File:Seki Kowa Katsuyo Sampo Bernoulli numbers.png

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Original file ‎(1,614 × 2,124 pixels, file size: 5.41 MB, MIME type: image/png)

Summary

This is a facsimile of the pages from the Katsuyo Sampo of <a href="https://en.wikipedia.org/wiki/Seki_Kowa" class="extiw" title="en:Seki Kowa">en:Seki Kowa</a> that tabulate the <a href="https://en.wikipedia.org/wiki/binomial_coefficients" class="extiw" title="en:binomial coefficients">binomial coefficients</a> and the <a href="https://en.wikipedia.org/wiki/Bernoulli_numbers" class="extiw" title="en:Bernoulli numbers">Bernoulli numbers</a>. Seki died in 1708, and the work was published in 1712, so it is in the public domain.

I obtained the file itself from the the web site of the <a href="https://en.wikipedia.org/wiki/Mathematical_Association_of_America" class="extiw" title="en:Mathematical Association of America">Mathematical Association of America</a>, <a rel="nofollow" class="external text" href="http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=2591&bodyId=3549">here</a>, then converted it from TIFF to PNG format.

The main portion of the table displays the <a href="https://en.wikipedia.org/wiki/binomial_coefficients" class="extiw" title="en:binomial coefficients">binomial coefficients</a> arranged in <a href="https://en.wikipedia.org/wiki/Pascal%27s_triangle" class="extiw" title="en:Pascal's triangle">Pascal's triangle</a>. The coefficients are written using <a href="https://en.wikipedia.org/wiki/counting_rods" class="extiw" title="en:counting rods">rod numerals</a>.

The bottommost row tabulates the <a href="https://en.wikipedia.org/wiki/Bernoulli_numbers" class="extiw" title="en:Bernoulli numbers">Bernoulli numbers</a>. The rightmost value is a special case, 全 ("everything"). Each of the other entries is either 空 (zero, literally, "empty") or a fraction n/d in the form d分之n. (See <a href="https://en.wikipedia.org/wiki/Chinese_numerals#Fractional_values" class="extiw" title="en:Chinese numerals">en:Chinese_numerals#Fractional_values</a> for an elaboration of this notation.) There is a small double mark 二 in the margin before the denominatord and a small single mark 一 in the margin after the numerator n. So for example the fraction in the third column from the right (representing <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199944d59dcc18842dfd1deab6000a1d1dadcbae" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:2.836ex; height:2.509ex;" alt="{\displaystyle B_{2}}">) is written as 六分之一, which means 1/6; the fraction in the seventh column ( <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6f54c8c73ba4f7fde4b4b911ec00739f0de1d0" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:2.836ex; height:2.509ex;" alt="{\displaystyle B_{6}}">) is written 四十二分之一, which means 1/42. The sign is given by the characters under the marginal 一 mark, with 爲加 ("add") indicating positive values, and 爲減 ("reduce") indicating negative values.

The complete reading of the bottom row, right to left, is: "everything", 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0.

—<a href="//commons.wikimedia.org/wiki/User:Dominus" title="User:Dominus">Dominus</a>

Licensing

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File history

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Date/TimeThumbnailDimensionsUserComment
current00:16, 13 January 2017Thumbnail for version as of 00:16, 13 January 20171,614 × 2,124 (5.41 MB)127.0.0.1 (talk)<p>This is a facsimile of the pages from the <i>Katsuyo Sampo</i> of <a href="https://en.wikipedia.org/wiki/Seki_Kowa" class="extiw" title="en:Seki Kowa">en:Seki Kowa</a> that tabulate the <a href="https://en.wikipedia.org/wiki/binomial_coefficients" class="extiw" title="en:binomial coefficients">binomial coefficients</a> and the <a href="https://en.wikipedia.org/wiki/Bernoulli_numbers" class="extiw" title="en:Bernoulli numbers">Bernoulli numbers</a>. Seki died in 1708, and the work was published in 1712, so it is in the public domain. </p> <p>I obtained the file itself from the the web site of the <a href="https://en.wikipedia.org/wiki/Mathematical_Association_of_America" class="extiw" title="en:Mathematical Association of America">Mathematical Association of America</a>, <a rel="nofollow" class="external text" href="http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=2591&bodyId=3549">here</a>, then converted it from TIFF to PNG format. </p> <p>The main portion of the table displays the <a href="https://en.wikipedia.org/wiki/binomial_coefficients" class="extiw" title="en:binomial coefficients">binomial coefficients</a> arranged in <a href="https://en.wikipedia.org/wiki/Pascal%27s_triangle" class="extiw" title="en:Pascal's triangle">Pascal's triangle</a>. The coefficients are written using <a href="https://en.wikipedia.org/wiki/counting_rods" class="extiw" title="en:counting rods">rod numerals</a>. </p> <p>The bottommost row tabulates the <a href="https://en.wikipedia.org/wiki/Bernoulli_numbers" class="extiw" title="en:Bernoulli numbers">Bernoulli numbers</a>. The rightmost value is a special case, 全 ("everything"). Each of the other entries is either 空 (zero, literally, "empty") or a fraction <i>n</i>/<i>d</i> in the form <i>d</i>分之<i>n</i>. (See <a href="https://en.wikipedia.org/wiki/Chinese_numerals#Fractional_values" class="extiw" title="en:Chinese numerals">en:Chinese_numerals#Fractional_values</a> for an elaboration of this notation.) There is a small double mark 二 in the margin before the denominator<i>d</i> and a small single mark 一 in the margin after the numerator <i>n</i>. So for example the fraction in the third column from the right (representing <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle B_{2}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199944d59dcc18842dfd1deab6000a1d1dadcbae" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:2.836ex; height:2.509ex;" alt="{\displaystyle B_{2}}"></span>) is written as 六分之一, which means 1/6; the fraction in the seventh column (<span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mn>6</mn></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle B_{6}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6f54c8c73ba4f7fde4b4b911ec00739f0de1d0" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:2.836ex; height:2.509ex;" alt="{\displaystyle B_{6}}"></span>) is written 四十二分之一, which means 1/42. The sign is given by the characters under the marginal 一 mark, with 爲加 ("add") indicating positive values, and 爲減 ("reduce") indicating negative values. </p> <p>The complete reading of the bottom row, right to left, is: "everything", 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0. </p> <p>—<a href="//commons.wikimedia.org/wiki/User:Dominus" title="User:Dominus">Dominus</a> </p>
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