Castelli, Benedetto, Of the mensuration of running waters, 1661

Table of figures

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              <s>
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              double, and the quantity increaſing nonuple, the height increa­
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              ſeth triple; ſo that, by adding to units all the odde numbers, ac­
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              cording to their Series, the heights increaſe according to the na­
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              tural progreſſion of all the numbers, from units. </s>
              <s>As for exam­
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              ple, there paſſing thorow a Regulator ſuch a certain quantity of
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              Water in one time; adding three of thoſe meaſures, the quick
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              height is two of thoſe parts, which at firſt was one; and con­
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              tinuing to adde five of thoſe ſaid meaſures, the height is three of
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              thoſe parts which at firſt were one; and thus adding ſeven, and
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              then nine, and then 11. and then 13, &c. </s>
              <s>the heights ſhall be 4.
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              then 5, then 6. then 7, &c. </s>
              <s>And for the greater facility of the
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              Work, we have deſcribed the following Table, of which we will
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              declare the uſe: The Table is divided into three Series or Pro­
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              greſſions of Numbers: the firſt Series containeth all the Num­
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              bers in the Natural Progreſſion, beginning at a Unit, and is called
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              the Series of the Heights; the ſecond containeth all the odde
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              numbers, beginning at an unit, and is called the Series of the
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              Additions: the third containeth all the ſquare numbers, begin­
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              ning at an unit, and is called the Series of Quantity.
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            <table>
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              <row>
                <cell>Heights.</cell>
                <cell>1</cell>
                <cell>2</cell>
                <cell>3</cell>
                <cell>4</cell>
                <cell>5</cell>
                <cell>6</cell>
                <cell>7</cell>
                <cell>8</cell>
                <cell>9</cell>
                <cell>10</cell>
                <cell>11</cell>
              </row>
              <row>
                <cell>Additions.</cell>
                <cell>1</cell>
                <cell>3</cell>
                <cell>5</cell>
                <cell>7</cell>
                <cell>9</cell>
                <cell>11</cell>
                <cell>13</cell>
                <cell>15</cell>
                <cell>17</cell>
                <cell>19</cell>
                <cell>21</cell>
              </row>
              <row>
                <cell>Quantities.</cell>
                <cell>1</cell>
                <cell>4</cell>
                <cell>9</cell>
                <cell>16</cell>
                <cell>25</cell>
                <cell>36</cell>
                <cell>49</cell>
                <cell>64</cell>
                <cell>81</cell>
                <cell>100</cell>
                <cell>121</cell>
              </row>
            </table>
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              <s>
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              The uſe of the afore-mentioned Table.
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              <s>Firſt, if we ſuppoſe the whole quick height of a River of Run­
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              ning Water to be divided into any number of equal parts, at
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              pleaſure, and would abate the ſame one fift, by means of a divi­
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              ſron; let there be found in the Table in the Series of heights the
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              number 5. the denominator of the part which the River is to a­
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              bate, and take the number that is immediately under it in the
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              row of Additions, which is 9. which let be ſubſtracted from the
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              number 25. placed underneath the ſame in the row of Quanti­
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              ties, the remainder 16. ſignifieth that of the 25. parts of Water
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              that ran in the River, whilſt it was 5 meaſures high, there do
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              onely run 16. parts; ſo that to make it abate 1/5 it is neceſſary to
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              take 9/25 from the Water that the whole River did carry; ſo that
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              with ſubſtracting ſomewhat more than one third of the Water of
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              the River, it is abated but only one fift.</s>
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              <s>2. And thus, in the ſecond place, if on the contrary, one would
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              know how much water is to be added to the ſaid River to make
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              it increaſe one fift more in height, ſo as that it may run in the </s>
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