\u83b1\u6602\u7eb3\u591a-\u76ae\u8428\u8bfa\uff08Leonardo Pisano\uff09\uff0c\u6216 “\u6590\u6ce2\u90a3\u5951”\uff0c\u662f\u5e0c\u814a\u795e\u5b66\u5bb6\u548c\u6570\u5b66\u5bb6\u6bd5\u8fbe\u54e5\u62c9\u65af\uff08\u7ea6\u516c\u5143\u524d569-475\u5e74\uff09\u81ea\u79f0\u7684\u827a\u672f\u5b66\u751f\uff0c\u4ed6\u521b\u9020\u4e86 “\u6570\u5b66 “\u4e00\u8bcd\uff08\u039c\u03ac\u03b8\u03b7\u03bc\u03b1, \u03b1\u03c4\u03bf\u03c2, \u03c4o, that which is learned\uff09\u6765\u4ee3\u8868\u7814\u7a76\u5f62\u72b6\u3001\u6570\u91cf\u548c\u7a7a\u95f4\u7684\u62bd\u8c61\u79d1\u5b66\uff08Donnegan 26\uff09\u3002 \u6bd5\u8fbe\u54e5\u62c9\u65af\u4e3b\u8981\u5bf9\u6570\u5b57\u7406\u8bba\u53ca\u5176\u5728\u97f3\u4e50\u4e2d\u7684\u5e94\u7528\u611f\u5174\u8da3\uff0c\u800c\u4e0d\u662f\u5728\u65e5\u5e38\u8ba1\u7b97\u4e2d\u4f7f\u7528\u6570\u5b57\uff08O’Shea\uff09\u3002 \u5e0c\u814a\u6570\u5b66\u5bb6\u6b27\u51e0\u91cc\u5f97\u4e5f\u5bf9\u6590\u6ce2\u90a3\u5951\u4ea7\u751f\u4e86\u5f88\u5927\u7684\u5f71\u54cd\uff1b\u5982\u524d\u6240\u8ff0\uff0c\u4ed6\u5728\u300a\u5143\u7d20\u300b\uff08\u7ea6\u516c\u5143\u524d300\u5e74\uff09\u4e2d\u768413\u672c\u51e0\u4f55\u5b66\u8457\u4f5c\uff08\u7ae0\u8282\uff09\u63d0\u4f9b\u4e86\u6590\u6ce2\u90a3\u5951\u719f\u77e5\u7684\u51e0\u4f55\u5b66\u5b9a\u4e49\u3001\u5047\u8bbe\u548c\u516c\u7406\u3002 \u5143\u7d20\u300b\u88ab\u8bb8\u591a\u4eba\u8ba4\u4e3a\u662f20\u4e16\u7eaa\u4ee5\u524d\u6700\u5177\u6709\u79d1\u5b66\u610f\u4e49\u7684\u6570\u5b66\u8457\u4f5c\uff08”\u6b27\u51e0\u91cc\u5f97”\uff09\u3002 “\u5373\u4f7f\u5728\u4eca\u5929\uff0c\u5f88\u5927\u4e00\u90e8\u5206\u7684\u6570\u5b66\u548c\u51e0\u4f55\u521d\u7b49\u6559\u80b2\u4e5f\u662f\u5efa\u7acb\u5728\u6b27\u51e0\u91cc\u5f97\u4f20\u7edf\u57fa\u7840\u4e0a\u7684”\uff08\u300a\u6b27\u51e0\u91cc\u5f97\u300b\uff09\u3002<\/p>\n\n\n\n
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\u53d7\u8fc7\u6559\u80b2\u5e76\uff08\u5f53\u7136\uff09\u7cbe\u901a\u62c9\u4e01\u8bed\uff0c\u6590\u6ce2\u90a3\u5951\u5728\u4e00\u672c\u540e\u6765\u88ab\u7ffb\u8bd1\u6210\u62c9\u4e01\u8bed\u5e76\u88ab\u547d\u540d\u4e3a\u300aAlgoritmi de Numero Indorum\u300b\uff08\u5173\u4e8e\u5370\u5ea6\u6559\u7b97\u672f\uff09\u7684\u4e66\u4e2d\u5b66\u4e60\u4e86al-Khwarizmi\u7684\u5370\u5ea6\u6559-\u963f\u62c9\u4f2f\u6559\u7b97\u672f\u8ba1\u7b97\u89c4\u5219\u6c47\u7f16\uff08Devlin, Man 24\uff09\u3002\u7b2c\u4e00\u6b21\u63a5\u89e6\u8fd9\u672c\u4e66\uff0c\u53ef\u80fd\u662f\u5728\u5e03\u5409\u4e9a\uff0c\u4e5f\u53ef\u80fd\u662f\u5728\u5730\u4e2d\u6d77\u65c5\u884c\u65f6\uff0c\u8fd9\u672c\u4e66\u6781\u5927\u5730\u5f71\u54cd\u4e86\u6590\u6ce2\u90a3\u5951\u5bf9\u5370\u5ea6\u6559-\u963f\u62c9\u4f2f\u6559\u7b97\u672f\u7684\u7406\u89e3\u548c\u5b9e\u8df5\u3002<\/p>\n\n\n\n
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\u53e6\u4e00\u4e2a\u539f\u56e0\u662f\uff0c\u7b97\u76d8\u4e3b\u4e49\u8005\uff08\u7b97\u76d8\u7684\u62e5\u62a4\u8005\uff09\u548c\u7b97\u672f\u5bb6\uff08\u8d5e\u6210\u4f7f\u7528\u5370\u5ea6\u6559-\u963f\u62c9\u4f2f\u6570\u5b57\u7684\u4eba\uff09\u4e4b\u95f4\u7684\u793e\u4f1a\u77db\u76fe\u4f7f\u8fd9\u4e00\u66f4\u65b0\u3001\u66f4\u6709\u6548\u7684\u7cfb\u7edf\u591a\u5e74\u6765\u4e00\u76f4\u6ca1\u6709\u88ab\u6b27\u6d32\u793e\u4f1a\u666e\u904d\u91c7\u7528\u3002\u4e8b\u5b9e\u4e0a\uff0c”\u5728\u6574\u4e2a\u4e2d\u4e16\u7eaa\uff0c\u5546\u4eba\u4eec\u7684\u8d26\u672c\u4e0a\u90fd\u6709\u7f57\u9a6c\u6570\u5b57\uff0c\u8fd9\u8868\u660e[\u6709\u4e9b\u4eba]\u4ecd\u7136\u662f\u575a\u5b9a\u7684\u7b97\u76d8\u4e3b\u4e49\u8005”\uff08Levy 117\uff09\u3002<\/p>\n\n\n\n
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\u6590\u6ce2\u90a3\u5951\u8ba4\u8bc6\u5230\u96f6\u548c\u5176\u4ed6\u963f\u62c9\u4f2f\u6570\u5b57\u7684\u4f18\u70b9\uff0c\u4ed6\u77e5\u9053\u597d\u5904\u8fdc\u8fdc\u5927\u4e8e\u5371\u5bb3\u3002\u540c\u610f\u4ed6\u7684\u610f\u5927\u5229\u5546\u4eba\u4e5f\u7ee7\u7eed\u4f7f\u7528\u5b83\u4eec\uff0c\u5373\u4f7f\u662f\u5728\u88ab\u7981\u6b62\u7684\u60c5\u51b5\u4e0b\uff08Seife 81\uff09\u3002\u6b64\u5916\uff0c\u65e0\u73b0\u91d1\u8d38\u6613\u793e\u4f1a\u7684\u666e\u53ca\uff08\u901a\u8fc7\u53d1\u884c\u6c47\u7968\u548c\u652f\u7968\uff09\uff0c\u4ee5\u53ca\u5229\u606f\u8ba1\u7b97\u7684\u53d1\u5c55\uff0c\u8feb\u4f7f\u94f6\u884c\u76f8\u5f53\u52a1\u5b9e\u5730\u63a5\u53d7\u6590\u6ce2\u90a3\u5951\u63a8\u884c\u7684\u65b0\u8ba1\u7b97\u65b9\u6cd5\u3002\u6700\u540e\uff0c\u5404\u56fd\u653f\u5e9c\u5c48\u670d\u4e8e\u5546\u4e1a\u538b\u529b\uff0c\u963f\u62c9\u4f2f\u8bb0\u6570\u6cd5\u5728\u610f\u5927\u5229\u84ec\u52c3\u53d1\u5c55\uff0c\u5e76\u5f88\u5feb\u4f20\u904d\u6574\u4e2a\u6b27\u6d32\uff08Seife 81\uff09\u3002<\/p>\n\n\n\n
