Harriot, Thomas, Mss. 6788

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              <s xml:space="preserve">[
                <emph style="bf">Commentary:</emph>
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            <p>
              <s xml:space="preserve"> This is a key page in which Harriot sets out his calculations of population under the conditions set out in Add MS 6782
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/HSPGZ0AE&start=60&viewMode=image&pn=62"> f. </ref>
              , and Add MS 6788
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/AYB35Z4D/&start=1070&viewMode=image&pn=1073"> f. </ref>
              , namely, that each male–female pair gives birth to a child each year, alternately male and female, and that those children begin to reproduce after 20 years. All survive indefinitely.
                <lb/>
              Harriot gives no verbal explanation of the calculations, and there are only a few surviving rough diagrams like that in Add MS 6788
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/AYB35Z4D/&start=1100&viewMode=image&pn=1107"> f. </ref>
              .
                <lb/>
              Take the calculations for 30 years as an example. Now there are 15 males born to the original couple; these are marked 'a'. Meanwhile, the 5 oldest males have begun to reproduce, giving birth to 5 + 4 + 3 + 2 + 1 = 15 male children. This sum can be calculated by the rule
                <math>
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                        <mi>n</mi>
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                        <mi>n</mi>
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                        <mn>2</mn>
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              , giving the
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                    <mi>n</mi>
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              th triangular number. This calculation is marked 'b'.
                <lb/>
              The calculations for subsequent years follow a similar pattern. At 50 years, the second generation, measured by triangular numbers, has begun to reproduce. The third generation is therefore measured by sums of triangular numbers, that is, pyramidal numbers, calculated by the rule
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                      <mrow>
                        <mi>n</mi>
                        <mo maxsize="1">(</mo>
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                        <mn>1</mn>
                        <mo maxsize="1">)</mo>
                        <mo maxsize="1">(</mo>
                        <mi>n</mi>
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                        <mn>2</mn>
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                      </mrow>
                      <mrow>
                        <mn>6</mn>
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              . This calculations is marked 'c'.
                <lb/>
              And so </s>
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          <p xml:lang="lat">
            <s xml:space="preserve"> progenies Maris et feminæ ad annos.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            the offspring of men and women for 100 ]</s>
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            [
              <emph style="bf">Translation: </emph>
            In years. ]</s>
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            [
              <emph style="bf">Translation: </emph>
            ]</s>
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            [
              <emph style="bf">Translation: </emph>
            ]</s>
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            <s xml:space="preserve"> a tribus
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            maribus
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            et </s>
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          <p xml:lang="lat">
            <s xml:space="preserve"> Ad annos.
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            [
              <emph style="bf">Translation: </emph>
            For 80 ]</s>
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