{"id":54035,"date":"2014-06-16T03:57:22","date_gmt":"2014-06-15T19:57:22","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=54035"},"modified":"2021-10-06T16:18:07","modified_gmt":"2021-10-06T08:18:07","slug":"%e5%b8%83%e9%87%8c%e6%a0%bc%e6%96%af%e7%9a%84%e3%80%8a%e5%b0%8d%e6%95%b8%e7%ae%97%e8%a1%93%e3%80%8b%e8%88%87%e5%b0%8d%e6%95%b8%e8%a1%a8%e7%9a%84%e8%a3%bd%e4%bd%9cii-briggs-arithmetica-loga","status":"publish","type":"post","link":"http:\/\/localhost\/%e5%b8%83%e9%87%8c%e6%a0%bc%e6%96%af%e7%9a%84%e3%80%8a%e5%b0%8d%e6%95%b8%e7%ae%97%e8%a1%93%e3%80%8b%e8%88%87%e5%b0%8d%e6%95%b8%e8%a1%a8%e7%9a%84%e8%a3%bd%e4%bd%9cii-briggs-arithmetica-loga\/","title":{"rendered":"\u5e03\u91cc\u683c\u65af\u7684\u300a\u5c0d\u6578\u7b97\u8853\u300b\u8207\u5c0d\u6578\u8868\u7684\u88fd\u4f5c(II)"},"content":{"rendered":"

\u5e03\u91cc\u683c\u65af\u7684\u300a\u5c0d\u6578\u7b97\u8853\u300b\u8207\u5c0d\u6578\u8868\u7684\u88fd\u4f5c(II) (Briggs’ Arithmetica Logarithmica and the creation of logarithmic table, part 2)<\/strong><\/span>
\n\u81fa\u5317\u5e02\u7acb\u897f\u677e\u9ad8\u4e2d\u8607\u60e0\u7389\u6559\u5e2b<\/strong><\/span><\/p>\n

\u9023\u7d50\uff1a\u5e03\u91cc\u683c\u65af\u7684\u300a\u5c0d\u6578\u7b97\u8853\u300b\u8207\u5c0d\u6578\u8868\u7684\u88fd\u4f5c(I)\u00a0<\/a><\/p>\n

\u300a\u5c0d\u6578\u7b97\u8853\u300b\u7b2c $$5$$\uff5e$$7$$ \u7ae0<\/p>\n

\u7b2c $$5$$ \u7ae0\u5230\u7b2c $$8$$ \u7ae0\u70ba\u8a08\u7b97\u4ee5 $$10$$ \u70ba\u5e95\u7684\u5c0d\u6578\u7684\u4e3b\u8981\u65b9\u6cd5\u3002\u5728\u7b2c $$5$$ \u7ae0\u4e2d\u6240\u63d0\u7684\u65b9\u6cd5\uff0c\u5e03\u91cc\u683c\u65af\u5c07\u5b83\u6b78\u529f\u65bc\u7d0d\u76ae\u723e\u3002\u4ed6\u4ee5 $$\\log 5$$ \u8207 $$\\log 7$$ \u70ba\u4f8b\uff0c\u8aaa\u660e\u5c0f\u4e00\u9ede\u7684\u8cea\u6578\u5982\u4f55\u6c42\u5176\u5c0d\u6578\u503c\u3002\u8003\u616e $$\\log 2$$\uff0c\u5148\u8a08\u7b97 $$2$$ \u7684\u6b21\u65b9\uff0c\u4e26\u6a19\u660e\u5176\u4f4d\u6578\u3002<\/p>\n