\u7b97\u76d8\u4e3b\u4e49\u8005\uff08\u6709\u65f6\u4e5f\u62fc\u6210abakists\uff09\u662f\u559c\u6b22\u4f7f\u7528\u4f20\u7edf\u7684\u7f57\u9a6c\u6570\u5b57\u548c\u673a\u68b0\u5de5\u5177\uff08abaci\u3001\u6728\u677f\u6216\u683c\u5b50\u56fe\u6848\u7684\u5e03\uff09\u6765\u8fdb\u884c\u8fd0\u7b97\u7684\u4eba\uff0c\u800c\u7b97\u6cd5\u5bb6\u662f\u63a5\u53d7\u4e66\u9762\u7684\u3001\u8c61\u5f81\u6027\u7684\u5370\u5ea6\u6559-\u963f\u62c9\u4f2f\u6559\u7684\u4f4d\u503c\u7b26\u53f7\uff08\u5305\u62ec\u96f6\uff09\u5e76\u4f7f\u7528\u7b97\u6cd5\u65b9\u6cd5\u6216\u516c\u5f0f\u8fdb\u884c\u8ba1\u7b97\u7684\u4eba\uff08\u5229\u7ef4112\uff09\u3002\u5927\u591a\u6570\u7b97\u6cd5\u5bb6\u653e\u5f03\u4e86\u7b97\u76d8\u7684\u4f7f\u7528\u3002<\/p>\n\n\n\n
\u54f2\u5b66\u5bb6Gregor Reisch\u7684\u8457\u4f5c\u300aMargarita Philosophica\u300b\uff08\u667a\u6167\u4e4b\u73e0\uff09\uff081503\uff09\u4e2d\u7684\u4e00\u5e45\u63d2\u56fe\u63cf\u7ed8\u4e86\u4f20\u7edf\u548c\u73b0\u4ee3\u7b97\u672f\u65b9\u6cd5\u4e4b\u95f4\u7684\u6597\u4e89\u3002\u8fd9\u5e45\u540d\u4e3a “\u7b97\u672f\u7684\u5bd3\u8a00 “\u7684\u6728\u523b\u7248\u753b\u63cf\u7ed8\u4e86\u559c\u6b22\u7f57\u9a6c\u6570\u5b57\u5e76\u575a\u6301\u4f20\u7edf\uff08\u4f7f\u7528\u7b97\u76d8\uff09\u7684\u4eba\u4e0e\u91c7\u7528\u7b97\u6cd5\u5e76\u7528\u7eb8\u7b14\u8ba1\u7b97\u7684\u4eba\u4e4b\u95f4\u7684\u7ade\u4e89\u3002\u6807\u6709 “\u6ce2\u63d0\u4e4c\u65af “\u548c “\u6bd5\u8fbe\u54e5\u62c9\u65af “\u7684\u6a2a\u5e45\u6807\u660e\u4e86\u56fe\u4e2d\u4eba\u7269\u7684\u8eab\u4efd\u3002\u53e4\u5e0c\u814a\u5b66\u8005\u6bd5\u8fbe\u54e5\u62c9\u65af(\u7ea6\u516c\u5143\u524d500\u5e74)\u5728\u63d2\u56fe\u7684\u53f3\u8fb9\uff0c\u4ed6\u6101\u7709\u82e6\u8138\uff0c\u4f7f\u7528\u4e00\u5757\u6570\u677f\uff0c\u4ed6\u4ee3\u8868\u7684\u662f\u963f\u5df4\u514b\u4e3b\u4e49\u8005\u3002\u5de6\u8fb9\u662f\u7f57\u9a6c\u54f2\u5b66\u5bb6\u6ce2\u7231\u4fee\uff08\u7ea6\u516c\u5143500\u5e74\uff09\uff0c\u4ed6\u4f7f\u7528\u5370\u5ea6-\u963f\u62c9\u4f2f\u6570\u5b57\uff0c\u4ee3\u8868\u963f\u5c14\u6208\u5c14\u4e3b\u4e49\u8005\uff08”\u6590\u6ce2\u90a3\u5951 “\u8457\u540d\uff1b”\u4e89\u8bae”\uff09\uff0c\u663e\u5f97\u5f88\u9ad8\u5174\u3002<\/p>\n\n\n\n
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\u5728\u4e24\u4e2a\u6570\u5b66\u5bf9\u624b\u4e4b\u95f4\uff0c\u5f98\u5f8a\u7740\u7b97\u672f\u7684\u7f2a\u65af\u5973\u795eArithmetica\uff0c\u5979\u8eab\u7a7f\u88c5\u9970\u7740\u963f\u62c9\u4f2f\u6570\u5b57\u7684\u88d9\u5b50\u3002\u5979\u7684\u670d\u9970\u88c5\u9970\u548c\u5979\u5bf9\u6ce2\u7231\u4fee\u65af\u5f62\u8c61\u7684\u9752\u7750\u8868\u660e\uff0c\u523014\u4e16\u7eaa\u672b\uff0c\u963f\u5c14\u6208\u91cc\u897f\u7c73\u6b63\u53d8\u5f97\u8d8a\u6765\u8d8a\u6d41\u884c\uff08O’Shea\uff1bO’Connor\u548cRobertson\uff09\u3002\u57281400\u5e74\u81f31700\u5e74\u4e4b\u95f4\u7684\u67d0\u4e2a\u65f6\u5019\uff0c\u5b83\u6700\u7ec8\u5360\u636e\u4e86\u4e0a\u98ce\u3002<\/p>\n\n\n\n
\u6b27\u6d32\u7ecf\u6d4e\u4f53\u7cfb\u5bf9\u65b0\u6570\u5b66\u7684\u91c7\u7528\u81f3\u5c11\u53ef\u4ee5\u8bf4\u662f\u8fdf\u7f13\u7684\uff0c\u5982\u679c\u7528Reisch\u4e66\u4e2d\u7684\u6728\u523b\u6765\u63cf\u7ed8\uff0c\u5b83\u53ef\u80fd\u662f\u4e00\u53ea\u8e52\u8dda\u7684\u4e4c\u9f9f\uff0c\u800c\u5370\u5ea6\u6559-\u963f\u62c9\u4f2f\u6570\u5b57\u5728\u5b66\u672f\u754c\u7684\u4f20\u64ad\u5219\u662f\u4e00\u53ea\u51b2\u523a\u7684\u5154\u5b50\u3002\u6590\u6ce2\u90a3\u5951\u5728\u300aLiber Abaci\u300b\u4e2d\u62e5\u62a4\u4e86al-Khwarizmi\u548cKamil\u7684\u5370\u5ea6\u6559-\u963f\u62c9\u4f2f\u6570\u5b57\u7cfb\u7edf\uff0c\u8be5\u4e66\u73b0\u5728\u88ab\u8ba4\u4e3a\u662f “\u5411\u897f\u65b9\u4f20\u64ad\u5370\u5ea6\u6559-\u963f\u62c9\u4f2f\u6570\u5b57\u4ee5\u53ca\u5982\u4f55\u7528\u5b83\u4eec\u8fdb\u884c\u52a0\u51cf\u4e58\u9664\u7684\u5f00\u521b\u6027\u8457\u4f5c”\u3002\u6bd4\u767e\u79d1\u5168\u4e66\u5f0f\u7684\u300aLiber Abaci\u300b\u66f4\u6709\u5f71\u54cd\u529b\u7684\u662f\u4ed6\u7684\u8f83\u5c0f\u7684\u3001\u66f4\u5bb9\u6613\u7406\u89e3\u7684\u6587\u6458\u300aLibro di Minor Guise\u300b(\u300a\u8f83\u5c0f\u7684\u4e66\u300b)\uff0c\u8be5\u4e66\u5728\u5546\u4eba\u4e2d\u5e7f\u6cdb\u6d41\u4f20\uff0c\u88ab\u6709\u79ef\u6781\u6027\u7684\u5546\u4eba\u3001\u5546\u4eba\u548c\u94f6\u884c\u5bb6\u590d\u5236\u4e86\u65e0\u6570\u6b21(\u5229\u7ef4117)\u3002<\/p>\n\n\n\n
\u5c3d\u7ba1\u6590\u6ce2\u90a3\u5951\u5411\u4eba\u4eec\u5c55\u793a\u4e86\u963f\u62c9\u4f2f\u6570\u5b57\u5bf9\u4e8e\u8fdb\u884c\u590d\u6742\u8ba1\u7b97\u7684\u4f5c\u7528\uff0c\u4f46\u5f53\u65f6\u8fd8\u6ca1\u6709\u53d1\u660e\u5370\u5237\u673a\uff1b\u6240\u4ee5\uff0c\u5728\u4e2d\u4e16\u7eaa\u7684\u5927\u90e8\u5206\u65f6\u95f4\u91cc\uff0c\u77e5\u8bc6\u4f20\u64ad\u5f97\u5f88\u6162\u3002”\u6559\u7687\u548c\u738b\u516c\u751a\u81f3\u4f1f\u5927\u7684\u5b97\u6559\u673a\u6784\u6240\u62e5\u6709\u7684\u4e66\u7c4d\u8fdc\u6bd4\u73b0\u65f6\u4ee3\u7684\u8bb8\u591a\u519c\u6c11\u5c11\u5f97\u591a”\uff08Smith and Karpinski\uff09\u3002\u7136\u800c\uff0c\u4e0e\u5927\u591a\u6570\u4f7f\u76c8\u5229\u6548\u7387\u63d0\u9ad8\u7684\u521b\u65b0\u548c\u7b56\u7565\u4e00\u6837\uff0c\u6590\u6ce2\u90a3\u5951\u4e66\u4e2d\u7684\u5b9e\u9645\u5e94\u7528\u4e0d\u80fd\u4e0d\u5728\u897f\u65b9\u4e16\u754c\u53d1\u5c55\u8d77\u6765\u7684\u5e02\u573a\u7ecf\u6d4e\u7684\u706b\u67f4\u76d2\u4e2d\u50cf\u91ce\u706b\u4e00\u6837\u8513\u5ef6\u3002<\/p>\n\n\n\n
