. 16
1, 2, 3,..., n. QR 1, QR 2, QR 3,..., QRn qE.
, , EqQ , . , , -, , P 1, P 2, 3,..., , qRQ A {110} QqE; -, P 1, P 2, P 3,... , . . , 1, 1 2 . ., ; , P 1, P 2, P 3..., , , . p 1, 2, 3, , , , EO 1, 1 2 . .
, . , , . , ; AQ ; .
[. 79, (1), (2), (3), (4)],
OA: OH 1 = QO: OH 1 = Qq: qO 1 = E 1 O 1 : O 1 R 1,
O 1 R 1 , O 1 E 1 , , , EO 1 FO 1, , , , , 1 1 O 1 R 1, E 1 O 1 O 1 R 1 1 FO 1:
: 1 = . EO 1 : . FO 1.
OH 1 , FO 1 , , 1, , 1; {111} , - 1 1 1. , , 1, , FO 1 , , 1, ; ,
(1) |
. FO 1 > P 1.
:
(2) |
. F 1 O 2 > 2; . F 2 O 3 > 3;...
1 1 O 1 R 1 1 FO 1, E 1 O 2 R 1 O 2. , , . .
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: 1 = . E 1 O 2: . R 1 O 2.
, R 1 2 , A, E 1 O 2, , H 1; , - 1 1 2 2 2. , , 1, R 1 O 2 , , 1, ; ,
(3) |
. R 1 O 2 < 2.
(2) (3),
(4) |
. F 1 O 2> P2 > R 1 O 2.
,
. F 2 O 3 > 3 > . R 2 O 3 . .
P 1, 2, 3,... EqQ, 1+ P 2+ P 3+... -{112} EqQ; , ,
P 1 + P 2 + P 3 +...= 1/3Δ EqQ.
,
. FO 1 + . F 1 O 2 + . F 2 O 3 >
1/3 . EqQ > . R 1 O 2 + . R 2 O 3
+ . R 3 O 4 +...
, : , , , qQ; , (. . ) qQF, Qq . , , , , 1/3 qQE, , .
, , ; , , , , , . , , , {113} , , , , .
, , (. 107108) , , , , ; , 1/3 Δ EqQ (. 16), : 1/3Δ EqQ, , EqQ. EqQ, , 1/3 Δ EqQ.
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, , , , . , , , , , .
, , , , , , , 1/8 {114} , , , 1/4 , , , 1/4 . . ,
l + 1/4 + (1/4)2 + (1/4)3 +... = 4/3.
(. . 19). , , 4/3 1/3 .
( ) : , , , D,... Z 1/4 .
(1) |
B + C + D +... Z + 1/3 B + 1/3 C + 1/3 D +...+ 1/3 Z = 4/3 B + 4/3 C +4/3 D +...+ 4/3 Z.
4/3 B = 1/3 A;
4/3 C = 1/3 B;
4/3 D = 1/3 C
. .,
(2) |
B + C + D +...+ Z + 1/3 B + 1/3 C + 1/3 D +...+ 1/3 Z = 1/3 A + 1/3 B + 1/3 C +...+ 1/3 Y.
1/3 B + 1/3 C + 1/3 D +...+ 1/3 Y, {115}
(3) |
B + C + D +...+ Z + 1/3 Z = 1/3 A,
, A,
A + B + C + D +...+ Z + 1/3 Z = 4/3 A.
= 1, B =1/4, =(1/4)2 . .,
l + 1/4 + (1/4)2 + (1/4)3 +...+ 1/3(1/4) n = 4/3.
, , , .
1/3 Z 1/3(1/4) n , , 4/3. , , , , , 4/3 , .
, , , . , . , , , , , , , ; , , , . , , ( , ), -{116} , , , ; , . , , , , , . (. 27) . 2 . XII , , , , , .
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. . , , , . , ; . , , , (resp. ) (resp. ) , . , reductio ad absurdum , , {117} . , , π rl π(Ö rl)2, . . , r l. π[Ö(r 1+ r 2) l ]2.
reductio ad absurdum ( ).
R, r, l, r l m. M m, S . , S = .
S , , . S > M.
, S: M. R R , , , . , R, R 1, 1, , R, R 2, a 2; S 1.
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R 1 : M 1 = r 2 : m 2 = r: l = R 1 : S 1,
M 1 = S 1.
M 1 : M 2 < S: M,
,
S 1 : M 2 < S: M. {118}
: S 1 > S, M 2 < ; , ; . , S > M .
, S < M ; S = M, .
. , , ,
sinp/ n + sin2p/ n + sin3p/ n +...,
2 , ¥.1
. , .
. 17 |
2 , . 17. , , AG. , : ABb GFf -{119} BbCc, CcDd . ., . Bb, Cc, Dd . . , bC, cD, dE . . b ABG
B b: b A = GB: BA.
( )
B b: b A = b b: b K = C g: g K:... = f j: j G.
Ut omnes ad omnes, ita unus ad unum (. . 25, . 5), . .
(B b + b b + C g + g c +...+ F j + j f)/(A b + b K + K g + g L +...+ N j + j G) = GB: BA,
, B b + b b = Bb, C g + g c = Cc . .
(Bb + Cc + Dd +...+ Ff): AG = GB: BA.
(1) |
, , , (. . 118) p rl p(r 1+ r 2) l , :
Abb = p × AB × B b1, {120}
. BbCc= p × BC (B b+ C g)=p AB (B b+ C g),
. CcDd= p × CD (C g+ DO) = p AB (C g+ DO) . .,
FfG= p × FG × F j = p × AB × F j.
, ,
(2) |
p × AB (2 B b + 2 C g) + 2 DO +...+ 2 F j) =
= p × AB (Bb + Cc + Dd +...+ Ff).
(1)
AB (Bb + Cc + Dd + Ff) = GB × AG,
p × GB × AG, , p AG 2 ( 4 p r 2).
, , , , , p × AG 2 ( 4 p r 2). ,
. 18
, reductione ad absurdum, p AG 2, , (. . 4p r 2).
, , ; (. 18). , -{121} 1 a 1, ( 1 1) , , 1 BD, , , .
, , , , (. . 14); , . , (p × Oa AC BD)/3.
:
(p aC 2 OC)/3.
(p aC 2 CB)/3.
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(p aC 2/3)(OC + CB) = (p aC aC OB)/3.
OBD (Ð BOD ; )
Oa: aC = OB: BD,
aC OB = Oa: BD,
,
(p aC aC OB)/3 = (p aC Oa BD)/3,
.
+ , MM 1 L ( ), ( ), PL, + , . . O. , {122} , . . .
, , . ,
(1) |
( ): ( ) = : .
, ,
(2) |
( MM 1 L): ( ) = PL: .
PL ; , ; , , . .
(3) |
( ) = ( LMM 1).
, = 1, 1 1, ,
(. 1 1): (. LMM 1) =
= (. . ): (. ) =
= : = O: BD = LP: EO 1.
, EA 1 a 1 LMM 1 1 LP, , . , (3), MM 1 L ; , 1 1, .
, (. 19). . 17, ABCDEFG AG. , , , , COD . ., , , . OFG , , , . {123} V, , Δ BOC, VCOc VBOb (. 20).