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庄子曰：“一尺之棰，日取其半，万世不竭。”

现在有一小球，在2s内匀速走了4m，可知其平均速度为$4\mathrm{m}/2\mathrm{s}=2 \mathrm{m} /\mathrm{s}$,那么如何求其在初始时刻的瞬时速度呢。

我们使用眼球技术可以得知其瞬时速度也为$2 \mathrm{m} /\mathrm{s}$，然而要怎么准确的描述这一数值。

瞬时，意味着在这一瞬间其移动的位移为0，移动的时间也为0，0不能作为除数，瞬时速度无法直接求解。

牛顿用一个动态的无穷小量来描述这个问题，即位移的无穷小量和时间的无穷小量的“终极比”来描述瞬时速度。

无穷小量的意思，就类似于“线段的长度无限缩小，最后会变成一个点，但是现在它还是一根线”。

一尺之棰是有长度的，日取一半，就算取成粉末也是有长度的，只有在无穷大的时间之后，它才会变成一个点，达到取尽的结果。

现在我们来日取其半,可知当位移取 $4/{2^{n}}$时，时间取$2/{2^{n}}$，当n趋近于无穷大的时候，位移和时间都变成一个“几乎可以忽略的值”，这就是无穷小量，一个动态趋近于无穷小的值。

他们的终极比，也就是瞬时速度，一直为2m/s。

以下为趋近的过程

|位移(m)|时间(s)|速度(m/s)|
|4.00000000000000000000|2.00000000000000000000|2.00000000000000000000|
|2.00000000000000000000|1.00000000000000000000|2.00000000000000000000|
|1.00000000000000000000|0.50000000000000000000|2.00000000000000000000|
|0.50000000000000000000|0.25000000000000000000|2.00000000000000000000|
|0.25000000000000000000|0.12500000000000000000|2.00000000000000000000|
|0.12500000000000000000|0.06250000000000000000|2.00000000000000000000|
|0.06250000000000000000|0.03125000000000000000|2.00000000000000000000|
|0.03125000000000000000|0.01562500000000000000|2.00000000000000000000|
|0.01562500000000000000|0.00781250000000000000|2.00000000000000000000|
|0.00781250000000000000|0.00390625000000000000|2.00000000000000000000|
|0.00390625000000000000|0.00195312500000000000|2.00000000000000000000|
|0.00195312500000000000|0.00097656250000000000|2.00000000000000000000|
|0.00097656250000000000|0.00048828125000000000|2.00000000000000000000|
|0.00048828125000000000|0.00024414062500000000|2.00000000000000000000|
|0.00024414062500000000|0.00012207031250000000|2.00000000000000000000|
|0.00012207031250000000|0.00006103515625000000|2.00000000000000000000|
|0.00006103515625000000|0.00003051757812500000|2.00000000000000000000|
|0.00003051757812500000|0.00001525878906250000|2.00000000000000000000|
|0.00001525878906250000|0.00000762939453125000|2.00000000000000000000|
|0.00000762939453125000|0.00000381469726562500|2.00000000000000000000|
|0.00000381469726562500|0.00000190734863281250|2.00000000000000000000|
|0.00000190734863281250|0.00000095367431640625|2.00000000000000000000|
|0.00000095367431640625|0.00000047683715820313|2.00000000000000000000|
|0.00000047683715820313|0.00000023841857910156|2.00000000000000000000|
|0.00000023841857910156|0.00000011920928955078|2.00000000000000000000|
|0.00000011920928955078|0.00000005960464477539|2.00000000000000000000|
|0.00000005960464477539|0.00000002980232238770|2.00000000000000000000|
|0.00000002980232238770|0.00000001490116119385|2.00000000000000000000|
|0.00000001490116119385|0.00000000745058059692|2.00000000000000000000|
|0.00000000745058059692|0.00000000372529029846|2.00000000000000000000|
|0.00000000372529029846|0.00000000186264514923|2.00000000000000000000|
|0.00000000186264514923|0.00000000093132257462|2.00000000000000000000|
|0.00000000093132257462|0.00000000046566128731|2.00000000000000000000|
|0.00000000046566128731|0.00000000023283064365|2.00000000000000000000|
|0.00000000023283064365|0.00000000011641532183|2.00000000000000000000|
|0.00000000011641532183|0.00000000005820766091|2.00000000000000000000|
|0.00000000005820766091|0.00000000002910383046|2.00000000000000000000|
|0.00000000002910383046|0.00000000001455191523|2.00000000000000000000|
|0.00000000001455191523|0.00000000000727595761|2.00000000000000000000|
|0.00000000000727595761|0.00000000000363797881|2.00000000000000000000|
|0.00000000000363797881|0.00000000000181898940|2.00000000000000000000|
|0.00000000000181898940|0.00000000000090949470|2.00000000000000000000|
|0.00000000000090949470|0.00000000000045474735|2.00000000000000000000|
|0.00000000000045474735|0.00000000000022737368|2.00000000000000000000|
|0.00000000000022737368|0.00000000000011368684|2.00000000000000000000|
|0.00000000000011368684|0.00000000000005684342|2.00000000000000000000|
|0.00000000000005684342|0.00000000000002842171|2.00000000000000000000|
|0.00000000000002842171|0.00000000000001421085|2.00000000000000000000|
|0.00000000000001421085|0.00000000000000710543|2.00000000000000000000|
|0.00000000000000710543|0.00000000000000355271|2.00000000000000000000|
|0.00000000000000355271|0.00000000000000177636|2.00000000000000000000|
|0.00000000000000177636|0.00000000000000088818|2.00000000000000000000|
|0.00000000000000088818|0.00000000000000044409|2.00000000000000000000|
|0.00000000000000044409|0.00000000000000022204|2.00000000000000000000|
|0.00000000000000022204|0.00000000000000011102|2.00000000000000000000|
|0.00000000000000011102|0.00000000000000005551|2.00000000000000000000|
|0.00000000000000005551|0.00000000000000002776|2.00000000000000000000|
|0.00000000000000002776|0.00000000000000001388|2.00000000000000000000|
|0.00000000000000001388|0.00000000000000000694|2.00000000000000000000|
|0.00000000000000000694|0.00000000000000000347|2.00000000000000000000|
|0.00000000000000000347|0.00000000000000000173|2.00000000000000000000|
|0.00000000000000000173|0.00000000000000000087|2.00000000000000000000|
|0.00000000000000000087|0.00000000000000000043|2.00000000000000000000|
|0.00000000000000000043|0.00000000000000000022|2.00000000000000000000|
|0.00000000000000000022|0.00000000000000000011|2.00000000000000000000|
|0.00000000000000000011|0.00000000000000000005|2.00000000000000000000|
|0.00000000000000000005|0.00000000000000000003|2.00000000000000000000|
|0.00000000000000000003|0.00000000000000000001|2.00000000000000000000|