\u4e00\u4e9b\u5386\u53f2\u5b66\u5bb6\u65ad\u8a00\uff0c\u5176\u4ed6\u4eba\u5173\u4e8e\u7b97\u672f\u7684\u8bba\u6587\uff0c\u5982Alexander de Villa Dei\uff08\u7ea6\u516c\u51431240\u5e74\uff09\u7684\u300aCarmen de Algorismo\u300b\u548cHalifax\u7684John\uff08Sacrobosco\uff0c\u7ea6\u516c\u51431250\u5e74\uff09\u7684\u300aAlgorismus Vulgaris\u300b\u6bd4Fibonacci\u7684\u8bba\u6587\u5f71\u54cd\u66f4\u5927\uff0c\u4f7f\u7528\u66f4\u5e7f\u6cdb\uff0c”\u65e0\u7591\u5bf9\u6570\u5b57\u5728\u666e\u901a\u4eba\u4e2d\u7684\u4f20\u64ad\u505a\u51fa\u4e86\u66f4\u5927\u7684\u8d21\u732e”\uff08Smith and Karpinski\uff1b”\u8fc7\u6e21”\uff09\u3002\u7136\u800c\uff0c\u6700\u8fd1\u7684\u7814\u7a76\u53d1\u73b0\u4e86\u6570\u4ee5\u767e\u8ba1\u88ab\u79f0\u4e3alibri d’abbaco\uff08”abbacus books”\uff09\u6216trattati d’abbaco\uff08”abbacus tracts”\uff09\u7684\u624b\u7a3f\uff0c\u8fd9\u4e9b\u624b\u7a3f\u660e\u786e\u6307\u51fa\u83b1\u6602\u7eb3\u591a\u7684\u300a\u81ea\u7531\u963f\u5df4\u5947\u300b\u662f\u4e2d\u4e16\u7eaa\u665a\u671f “\u70b9\u71c3\u73b0\u4ee3\u5546\u4e1a\u4e16\u754c\u4e4b\u706b\u7684\u67aa\u58f0 “\u6216 “\u706b\u82b1”\uff0c”\u56e0\u4e3a\u5b83\u662f\u4e00\u4e2a\u9ad8\u5ea6\u6613\u71c3\u7684\u666f\u89c2”\uff0c\u5df2\u7ecf\u7ecf\u5386\u4e86\u5feb\u901f\u7684\u5546\u4e1a\u6269\u5f20\uff08Devlin Finding 25\uff0c27\uff0c33\uff09\u3002<\/p>\n\n\n\n
\u73b0\u5728\u770b\u6765\uff0c\u897f\u65b9\u4e16\u754c\u5bf9\u91c7\u7528\u83b1\u6602\u7eb3\u591a-\u76ae\u8428\u8bfa\u5982\u6b64 “\u4e25\u683c “\u5730\u62e5\u62a4\u7684\u65b0\u6570\u5b57\u4f3c\u4e4e\u96be\u4ee5\u60f3\u8c61\uff1b\u8fd9\u4e9b\u6570\u5b57\u663e\u7136\u6bd4\u5f53\u65f6\u5728\u57fa\u7763\u6559\u6b27\u6d32\u76db\u884c\u7684\u8ba1\u7b97\u65b9\u6cd5\u8981\u4f18\u8d8a\u5f97\u591a! \u752821\u4e16\u7eaa\u7684\u8bdd\u6765\u8bf4\uff0cFibonacci\u7684Liber Abaci\u662f\u4e00\u79cd\u65b0\u7684\u5e02\u573a\u98a0\u8986\u5de5\u5177\uff0c\u56e0\u4e3a\u5b83\u9002\u5408\u4e8e\u4e00\u4e2a\u65b0\u5174\u7684\u5e02\u573a\u9886\u57df\uff08\u56fd\u9645\u8d38\u6613\uff09\uff0c\u800c\u8be5\u884c\u4e1a\u73b0\u6709\u7684\u5de5\u5177\uff08\u7f57\u9a6c\u6570\u5b57\u548c\u7b97\u76d8\uff09\u5e76\u6ca1\u6709\u63d0\u4f9b\u8db3\u591f\u7684\u670d\u52a1\u3002<\/p>\n\n\n\n
\u6700\u521d\uff0c\u5f53\u7136\u4e5f\u662f\u5728\u4ed6\u6d3b\u7740\u7684\u65f6\u5019\uff0c\u6590\u6ce2\u90a3\u5951\u7684\u8457\u4f5c\u5728\u610f\u5927\u5229\u5f97\u5230\u4e86\u6df1\u5165\u7684\u7814\u7a76\u548c\u8d5e\u8d4f\u3002\u8be5\u4e66\u4e2d\u5b9e\u7528\u7684 “\u7ecf\u6d4e “\u90e8\u5206\u88ab\u624b\u5199\u6210\u624b\u6284\u672c\uff0c\u5e76\u4ee5\u6570\u5343\u4efd\u7684\u6570\u91cf\u5206\u53d1\uff0c\u636e\u63a8\u6d4b\uff0c\u8fd9\u4e0d\u4ec5\u662f\u5546\u4eba\u548c\u8d38\u6613\u5546\u7684\u884c\u4e3a\uff0c\u4e5f\u662f13\u4e16\u7eaa\u4e0b\u534a\u53f6\u7a81\u7136\u51fa\u73b0\u7684\u8bb8\u591a\u610f\u5927\u5229\u767d\u8bdd\u5b66\u6821\u7684\u5b66\u751f\u7684\u884c\u4e3a\u3002\u8fd9\u4e9b\u5b66\u6821\u9664\u4e86\u6559\u6388\u6587\u5b66\u5916\uff0c\u8fd8\u6559\u6388\u5546\u4e1a\u6570\u5b66\uff08abbaco\uff09\u548c\u590d\u6742\u7684\u8bb0\u8d26\u6280\u80fd\u3002\u56e0\u6b64\uff0cLiber Abaci\u4e0d\u4ec5\u5bf9Liber Abaci\u4e4b\u540e\u51fa\u7248\u7684\u5927\u91cf\u7b97\u672f\u5c0f\u518c\u5b50(trattati d’abaco)\u4ea7\u751f\u4e86\u91cd\u5927\u5f71\u54cd\uff0c\u800c\u4e14\u5bf914\u4e16\u7eaa\u84ec\u52c3\u53d1\u5c55\u7684abbaco\u5b66\u6821\u4e5f\u4ea7\u751f\u4e86\u91cd\u5927\u5f71\u54cd(“\u6559\u80b2”)\u3002<\/p>\n\n\n\n
Liber Abaci\u7684\u89c4\u6a21\u4e4b\u5927\uff0c\u51e0\u4e4e\u4e0d\u53ef\u80fd(\u5f53\u7136\u4e0d\u5207\u5b9e\u9645)\u5b8c\u6574\u5730\u590d\u5236\u3002Liber Abaci\u7684\u82f1\u6587\u7248\uff08\u7531\u7f8e\u56fd\u4ebaLaurence Sigler\u5f00\u59cb\u7ffb\u8bd1\uff0c\u5e76\u7531\u4ed6\u7684\u59bb\u5b50Joan\u5728\u6b7b\u540e\u5b8c\u6210\uff09\uff0c”\u6709600\u591a\u9875\uff0c\u7528\u76f8\u5f53\u5c0f\u7684\u5b57\u4f53\u8bbe\u7f6e\uff1b\u4f8b\u5b50\u662f\u5360\u636e\u5927\u90e8\u5206\u9875\u9762\u7684\u5185\u5bb9”\uff08Devlin\uff0cFinding 88\uff09\u3002\u8fd9\u672c\u4e66\u4e00\u5f00\u59cb\u5c31\u5bf9\u5982\u4f55\u4e66\u5199\u548c\u64cd\u4f5c\u5370\u5ea6-\u963f\u62c9\u4f2f\u6570\u5b57\u8fdb\u884c\u4e86\u89e3\u91ca\u548c\u8bf4\u660e\uff0c\u7136\u540e\u6590\u6ce2\u90a3\u5951\u7ee7\u7eed\u63d0\u4f9b\u4e86\u5370\u5ea6-\u963f\u62c9\u4f2f\u7b97\u672f\u7684\u57fa\u672c\u539f\u7406\uff0c\u4ed6 “\u7528\uff08\u8bb8\u591a\uff09\u5177\u4f53\u7684\u6570\u5b57\u4f8b\u5b50\u6765\u89e3\u91ca\uff0c\u5f88\u50cf\u4eca\u5929\u5c0f\u5b66\u751f\u7684\u5b66\u4e60\u65b9\u5f0f”\uff08Devlin, Finding 116\uff09\u3002\u5728\u968f\u540e\u7684\u7ae0\u8282\u4e2d\uff0c\u4ed6\u63d0\u4f9b\u4e86\u73b0\u5b9e\u4e16\u754c\u7684\u4f8b\u5b50\uff0c\u5e76\u5c55\u793a\u4e86\u89e3\u51b3\u4e0e\u5546\u4e1a\u548c\u516c\u53f8\u7279\u522b\u76f8\u5173\u7684\u95ee\u9898\u7684\u5b9d\u8d35\u65b9\u6cd5\u3002\u5e9e\u5927\u7684\u7b2c\u5341\u4e8c\u7ae0\u5305\u542b\u4e86259\u4e2a\u5de5\u4f5c\u5b9e\u4f8b\uff08\u5728Sigler\u7684\u7ffb\u8bd1\u4e2d\uff0c\u8be5\u7ae0\u5360\u4e86187\u9875\u7684\u5370\u5237\u9875\uff09\uff08Devlin\uff0cFinding 119\uff09\u3002<\/p>\n\n\n\n
\u6590\u6ce2\u90a3\u5951\u5728\u300a\u81ea\u7531\u7684\u963f\u5df4\u5947\u300b\u4e2d\u5f15\u5165\u4e86\u963f\u62c9\u4f2f\u6570\u5b57\uff0c\u5e76\u4f5c\u4e86\u7b80\u5355\u7684\u8bf4\u660e\uff1a “\u4e5d\u4e2a\u5370\u5ea6\u6570\u5b57\u662f\uff1a 9 8 7 6 5 4 3 2 1.\u201d “\u7136\u540e\u4ed6\u65ad\u8a00\uff0c\u7528\u8fd9\u4e5d\u4e2a\u6570\u5b57\u548c\u7b26\u53f70.\u53ef\u4ee5\u5199\u51fa\u4efb\u4f55\u6570\u5b57\uff08Horadam\uff09\u3002\u83b1\u6602\u7eb3\u591a\u4f30\u8ba1\uff0c\u5927\u591a\u6570\u4eba\u5bf9\u7406\u8bba\u7684\u3001\u62bd\u8c61\u7684\u95ee\u9898\u4e0d\u4f1a\u6709\u4ec0\u4e48\u5174\u8da3\uff0c\u4ed6\u4eec\u4f1a\u5bf9\u5b9e\u9645\u5e94\u7528\u611f\u5174\u8da3\u3002\u56e0\u6b64\uff0c\u83b1\u6602\u7eb3\u591a “\u60f3\u65b9\u8bbe\u6cd5\u628a\u62bd\u8c61\u7684\u4e1c\u897f\u7a7f\u4e0a\u719f\u6089\u7684\u3001\u65e5\u5e38\u7684\u8863\u670d”\u3002\u4ed6\u7528 “\u5a31\u4e50\u6570\u5b66 “\u628a\u963f\u62c9\u4f2f\u6570\u5b57\u548c\u5370\u5ea6\u6559-\u963f\u62c9\u4f2f\u6559\u7684\u4f4d\u503c\u5341\u8fdb\u4f4d\u5236\u90fd\u4ecb\u7ecd\u7ed9\u4e86\u6b27\u6d32\uff08\u5fb7\u592b\u6797\uff0c\u658769\uff1b\u5965\u5eb7\u7eb3\u548c\u7f57\u4f2f\u900a\uff09\u3002\u7531\u4e8e\u4ed6\u66fe\u5e7f\u6cdb\u5730\u65c5\u884c\u8fc7\uff0c\u5e76\u4e14\u77e5\u9053 “\u4ed6\u7684\u8bb8\u591a\u540c\u80de\u90fd\u662f\u7ecf\u5e38\u65c5\u884c\u7684\u4eba”\uff0c\u83b1\u6602\u7eb3\u591a\u76f8\u4fe1\u5173\u4e8e\u65c5\u884c\u7684\u91d1\u94b1\u95ee\u9898\u4e00\u5b9a\u4f1a\u5f15\u8d77\u5e7f\u6cdb\u7684\u5174\u8da3\uff0c\u6240\u4ee5\u8fd9\u4e9b\u95ee\u9898\u6784\u6210\u4e86\u4ed6\u63a5\u4e0b\u6765\u7684\u4e00\u7ec4\u4f8b\u5b50\u3002\u5bf9\u4e8e\u4ed6\u7684\u7b2c\u4e00\u4e2a\u65c5\u884c\u8005\u95ee\u9898\uff0c\u4ed6\u5199\u9053\u3002\u4e00\u4e2a\u4eba\u53bb\u5362\u5361\u505a\u751f\u610f\uff0c\u60f3\u8d5a\u53d6\u53cc\u500d\u7684\u5229\u6da6 \u4ed6\u5728\u90a3\u91cc\u82b1\u4e8612\u8fea\u7eb3\u91cc\u3002\u7136\u540e\u4ed6\u79bb\u5f00\uff0c\u7ecf\u8fc7\u4f5b\u7f57\u4f26\u8428\uff0c\u5728\u90a3\u91cc\u4ed6\u7684\u94b1\u7ffb\u4e86\u4e00\u756a\uff0c\u82b1\u4e8612\u7b2c\u7eb3\u5c14\u3002\u7136\u540e\u4ed6\u53c8\u56de\u5230\u6bd4\u8428\uff0c\u628a\u94b1\u7ffb\u4e86\u4e00\u500d\uff0c\u82b1\u4e8612\u7b2c\u7eb3\u5c14\uff0c\u6709\u4eba\u63d0\u51fa\u4ed6\u6700\u540e\u4ec0\u4e48\u4e5f\u6ca1\u6709\u4e86\u3002\u6709\u4eba\u5bfb\u6c42\u4ed6\u4e00\u5f00\u59cb\u6709\u591a\u5c11\u94b1\uff08\u5fb7\u592b\u6797\uff0c\u53d1\u73b0124\uff09\u3002\u83b1\u6602\u7eb3\u591a\u5728Liber Abaci\u4e2d\u6d89\u53ca\u7684\u5176\u4ed6\u4e3b\u9898\u6709\uff1a\u4e58\u6cd5\u548c\u52a0\u6cd5\uff1b\u51cf\u6cd5\uff1b\u9664\u6cd5\uff1b\u5206\u6570\uff1b\u8d38\u6613\u548c\u8d27\u5e01\u7684\u5b9e\u9645\u4efb\u52a1\u548c\u89c4\u5219\uff1b\u4f1a\u8ba1\uff1b\u4e8c\u6b21\u6839\u548c\u7acb\u65b9\u6839\uff1b\u4e8c\u6b21\u65b9\u7a0b\uff1b\u4e8c\u9879\u5f0f\uff1b\u6bd4\u4f8b\uff1b\u4ee3\u6570\u89c4\u5219\uff1b\u901a\u8fc7\u629b\u51fa\u4e5d\u6765\u68c0\u67e5\u8ba1\u7b97\u7ed3\u679c\uff1b\u8fdb\u4f4d\uff1b\u4ee5\u53ca\u5e94\u7528\u4ee3\u6570\uff08\u300a\u4f20\u8bb0\u300b\uff09\u3002<\/p>\n\n\n\n
\u5728\u73b0\u5b58\u76841228\u5e74\u7b2c\u4e8c\u7248\u7684\u7b2c123-4\u9875\u4e0a\uff0c\u53d1\u73b0\u4e86\u8fd9\u4f4d\u5e74\u8f7b\u7684\u3001\u6770\u51fa\u7684\u6570\u5b66\u5bb6 “\u5728\u5934\u8111\u4e2d\u60f3\u8c61\u51fa\u7684’abracadabric’\u5154\u5b50\u7684\u7406\u8bba\u5bb6\u65cf”\uff08\u7b2c6\u884c\u3001\u7b2c19\u884c\uff09\u3002\u8fd9\u4e2a\u95ee\u9898\u5bfc\u81f4\u4e86\u6590\u6ce2\u90a3\u5951\u6570\u548c\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u5f15\u5165\uff0c\u800c\u83b1\u6602\u7eb3\u591a\u4eca\u5929\u6700\u88ab\u4eba\u8bb0\u4f4f\u7684\u5c31\u662f\u8fd9\u4e2a\u95ee\u9898\u3002\u4ed6\u63d0\u51fa\u4e86\u4ee5\u4e0b\u96be\u9898\uff08\u8bd1\u6ce8\uff09\u3002<\/p>\n\n\n\n
