592575CORPORUM FIRMORUM.
PROPOSITIO LII.
Tab.
XXV.
fig.
4.
&
6.
Sit priſma Triangulare A A B C D rectan-
gulum in B, & horizontale, cujus latus planum A A B inferius &
horizonti parallelum, angulus ſolidus C C ſuperior, erit Cohæren-
tia ejus reſpectiva ad eam parallelopipedi baſin A D B C duplo ma-
jorem habentis, uti ſumma omnium quadratorum linearum f f,
g g, h h, i i, B C perpendicularium in latus A B, ad ſummam
omnium quadratorum A D, f F, g G, h H, i i. B C.
gulum in B, & horizontale, cujus latus planum A A B inferius &
horizonti parallelum, angulus ſolidus C C ſuperior, erit Cohæren-
tia ejus reſpectiva ad eam parallelopipedi baſin A D B C duplo ma-
jorem habentis, uti ſumma omnium quadratorum linearum f f,
g g, h h, i i, B C perpendicularium in latus A B, ad ſummam
omnium quadratorum A D, f F, g G, h H, i i. B C.
Sit latus A B diviſum in partes infinite parvas A f, f g, g h, h i,
i B: concipiantur ductæ ex his omnibus punctis rectæ f f, g g, h h,
i i, B C, perpendiculares in A B; completoque quadrato A D C B,
producantur rectæ uſque in F. G, H, I, C; erit Cohærentia rectæ
f f in Priſmate, ad Cohærentiam rectæ f F in parallelopipedo, in
ratione duplicata altitudinis f F ad altitudinem f F per Prop. XXII.
eodem modo erit Cohærentia g g ad g G in ratione duplicata g g ad
g G, atque ita porro comparata erit Cohærentia aliarum rectarum
h h, i i, B C in Priſmate, ad Cohærentiam g G, h H, i I, B C in
parallelopipedo A D B C. Quare uti ſumma quadratorum omnium
rectarum f f, g g, h h, i i, B C, ad ſummam omnium quadratorum
A D, f F, G g, H h, I i, C B, ita eſt Cohærentia reſpectiva Priſma-
tis Trigoni, ad Cohærentiam parallelopipedi, baſin planam A D B C
habentis duplo majorem.
i B: concipiantur ductæ ex his omnibus punctis rectæ f f, g g, h h,
i i, B C, perpendiculares in A B; completoque quadrato A D C B,
producantur rectæ uſque in F. G, H, I, C; erit Cohærentia rectæ
f f in Priſmate, ad Cohærentiam rectæ f F in parallelopipedo, in
ratione duplicata altitudinis f F ad altitudinem f F per Prop. XXII.
eodem modo erit Cohærentia g g ad g G in ratione duplicata g g ad
g G, atque ita porro comparata erit Cohærentia aliarum rectarum
h h, i i, B C in Priſmate, ad Cohærentiam g G, h H, i I, B C in
parallelopipedo A D B C. Quare uti ſumma quadratorum omnium
rectarum f f, g g, h h, i i, B C, ad ſummam omnium quadratorum
A D, f F, G g, H h, I i, C B, ita eſt Cohærentia reſpectiva Priſma-
tis Trigoni, ad Cohærentiam parallelopipedi, baſin planam A D B C
habentis duplo majorem.
Corol.
1.
Sed omnes rectæ f f, g g, h h, i i, B C infinite parvo in-
tervallo remotæ complent integrum Triangulum A B C, veluti eæ-
dem productæ uſque ad D, F, G, H I, C complent Quadratum A D
C B, quare erit Cohærentia Triangularis baſeos A B C, ad eam Qua-
dratæ baſeos A D B C, uti quadratum Trianguli A B C, ad quadra-
tum quadrati A D B C. eſt vero Triangulum A B C dimidium qua-
drati A D B C, ſive uti 1 ad 2. quorum quadrata ſunt uti 1 ad 4.
quare Cohærentia baſeos triangularis A B C erit ad Cohærentiam
baſeos A D B C, uti 1 ad 4.
tervallo remotæ complent integrum Triangulum A B C, veluti eæ-
dem productæ uſque ad D, F, G, H I, C complent Quadratum A D
C B, quare erit Cohærentia Triangularis baſeos A B C, ad eam Qua-
dratæ baſeos A D B C, uti quadratum Trianguli A B C, ad quadra-
tum quadrati A D B C. eſt vero Triangulum A B C dimidium qua-
drati A D B C, ſive uti 1 ad 2. quorum quadrata ſunt uti 1 ad 4.
quare Cohærentia baſeos triangularis A B C erit ad Cohærentiam
baſeos A D B C, uti 1 ad 4.
Corol.
2.
Si in fig.
6.
Angulus Priſmatis Triangularis D A C in-
ferne ponatur, & latus planum D C ſuperius, erit Cohærentia re-
ſpectiva eadem Priſmatis A B C.
ferne ponatur, & latus planum D C ſuperius, erit Cohærentia re-
ſpectiva eadem Priſmatis A B C.
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