以上结论有什么意义呢，当瞬时速度是一个变化的量的时候，我们也可以求出这种复杂的解。

如速度的函数是$c2^t$,在2s内走过了4m，求c。

用0.01s来充当无穷小的时间间隔，虽然不准确但是可以看看精度如何。

|时间（间隔0.01s）|瞬时速度除以C=$2^t$|瞬时速度除以C后求和$\Sigma 2^t$|总位移除以C$\Sigma 2^t \cdot 0.01$|
|0.01000000000000000000|1.00695555005672000000|1.00695555005672000000|0.01006955550056720000|
|0.02000000000000000000|1.01395947979003000000|2.02091502984675000000|0.02020915029846750000|
|0.03000000000000000000|1.02101212570719000000|3.04192715555394000000|0.03041927155553940000|
|0.04000000000000000000|1.02811382665607000000|4.07004098221001000000|0.04070040982210010000|
|0.05000000000000000000|1.03526492384138000000|5.10530590605139000000|0.05105305906051390000|
|0.06000000000000000000|1.04246576084112000000|6.14777166689251000000|0.06147771666892510000|
|0.07000000000000000000|1.04971668362307000000|7.19748835051557000000|0.07197488350515580000|
|0.08000000000000000000|1.05701804056138000000|8.25450639107695000000|0.08254506391076950000|
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|1.47000000000000000000|2.77021893622185000000|256.27474004334600000000|2.56274740043346000000|
|1.48000000000000000000|2.78948733270081000000|259.06422737604700000000|2.59064227376047000000|
|1.49000000000000000000|2.80888975147600000000|261.87311712752300000000|2.61873117127523000000|
|1.50000000000000000000|2.82842712474619000000|264.70154425226900000000|2.64701544252269000000|
|1.51000000000000000000|2.84810039119414000000|267.54964464346300000000|2.67549644643463000000|
|1.52000000000000000000|2.86791049603166000000|270.41755513949500000000|2.70417555139495000000|
|1.53000000000000000000|2.88785839104499000000|273.30541353054000000000|2.73305413530540000000|
|1.54000000000000000000|2.90794503464062000000|276.21335856518000000000|2.76213358565180000000|
|1.55000000000000000000|2.92817139189125000000|279.14152995707100000000|2.79141529957071000000|
|1.56000000000000000000|2.94853843458220000000|282.09006839165400000000|2.82090068391654000000|
|1.57000000000000000000|2.96904714125810000000|285.05911553291200000000|2.85059115532912000000|
|1.58000000000000000000|2.98969849726988000000|288.04881403018200000000|2.88048814030182000000|
|1.59000000000000000000|3.01049349482214000000|291.05930752500400000000|2.91059307525004000000|
|1.60000000000000000000|3.03143313302080000000|294.09074065802500000000|2.94090740658025000000|
|1.61000000000000000000|3.05251841792112000000|297.14325907594600000000|2.97143259075946000000|
|1.62000000000000000000|3.07375036257603000000|300.21700943852200000000|3.00217009438522000000|
|1.63000000000000000000|3.09512998708478000000|303.31213942560700000000|3.03312139425607000000|
|1.64000000000000000000|3.11665831864200000000|306.42879774424900000000|3.06428797744249000000|
|1.65000000000000000000|3.13833639158701000000|309.56713413583600000000|3.09567134135836000000|
|1.66000000000000000000|3.16016524745351000000|312.72729938328900000000|3.12727299383289000000|
|1.67000000000000000000|3.18214593501968000000|315.90944531830900000000|3.15909445318309000000|
|1.68000000000000000000|3.20427951035849000000|319.11372482866700000000|3.19113724828667000000|
|1.69000000000000000000|3.22656703688851000000|322.34029186555600000000|3.22340291865556000000|
