Harriot, Thomas, Mss. 6789

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      <text xml:lang="eng" type="free">
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          <pb file="0421.jpg" o="211r" n="421"/>
          <div type="page_commentary" level="0" n="0">
            <p>
              <s xml:space="preserve">[
                <emph style="bf">Commentary:</emph>
              </s>
            </p>
            <p>
              <s xml:space="preserve">This page is part of Harriot's account of the
                <ref target="http://echo.mpiwg-berlin.mpg.de/content/scientific_revolution/harriot/maps/9.2.5_prism.pt"> refractive properties of the triangular prism</ref>
              . </s>
              <s xml:space="preserve">Here, Harriot sets out the basic relations that hold among the angles of incidence and refraction, simply by reasons of geometry (i.e., independently of how one calculates the amount of refraction). He will use these identities throughout his treatment both of the simple refraction of light through the prism, and his
                <ref target="http://echo.mpiwg-berlin.mpg.de/content/scientific_revolution/harriot/maps/9.2.3_colour.pt"> experiments on refractive colours</ref>
              , also conducted using triangular media. </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat">Lemma. Propositio.
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Lemma. Proposition ] </head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Duae perpendiculares super
              <lb/>
            duae lineas inclinantes
              <lb/>
            faciunt angulum aequalem
              <lb/>
            angulo inclinantium
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Two perpendicular lines constructed upon two intersecting lines make an angle to each other equal to the angle between the intersecting lines. </s>
          </p>
          <head xml:space="preserve" xml:lang="lat"> Per angulum/prisma vitreum
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Through a glass angle/prism] </head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Adfractivus et Abfractivus aerei sunt eiusdem amplitudinis
              <lb/>
            [???] Adfractus et Abfractivus vitrei sunt etiam aequales amplitudine.
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            The adfractive and abfractive rays in air are of the same angular size.
              <lb/>
            The adfracted and abfractive rays in glass are also equal in angular size. ]
              <lb/>
            [
              <emph style="bf">Commentary: </emph>
            Here Harriot uses a variation of his usual terminology for the multiple angles of refraction through an interface and back out the other side. [Diagram here??]
              <lb/>
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>1</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            :
              <foreign xml:lang="lat">adfractivus</foreign>
            (or
              <foreign xml:lang="lat">incidens</foreign>
            ); translated 'adfractive'.
              <lb/>
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>2</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            :
              <foreign xml:lang="lat">adfractus</foreign>
            ; translated 'adfracted'.
              <lb/>
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>3</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            :
              <foreign xml:lang="lat">[incidens] abfractivus</foreign>
            ; translated 'abfractive'.
              <lb/>
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>4</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            :
              <foreign xml:lang="lat">abfractus</foreign>
            ; translated 'abfracted'.
              <lb/>
            Harriot states here, then, that
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>1</mn>
                    </msub>
                  </mrow>
                  <mo>=</mo>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>4</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>2</mn>
                    </msub>
                  </mrow>
                  <mo>=</mo>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>3</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            — which is not true in general, but he will explain further below the circumstance in which this will be true. ] </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Dato igitur
              <emph style="st">triangulo</emph>
              <emph style="super">angulo</emph>
            (prismatis); dabitur talis
              <emph style="super">incidens</emph>
            adfractivus aereus
              <lb/>
            qui faciet abfractum aereum sibi amplitudine æqualem.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            And so, given the
              <emph style="st">triangle</emph>
            angle (of the prism), just such an incident, adfractive ray in air will also be given, which will make the abfracted ray in air equal to it in angular size. ] </s>
          </p>
          <head xml:space="preserve" xml:lang="latin"> De medio densiori
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Concerning a denser interposed medium. ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Illud medium est superficierum parallelarum;
              <lb/>
            vel
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            This medium has either parallel surfaces, or inclined ones. ] </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> In medio planorum parallelorum, angulus adfractus
              <lb/>
            et abfractivus sunt
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            In a medium [bounded by] plane parallel sides, the adfracted and abfractive angles are equal. ]
              <lb/>
            [
              <emph style="bf">Commentary: </emph>
            That is, the two angles within the glass are equal, or
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>2</mn>
                    </msub>
                  </mrow>
                  <mo>=</mo>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>3</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> In medio planorum inclinantium, differentia
              <emph style="super">vel summa</emph>
            angulorum
              <lb/>
            inclinationis, et adfracti, est amplitudo
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            In a medium [bounded by] inclined plane sides, the difference
              <emph style="super">or sum</emph>
            of the angle of inclination and the adfracted angle, is the angular size of the abfractive ray. ]
              <lb/>
            [
              <emph style="bf">Commentary: </emph>
            That is, if the sides are inclined at some angle (say
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            ), then
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mo>±</mo>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>2</mn>
                    </msub>
                  </mrow>
                  <mo>=</mo>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>3</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            . </s>
          </p>
          <head xml:space="preserve" xml:lang="lat"> Angulus inclinationis
            <lb/>
            <emph style="st">minor adfracto.</emph>
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Angle of inclination
            <emph style="st">less than the adfracted [angle].</emph>
          </head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Summa. Si perpendiculares
              <lb/>
            secant se mutuo extra
              <lb/>
            angulum
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            A sum: if the perpendiculars cut each other beyond the angle of inclination. ]
              <lb/>
            </s>
            <s xml:space="preserve"> vide exempla magis
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            See the more salient examples. ]
              <lb/>
            [
              <emph style="bf">Commentary: </emph>
            These more salient examples are found on
              <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/mpiwg/online/permanent/library/0VGM2B80&viewMode=image&characterNormalization=orig&viewLayer=dict&pn=431"> f. 216</ref>
            . </s>
          </p>
          <head xml:space="preserve" xml:lang="lat">
            <emph style="st">maior adfracto.</emph>
            <lb/>
          [
            <emph style="bf">Translation: </emph>
            <emph style="st">greater than the adfracted [angle].</emph>
          ] </head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Differentia. Se perpendiculares
              <lb/>
            secant se intra
              <lb/>
            angulum
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            A difference: if the perpendiculars cut each other within the angle of inclination. ]</s>
          </p>
          <head xml:space="preserve" xml:lang="lat">aequalis
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          equal to the adfracted [angle]. </head>
          <p xml:lang="lat">
            <s xml:space="preserve"> 3 casus. secantes se in una
              <lb/>
            linearum continentium angulum inclinationis.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Third case. Cutting each other in a single one of the line containing the angle of inclination. ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> nota:
              <lb/>
            In medio rariori iidem canones
              <lb/>
            mutatis solum ad in ab &
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Note: In a rarer medium, the same rules [apply], just changing 'ad' to 'ab' and vice versa. ]
              <lb/>
            [
              <emph style="bf">Commentary: </emph>
            That is, swap 'abfractus' and 'adfractus', and 'abfractivus' and 'adfractivus', or
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>2</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>4</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>1</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>3</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            ; so that, for example, the rule for a medium with sides inclined at angle
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            becomes
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mo>±</mo>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>4</mn>
                    </msub>
                  </mrow>
                  <mo>=</mo>
                  <mrow>
                    <msub>
                      <mi>θ</mi>
                      <mn>1</mn>
                    </msub>
                  </mrow>
                </mstyle>
              </math>
            . </s>
          </p>
        </div>
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