\u67d0\u4eba\u628a\u4e00\u5bf9\u521a\u51fa\u751f\u7684\u5154\u5b50\uff0c\u4e00\u516c\u4e00\u6bcd\uff0c\u653e\u5728\u4e00\u4e2a\u7528\u5899\u56f4\u8d77\u6765\u7684\u82b1\u56ed\u91cc\u3002\u5154\u5b50\u5728\u4e00\u4e2a\u6708\u5927\u65f6\u5c31\u80fd\u4ea4\u914d\uff0c\u8fd9\u6837\u5728\u5b83\u7b2c\u4e8c\u4e2a\u6708\u7ed3\u675f\u65f6\uff0c\u4e00\u53ea\u6bcd\u5154\u5c31\u80fd\u518d\u751f\u4ea7\u51fa\u4e00\u5bf9\u5154\u5b50\u3002\u5982\u679c\u8fd9\u5bf9\u5154\u5b50\u6c38\u8fdc\u4e0d\u6b7b\uff0c\u800c\u4e14\u6bcf\u4e2a\u6708\u6bcf\u5bf9\u5154\u5b50\u90fd\u4f1a\u751f\u51fa\u4e00\u5bf9\u65b0\u7684\u5154\u5b50\uff0c\u4ece\u7b2c\u4e8c\u4e2a\u6708\u5f00\u59cb\uff0c\u8fd9\u5bf9\u5154\u5b50\u5c31\u4f1a\u6210\u4e3a\u751f\u4ea7\u8005\uff0c\u90a3\u4e48\u8fd9\u5bf9\u5154\u5b50\u4e00\u5e74\u80fd\u751f\u4ea7\u591a\u5c11\u5bf9\u5462\uff1f\u4ed6\u63a5\u7740\u89e3\u91ca\u9053 “\u56e0\u4e3a\u4e0a\u8ff0\u7684\u4e00\u5bf9\u5154\u5b50\u5728\u7b2c\u4e00\u4e2a\u6708\u5c31\u51fa\u751f\u4e86\uff0c \u4f60\u53ef\u4ee5\u628a\u5b83\u7ffb\u500d\uff0c\u8fd9\u6837\u4e00\u4e2a\u6708\u540e\u5c31\u4f1a\u6709\u4e24\u5bf9\u5154\u5b50\u3002\u5176\u4e2d\uff0c\u6709\u4e00\u5bf9\uff0c\u4e5f\u5c31\u662f\u7b2c\u4e00\u5bf9\uff0c\u5728\u7b2c\u4e8c\u4e2a\u6708\u751f\u4e86\u5b69\u5b50\uff0c\u6240\u4ee5\u7b2c\u4e8c\u4e2a\u6708\u6709\u4e09\u5bf9\u3002\u5176\u4e2d\u4e24\u53ea\u5728\u4e00\u4e2a\u6708\u5185\u53c8\u6000\u5b55\u4e86\uff0c\u8fd9\u6837\u7b2c\u4e09\u4e2a\u6708\u5c31\u6709\u4e24\u5bf9\u5154\u5b50\u51fa\u751f\uff0c\u6240\u4ee5\u8fd9\u4e2a\u6708\u5c31\u6709\u4e94\u5bf9\u3002\u5176\u4e2d\u6709\u4e09\u5bf9\u5728\u540c\u4e00\u4e2a\u6708\u6000\u5b55\uff0c\u6240\u4ee5\u7b2c\u56db\u4e2a\u6708\u5c31\u6709\u516b\u5bf9\u3002\u5176\u4e2d\uff0c\u6709\u4e94\u5bf9\u592b\u5987\u53c8\u751f\u4e86\u4e94\u5bf9\uff0c\u5982\u679c\u518d\u52a0\u4e0a\u516b\u5bf9\uff0c\u7b2c\u4e94\u4e2a\u6708\u5c31\u6709\u5341\u4e09\u5bf9\u3002\u5176\u4e2d\uff0c\u8fd9\u4e2a\u6708\u751f\u7684\u4e94\u5bf9\u592b\u5987\u6ca1\u6709\u5728\u540c\u4e00\u4e2a\u6708\u4ea4\u914d\uff0c\u4f46\u5176\u4ed6\u516b\u5bf9\u592b\u5987\u5374\u6000\u5b55\u4e86\uff1b\u6240\u4ee5\u5728\u7b2c\u516d\u4e2a\u6708\u6709\u4e8c\u5341\u4e00\u5bf9\u3002\u5982\u679c\u518d\u52a0\u4e0a\u7b2c\u4e03\u4e2a\u6708\u51fa\u751f\u7684\u5341\u4e09\u5bf9\u592b\u5987\uff0c\u8fd9\u4e2a\u6708\u5c31\u6709\u4e09\u5341\u56db\u5bf9\u592b\u5987\u3002\u5982\u679c\u518d\u52a0\u4e0a\u7b2c\u516b\u4e2a\u6708\u51fa\u751f\u7684\u4e8c\u5341\u4e00\u5bf9\uff0c\u8fd9\u4e2a\u6708\u5c31\u6709\u4e94\u5341\u4e94\u5bf9\u3002\u5982\u679c\u518d\u52a0\u4e0a\u7b2c\u4e5d\u4e2a\u6708\u51fa\u751f\u7684\u4e09\u5341\u56db\u5bf9\uff0c\u8fd9\u4e2a\u6708\u5c31\u4f1a\u6709\u516b\u5341\u4e5d\u5bf9\u3002\u518d\u52a0\u4e0a\u7b2c\u5341\u4e2a\u6708\u51fa\u751f\u7684\u4e94\u5341\u4e94\u5bf9\uff0c\u8fd9\u4e2a\u6708\u5c31\u4f1a\u6709\u4e00\u767e\u56db\u5341\u56db\u5bf9\u3002<\/p>\n\n\n\n
\u52a0\u4e0a\u5341\u4e00\u6708\u4efd\u51fa\u751f\u7684\u516b\u5341\u4e5d\u5bf9\uff0c\u8fd9\u4e2a\u6708\u5c31\u6709233\u5bf9\u3002\u800c\u5982\u679c\u6700\u540e\u518d\u52a0\u4e0a\u4e0a\u4e2a\u6708\u51fa\u751f\u7684\u4e00\u767e\u56db\u5341\u56db\u5bf9\uff0c\u6700\u540e\u5c31\u6709\u4e09\u767e\u4e03\u5341\u4e03\u5bf9\u3002\u800c\u5982\u6b64\u591a\u7684\u592b\u59bb\u5c06\u5728\u5e74\u672b\u6240\u8ff0\u7684\u5730\u65b9\u751f\u4e0b\u4e0a\u8ff0\u592b\u59bb”<\/p><\/blockquote>\n\n\n\n
(\u300a\u5154\u5b50\u300b)\u3002<\/p>\n\n\n\n
\u6211\u4eec\u5047\u8bbe\u30021. \u4e00\u5bf9\u5154\u5b50\u6bcf\u5e74\u90fd\u6709\u4e00\u5bf9\u513f\u5973\u30022. \u8fd9\u4e9b\u5b69\u5b50\u592a\u5c0f\uff0c\u8981\u5230\u4e24\u5e74\u540e\u624d\u4f1a\u6709\u81ea\u5df1\u7684\u5b69\u5b50\u30023. \u5154\u5b50\u6c38\u8fdc\u4e0d\u4f1a\u6b7b\u3002Eppstein\u89c2\u5bdf\u5230\uff0c\u6700\u540e\u4e00\u4e2a\u5047\u8bbe\u662f\u4e0d\u73b0\u5b9e\u7684\uff0c\u4f46\u5374\u4f7f\u95ee\u9898\u53d8\u5f97\u66f4\u7b80\u5355\uff1a”\u5728\u6211\u4eec\u5206\u6790\u4e86\u66f4\u7b80\u5355\u7684\u7248\u672c\u4e4b\u540e\uff0c\u6211\u4eec\u53ef\u4ee5\u56de\u8fc7\u5934\u6765\u6dfb\u52a0\u4e00\u4e2a\u5047\u8bbe\uff0c\u4f8b\u5982\u5154\u5b50\u5341\u5e74\u540e\u6b7b\u4ea1\uff0c\u4f46\u8fd9\u5e76\u4e0d\u4f1a\u5bf9\u95ee\u9898\u7684\u6574\u4f53\u884c\u4e3a\u4ea7\u751f\u5f88\u5927\u7684\u6539\u53d8\u3002”<\/p>\n\n\n\n
\u7136\u540e\uff0c\u6211\u4eec\u5c06\u5154\u5b50\u7684\u5bf9\u6570\u8868\u793a\u4e3a\u65f6\u95f4\u7684\u51fd\u6570\uff08\u4ee5\u5b9e\u9a8c\u5f00\u59cb\u540e\u7684\u5e74\u6570\u6765\u8861\u91cf\uff09\uff08Eppstein\uff09\u3002<\/p>\n\n\n\n
F(1) = 1 – \u6211\u4eec\u4ece\u4e00\u5bf9\u5154\u5b50\u5f00\u59cb F(2) = 1 – \u4ed6\u4eec\u592a\u5e74\u8f7b\uff0c\u7b2c\u4e00\u5e74\u4e0d\u80fd\u751f\u5b69\u5b50 F(3) = 2 – \u5728\u7b2c\u4e8c\u5e74\uff0c\u4ed6\u4eec\u751f\u4e86\u4e00\u5bf9\u5b69\u5b50 F(4) = 3 – \u5728\u7b2c\u4e09\u5e74\uff0c\u4ed6\u4eec\u53c8\u751f\u4e86\u4e00\u5bf9 F(5) = 5 – \u6211\u4eec\u5f97\u5230\u7b2c\u4e00\u5bf9\u5b59\u5b50\u5b59\u5973\u3002<\/p>\n\n\n\n
