIII. THEOREM. If a Plane be perpendicular to a Plane, and a Line be drawn from a Point in one of the Planes perpendicular to the other Plane, that Perpendicular shall fall in the common Section of the Planes. LET the Plane CD be perpendicular to the Plane AB, let their common Section be AD, and let fome Point Ebe taken in the Plane CD. I say, a Perpendicular, drawn from the Point E to the Plane A B, falls on AD. For if it does not, let it fall without the fame, as EF meeting the Plane AB in the Point F, and from the Point Filet FG be drawn in the Plane A B perpendi* Def. 4 of cular to AD; this shall be * perpendicular to the Plane CD; and join EG. Then because F G is perpendicular to the Plane CD, and the Right Line EG in the Plane of CD touches it: The Angle FGE shall + Def. 3. of be + a Right Angle. But EF is also at Right Angles to the Plane Angle AB; therefore the Angle EFG is a Right Angle. And so two Angles of the Triangle 17.1. EFG, are equal to two Right Angles; which is ‡ abfurd. Wherefore if a Right Line, drawn from the Point E perpendicular to the Plane AB, does not fall without the Right Line AD: And so it mustineceffarily fall on it. Therefore, if a Plane be perpendicular to a Plane, and a Line be drawn from a Point in one of the Planes perpendicular to the other Plane, that Perpendicular shall fall in the common Section of the Planes; which was to be demonftrated. this. this. PRO 1 PROPOSITION ΧΧΧΙΧ. THEOREM. If the Sides of the oppofite Planes of a folid Parallelepi- LET the Sides of CF, AH, the oppofite Planes of 29. Ι. For join DY, YE, BS, SG. Then because D X is parallel to OÉ, the Alternate Angles DXY, YOE are * equal to one another. And because DX is * equal OE, and Y X to YO, and they contain equal Angles, the Bafe DY shall be f equal to the Base YE; 14. 1. and the Triangle DXY to the Triangle YOE, and the other Angles equal to the other Angles: Therefore the Angle XYD is equal to the Angle OYE; and fo DYE is † a Right Line. For the fame Rea- † 14. 1. fon BSG is also a Right Line, and BS is equal to SG. Then because CA is equal and parallel to DB, as also to EG, DB shall be equal and parallel to EG; and the Right Lines DE, GB, join them: Therefore DEis * parallel to BG, and D, Y, G, S, are Points * 33. 1. taken in each of them, and DG, Ý S, arejoined. Therefore DG, YS, are † in one Plane. And fince DE is + 7 of this. parallel to BG; the Angle EDT shall be * equal to 29.1. the Angle BGT, for they are Alternate. But the Angle DTY, is equal to the Angle GTS. Therefore ≠ 15.1. DTY, GTS are two Triangles, having two Angles of the one equal to two Angles of the other, as likewife one Side of the one equal to one Side of the other, viz. the Side DY equal to the Side GS: For they are Halves of DE, BG: Therefore they shall have the other Sides of one equal to the other Sides of * 14. 1. of the other; and so DT is equal to TG, and YT to TS. Wherefore, if the Sides of the opposite Planes of a folid Parallelepepidon be divided into two equal Parts, and Planes be drawn thro' their Sections; the common Section of the Planes, and the Diameter of the folid Parallelepipedon, shall divide each other into two equal Parts; which was to be demonstrated. PROPOSITION XL. THEOREM. If of two triangular Prisms, one standing on a Base, which a Parallelogram, and the other on a Triangle, if their Altitudes from these Bases are equal, and the Parallelogram double to the Triangle; then those Prisms are equal to each other. I ET ABCDEF, GHKLMN be two Prisms of equal Altitude. The Base of one of which is the Parallelogram AF, and that of the other, the Triangle GHK, and let the Parallelogram AF be double to the Triangle GHK. I say the Prism A B CDEF is equal to the Prism GHKLMN. For compleat the Solids AX, GO. Then because the Parallelogram AF is double to the Triangle GHK, and fince the Parallelogram HK is * double to the Triangle GHK, the, Parallelogram AF shall be equal to the Parallelogram HK. But folid Parallelepipedons. that stand upon equal Bases, and 131 of this. have the fame Altitude are † equal to one another. Therefore the folid A X is equal to the Solid GO. #28 of this. But the Prifm ABCDEF is half the Solid AX, and the Prism GHKLMN is half the Solid GO. Therefore the Prism ABCDEF is equal to the Prism GH KLMN. Wherefore, if there be two Prisms having equal Altitudes, the Base of one of which is a Parallelogram, and that of the other a Triangle, and if the Parallelogram be double to the Triangle, the said Prisms shall be equal to each other. The END of the ELEVENTH BOOK. EUCLID's |