$6 GIDEONIS JANI VERDAM 



Determinemus primum inclinationes planorum ABGF et AIIEF in planum PQ. 

 Vocetur latus cubi AB = BC = BG r= caet. = a, erit 



A* = AB Cos. BA = a Cos. <p ; AC = V(AB a + BC 3 ) = V*a* = a\/i, 

 Ac = AC Cos. CAc = a\/z Cos. %. 



B* = AB Sin. BA = a Sin. <J>; Cc = AC Sin, CAc = |/a . Sin. x ; ducatur B/ 

 jwralkla lineae be , erit B/=: be , et C/=r Cc C/ = Cc B =: <* |/ 2 &' % * &'" <? 

 B/ = t/(BC 2 C/ 2 ) = t/O 2 a 2 ( Va. J/. J/. $)*] Vl^ a * 

 (2 .S/V;. 2 2 I/a . Sin. % Sin. -f- Aw. 2 ) ] = V/C ( 2 a 2 Sin.* Q ) za 2 Sin. % 

 (Sin.% \/a.Sin. $)] = a |/L C ^ 2 Sin ' % ($* % I/a.*'". <?>)] = ^ 

 Est autem 



16. ( /fr^) 2 trianguli Abe = [(A^ + A*) a ^ 2 ] . [fc 2 - (Ac A) 2 ] (6). 

 Habemus autem , 



Ac -f M = a ( Co;. (J) + 1/2 . Cb;. ^ ) ; Ac ^A = a ( Cos. (f> 1/a . GOJ. ^ ) > 

 ( Ac + A*) 2 = a (Cw. 2 $ + 2 \/a . Co;. <p Coj. & + 2 Cw. 2 ); 

 ( Ac . A) a = a 2 (C<w. 2 - a V* . Cos. <p Co/. % + 2 Cw. 2 a$ ) ; 



fc 2 = 2 ( Cw. 2 2 J/fl. 2 JK + 2 V2 . &. cp J/w. ^ ) , 

 (Ac + A*) a *c 2 = c 2 [Co;. 2 Q + zyn.Cos.Q Cos.% + 2 Cos. 2 % Gw 



2 Va <?> *' ; 3 = 2fl 2 [ i + 1/2 . Co;, (aj + <|))] ; 

 fc a (Ac A*) 2 = a* [Co;. 3 <|> 2 ,. + a v'a . J/. <p &. 



Co;. Co;, aj a Co;. 2 ^] = 2<z 2 [V/a . Co*. (^ <J>) I ]. 

 Ergo 16 X <Xr*O a s= 44 [ i + |/ a . Co;. (35 + $) ] . [ |/a . Co;. ( 



Co;. 



sed Co;, (^ <p ) Co;. (# + <p ) = 2 &'. % Sin - $ 5 



2 Co;, (x <p) Co;, (x -f $) = a Co; J (2^ a$) Co;. J ( 2 X + c< ?) - Cw - a ^ + G^ *$' C?) 

 Co;.2% = i a SV'. a ^ ; Co;.:$ = 12 iSV.-$ ; Co;.2# + Cos.iQ>i \_i-Sin.~x, SiriSQ] , 

 si haec subsituantur habemus: 



16 ( drca ) 2 = 4* [ a ( a -ft'/;. 3 ^ a V 7 ^ ^' /; - $ *' X + 2 J/. 2 <$ ) I ] 

 Q (a Sin. a % a|/2.iSV'. (?) iS 1 //;. % + &'/;. 2 $) ] = 4*4 [Co;. 2 <J) 



Ergo ^a = la* \/ [ Co;. 2 $ ( J/a . S/. % >5 ' i; "' $ ) 2 ] 

 Pvoducattir planum laterale ABCH, donee planum PQ secet, secundum lincam NL , 

 ct vocetur inclinatio planorum PQ et NLCB , x, erit: 



Area trianguli \bc ^r triangulo ABC X Co;, x (8); 

 sed area trianguli ABC = JAB . BC = Jo a . 



er- 



( 6) Vid. Cl. dc Gelder, J3<g/j. </ ^/r/*. , III Bock, XIII Stell. . 234. 

 (7) Vid. de Gelder, op. mci'o citato, VIII 15oek, VI Stelling. 

 (8 ) Vid. de Gelder, cod. op. XV Boek , II Hdp Stell. . 15^0. 