\u8fd9\u4e2a\u95ee\u9898\u4ea7\u751f\u4e86 “\u6590\u6ce2\u90a3\u5951\u5e8f\u5217”\u30021, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . . .<\/p>\n\n\n\n
Fibonacci\u5728Liber Abaci\u4e2d\u7701\u7565\u4e86\u7b2c\u4e00\u9879\uff081\uff09\u3002\u8fd9\u4e9b\u6570\u5b57\u7684\u9012\u63a8\u516c\u5f0f\u662f\uff1a\u3002F(0) = 0 F(1) = 1 F(n) = F(n – 1) + F(n – 2) n > 1 \u3002\u867d\u7136\u6590\u6ce2\u90a3\u5951\u53ea\u7ed9\u51fa\u4e86\u5e8f\u5217\uff0c\u4f46\u4ed6\u663e\u7136\u77e5\u9053\u4ed6\u5e8f\u5217\u4e2d\u7684\u7b2cn\u4e2a\u6570\u662f\u524d\u9762\u4e24\u4e2a\u6570\uff08Scotta\u548cMarketos\uff09\u4e4b\u548c\u3002”\u8fd9\u4e2a\u5e8f\u5217\uff0c\u5176\u4e2d\u6bcf\u4e2a\u6570\u5b57\u90fd\u662f\u524d\u9762\u4e24\u4e2a\u6570\u5b57\u7684\u548c\uff0c\u51fa\u73b0\u5728\u8bb8\u591a\u4e0d\u540c\u7684\u6570\u5b66\u548c\u79d1\u5b66\u9886\u57df”\uff08O’Connor\u548cRobertson\uff09\u3002<\/p>\n\n\n\n
Fibonacci\u53ef\u80fd\u5e76\u6ca1\u6709\u53d1\u660e\u5154\u5b50\u95ee\u9898\uff0c\u800c\u662f\u628a\u4ed6\u81ea\u5df1\u4ece\u6469\u5c14\u4eba\u90a3\u91cc\u6216\u5728\u65c5\u884c\u65f6\u5b66\u5230\u7684\u5154\u5b50\u95ee\u9898\u5305\u62ec\u5728\u5185\uff08Knott\uff09\u3002\u4ed6\u751a\u81f3\u53ef\u80fd\u4f9d\u8d56\u4e8e “\u514b\u96f7\u83ab\u7eb3\u7684\u6770\u62c9\u5fb7\uff08Gerard of Cremona\uff0c\u516c\u51431114-1187\u5e74\uff09\u5bf9Al-Khw\u02c6arizm\u02c6\u0131\u4f5c\u54c1\u7684\u7ffb\u8bd1\uff0c\u540e\u8005\u662f\u4ee5\u897f\u73ed\u7259\u6258\u83b1\u591a\u4e3a\u57fa\u5730\uff0c\u4e3a\uff08\u57fa\u7763\u6559\uff09\u6b27\u6d32\u5c06\u963f\u62c9\u4f2f\u6587\u8457\u4f5c\u7ffb\u8bd1\u6210\u62c9\u4e01\u6587\u7684\u4e3b\u8981\u52aa\u529b\u7684\u5148\u9a71”\uff08Scotta and Marketos\uff09\u3002\u5e8f\u5217F(n)\u5df2\u7ecf\u88ab\u5370\u5ea6\u6570\u5b66\u5bb6\u6240\u719f\u77e5\u548c\u8ba8\u8bba\uff0c”\u4ed6\u4eec\u5bf9\u7531\u4e00\u62cd\u548c\u4e24\u62cd\u97f3\u7b26\u5f62\u6210\u7684\u8282\u594f\u6a21\u5f0f\u5f88\u611f\u5174\u8da3\u3002\u8fd9\u6837\u7684\u8282\u594f\u4e00\u5171\u6709n\u4e2a\u62cd\u5b50\uff0c\u5176\u6570\u91cf\u5c31\u662fF(n+1)\u3002\u56e0\u6b64\uff0cGospala(1135\u5e74\u524d)\u548cHemachandra(\u7ea61150\u5e74)\u90fd\u660e\u786e\u63d0\u5230\u4e861\u30012\u30013\u30015\u30018\u300113\u300121\u2026\u2026\u8fd9\u4e9b\u6570\u5b57\u3002<\/p>\n\n\n\n
“\u6590\u6ce2\u90a3\u5951\u672c\u4eba\u4f3c\u4e4e\u5e76\u6ca1\u6709\u628a\u5b83\u4eec\u8054\u60f3\u5230\u90a3\u4e48\u91cd\u8981\u7684\u5730\u4f4d\uff1b\u5154\u5b50\u95ee\u9898\u4f3c\u4e4e\u662f\u4ed6\u5de5\u4f5c\u4e2d\u7684\u4e00\u4e2a\u5c0f\u7ec3\u4e60”\uff08Scotta\u548cMarketos\uff09\u3002\u76f4\u523019\u4e16\u7eaa\uff0c\u8fd9\u4e2a\u6570\u5217\u624d “\u7531\u4e8e\u6cd5\u56fd\u6570\u5b66\u5bb6\u7231\u5fb7\u534e-\u5362\u5361\u65af\u7684\u5de5\u4f5c\u800c\u83b7\u5f97\u4e86\u4e3b\u8981\u7684\u91cd\u8981\u6027\u548c\u8ba4\u53ef”\u3002\u4ece\u90a3\u65f6\u8d77\uff0c\u6570\u5b66\u53f2\u5b66\u5bb6\u5c31\u60f3\u77e5\u9053\u8fd9\u4e9b\u6570\u5b57\u80cc\u540e\u7684\u771f\u6b63\u7075\u611f\uff0c\u4ee5\u53caFibonacci\u662f\u5426\u5b8c\u5168\u610f\u8bc6\u5230\u5b83\u4eec\u7684\u91cd\u8981\u6027\uff08Scotta\u548cMarketos\uff09\u3002\u867d\u7136\u6590\u6ce2\u90a3\u5951\u6db5\u76d6\u4e86\u4f17\u591a\u7684\u6570\u5b66\u8bfe\u9898\uff0c\u4f46\u4ed6\u6700\u8457\u540d\u7684\u662f\u8fd9\u4e2a\u6570\u5b57\u5e8f\u5217\uff0c\u540e\u6765\u57281838\u5e74\u88ab\u7eaa\u5c27\u59c6-\u5229\u5e03\u91cc\u4ee5\u4ed6\u7684\u540d\u5b57\u547d\u540d\uff0c\u81f3\u4eca\u4ecd\u5728\u88ab\u79ef\u6781\u7814\u7a76\uff08”\u5154\u5b50\u95ee\u9898”\uff09\u3002\u867d\u7136 “\u5154\u5b50\u95ee\u9898 “\u5f88\u6709\u610f\u601d\uff0c\u4e5f\u662f\u4ed6\u4eca\u5929\u6700\u8457\u540d\u7684\u4e00\u4e2a\u95ee\u9898\uff0c\u4f46\u8fd9\u7edd\u4e0d\u662fLiber Abaci\u4e2d\u63d0\u51fa\u7684\u552f\u4e00\u91cd\u8981\u7684\u6570\u5b66\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u501f\u7528\u4e5d\u4e16\u7eaa\u9a6c\u54c8\u7ef4\u62c9\uff08Mahavira\uff09\uff08\u7ea6800-870\u5e74\uff09\u6240\u8457\u7684\u300aGanita Sara Sangraha\u300b\u4e00\u4e66\u4e2d\u7684\u4e00\u4e2a\u573a\u666f\uff0c\u6590\u6ce2\u90a3\u5951\u63d0\u51fa\u4e86\u4e00\u7cfb\u5217\u7684 “\u94b1\u5305\u95ee\u9898”\uff0c\u4ee5\u5e2e\u52a9\u90a3\u4e9b\u53ef\u80fd\u60f3\u628a\u94b1\u5206\u7ed9\u4e24\u4e2a\u6216\u66f4\u591a\u4eba\u7684\u4eba\u3002\u5728\u65e5\u5e38\u7528\u8bed\u4e2d\uff0c\u4ed6\u9610\u660e\u4e86\u5e73\u7b49\u516c\u5e73\u5730\u5206\u914d\u67d0\u7269\uff08\u5982\u91d1\u94b1\uff09\u6240\u5e94\u9075\u5faa\u7684\u89c4\u5219\u3002\u7b2c\u4e00\u4e2a “\u94b1\u5305\u95ee\u9898 “\u7684\u89e3\u51b3\u65b9\u6848\u586b\u6ee1\u4e86\u534a\u5f20\u7f8a\u76ae\u7eb8\uff0c\u7136\u540e\u4ed6\u63d0\u4f9b\u4e86\u8bb8\u591a\u66f4\u590d\u6742\u7684\u540c\u4e00\u95ee\u9898\u7684\u53d8\u4f53\u53ca\u5176\u89e3\u51b3\u65b9\u6848\uff0c\u5305\u62ec\u5982\u4f55\u5728\u4e09\u4e2a\u4eba\u800c\u4e0d\u662f\u4e24\u4e2a\u4eba\u53d1\u73b0\u7684\u94b1\u5305\u4e2d\u5206\u914d\u76f8\u540c\u6570\u91cf\u7684\u94b1\uff0c\u56db\u4e2a\u4eba\u53d1\u73b0\u7684\u94b1\u5305\uff0c\u6700\u540e\u662f\u4e94\u4e2a\u4eba\u53d1\u73b0\u7684\u94b1\u5305\u3002\u51e0\u9875\u4e4b\u540e\uff0c\u6590\u6ce2\u90a3\u5951\u5df2\u7ecf\u5305\u542b\u4e86\u5341\u516b\u4e2a\u4e0d\u540c\u7684\u94b1\u5305\u95ee\u9898\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u6bcf\u4e2a\u95ee\u9898\u90fd\u6709 “\u72ec\u7279\u7684\u8f6c\u6298\uff0c\u6bcf\u4e2a\u95ee\u9898\u90fd\u4f7f\u7528\u4e86\u7a0d\u5fae\u4e0d\u540c\u7684\u6570\u5b57”\u3002\u5fb7\u592b\u6797\uff0c\u53d1\u73b0121\uff09\u3002<\/p>\n\n\n\n
FIBONACCI\u7684\u6570\u5b66\u8d21\u732e<\/h2>\n\n\n\n
\u4e00\u3001\u6570\u5b57<\/h3>\n\n\n\n
\u6570\u5b57\u4e66\u5199\u7684\u4f4d\u503c\u7cfb\u7edf\u6bd4\u4ee5\u5b57\u6bcd\u4e3a\u57fa\u7840\u7684\u7f57\u9a6c\u6570\u5b57\u7cfb\u7edf\u8981\u5bb9\u6613\u5f97\u591a\uff1b\u6570\u5b57\u7684\u4f4d\u7f6e\u51b3\u5b9a\u4e86\u5b83\u76f8\u5bf9\u4e8e\u6570\u5b57\u4e2d\u5176\u4ed6\u6570\u5b57\u7684\u5927\u5c0f\uff08\u4f8b\u5982\uff0c19\u4e2d\u76841\u662f “\u5341 “\u4f4d\u4e0a\u7684\u7b2c\u4e8c\u4f4d\u6570\u5b57\uff0c\u8868\u793a\u4ef7\u503c\u662f\u5176\u540d\u4e49\u4ef7\u503c\u7684\u5341\u500d\u621610\u00d71\uff09\u3002\u6590\u6ce2\u90a3\u5951\u5411\u5546\u4eba\u548c\u5546\u4eba\u5c55\u793a\u4e86\u5982\u4f55\u4f7f\u7528\u7b97\u672f\u7684\u4f4d\u503c\u7cfb\u7edf\u3002<\/p>\n\n\n\n
\u4e8c\u3001\u6570\u5b57\u4e0e\u5c0f\u6570<\/h3>\n\n\n\n
\u51e0\u4e2a\u4e16\u7eaa\u4ee5\u6765\uff0c\u6b27\u6d32\u4e00\u76f4\u4f7f\u7528\u7f57\u9a6c\u6570\u5b57\u7cfb\u7edf\uff0c7\u4e2a\u7b26\u53f7\u4ee3\u88687\u4e2a\u4e0d\u540c\u7684\u6570\u503c\uff0c\u7f57\u9a6c\u6570\u5b572018\u53ef\u4ee5\u5199\u6210MMXVVIII\u6216IIIXVMM–\u5b57\u6bcd\u987a\u5e8f\u5e76\u4e0d\u91cd\u8981\uff0c\u56e0\u4e3a\u5b57\u6bcd\u7684\u6570\u503c\u76f8\u52a0\u624d\u662f\u6570\u5b57\u3002<\/p>\n\n\n\n
I=1 V=5 X=10 L=50 C=100 D=500 M=1000<\/p>\n\n\n\n
\u5728\u5370\u5ea6\u6559-\u963f\u62c9\u4f2f\u6559\u7cfb\u7edf\u4e2d\uff0c\u6570\u5b57\u7684\u987a\u5e8f\u603b\u662f\u5f88\u91cd\u8981\uff0c\u56e0\u4e3a\u6bcf\u4e2a\u6570\u5b57\u7684\u4f4d\u7f6e\u51b3\u5b9a\u4e86\u5b83\u7684\u4ef7\u503c\uff1b\u6570\u5b572018\u4e0e8102\u662f\u5b8c\u5168\u4e0d\u540c\u7684\u3002\u6590\u6ce2\u90a3\u5951\u8feb\u4f7f\u4eba\u4eec\u5728\u5546\u4e1a\u4e0a\u4f7f\u7528\u963f\u62c9\u4f2f\u7b26\u53f7–1\u30012\u30013\u30014\u30015\u30016\u30017\u30018\u30019–\u8fd9\u4e9b\u7b26\u53f7\u5728\u6b27\u6d32\u5df2\u7ecf\u88ab\u4eba\u4eec\u6240\u719f\u77e5\uff0c\u4f46\u5728\u65e5\u5e38\u5b9e\u8df5\u4e2d\u8fd8\u6ca1\u6709\u5b9e\u65bd\uff1b\u6700\u91cd\u8981\u7684\u662f\uff0c\u8fd9\u4e2a\u6570\u5b57\u7cfb\u7edf\u5305\u62ec\u4e00\u4e2a\u96f6\u7684\u7b26\u53f7\u3002\u96f6\u662f\u9700\u8981\u4f5c\u4e3a\u5360\u4f4d\u7b26\u7684\uff0c\u56e0\u4e3a\u5b83\u786e\u4fdd\u4e86\u6570\u5b57\u88ab\u653e\u7f6e\u5728\u9002\u5f53\u7684\u4f4d\u7f6e\uff08\u5217\uff09\uff1b\u4f8b\u5982\uff0c2009\u5e74\u6ca1\u6709\u5341\u548c\u767e\u3002\u7f57\u9a6c\u7cfb\u7edf\u4f1a\u628a2009\u5199\u6210MMIX\uff0c\u7701\u7565\u4e0d\u7528\u7684\u6570\u503c\u3002\u7f57\u9a6c\u4eba\u7684\u7b97\u672f\u5e76\u4e0d\u7b80\u5355\uff0c\u4f8b\u5982\uff0cMXVII\u52a0\u5230LI\u4e0a\u662fMLVIII\uff0cXLI\u51cf\u53bbIV\u662fXXXVII\uff08Knott\uff0c”\u7b80\u4ecb”\uff09\u3002\u5728Liber Abaci (1228)\u4e2d\uff0cFibonacci\u627f\u8ba4\u4ed6\u5728\u51fa\u5dee\u65f6\u5e7f\u6cdb\u7814\u7a76\u4e86\u7b97\u672f\uff1b\u56de\u5230\u610f\u5927\u5229\u540e\uff0c\u4ed6\u70ed\u60c5\u5730\u6559\u6388\u4e86\u4ed6\u4ece\u963f\u62c9\u4f2f\u6570\u5b66\u5bb6\u90a3\u91cc\u5b66\u5230\u7684\u7b97\u672f\u89c4\u5219\uff0c\u5e76\u5728\u6b27\u6d32\u63d0\u4f9b\u4e86\u7b2c\u4e00\u4e2a\u7cfb\u7edf\u7684\u5341\u8fdb\u5236\u7cfb\u7edf\u8868\u793a\u6cd5\uff08Knott\uff0c”\u7b80\u4ecb”\uff09\u3002<\/p>\n\n\n\n
\u6590\u6ce2\u90a3\u5951\u4e0e\u4ee3\u6570<\/h3>\n\n\n\n
