㐊 ☦ ᦀ ళ
㑦♽౽ƖU\DEKDW
·D ,ߨ߉476ύᓴࣾတዱ⩶ᐂⲃ◵.X VXPDSXUD 柋࣯ᓴߨ߉ 550 ώᓄฎ ύ಄ᓑฎ ώ
㑦♽౽לݴⷍध㐩7KH 0DF7XWRU +LVWRU\ RI 0DWKHPDWLFV DUFKLYH
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㐊 ☦ ᦀ ళ
㔉 ت ᅂ
ᭇॠ⌋ጦⲣ᷹⊣
㑦♽౽ƖU\DEKDW
·D ,ߨ߉476ύᓴࣾတዱ⩶ᐂⲃ◵.X VXPDSXUD 柋࣯ᓴߨ߉ 550 ώᓄฎ ύ಄ᓑฎ ώ
㓨⛴ᩂ೬ᖺㄡڌዷ⇉ᙆᖆ⃞६ႃᔷວ цڒ⃞ῳ፣⫱ᑵ
⳾♭ .XVXPDSXUD Ἢڌ⃞ဓḚፕ 3DWQD سㆉ цဓḚፕா‧ᗆ প⥩ᦧఖ 3ƗWDOLSXWUD 栾ᖺتႄᙌু⃞তఖ ц㊝ㄶḈᗓၣᙄⲟ ᔢ⠌ᤴ ц⥩ᦧఖ࠻ᄞᖺ㔽ộᙙ х⏄೬ộᙙ⃞ㇺఖ цࡕ࠶ٯغ⓿ ࢩ 栾९㓨⛴ᩂ೬ῳࣆㄩتݗغ⓿فம⃞㞖߱ᨷ㙝ᙄா⸋ఖ⃞ۥᔢ ໘㓾ⴂᄨ٨ⲣ᪃ऑ ц
㓨⛴ᩂ೬ா⥩ᦧఖਘ፣⫱ᑵ⳾♭⦰ᙀ⎧⸲ 栾༝ᕧ፣⫱ᑵ⳾
♭᪆ цڒ⃞ى⦰ۦتࡘᙌࡒᙥ 桌تᙥᖺъ㓨⛴ᩂ೬ᘟᙀы ƖU\DEKDWƯ\D 栾ᙀᕧࡕ࠶ 499 ၣ 栾দتᙥೳ␡ᙀထ╲ߊ ц ъ㓨⛴ᩂ೬ᘟᙀыभ፩ ³ೳᕄⲯ㕃´ 'DĞDJƯWLNƗ х³␡ⲣ´*DQL WDSƗGD 栾³ᗆ㒙⃞ႃ㊡´ .ƗODNUL\ƗSƗGD 栾³Ἦ´*RODSƗGD ⏯
࢛ ц⸋ᙀᕧࡕ࠶ 800 ၣညলⳝ⻌ፕثᕄ 栾ᙌダೲ⃞ᄉ㘐 ц ъ㓨
⛴ᩂ೬ᘟᙀыᙄⳝ೬ᤄⷷⷤ 栾Ḛࢭᖺாॠ६ႃ 栾೬⛓⸋ᙀ ㅪⲟㅻᬂࡏ⃞⇩⍭ ц
㓨⛴ᩂ೬ᔷۦ٦೬ᕥ㖭⃞⼽ỉ ц࡚ف π ݽ хᤳოⲯਘت ᤄسຖᕥ⌫⃞ⶫᨷᖺڒ⃞ᙆᙌڜⲯᆄ⃞ᚶ ц
ாᔷذ 栾π ݽ९லৼổ⃞ⷆ␡۞ᙌ㊟⃞ۗ цாᛖ⍁
⌫ႃذ 栾ছᖯتݗமວᔷ⃘།⃞ᧃ᭭ цفம㟧ᗋᗆᙛ 栾࣬
ᄽㅹΌ ³ࣛலⲣ´ ᄣ 3.14 + 64
625 < π < 3.14 + 169
625 цᎃᵥ ъٚ⎮␡ⲣыᕥ ⎮ᄞ㖭⃞ⷤᕄ 栾π = 3927
1250 栾९ 3.1416 цے ㅚتⷤᕄᖺ࣬ᄽዷࣺ 栾ᴸຖ⹐ цாॠऻᙙᗆ 栾⊸ᨌِ
ᄣ 3.1415926 < π < 3.1415927 栾ؿᄣࡒݗ㊟⃞ㄩیݽ 桌
┃ổ 22
7 栾ັổ 355
