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The two sides KN, KL of the triangle Ken, being, therefore, equal to the two sides AD, AB of the triangle ABD, and the angle NKL to the angle DAB, the triangle KLN will be equal and like the triangle ABD (I. 4.)
And in the same manner it may be shewn, that the triangle LMN is equal and like to the triangle BCD.
But the triangles KLN, LMN are, together, equal to the section KLMN; and the triangles ABD, BCD to the section ABCD; whence the section KLMN is equal and like to the section ABCD.
Q. E. D.
Prisms of equal bases and altitudes are equal to each other.
Let Am, Es be any two prisms, standing upon the equal bases ABCD, Ergh, and having equal altitudes; then will am be equal to es.
For parallel to the bases, and at equal distances from them, draw the planes mp and vw.
Then, by the last propofition, the section mnps will be equal to the base ABCD, and the section vowr to the base
But the base ABCD is equal to the base EFGH by hypothesis ; whence the section mnps is, also, equal to the section vowr.
And in the same manner it may be shewn, that any other fe&tions, at equal distances from the bases, are equal to each other.
Since therefore every section in the prison Am is equal to its corresponding section in the prism Es, the prifms themselves, which are composed of those sections, must also be equal.
Q. E. D. Cor. Every prism is equal to a rectangular parallelepipedon of an equal base and altitude.
PRO P. X. THEOREM.
Rectangular parallelepipedons, of equal altitudes, are to each other as their bases.
Let AC, MP be two rectangular parallelepipedons, having the equal altitudes ED, QR; then will ac be to MP as the base be is to the base no.
For in AB, produced, take any number of right lines AF, FL each equal to AB; and in mn, produced, take any number of right lines nt, tx each equal to mn.
Complete the parallelograms Fe, FK, MV, Tz, and make the upright solids AG, FH, NW, Ty of equal altitudes with ac or MP.
Then, because AF, FL are each equal to AB, and nt, TX are each equal to MN (by Const.), the parallelograms FE, FK will be each equal to be, and the parallelograms NV, Tz to nQ (II. 5.)
And, since the folids AG, FH have equal bases and altitudes with the folid Ac, they will be each equal to AC (VIII. 9.); and, for the same reason, the folids nw, TY, will be each equal to NR.
Whatever multiple, therefore, the base ek is of the base be, the same multiple will the folid bh be of the folid Ac; and, for the same reason, whatever multiple the base mz is of the base no, the same multiple will the folid my be of the folid NR.
If, therefore, the base BK be equal to the base mz, the folid Bh will be equal to the folid My; and if greater, greater; and if less, less; whence the base be is to the base ng, as the solid ac is to the solid nr (V. Def. 5.)
Q. E. D. COR. From this demonftration, and the Cor. to the laft Prop. it appears that all prisms of equal altitudes, are to each other as their bases ; every prism being equal to a rectangular parallelepipedon of an equal base and altitude.
Rectangular parallelepipedons of equal bases are to each other as their altitudes.
Let AK, EP be two rectangular parallelepipedons standing on the equal bases AC, EG; then will AK be to EP as the altitude AM is to the altitude Es.
For let aw be a rectangular parallelepipedon on the base Ac, whose altitude Av is equal to the altitude Es of the parallelepipedon EP:
Then, since the base Ac is equal to the base EG (by Hyp.), and the altitude av is equal to the altitude Es (by Conf.), the solid Aw will be equal to the solid ep (VIII. 9.)
And if AL, AY be considered as bases, the solid ak will be to the solid Aw as the base al is to the base AY (VIII. 10.)
But the base al is to the base Ay as the side AM is to the side av (VI. 1.); whence by equality the solid ak will be to the solid Aw as the altitude AM is to the alti. tude av (V. 11.)
Since, therefore, the solid Aw is equal to the solid EP, and the altitude av to the altitude Es, the folid AK will also be to the solid EP as Am is to ES (V.9.)
Q. E. D. Cor. From the reason given in the Cor. to the last Prop. it follows, that all prisms of equal bases, are to each other as their altitudes.
PRO P. XII. THEOREM.
The bases and altitudes of equal rectangular parallelepipedons are reciprocally proportional; and if the bases and altitudes be reciprocally proportional, the parallelepipedons will be equal.
Let the rectangular parallelepipedon Ar be equal to the rectangular parallelepipedon Ey; then will the base AC be to the base &G, as the altitude eo is to the alti
For let Al be a rectangular parallelepipedon on the base AC, whose altitude AP is equal to eo, the altitude of the parallelepipedon Ey.
'Then fince the altitudes AP, Eo are equal to each other (by Conft.), the solid AL will be to the solid ex as the base ac is to the base eG (VIII. 10.)
And because the solid ar is equal to the solid EY (by Hyp.), the solid AL will be to the folid ar as ac is to EG (V. 9.)