|1.70000000000000000000|3.24900958542494000000|325.58930145098100000000|3.25589301450981000000|
|1.71000000000000000000|3.27160823423113000000|328.86090968521200000000|3.28860909685212000000|
|1.72000000000000000000|3.29436406907029000000|332.15527375428200000000|3.32155273754282000000|
|1.73000000000000000000|3.31727818325777000000|335.47255193754000000000|3.35472551937540000000|
|1.74000000000000000000|3.34035167771348000000|338.81290361525300000000|3.38812903615253000000|
|1.75000000000000000000|3.36358566101486000000|342.17648927626800000000|3.42176489276268000000|
|1.76000000000000000000|3.38698124945011000000|345.56347052571800000000|3.45563470525718000000|
|1.77000000000000000000|3.41053956707183000000|348.97401009279000000000|3.48974010092790000000|
|1.78000000000000000000|3.43426174575102000000|352.40827183854100000000|3.52408271838541000000|
|1.79000000000000000000|3.45814892523146000000|355.86642076377300000000|3.55866420763773000000|
|1.80000000000000000000|3.48220225318450000000|359.34862301695700000000|3.59348623016957000000|
|1.81000000000000000000|3.50642288526414000000|362.85504590222100000000|3.62855045902221000000|
|1.82000000000000000000|3.53081198516262000000|366.38585788738400000000|3.66385857887384000000|
|1.83000000000000000000|3.55537072466628000000|369.94122861205000000000|3.69941228612050000000|
|1.84000000000000000000|3.58010028371189000000|373.52132889576200000000|3.73521328895762000000|
|1.85000000000000000000|3.60500185044332000000|377.12633074620500000000|3.77126330746205000000|
|1.86000000000000000000|3.63007662126865000000|380.75640736747400000000|3.80756407367474000000|
|1.87000000000000000000|3.65532580091760000000|384.41173316839200000000|3.84411733168392000000|
|1.88000000000000000000|3.68075060249950000000|388.09248377089100000000|3.88092483770891000000|
|1.89000000000000000000|3.70635224756149000000|391.79883601845300000000|3.91798836018453000000|
|1.90000000000000000000|3.73213196614723000000|395.53096798460000000000|3.95530967984600000000|
|1.91000000000000000000|3.75809099685605000000|399.28905898145600000000|3.99289058981456000000|
|1.92000000000000000000|3.78423058690239000000|403.07328956835800000000|4.03073289568358000000|
|1.93000000000000000000|3.81055199217575000000|406.88384156053400000000|4.06883841560534000000|
|1.94000000000000000000|3.83705647730106000000|410.72089803783500000000|4.10720898037835000000|
|1.95000000000000000000|3.86374531569939000000|414.58464335353500000000|4.14584643353535000000|
|1.96000000000000000000|3.89061978964915000000|418.47526314318400000000|4.18475263143184000000|
|1.97000000000000000000|3.91768119034771000000|422.39294433353100000000|4.22392944333531000000|
|1.98000000000000000000|3.94493081797344000000|426.33787515150500000000|4.26337875151505000000|
|1.99000000000000000000|3.97236998174815000000|430.31024513325300000000|4.30310245133253000000|
|2.00000000000000000000|4.00000000000000000000|434.31024513325300000000|4.34310245133253000000|
| | |结论：$c=4/(\Sigma 2^t$*0.01)|0.92100060839521300000|

实际$c=4/3 \ln 2 = 0.924196$

小数点后面已经非常接近啦！



参考：[[https://newtonproject.ox.ac.uk/view/texts/normalized/NATP00356|牛顿流数术的历史]]

{{ :数学分析:history_of_the_method_of_fluxions_normalized_.pdf |}}
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