113 ц࡚ف┃ổ 22
7 ᖆထသ⟷ᔷວ㓨ళ⒮
ᄹ $UFKLPHGHV ዷ⇉ 栾ڒࢯΌல⃞೩࢜⠛ࡐᏮᤳ ㆬჽ 栾ᙄ
␡ᄣ 3 10
71 < π < 3 1
7 ц㔀فமڞ೩ 栾㓊ᕧ π ݽ 3.1416 ⃞ⷒ ヅ 栾ٜᕧ㓨⛴ᩂ೬⃞⦰ۦف ц㓨⛴ᩂ೬ᎁ 桌³100 ࣺ 4 ࡣ
ٗ 8 栾ࡣࣺ 62000 栾ᄣࢲΩᄠᖺ 20000 ⃞லৼ㒆ㄩیݽ´ ц९ π = 104 × 8 + 62000
20000 = 3.1416 цㅚݗ π ݽᄞ܊⃞೬६ႃᔷ
ວዷᏪΌ 栾ථڃㄶ♭ %KƗVNDUD ,, ᘽጮ໎ 3927
1250 ц⍭⎭ᖺ 㓨⛴ᩂ೬⠇တẺ⎧Όၬ۟ᕥᨷᄣ⃞ 栾ㆧᖺ⠛فம⃞ π = 3927
1250 ᙌᛖ⍁ဳጥ㓊ܫ 栾ᄗㅪتᤵ⇩⍭ ц
ாدᕥ㖭 栾㓨⛴ᩂ೬ڞڒࢷۦ⃞ᤳოⲯ⛘✈ুᕧغ цသ⟷
ڂገएັ 3WROHP\ ᖆࢷۦㅻᄨ 0o ࢲ 90o 㔡ख़ႃ⃞ოⲯ 栾ڒ ጮலৼ࢛ 360 ⏯ڪ 栾⏯ڪ☟☪࢛ 60 ⏯ڪ цদጮख़ᄠ࢛
60 ⏯ڪ 栾ت═ຖலპ 栾ኩოΌख़ᄠ⃞ 1
60 㒆ႃۗ
܊ⲯ⊡⃞㒆ႃ ц६ႃထߊ⃞ೳᕄ⦰ۦъ⫱ࢯ⛴ᘟᙀы6njU\D
6LGGKƗQWD فᓆ⸲ٜヅᙌᤳოⲯ 栾㓨⛴ᩂ೬⃞ᤳოⲯᄚম❫ᖺாᤴ
ⲯ⃞ళ⊃ذᔌㅪ⛘⃞ цாࢷۦㅻ⌫ف 栾ڒೲ៷Ό٦ၬ۟ጦဋਘ
ㄩیㅹ␡⏯ᔷ⇉⺼ ц㓨⛴ᩂ೬ᤳოⲯभᄨ 0◦ ࢲ 90◦ 㔡 3◦45′
⃞ᤳოݽ 栾ダㅻঔသ⟷ڂ⃞ოⲯ 栾ᙌࡒ㦼ᖣ㙝⃞॑ࢭ 桌࡚
ت 栾ጮலৼ࢛ 360 ⏯ڪ 栾ڪ☟☪࢛ 60 ⏯ڪ 栾ख़ᄠ r ٜ
ীலৼتᡥႃ㊡ цᕧᖺ 栾ᄨலৼ㒆 = 360 × 60 = 21600 ࢛ঘல ৼ㒆 2πr 栾ᄣख़ᄠ r = 3437.746 ц‟ঔᔷ࢛ 栾ঝㄩیݽᄣ r = 3438 цسࡣߡသ⟷ڂㆽᡥ 栾ጮலৼ࢛ 360 ڪ 栾⛘ጮख़ᄠদ
࢛ 60 ڪ ц㓨⛴ᩂ೬㦷⸟ᘻ▤ਘΩ▤মΌীتۗႃ㊡ 栾ㅚᴸ
‹ᖺتೲㅪᤵ цᎃᵥㅚ⍁╚ت⃞ႃ㊡ᨷ 栾९ᙌ VLQ7◦20′ = 449 х VLQ30◦ = 1719 ⏯⏯ ц࡚٩ 栾㓨⛴ᩂ೬ᖺⷆ␡ख़ო ℩‧ᕧἪா
⃞ᤳო▤ ⛘سᖺࡑო⃞㒆 栾ㅚٜᖺ⠛သ⟷ڂسী⃞ 栾㓨⛴ᩂ ೬⍄ख़ო MvYD 栾⸋ⷽউሣỄڂ⃞ბო ц㓨ፕۄڂໜ⻌
GVFKvED цᄞ܊খ⸭ჽṊ℩ی⃞ GVFKDLE 栾ㅚݗⷽ⃞উሣ❦