\u70ba\u4e86\u4f7f\u5c0d\u6578\u503c\u7cbe\u78ba\u5230\u5c0f\u6578\u9ede\u5f8c\u7b2c $$14$$ \u4f4d\uff0c\u5e03\u91cc\u683c\u65af\u8a08\u7b97\u5230\u4e86 $$2^{10^{14}}$$\uff1b\u4e0d\u904e\uff0c\u4ed6\u4e5f\u4e0d\u662f\u6bcf\u500b\u90fd\u7b97\uff0c\u800c\u662f\u4ee5\u56db\u500b\u6578\u4e00\u7d44\uff0c\u6bcf\u6b21\u90fd\u8a08\u7b97\u6b21\u65b9\u70ba $$2\\times 10^k,4\\times 10^k,8\\times 10^k,10\\times 10^k$$\u00a0\u7684\u56db\u500b\u6578\u7684\u4f4d\u6578\uff0c\u5982\u4e0b\u5716\u4e00\u3002\u5728\u8a08\u7b97\u4f4d\u6578\u6642\uff0c\u5e03\u91cc\u683c\u65af\u4e26\u6c92\u6709\u5c07\u6bcf\u500b\u6578\u5b8c\u6574\u7b97\u51fa\u5f8c\u8a08\u7b97\uff0c\u4ed6\u5229\u7528\u4e86\u4e0b\u9762\u9019\u500b\u6027\u8cea\uff1a\u5982\u8981\u8a08\u7b97\u5169\u6578\u76f8\u4e58\u5f8c\u7684\u4f4d\u6578\uff0c\u8003\u616e\u9019\u5169\u6578\u7684\u9996\u5e7e\u4f4d\u6578\u5b57\uff0c\u76f8\u4e58\u5f8c\u7684\u4f4d\u6578\u4e0d\u662f\u5169\u8005\u4f4d\u6578\u76f8\u52a0\uff0c\u5c31\u662f\u5169\u8005\u4f4d\u6578\u76f8\u52a0\u518d\u6e1b $$1$$\uff0c\u5982\u4e0b\u5716\u4e8c\u3002<\/p>\n

\"54035_p1\"<\/a>

\u5716\u4e00<\/p><\/div>\n

\"54035_p2\"<\/a>

\u5716\u4e8c<\/p><\/div>\n

\u7576 $$2^{10^{14}}$$\u00a0\u7684\u4f4d\u6578\u70ba $$M$$\u00a0<\/i>\u6642\uff0c\u56e0\u70ba $$\\log {2^{{{10}^{14}}}} = {10^{14}} \\cdot \\log 2 = (M – 1) + \\log N$$\uff0c<\/p>\n

\u6240\u4ee5 $$\\log 2 = \\frac{{M – 1}}{{{{10}^{14}}}} + \\frac{{\\log N}}{{{{10}^{14}}}}$$\u00a0\uff0c\u5f8c\u9762\u7684 $$\\frac{{\\log N}}{{{{10}^{14}}}}$$\u00a0\u592a\u5c0f\u5ffd\u7565\u4e0d\u8a08\uff0c\u56e0\u6b64\u53ef\u5f97 $$\\log 2 \\approx \\frac{{M – 1}}{{{{10}^{14}}}}$$\u00a0\u3002<\/p>\n

\u5e03\u91cc\u683c\u65af\u8a08\u7b97\u6240\u5f97\u7684 $$M$$\u00a0<\/i>\u70ba $$3010,29995,66399$$\uff0c<\/p>\n

\u6545\u53ef\u5f97 $$\\log 2 \\approx 0.030102999566398$$\u3002<\/p>\n

 <\/p>\n

\u5728\u7b2c $$6$$ \u7ae0\u8207\u7b2c $$7$$ \u7ae0\u4e2d\uff0c\u4ed6\u4f7f\u7528\u6240\u8b02\u7684\u9023\u7e8c\u958b\u65b9\u6cd5\uff08continued mean number\uff09\u3002<\/p>\n

\u5728\u7b2c $$6$$ \u7ae0\u4e2d\uff0c\u4ed6\u5148\u5229\u7528\u9019\u500b\u65b9\u6cd5\u627e\u51fa $$1+x$$\u00a0\u7684\u5c0d\u6578\u503c\uff0c\u5176\u4e2d $$x>0$$\u00a0\u4f46\u5f88\u63a5\u8fd1\u65bc $$0$$\u3002<\/p>\n

\u4ed6\u5148\u5c07 $$10$$ \u9023\u7e8c\u958b\u5e73\u65b9\u6839\uff0c\u5373\u5728 $$1$$ \u548c $$10$$ \u4e4b\u9593\u4e00\u76f4\u6c42\u5e7e\u4f55\u5e73\u5747\u6578\uff1a<\/p>\n

$$\\sqrt {1 \\cdot 10}= {10^{\\frac{1}{2}}},\\sqrt{1\\cdot\\sqrt{10}}={10^{\\frac{1}{4}}},\\sqrt {1\\cdot 10^{1\/4}}={10^{\\frac{1}{8}}}\\cdots$$\uff1b<\/p>\n

\u4ed6\u70ba\u4e86\u8981\u4f7f\u5f97\u5230\u7684\u5c0d\u6578\u6b63\u78ba\u5230\u5c0f\u6578\u9ede\u5f8c $$14$$ \u4f4d\uff0c<\/p>\n

\u56e0\u6b64\u4ed6\u8a08\u7b97\u5230\u5c0f\u6578\u9ede\u5f8c\u7684 $$0$$ \u8207\u9019\u4e9b $$0$$ \u4e4b\u5f8c\u7684\u6578\u5b57\u6709\u76f8\u540c\u7684\u4f4d\u6578\uff0c<\/p>\n

