【関孝和『発微算法』第1問】

<BODY bgcolor="#fffff0" background="img006.gif" text="#008000" link="#ff0000" vlink="#bc8f8f" alink="#1e90ff"> <A name="_pageHead"></A> <CENTER> <BR> </CENTER> <CENTER> 【関孝和『発微算法』第1問】                                                         <BR> </CENTER> <CENTER> <BR> </CENTER> <DIV align="left"> 　【原文の要約】<BR> </DIV> <DIV align="left"> 　図のように大円の内側に中円1箇と小円2箇を互いに接するように内接させます。いま、中円1箇と小円2箇の面積を除く,<BR> </DIV> <DIV align="left"> 外余積が120歩、小円の直径が中円の直径より5寸短いとするとき、大、中、小円の直径をそれぞれ求めなさい。<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">                                          <IMG src="img055.jpg" border="0"><BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">   大円径 = L、中円径 = M、小円径 = Sとする。　中円径と小円径の差 = M － S 、外余積 = π/4(L^2－M^2－S^2) とする。<BR> </DIV> <DIV align="left"> π/4(円周率/4)は円積率と呼ばれますが、これは円の面積を直径で求めたことに関係します。<BR> </DIV> <DIV align="left"> 外余積は、大円の面積から、中円と小円の面積を引いた残り。<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 円の直径の2乗から円の面積を求める。 <BR> </DIV> <DIV align="left"> 円の面積は半径×半径×π<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 直径の2乗を計算してみましょう。<BR> </DIV> <DIV align="left">  直径×直径 = (半径×2)×(半径×2) = 半径×半径×4<BR> </DIV> <DIV align="left"> 結果を比べると、半径×半径の係数がπか4かの違いだけのようですね。<BR> </DIV> <DIV align="left"> πは定数ですから、直径の2乗が円の面積と比例し、π/4だけ違ってる。<BR> </DIV> <DIV align="left"> π/4は円積率と呼ばれますが、これは円の面積を直径で求めたことに関係します。<BR> </DIV> <DIV align="left">                            <IMG src="img056.jpg" border="0"><BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">  図で、2つの直角三角形の辺の長さに着目して、<BR> </DIV> <DIV align="left">  <BR> </DIV> <DIV align="left">  a = L/2 ― S/2 d = M/2 + S/2 ∴ 2d = M + S  <BR> </DIV> <DIV align="left">  　【 AD(b)の長さを求める】 <BR> </DIV> <DIV align="left">  b^2 = a^2 - (S/2)^2 = (L/2 ― S/2)^2 - (S/2)^2 = (L/2)^2 - 2(L/2)(S/2) = L^2/4 - 2LS／4) = (L^2 - 2LS)／4)<BR> </DIV> <DIV align="left">   ＿＿＿＿＿＿<BR> </DIV> <DIV align="left">   b = (√(L^2 - 2LS))／2　 <BR> </DIV> <DIV align="left">  は、 4b^2 = (2a)^2 - S^2 = (L ― S)2 - S2 = L^2 - 2LS 　 4b^2 = L^2 - 2LS　  <BR> </DIV> <DIV align="left"> 　((L - S)／2)^2 = b^2 + (S/2)^2 <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 　　【 BA(c)の長さを求める】<BR> </DIV> <DIV align="left">   　　　　　　　　　　　　　　　　　　＿＿＿＿＿＿　　　　　　＿＿＿＿＿＿<BR> </DIV> <DIV align="left">  c = L/2 - M/2 = BD - b = (√(M^2 + 2MS))／2 - (√(L^2 - 2LS))／2       <BR> </DIV> <DIV align="left"> ∴　2c = L - M          4c^2 = L^2 - 2LM + M^2<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 　　【 BD(c + b)の長さを求める】 <BR> </DIV> <DIV align="left">  BD^2 = (c + b)^2 = d^2 - (S/2)^2 = (M/2 + S/2)^2 - (S/2)^2 = (M/2)^2 + 2(M/2)(S/2) = M^2/4 + 2MS／4 =<BR> </DIV> <DIV align="left">                                                  ＿＿＿＿＿＿<BR> </DIV> <DIV align="left">  (M^2 + 2MS)／４　     BD（c + b) = (√(M^2 + 2MS))／2    <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">  KM = AM - AK = KB -MB　であるから　　【 BA(c)の長さを求める】<BR> </DIV> <DIV align="left">   　　　　　　　　　　　　　　　　　　＿＿＿＿＿＿　　　　　　＿＿＿＿＿<BR> </DIV> <DIV align="left">  c = L/2 - M/2 = BD - b = (√(M^2 + 2MS))／2 - (√(L^2 - 2LS))／2 　<BR> </DIV> <DIV align="left">   __________________ ________________ __________________ _________________<BR> </DIV> <DIV align="left">  L - M = (√(M^2 + 2MS)) - (√(L^2 - 2LS)) (L - M)^2 = ((√(M^2 + 2MS)) - (√(L^2 - 2LS)))^2<BR> </DIV> <DIV align="left">   _________________________________<BR> </DIV> <DIV align="left">  L^2 - 2LM + M^2 = M^2 + 2MS - 2(√((M^2 + 2MS)(L^2 - 2LS))) + L^2 - 2LS<BR> </DIV> <DIV align="left">   _________________________________<BR> </DIV> <DIV align="left">  L^2 - 2LM + M^2 - M^2 - 2MS - L^2 + 2LS = - 2(√((M^2 + 2MS)(L^2 - 2LS))) <BR> </DIV> <DIV align="left">   _________________________________<BR> </DIV> <DIV align="left">  - 2LM - 2MS + 2LS = - 2(√((M^2 + 2MS)(L^2 - 2LS)))<BR> </DIV> <DIV align="left">   __________________________________<BR> </DIV> <DIV align="left">  LM + MS - LS = √((M^2 + 2MS)(L^2 - 2LS)) <BR> </DIV> <DIV align="left">   ___________________________________<BR> </DIV> <DIV align="left">  ((LM + MS) - LS)^2 = (√((M^2 + 2MS)(L^2 - 2LS))) ^2<BR> </DIV> <DIV align="left">  <BR> </DIV> <DIV align="left">  ((LM + MS)^2 - 2LS(LM + MS)) + LS^2 = (M^2 + 2MS)(L^2 - 2LS)<BR> </DIV> <DIV align="left">  ((LM + MS)^2 - 2LS(LM + MS)) + LS^2 = M^2(L^2 - 2LS) + 2MS(L^2 - 2LS)<BR> </DIV> <DIV align="left">  L^2M^2 + 2LM*MS + M^2S^2 - 2L^2MS - 2LMS^2 + L^2S^2 = L^2M^2 - 2LM^2S + 2L^2MS - 4LMS^2 <BR> </DIV> <DIV align="left">  2LM^2S + M^2S^2 - 2L^2MS - 2LMS^2 + L^2S^2 = - 2LM^2S + 2L^2MS - 4LMS^2 <BR> </DIV> <DIV align="left">  2LM^2S + 2LM^2S + M^2S^2 - 2L^2MS - 2L^2MS - 2LMS^2 + L^2S^2 + 4LMS^2 = 0<BR> </DIV> <DIV align="left">  4LM^2S + M^2S^2 - 4L^2MS + 4LMS^2 - 2LMS^2 + L^2S^2 = 0<BR> </DIV> <DIV align="left">  4LM^2S + M^2S^2 - 4L^2MS + 2LMS^2 + L^2S^2 = 0<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> S^2(M^2 + 2LM + L^2) = 4S(L^2M - LM^2) ・・・　両辺を S で割る。<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">  S(M^2 + 2LM + L^2) = 4(L^2M - LM^2)<BR> </DIV> <DIV align="left">  <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">    　　　　　■ 条件を整理し、解を得る ■<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">  (1) S(M^2 + 2LM + L^2) = 4(L^2M - LM^2)<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">  (2) M - S = 5 M = S + 5      <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">  (3) ４外余積(4 * 120 = 480) = V = π(L^2 - M^2 - 2S^2)   V/π = π/π(L^2 - M^2 - 2S^2) = L^2 - M^2 - 2S^2<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">  (3)　より　　L^2 = α = V/π + M^2 + 2S^2 = V/π + (S + 5)^2 + 2S^2 = V/π + 3S^2 + 10S + 25 <BR> </DIV> <DIV align="left">    <BR> </DIV> <DIV align="left">   【 (1) の処理を行う】 .           <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> S((S + 5)^2 + 2L(S + 5) + L^2) = 4(L^2(S + 5) - L(S + 5)^2) <BR> </DIV> <DIV align="left"> S((S^2 + 10S + 25) + 2L(S + 5) + L^2) = 4(L^2(S + 5) - L(S^2 + 10S + 25)) <BR> </DIV> <DIV align="left"> S^3 + 10S^2 + 25S + 2LS^2 + 10LS + L^2S = 4L^2S + 20L^2 - 4LS^2 - 40LS - 100L <BR> </DIV> <DIV align="left"> S^3 + 10S^2 + 25S + 2LS^2 + 4LS^2 + 10LS + 40LS + L^2S - 4L^2S = 20L^2 - 100L <BR> </DIV> <DIV align="left"> S^3 + 10S^2 + 25S + 6LS^2 + 50LS - 3L^2S = 20L^2 - 100L <BR> </DIV> <DIV align="left"> 6LS^2 + 50LS + 100L = 20L^2 + 3L^2S - S^3 - 10S^2 - 25S<BR> </DIV> <DIV align="left"> 6LS^2 + 50LS + 100L = L^2(20 + 3S) - S(S + 5)^2 ・・・　両辺を2乗する。<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> ((6LS^2 + 50LS) + 100L)^2 = ( (L^2(20 + 3S)) - (S(S + 5)^2) )^2<BR> </DIV> <DIV align="left"> ((6LS^2 + 50LS)^2 + 2*100L(6LS^2 + 50LS) + (100L)^2 = (L^2(20 + 3S))^2 - 2S((S + 5)^2*(L^2(20 + 3S)) + <BR> </DIV> <DIV align="left"> (S(S + 5)^2)^2<BR> </DIV> <DIV align="left"> (36L^2S^4 + 2*(6LS^2)(50LS) + (50LS)^2 + 1200L^2S^2 + 10000L^2S + 10000L^2 = <BR> </DIV> <DIV align="left"> L^4(20^2 + 2*20*3S + 9S^2) - 2L^2S((S^2 + 10S + 25)*(20 + 3S)) + (S(S^2 + 10S + 25))^2    <BR> </DIV> <DIV align="left">  <BR> </DIV> <DIV align="left"> 36L^2S^4 + 600L^2S^3 + 3700L^2S^2 + 10000L^2S + 10000L^2 = <BR> </DIV> <DIV align="left"> L^4(400 + 120S + 9S^2) - 2L^2S((20 + 3S)(S^2 + 10S + 25)) + ((S^3 + 10S^2) + 25S)^2<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 36L^2S^4 + 600L^2S^3 + 3700L^2S^2 + 10000L^2S + 10000L^2 = 400L^4 + 120L^4S + 9L^4S^2 - 2L^2S(20S^2 +<BR> </DIV> <DIV align="left"> 200S +500 + 3S^3 + 30S^2 + 75S) + ((S^3)^2 + 2S^3*10S^2 + (10S^2)^2 + 2*25S(S^3 + 10S^2) + (25S)^2         <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 36L^2S^4 + 600L^2S^3 + 3700L^2S^2 + 10000L^2S + 10000L^2 = 400L^4 + 120L^4S + 9L^4S^2 - 40L^2S^3 - <BR> </DIV> <DIV align="left"> 60L^2S^3 - 400L^2S^2 - 150L^2S^2 -1000L^2S - 6L^2S^4+ S^6 + 20S^5 + 100S^4 + 50S^4 + 500S^3 + 625S^2  <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 36L^2S^4 + 600L^2S^3 + 3700L^2S^2 + 10000L^2S + 10000L^2 - S^6 - 20S^5 - 150S^4 - 500S^3 - 625S^2<BR> </DIV> <DIV align="left"> = 400L^4 + 120L^4S + 9L^4S^2 - 100L^2S^3 - 550L^2S^2 -1000L^2S - 6L^2S^4  <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> L^2(36S^4 + 600S^3 + 3700S^2 + 10000S + 10000) - S^6 - 20S^5 - 150S^4 - 500S^3 - 625S^2 <BR> </DIV> <DIV align="left"> = L^2(400L^2 + 120L^2S + 9L^2S^2 - 100S^3 - 550S^2 -1000S - 6S^4)  <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> (3) より L^2 = α = V/π + 3S^2 + 10S + 25 V/π = 480/π ≒ 152.7887　とする。 ∴　 L^2 = 3S^2 + 10S + 177.7887　　<BR> </DIV> <DIV align="left"> 　<BR> </DIV> <DIV align="left"> (3S^2 + 10S + 177.7887)(36S^4 + 600S^3 + 3700S^2 + 10000S + 10000) - S^6 - 20S^5 - 150S^4 - 500S^3 - 625S^2 <BR> </DIV> <DIV align="left"> = (3S^2 + 10S + 177.7887)(400(3S^2 + 10S + 177.7887) + 120S(3S^2 + 10S + 177.7887) + 9S^2(3S^2 + 10S + 177.7887) <BR> </DIV> <DIV align="left"> - 100S^3 - 550S^2 -1000S - 6S^4) <BR> </DIV> <DIV align="left"> (3S^2 + 10S + 177.7887)(36S^4 + 600S^3 + 3700S^2 + 10000S + 10000) - S^6 - 20S^5 - 150S^4 - 500S^3 - 625S^2 <BR> </DIV> <DIV align="left"> = (3S^2 + 10S + 177.7887)(27S^4 - 6S^4 + 90S^3 + 360S^3 - 100S^3 + 1200S^2 + 1200S^2 + 1600.0983S^2 - 550S^2 <BR> </DIV> <DIV align="left"> -1000S + 4000S + 21334.644S + 71115.48)<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> (3S^2 + 10S + 177.7887)(36S^4 + 600S^3 + 3700S^2 + 10000S + 10000) - S^6 - 20S^5 - 150S^4 - 500S^3 - 625S^2 <BR> </DIV> <DIV align="left">  = (3S^2 + 10S + 177.7887)(21S^4 + 350S^3 + 3450.0983S^2 + 24334.644S + 71115.48)<BR> </DIV> <DIV align="left">  <BR> </DIV> <DIV align="left">   【左辺の処理を行う】  <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> (3S^2(36S^4 + 600S^3 + 3700S^2 + 10000S + 10000) + 10S(36S^4 + 600S^3 + 3700S^2 + 10000S + 10000) <BR> </DIV> <DIV align="left"> + 177.7887(36S^4 + 600S^3 + 3700S^2 + 10000S + 10000) - S^6 - 20S^5 - 150S^4 - 500S^3 - 625S^2    <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 108S^6 + 1800S^5 + 360S^5 + 11100S^4 + 6000S^4 + 6400.3932S^4 + 30000S^3 + 106673.22S^3 + 37000S^3 + <BR> </DIV> <DIV align="left"> 30000S^2 + 100000S^2 + 657818.19S^2 + 100000S + 1777887S + 1777887 - S^6 - 20S^5 - 150S^4 - 500S^3 - 625S^2<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 107S^6 + 2140S^5 + 23350.3932S^4 + 173173.22S^3 + 787193.19S^2 + 1877887S + 1777887<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">   【右辺の処理を行う】<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">  (3S^2 + 10S + 177.7887)(21S^4 + 350S^3 + 3450.0983S^2 + 24334.644S + 71115.48)   <BR> </DIV> <DIV align="left">  = 3S^2( 21S^4 + 350S^3 + 3450.0983S^2 + 24334.644S + 71115.48) + 10S( 21S^4 + 350S^3 + 3450.0983S^2 +<BR> </DIV> <DIV align="left">  24334.644S + 71115.48) + 177.7887(21S^4 + 350S^3 + 3450.0983S^2 + 24334.644S + 71115.48) <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">  = 63S^6 + 1260S^5 + 17583.8576S^4 + 169730.96S^3 + 1070081.37163S^2 +5037579.52172S + 12643528.7391 <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">   【上記の解より、両辺の処理を行う】<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 107S^6 + 2140S^5 + 23350.3932S^4 + 173173.22S^3 + 787193.19S^2 + 1877887S + 1777887<BR> </DIV> <DIV align="left"> = 63S^6 + 1260S^5 + 17583.8576S^4 + 169730.96S^3 + 1070081.37163S^2 +5037579.52172S + 12643528.7391 <BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 107S^6 - 63S^ + 2140S^5 - 1260S^5 + 23350.3932S^4 - 17583.8576S^4 + 173173.22S^3 - 169730.96S^3 + <BR> </DIV> <DIV align="left"> 787193.19S^2 - 1070081.37163S^2 + 1877887S - 5037579.52172S + 1777887 - 12643528.7391 = 0<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> 44S^6 + 880S^5 + 5766.5356S^4 + 3442.26S^3 - 282888.18163S^2 - 3159692.52172S -10865641.7391 = 0<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left">   【 <A href="page102.html">六次方程式</A> で、S（小円）を求め,解を得る】<BR> </DIV> <DIV align="left">                                                                                                                                       <BR> </DIV> <DIV align="left">  S = 7.586875355　　　　　M = S + 5　より　　M = 7.586875355 + 5 = 12.586875355<BR> </DIV> <DIV align="left">   480/3.1415926 = 152.788747975 152.788745368<BR> </DIV> <DIV align="left"> (3) より L^2 = α = V/π + 3S^2 + 10S + 25 V/π = 480/π ≒ 152.7887　とする。 ∴　 L^2 = 3S^2 + 10S + 177.7887　<BR> </DIV> <DIV align="left"> 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 _____________________<BR> </DIV> <DIV align="left">  L^2 = 3(7.586875355)^2 + 10(7.586875355) + 177.7887 = 426.339486507 　L = √426.339486507 = 20.6479898902<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> ４外余積(4 * 120 = 480) より解を検証する。 π = 3.1415926・・・とする。　<BR> </DIV> <DIV align="left"> V(４外余積) = π(L^2 - M^2 - 2S^2) = 3.1415926（20.6479898902^2 - 12.586875355^2 - 2*7.586875355^2) = 479.99984928<BR> </DIV> <DIV align="left"> <BR> </DIV> <DIV align="left"> ∴ 解は　　　大円(L) = 20.6479898902 中円(M) = 12.586875355 小円(S) = 7.586875355<BR> </DIV> <DIV align="left">          <BR> </DIV> <CENTER> <BR> </CENTER> <CENTER> <BR> </CENTER> <CENTER> <BR> </CENTER> <CENTER> <BR> </CENTER> <CENTER> <BR> </CENTER> <CENTER> <BR> </CENTER> <CENTER> <A href="index.html" target="_top"><IMG src="img051.gif" height="28" width="70" border="0"></A><BR> </CENTER> <CENTER> <BR> </CENTER> <CENTER> <BR> </CENTER> <CENTER> <BR> </CENTER> <CENTER> <BR> </CENTER> <CENTER> <BR> </CENTER> <CENTER> <BR> </CENTER> <DIV align="right"> <BR> </DIV> <DIV align="right"> </DIV> <A name="_pageTail"></A> </BODY>