⟉ х᪤ዜ⬧ ц॒٩غ⓿ᗆ 栾ⳝ⧏᧺ࢯ ⚉ೲࢯف 栾♭㛦
ِᚑ ᕥ⃞ᛕፕழ 3ODWR RI 7LYROL ሣ⻌ፕثᕄ VLQXV 栾³ᤳო´
تⷽ९܊᭩ᕧᤴ ц
سຖᕥ⌫মڞ⸲ᖺ㓨⛴ᩂ೬⼽ỉᙆೲ⃞تݗ㘥ఝ цڒᐋ 桌ഫ
۟᧳ຖتݗᔳᔷN 栾܃N 㔀ڞᔳᔷa 㚣r1 栾㔀ڞᔳᔷb㚣r2 栾९ N = ax + r1 = by + r2 ዜ by − ax = c 栾࡚ف c = r1 − r2 цㅛㅻ
⇩⍭ㅚ㙒ક㙃 栾㓨⛴ᩂ೬Ⴣ⎧٦تᤄ▤ᆄسຖᕥ⌫ by − ax = c a хb хc ㇺᖺᔳᔷ ⃞ᤳᔳᔷㅛⶫ⃞ᨷࣁ 栾ؿໜᤴᨷࣁᏰႵࢲⶫ تᤄ✌⎧سຖᕥ⌫┺ цㅚ㘖ဉۦᖺ⾣ா‧ᗆغ‐ࣆࢧ⃞ ц㓨⛴ᩂ ೬⃞ᨷࣁ້㔦ذᖺロヷ℩㔀ᨷ 栾६ႃڂ⍄ⶫتᤄسຖᕥ⌫
ႇ॥ .XWWDND 栾ሣᅼ≏┨ ц㓨⛴ᩂ೬㒑ႇ॥⃞࠻ᨙ цᎃ
ᵥڒ⃞ῳථڃㄶ♭ %KƗVNDUD , ⏯ڂ⃞ⶫ㊝ 栾ΌἪڜᔷ⸩ⷁ ⲯㆁ 栾 by − ax = c , سൄⷝ a хb ٭⽶ ц㓨⛴ᩂ೬⃞ⶫᨷഫ ر 桌
ۦロヷ㔀ᨷ 栾মᄣࢲت⓼ࢧ⃞એਘ㚣ᔷ 桌
q хq1 хq2 хq3 х· · · хqm хr1 хr2 хr3 х· · · хrm+1 ц
࡚ف 栾
a = bq + r1
b = r1q1 + r2 栾 r1 = r2q2 + r3 栾 r2 = r3q3 + r4 栾
· · · ·
rm−2 = rm−1qm−1 + rm 栾 rm−1 = rmqm + rm+1 ц ڞ a = bq + r1 ڜࡏᕥ⌫ , ف 栾মᄣ
by = (bq + r1)x + c ц ᔓ
y = qx + y1 栾 by1 = r1x + c ц , ໜ b = r1q1 + r2 ڜࡏ , ف 栾ᄣ
x = q2y1 + x1 栾 r1x1 = r2y1 − c ц , ᎃذᨷㅹ␡رঔ 栾ؿጮዷᄣ⃞Ꮮࡒᣲ 栾ᙌ
1 y qx + y1 栾 by1 r1x + c 栾 ,1 2 x q1y + x1 栾 r1x1 r2y1 − c 栾,2 3 y1 q2x1 + y2 栾 r2y2 r3x1 + c 栾,3 4 x1 q3y2 + x2 栾 r3x2 r4y2 − c 栾,4 5 y2 q4x2 + y3 栾 r4y3 r5x2 + c 栾,5 6 x2 q5y3 + x3 栾 r4x3 r6y3 − c 栾,6