\u56e0\u6b64\u9023\u7e8c\u958b\u65b9 $$54$$ \u6b21\uff0c\u8a08\u7b97\u5230 $$10^{1\/2^{54}}=r=1+\\Delta$$\uff0c<\/p>\n

\u5176\u4e2d $$\\Delta=0.0(15)12781,91493,20032,3442$$
\n\uff08\u8a3b\uff1a$$0(15)$$ \u4e2d\u7684 $$15$$ \u8868\u793a\u5728\u5c0f\u6578\u9ede\u5f8c\u6709 $$15$$ \u500b $$0$$\uff09\uff1b<\/p>\n

\u540c\u6642\u4e5f\u5f9e $$1$$ \u958b\u59cb\u9023\u7e8c\u9664\u4ee5 $$2$$\uff0c\u8a08\u7b97\u4e86 $$54$$ \u6b21\uff0c<\/p>\n

\u5f97\u5230 $$\\frac{1}{2^{54}}=0.0(16)555111512312578270212$$\uff0c\u4ee4\u5176\u503c\u70ba $$l$$\uff0c<\/p>\n

\u53d6\u5c0d\u6578\u4e4b\u5f8c\uff0c\u53ef\u5f97 $$\\log r=\\log(1+\\Delta)=l$$\u00a0\u70ba\u5df2\u77e5\uff0c\u6b64\u6642 $$\\Delta$$ \u662f\u500b\u5f88\u5c0f\u7684\u6578\u3002<\/p>\n

\u5f9e\u524d\u9762\u7b2c $$2$$ \u7ae0\u7684\u5b9a\u7406\u77e5\u9053\u771f\u6578\u7684\u6b21\u65b9 $$(r^n)$$ \u8207\u5c0d\u6578\u503c $$(nl)$$ \u9593\u6709\u6bd4\u4f8b\u95dc\u4fc2\uff1b<\/p>\n

\u53c8\u56e0\u70ba $$\\Delta$$ \u8db3\u5920\u5c0f\uff0c\u56e0\u6b64\u6709\u4e0b\u9762\u7684\u5c0d\u61c9\u95dc\u4fc2\uff1a<\/p>\n

\"54035_c1\"<\/a><\/p>\n

\u53ef\u77e5\u771f\u6578\u53bb\u6389 $$1$$ \u4e4b\u5f8c\u7684\u5c0f\u6578\u8207\u5c0d\u6578\u503c\u4e4b\u9593\u4ea6\u6709\u540c\u6a23\u7684\u6bd4\u4f8b\u95dc\u4fc2\u3002<\/p>\n

\u4ed6\u8209\u4f8b\u8aaa\u660e\u4e86\u9019\u6a23\u7684\u6bd4\u4f8b\u95dc\u4fc2\u5982\u4f55\u4f7f\u7528\uff1a<\/p>\n

\"54035_c2\"<\/a><\/p>\n

\u6309\u7167\u6bd4\u4f8b\u95dc\u4fc2\uff0c$$P$$\u00a0<\/i>\u8207 $$X$$\u00a0<\/i>\u53bb\u6389 $$1$$ \u4e4b\u5f8c\u7684\u503c\u8207\u5176\u5c0d\u6578\u503c\u6210\u6bd4\u4f8b\uff0c\u5373<\/p>\n

$$0.0000000000000001278191493200323442:0.0000000000000001=0.0000000000000000555111512312578270212:\\log X$$<\/p>\n

\u56e0\u6b64\u53ef\u5f97 $$\\log X=0.0000000000000000434294481903251804$$<\/p>\n

\u56e0\u6b64\uff0c\u7576\u771f\u6578\u70ba $$1+\\delta$$\uff0c$$\\delta$$\u00a0\u662f\u500b\u5f88\u5c0f\u7684\u6b63\u6578\u6642\uff0c\u82e5\u8a2d $$\\delta=t\\Delta$$\uff08\u5373\u00a0$$t=\\frac{\\delta}{\\Delta}$$\uff09\uff0c<\/p>\n

\u53ef\u5f97 $$\\log(1+\\delta)\\approx tl\\approx \\frac{l}{\\Delta}\\delta$$\uff1b<\/p>\n

\u82e5\u518d\u4ee4 $$k=\\frac{l}{\\Delta}$$\uff0c$$k$$\u00a0<\/i>\u5c31\u6210\u4e86\u4e00\u5df2\u77e5\u7684\u5e38\u6578\u503c\uff0c\u53ef\u7528\u65bc\u985e\u4f3c\u7684\u5c0d\u6578\u6c42\u503c\uff0c<\/p>\n