1225\u5e74\uff0c\u5f53\u6590\u6ce2\u90a3\u5951\u5728\u5411\u8153\u7279\u70c8\u7687\u5e1d\u7684\u76ae\u6851\u5bab\u5ef7\u8fdb\u884c\u6f14\u793a\u65f6\uff0c\u4ed6\u5c55\u793a\u4e86\u81ea\u5df1\u7684\u6570\u5b66\u80fd\u529b\uff0c\u5e76\u89e3\u91ca\u4e86\u4ed6\u5c06\u5982\u4f55\u89e3\u51b3\u4ee5\u4e0bDiophantine\u4ee3\u6570\u95ee\u9898\uff1a\u89e3x3+2\u00d72+10x=20\u3002\u7531\u4e8e\u8ba4\u8bc6\u5230\u6b27\u51e0\u91cc\u5f97\u7528\u5e73\u65b9\u6839\u89e3\u65b9\u7a0b\u7684\u65b9\u6cd5\u884c\u4e0d\u901a\uff0c\u6590\u6ce2\u90a3\u5951\u4f7f\u7528\u4e86 “\u4ed6\u81ea\u5df1\u7684\u72ec\u521b\u65b9\u6cd5\uff0c\u7528\uff08\u5df4\u6bd4\u4f26\uff09\u6027\u5c0f\u6570\u7b26\u53f7\u7ed9\u51fa\u4e86\u7b54\u6848\u3002\u4ed6\u7684\u8fd1\u4f3c\u65b9\u6cd5\u6bd4\u4ed6\u540c\u65f6\u4ee3\u7684\u963f\u62c9\u4f2f\u4eba\u7684\u8fd1\u4f3c\u65b9\u6cd5\u8981\u51c6\u786e\u5f97\u591a”\uff0c\u4ee4\u542c\u4f17\u9707\u60ca\u4e0d\u5df2\uff08Horadam\uff09\u3002\u5728\u8fd9\u6b21\u4e8b\u4ef6\u540e\u4e0d\u4e45\uff0c\u6590\u6ce2\u90a3\u5951\u53c8\u5199\u4e86\u4e00\u7bc7\u7b80\u77ed\u7684\u8bba\u6587\u300aFlos\u300b\uff081225\u5e74\uff09\uff08\u300a\u82b1\u300b\uff0c\u8fd9\u4e2a\u540d\u5b57\u4ee4\u4eba\u8d39\u89e3\uff0c\u56e0\u4e3a\u5b83\u4e0e\u5f00\u82b1\u690d\u7269\u6ca1\u6709\u4efb\u4f55\u660e\u663e\u7684\u5173\u7cfb\uff09\uff0c\u89e3\u91ca\u4e86\u4ed6\u662f\u5982\u4f55\u5f97\u51fa\u8fd9\u4e2a\u4ee3\u6570\u95ee\u9898\u548c\u53e6\u4e00\u4e2a\u95ee\u9898\u7684\u89e3\u51b3\u65b9\u6848\u7684\u3002<\/p>\n\n\n\n
\u5728\u5199Flos\u4e4b\u524d\uff0c\u6590\u6ce2\u90a3\u5951\u51fa\u7248\u4e86\u4e00\u672c\u5305\u542b\u4ee3\u6570\u65b9\u7a0b\u89e3\u6cd5\u7684\u8bba\u6587\u300a\u4e66\u4fe1\u96c6\u300b\uff08Epistola ad Magistrum Theodorum\uff09\uff0c\u4e5f\u662f\u4eca\u5929\u6570\u5b66\u5bb6\u8ba4\u4e3a\u4ed6\u6700\u91cd\u8981\u7684\u8457\u4f5c\u3002Liber Quadratorum, the Book of Squares (1225), \u4e00\u672c\u4ed6\u732e\u7ed9\u7687\u5e1d\u7684\u6570\u8bba\u4e66. \u5728\u8fd9\u672c\u4e66\u4e2d\uff0c\u4ed6\u63cf\u8ff0\u4e86\u5e73\u65b9\u7684\u7279\u6027\uff08\u5982\u4e8c\u3001\u4e09\u3001\u56db\u5e73\u65b9\u6570\u4e4b\u548c\uff0c\u6216\u5e73\u65b9\u5206\u6570\uff09\u548c\u5bfc\u81f4\u4e8c\u6b21\u65b9\u7a0b\u7684\u4efb\u52a1\uff08McClenon\uff09\u3002\u4f7f\u6590\u6ce2\u90a3\u5951\u7684\u6210\u5c31\u66f4\u52a0\u4ee4\u4eba\u5370\u8c61\u6df1\u523b\u7684\u662f\uff0c\u4ed6\u6ca1\u6709\u50cf\u6211\u4eec\u4eca\u5929\u8fd9\u6837\u4f7f\u7528\u4ee3\u6570\u7b26\u53f7\uff0c\u56e0\u4e3a\u4ed6\u6ca1\u6709\u8fd9\u6837\u7684\u4ee3\u6570\u7b26\u53f7\u6765\u5e2e\u52a9\u4ed6\u3002\u76f8\u53cd\uff0c\u4ed6\u7528\u7ebf\u6bb5\u7684\u51e0\u4f55\u5f62\u5f0f\u6765\u8868\u793a\u6570\u5b57\uff0c\u5c31\u50cf\u6b27\u51e0\u91cc\u5f97\u90a3\u6837\u3002\u4e0d\u8fc7\uff0c\u4ed6\u5bf9\u8fc7\u7a0b\u548c\u7b97\u6cd5\u7684\u63cf\u8ff0\u8fd8\u662f\u51fa\u5947\u7684\u6e05\u6670\u3002\u4f8b\u5982\uff0c\u4ed6\u5bf9\u672a\u77e5\u7684x\u4f7f\u7528\u4e86res(\u4e8b\u7269)\u8fd9\u6837\u7684\u77ed\u8bed\uff0c\u5bf9x2\u4ed6\u5199\u4e86quadratus numerus(\u5e73\u65b9\u6570)\u3002\u4e0b\u9762\u7684\u95ee\u9898\u662f\u4ed6\u89e3\u51b3\u7684\u8ba1\u7b97\u7c7b\u578b\u7684\u4ee3\u8868\uff0c\u4ed6\u7684\u89e3\u91ca\u65b9\u5f0f\u51e0\u4e4e\u4f18\u4e8e\u4ed6\u4e4b\u524d\u6240\u6709\u7684\u6570\u5b66\u6559\u79d1\u4e66\u4f5c\u8005\u3002”\u627e\u51fa\u4e00\u4e2a\u5e73\u65b9\u6570\uff0c\u5f535\u88ab\u52a0\u51cf\u65f6\uff0c\u603b\u4f1a\u4ea7\u751f\u4e00\u4e2a\u5e73\u65b9\u6570\u3002” \u6b63\u5982\u970d\u62c9\u4e39\u6240\u8bf4\uff1a”\u4ed6\u80fd\u5728\u8fd9\u6709\u9650\u7684\u6570\u5b66\u8bbe\u5907\u4e0a\u53d6\u5f97\u5982\u6b64\u5927\u7684\u8fdb\u6b65\uff0c\u5b9e\u5728\u662f\u4e86\u4e0d\u8d77\u3002\u4ed6\u5728\u672c\u4e66\u4e2d\u7684\u6210\u5c31\uff0c\u6b63\u597d\u8bc1\u5b9e\u4e86\u4ed6\u662f\u4e2d\u4e16\u7eaa\u6570\u8bba\uff0c\u7279\u522b\u662f\u4e0d\u786e\u5b9a\u5206\u6790\u7684\u6700\u4f1f\u5927\u7684\u9610\u91ca\u8005”\uff08\u300a\u970d\u62c9\u8fbe\u59c6\u300b\uff0c\u4e66\uff09\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u83b1\u6602\u7eb3\u591a-\u76ae\u8428\u8bfa\uff08Leonardo Pisano\uff09\uff0c\u6216 “\u6590\u6ce2\u90a3\u5951”\uff0c\u662f\u5e0c\u814a\u795e\u5b66\u5bb6\u548c\u6570 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[8],"tags":[17,18],"_links":{"self":[{"href":"https:\/\/www.fibonance.com\/index.php\/wp-json\/wp\/v2\/posts\/171"}],"collection":[{"href":"https:\/\/www.fibonance.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fibonance.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fibonance.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fibonance.com\/index.php\/wp-json\/wp\/v2\/comments?post=171"}],"version-history":[{"count":1,"href":"https:\/\/www.fibonance.com\/index.php\/wp-json\/wp\/v2\/posts\/171\/revisions"}],"predecessor-version":[{"id":172,"href":"https:\/\/www.fibonance.com\/index.php\/wp-json\/wp\/v2\/posts\/171\/revisions\/172"}],"wp:attachment":[{"href":"https:\/\/www.fibonance.com\/index.php\/wp-json\/wp\/v2\/media?parent=171"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fibonance.com\/index.php\/wp-json\/wp\/v2\/categories?post=171"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fibonance.com\/index.php\/wp-json\/wp\/v2\/tags?post=171"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}