· · · ·
2n − 1yn−1 q2n−2xn−1 r2n−2yn r2n−1xn−1
+yn 栾 +c 栾 ,2n − 1
2n xn−1 q2n−1yn r2n−1xn r2nyn
+xn 栾 −c 栾 ,2n
2n + 1 yn q2nxn r2nyn+1 r2n+1xn
+yn+1 栾 +c 栾 ,2n + 1
٭㔀মڞㅪⲟࢲ 0 栾ٜমڞㅪⲟࢲᛖتᤵᤲ цࡣ࢛رࢧၬ⍁
ᇭᨦⷉ⹐ 桌
1 ރຖ٭㔀ㅪⲟࢲ 0 栾ࣁங a хb ٭⽶ 栾ᔓݝᔷ⏜٩ݗ㚣 ᔷᖺ 1 ц⢵ၷᔷᖺިᔷ 栾ࣁᙈᙌ r2n = 1 хr2n+1 = 0 ڞঘ q2n = r2n−1 ц ,2n ਘ,2n+1 ࢛ࢭ yn = q2nxn+c хyn+1 = c ц
═ xn ڞکتᔳᔷݽ t মᄣ yn ⃞تᔳᔷݽ ц 2n 栾খᄣࢲ xn−1
⃞ݽ 栾تᤵᤵᄔᏰ 栾ᙆᄞমᄣࢲ x хy ⃞ᔳᔷݽ 桍⢵ၷᔷᖺം
ᔷ 栾ࣁম ,2n − 1 ਘ ,2n ⏯ⶫ ц
2 ރຖ٭㔀ாᛖتᤵޒᤲ ц⢵ၷᔷᖺިᔷ 栾ࣁᙌ r2nyn+1 = r2n+1xn+c 栾ዜyn+1 = r2n+1xn + c
r2n
ц═xn تݗ‧⃞ᔳᔷݽ 栾
܃ yn+1 ٜᔳᔷ ц (2n + 1) 栾ᄣ yn ⃞ᔳᔷݽ цࡣتᤵᤵ☟☪
ᄔᏰ 栾মᄣ x хy ⃞ᔳᔷݽ 桍⢵ၷᔷᖺംᔷ 栾ࣁᙌ r2n−1xn = r2nyn −c 栾ዜ xn = r2nyn − c
r2n − 1 цڝ yn ت‧⃞ᔳᔷ 栾܃ xn ٜ
ᔳᔷ ц (2n) 栾ᄣ xn−1 ⃞ᔳᔷݽ цㅎᏰমⶫ x хy ⃞ᔳᔷ ݽ ц
㙝ᵃ 栾⢵ x = α хy = β ᖺᕥ⌫ by − ax = c ⃞ᙆᔳᔷ ⶫ 栾ࣁ x = bm + α хy = am + β m کሣᔳᔷ ٜᖺᕥ⌫⃞
ⶫ 栾ㅚᖺᕥ⌫⃞ㅛⶫ ц
㓨⛴ᩂ೬⃞ᨷࣁ 栾ⳝڒ⃞ῳථڃㄶ♭ᏰႵࢲⶫᕥ⌫ by−ax =
−c 栾ᄞ܊⃞६ႃᔷວ☟☪⇩⍭٦ㅚ㙒سຖᕥ⌫ક㙃 栾ᄣࢲ٦࡚
ڒتٲ╁ᚶ ц॒غ⓿ف 栾㓨⛴ᩂ೬,, ƖU\DEKDW
·D ,, ㅪتᤵᔌㅪ٦ 㓨⛴ᩂ೬⃞ᨷࣁ 栾ؿᎁㅹ␡মڞ⑹ऺঘᨷࣁম❫ᔕ⃞ᇭᨦ ц ᔷ⃚ၣ܊⍕┧⃞ㅚٲᚶ 栾ჽ٦६ႃᔷفᙌু⃞ႇ॥ἱ