\u56e0\u6b64\u5e03\u91cc\u683c\u65af\u53ef\u5f97\u5230\u4e00\u500b\u7c21\u55ae\u597d\u7528\u7684\u5de5\u5177\uff1a$$\\log(1+\\delta)=k\\delta$$\u3002<\/p>\n

\u63a5\u8457\u5728\u7b2c $$7$$ \u7ae0\u4e2d\uff0c\u5e03\u91cc\u683c\u65af\u5229\u7528\u9019\u500b\u7d50\u679c\u4f86\u8a08\u7b97 $$\\log 2,\\log 3$$ \u7684\u503c\u3002<\/p>\n

\u4ee5 $$\\log 2$$ \u70ba\u4f8b\uff0c\u4ed6\u9996\u5148\u8a08\u7b97 $$\\log 1.024$$ \u7684\u503c\uff1a<\/p>\n

\u56e0\u70ba\u82e5 $$\\log 1.024=P$$\u00a0\u70ba\u5df2\u77e5\uff0c<\/p>\n

\u5247\u6709 $$\\log 1.024 = \\log \\frac{{1024}}{{1000}} = 10\\log 2 – 3 = P$$\uff0c\u90a3\u9ebc $$\\log 2 = \\frac{{P + 3}}{{10}}$$\u3002<\/p>\n

\u5229\u7528\u7b2c $$6$$ \u7ae0\u7684\u9023\u7e8c\u958b\u65b9\u6cd5\uff0c<\/p>\n

\u4ed6\u5c07 $$1.024$$ \u9023\u7e8c\u958b\u65b9\uff0c\u4f5c\u5230\u7b2c $$47$$ \u6b21\uff0c\u624d\u4f7f\u5f97\u5176\u503c\u5c0f\u6578\u9ede\u5f8c\u6709 $$15$$ \u500b $$0$$\uff0c<\/p>\n

\u5373 $$(1.024)^{1\/2^{47}}=1.0(15)16851605705394977$$\uff0c<\/p>\n

\u4ee4\u5176\u503c\u70ba $$1+\\delta~(\\delta=0.0(15)16851605705394977)$$\uff0c<\/p>\n

\u56e0\u6b64 $$\\log(1.024)^{1\/2^{47}}=\\log(1+\\delta)\\approx k\\delta$$\uff0c\u5176\u4e2d $$k$$\u00a0<\/i>\u662f\u7b2c $$6$$ \u7ae0\u7b97\u51fa\u4f86\u7684\u5df2\u77e5\u5e38\u6578\u3002<\/p>\n

\u5982\u6b64\u4e00\u4f86\u5c31\u53ef\u6c42\u5f97 $$\\log 1.024=2^{47}\\times k\\delta$$\uff0c<\/p>\n

\u6700\u5f8c\u4ed6\u8a08\u7b97\u5f97 $$\\log 2=0.30102,99956,63981,195$$\u3002\u540c\u7406\u53ef\u7b97\u5f97 $$\\log 3$$ \u7684\u503c\u3002<\/p>\n

\u5728\u9019\u4e00\u7ae0\u6700\u5f8c\uff0c\u4ed6\u5229\u7528\u524d\u5e7e\u7ae0\u6240\u5f97\u5230\u7684\u5c0d\u6578\u6027\u8cea\u8a08\u7b97 $$\\log 5$$ \u8207 $$\\log 6$$\uff0c<\/p>\n

\u7279\u5225\u5728\u8a08\u7b97 $$\\log 6$$ \u6642\u7528\u4e86 $$\\frac{6^9}{10^7}$$\uff0c\u9019\u500b\u6578\u95dc\u806f\u5230\u4e0b\u4e00\u7ae0\u7684\u65b9\u6cd5\u3002<\/p>\n

\u53e6\u5916\uff0c\u503c\u5f97\u4e00\u63d0\u7684\u662f\u5e03\u91cc\u683c\u65af\u9078\u64c7\u4e00\u500b\u8207 $$1$$ \u8db3\u5920\u63a5\u8fd1\u7684\u6578 $$10^{\\frac{1}{2^{54}}}$$\uff0c<\/p>\n

\u6839\u64da\u4ed6\u7684\u7b97\u6cd5\u8ddd\u96e2\uff0c\u6211\u5011\u767c\u73fe\u81ea\u7136\u5c0d\u6578\u7684\u5e95 $$e$$ \u9019\u500b\u5e38\u6578\u53ea\u5269\u4e00\u6b65\u4e4b\u9059\u800c\u5df2\u3002<\/p>\n