⹐ ц
ாغ‐তڜᔷذ 栾سຖᕥ⌫ٜঞࢲفம хသ⟷⏯ம
⛓⃞ᩆሣ цفமতڜᔷু⦰ъٚ⎮␡ⲣыⷉ⹐٦سຖᕥ⌫┺
ક㙃 栾ؿᎁⶫᨷ 桌³ഫᕥ⌫ 栾ڞᤳ⼻ⲣࡏِʌ ц९ᎃ▤ᆄᕥ⌫
┺܊ⶫ цতသ⟷⛓ؾ‡ழ 'LRSKDQWXV ங⇩⍭سຖᕥ⌫ᄚᙌ
栾ڞ⠌ᄞڂጮᔳܫᔷسຖᕥ⌫⃞ᔳᔷⶫ⍄ⶫ ³ؾ‡ழᕥ
⌫´ цؾ‡ழ⇩⍭⃞ىᖺ㞖ᤄسຖᕥ⌫ 栾ڒⶫᕥ⌫ᗆ㓳ᕧᤳ
栾⸟⼻Ἢࣁⲯᖣᕥ⌫س়ἱ цⶫ٩ᤄᕥ⌫⃞ᗆݤ 栾९܃
ࡒݗㇺᖺᤳ 栾ڒٜঝت цသ⟷⛓ாㅚᕥ㖭⃞☻㔍 栾ⳝ 㓨⛴ᩂ೬ঘᄞ܊⃞६ႃᔷວჳ⳾٦ ц
㓨⛴ᩂ೬ㆧᙌ࡚೬ᔷᚶ 栾ഫ६ႃ⃞ⷒᔷᨷ 栾㒑
ၢᕥ х㒑⎧ᕥᨷࣁ · · · ⏯⏯ цڒㆧდࡏ٦تٲ␡ⲣ┙ᔷ 栾ݜ
ாㅻঔ⃞६ႃ⒗فᨈᙌ⃘Ἢㅻ цےᖺ 栾ڒ㓊ᕧல㖭⍕⃞ࡕ
㙝ᵃঝ⠇ᖆᙛ⃞६ႃೳ␡⦰ۦ цᕧख़ᄠ r ⃞Ἦ⃞㞓⍕ 栾㓨⛴
ᩂ೬⸭ πr2√
πr2 栾د⌹㎄⃞㞓⍕ࣁ⸭ၸدჽ⃞㖭⍕ × 㞖
× 1 2 ц
ъ㓨⛴ᩂ೬ᘟᙀыᖺ६ႃ⏜ت㊟ೳ␡⦰ۦ цாᙀف 栾㓨⛴
ᩂ೬ㅹΌڒᐋ⃞ᔷᕥᨷ 栾ⷆ␡٦㦫ㆀ х⃙ㆀ⃞ॖٷ㦼ਘ㓰ٷ 㦼⃞ㅹऑ 栾ⷉ⹐٦ᖃᙋٯᖮ⃞ᙆ㆕㦼ঘ࡚㆕ㅠㅹऑ 栾Ᏸ␡٦ᖃᙋ
⯟⃞⃘ῳᗆ㒙 栾ؿߡفமڂㆽᡥঔᏰ␡ذ࠶⍕ၣ цڒㆧᐋㅻ
Ἦ⠇ヷ⃞࠻ㅪᅼላ 栾মሀᙣⳝᄞ܊⃞ೳᕄວዷጥ⸟ ц
㓨⛴ᩂ೬ா६ႃ⌋ذᖺᙌ㊟ᄉ㘐⃞ڂḎ 栾1975 ၣ 4 ᙋ 19 ᖃ६ႃ⃘⃞⏜ت㘿ڂㅡⲩᖮু㓨⛴ᩂ೬⬭ 栾ᖺ٦⓿ᅡ ڒ⃞ ц
ᒄ Ḃ
ࣕತᒄḂ [1] ƖU\DEKDW
·D , ƖU\DEKDW
·Ư\D /HLGHQ 1874 ц
℠⊤ᒄḂ
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[3] + .HUQ 2Q VRPH IUDJPHQWV RI ƖU\DEKDW
·D -RXUQDO RI WKH 5R\DO
$VLDWLF 6RFLHW\ 201863 371 ± 387 ц
[4] %KƗX 'ƗMƯ %UƯHI QRWHV RQ WKH DJH DQG DXWKHQWLFLW\ RI WKH ZRUNV RI ƖU\DEKDW
·D 9DUƗKDPLKLUD %UDKPDJXSWD %KDW
·W
· RWSDOD DQG