\u8a2d $$10^{\\frac{1}{2^n}}=1+x$$\uff0c\u5176\u4e2d $$x$$\u00a0<\/i>\u662f\u500b\u5f88\u5c0f\u7684\u6578\u3002<\/p>\n

\u7531\u65bc $$\\ln(1+x)=x-\\frac{x^2}{2}+\\frac{x^3}{3}-\\cdots$$\uff0c\u6545\u7576 $$x$$\u00a0<\/i>\u5f88\u5c0f\u6642\uff0c$$\\ln(1+x)\\approx x=\\frac{1}{2^n}\\times \\ln 10$$\u3002<\/p>\n

\u5728\u5e03\u91cc\u683c\u65af\u7684\u8a08\u7b97\u4e2d\uff0c$$10^{\\frac{1}{2^{54}}}=1+\\Delta$$\uff0c\u4ee4 $$\\frac{1}{2^{54}}=l$$\uff0c\u56e0\u6b64 $$\\ln(1+\\Delta)\\approx \\Delta=l\\times \\ln 10$$\uff0c<\/p>\n

\u4ea6\u5373\u5e03\u91cc\u683c\u65af\u8a08\u7b97\u7684\u5e38\u6578\u00a0$$\\frac{l}{\\Delta } = \\frac{1}{{\\ln 10}} = \\log e$$\u3002<\/p>\n

\u7576\u7136\u6b64\u6642\u7684\u5e03\u91cc\u683c\u65af\u4e26\u4e0d\u77e5\u9053 $$\\ln(1+x)$$\u00a0\u7684\u5c55\u958b\u5f0f\u3002\u96d6\u8aaa\u662f\u4e00\u6b65\uff0c\u537b\u4e5f\u662f\u9700\u8981\u8de8\u904e\u4e00\u9053\u574e\u7684\u4e00\u6b65\u554a\u3002<\/p>\n

\u9023\u7d50\uff1a\u5e03\u91cc\u683c\u65af\u7684\u300a\u5c0d\u6578\u7b97\u8853\u300b\u8207\u5c0d\u6578\u8868\u7684\u88fd\u4f5c(III)\u00a0<\/a><\/p>\n

\u53c3\u8003\u8cc7\u6599\uff1a<\/p>\n

    \n
  1. Briggs’ Arithmetica Logarithmica<\/i>, translated and annotated by Ian Bruce, http:\/\/www.17centurymaths.com\/contents\/albriggs.html<\/a><\/li>\n
  2. \u6797\u5009\u5104\uff082011\uff09\uff0c\u3008\u600e\u9ebc\u7b97log2\u3009\uff0c\u6578\u5b78\u5b78\u79d1\u4e2d\u5fc3\u96fb\u5b50\u5831\u7b2c52\u671f\u3002<\/li>\n
  3. \u8607\u4fca\u9d3b\uff082003\uff09\uff0c\u3008\u6578\u5b78\u53f2\u878d\u5165\u6559\u5b78\uff0d\u4ee5\u5c0d\u6578\u70ba\u4f8b\u3009\uff0c\u300aHPM\u901a\u8a0a\u300b\u7b2c6\u537723\u671f\u5408\u520a\u3002<\/li>\n
  4. \u66f9\u4eae\u5409\uff082010\uff09\uff0c\u3008\u5c0d\u6578\u8868\u7684\u88fd\u4f5c\u3009\uff0c\u7db2\u5740\uff1ahttp:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=14488<\/a><\/li>\n
  5. \u7db2\u8def\u8cc7\u6e90\uff1aThe Difference Method of Henry Briggs\uff0c\u7db2\u5740\uff1ahttp:\/\/www.jacques-laporte.org\/The%20method%20of%20Henry%20briggs.htm<\/a><\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"

    \u5e03\u91cc\u683c\u65af\u7684\u300a\u5c0d\u6578\u7b97\u8853\u300b\u8207\u5c0d\u6578\u8868\u7684\u88fd\u4f5c(II) (Briggs’ Arithmetica Logar…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[221,111,234],"tags":[426,6656,382],"class_list":["post-54035","post","type-post","status-publish","format-standard","hentry","category-math03-02","category-mathematics00","category-math09","tag-426","tag-6656","tag-382","loop-entry","cat-221","cat-111","cat-234","no-thumbnail"],"views":3085,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/54035","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=54035"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/54035\/revisions"}],"predecessor-version":[{"id":87024,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/54035\/revisions\/87024"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=54035"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=54035"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=54035"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}