%KƗVN ƗUDFKƗU\D -RXUQDO RI WKH 5R\DO $VLDWLF 6RFLHW\ 1865 392 ± 418 ц
[5] / 5RGHW 6XU OD YpULWDEOH VLJQLILFDWLRQ GH OD QRWDWLRQ QXPpULTXH LQYHQWpH SDU ƖU\DEKDW
·D -RXUQDO $VLDWLTXH 6HU 7 161880 440 ± 485 ц
[6] *5 .D\H7ZR ƖU\DEKDW
·DV %LEOLRWKHFD 0DWKHPDWLFD 101910 289 ± 292 ц
[7] -) )OHHW ƖU\DEKDW
·D¶V 6\VWHP RI H[SUHVVLQJ QXPEHUV -RXUQDO RI WKH 5R\DO $VLDWLF 6RFLHW\P 1911 109 ± 126 ц
[8] 1. 0D]XPGDU ƖU\DEKDW
·D¶V UXOH LQ UHODWLRQ WR LQGHWHUPLQDWH HTXDWLRQV RI WKH ILUVW GHJUHH %XOOHWLQ RI WKH &DOFXWWD 0DWKHPDW LFDO 6RFLHW\ 31911 ± 1912 11 ± 19 ц
[9] 5 6HZHOO 7KH ILUVW $U\D 6LGGKDQWD (SLJUDSKLD ,QGLFD 161921
± 1922 100 ± 144 桍171923 ± 1924 17 ± 104 ц
[10] $ $ .ULVKQDVZDPL $\\DQJDU 7KH PDWKHPDWLFV RI ƖU\DEKDW
·D 4XDUWHUO\ -RXUQDO RI WKH 0\WKLF 6RFLHW\ 161926 158 ± 179 ц [11] Ⱥ ɂ ȼɨɥɨɞɚɪɫɤɢɣ Ɉɱɟɪɤɢ ɢɫɬɨɪɢɢ ɫɪɟɞɧɟɜɟɤɨɜɨɣ ɢɧɞɢɣɫɤɨɣ
Ɇɚɬɟɦɚɬɢɤɢ ɂɷɞɚɬɟɥɶɫɬɜɨъɇɚɭɤɚы Ɇɨɫɤɜɚ 1977 ц [12] :( &ODUN 7KH ƖU\DEKDW
·Ư\D RI ƖU\DEKDW
·D 7KH 8QLYHUVLW\ RI
&KLFDJR 3UHVV 1930 ц
