Harriot, Thomas, Mss. 6787

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          <pb file="add_6787_f221" o="221" n="440"/>
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            <p>
              <s xml:space="preserve">[
                <emph style="bf">Commentary:</emph>
              </s>
            </p>
            <p>
              <s xml:space="preserve"> At the end of Chapter XIX of
                <emph style="it">Variorum responsorum liber VIII</emph>
              (1593), under the heading 'ALIUD', Viète listed sixteen propositions connecting sines, tangents, and secants
                <ref id="Viete_1593d" target="http://www.e-rara.ch/zut/content/pageview/2684282"> (Viete 1593d, Chapter 19, </ref>
              . In the first edition of the
                <emph style="it">Responsorum</emph>
              , the pages were numbered only on the recto side. However, the pagination went badly wrong, so that in Chapter XIX we have the sequence: 37, 38, 39, 38. This sometimes makes it difficult to follow Harriot's references correctly. Here it seems that he has seen the number '38' on the right-hand (recto) page, and thus inferred that the left-hand (verso) page must be 37v, whereas it is in fact 39v. In the 1646 edition of Viète's
                <emph style="it">Opera mathematica</emph>
              the sixteen propositions are to be found on pages </s>
              <lb/>
              <s xml:space="preserve"> On this and the following pages, Harriot worked through the sixteen propositions systematically. On this page he lists the first six. Note that for Harriot, as for Viète, trigonometrical relationships arose from astronomy. Thus the concepts of sine, tangent, and secant related not to angles defined by a pair of lines meeting at a point, but to arcs of a circle with a given radius, and therefore only by implication to the angles subtended by </s>
              <lb/>
              <s xml:space="preserve"> At the top of the page Harriot listed the relevant quantities: sine, tangent, secant, radius (or whole sine), and the symbols he regularly used for them. The equivalent names used by Viète were sinus, prosinus, transinuosa, totus. Harriot had no words for cosine, cotangent, or coseceant. Where we would use 'cosine', for example, he spoke of the sine of the complement. Thus he wrote
                <math>
                  <mstyle>
                    <mi>υ</mi>
                  </mstyle>
                </math>
              BC for sine(arc BC) but
                <math>
                  <mstyle>
                    <mi>υ</mi>
                  </mstyle>
                </math>
                <emph style="st">BC</emph>
              for the sine of the complement of BC, that is, cosine(BC). </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> vieta in lib. 8. respons.
            <lb/>
          pag. 37.
            <lb/>
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Viète, in Responsorum liber VIII, page 37, ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> sinus
              <lb/>
            tangens
              <lb/>
            secans
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            sine
              <lb/>
            tangent
              <lb/>
            secans
              <lb/>
            ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> 1. Sinus peripheriæ, Radius: Radius, Secans complementi.
              <lb/>
            2. Sinus comp. peripheriæ. Radius. Radius. Secans peripheriæ.
              <lb/>
            3. Tangens peripheriæ. Radius. Radius. Tangens
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            1. Sine of the arc : Radius = Radius : Secant of the complement.
              <lb/>
            2. Sine of the compplement of the arc : Radius = Radius : Secant of the arc.
              <lb/>
            3. Tangent of the arc : Radius = Radius : Tangent of the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Ergo: cum proponuntur duæ peripheriæ
              <lb/>
            4. Sinus peripheriæ primæ. Sinus secundæ. Secans. compl. secundæ. Secans compl. primæ.
              <lb/>
            5. Sinus compl. primæ. Sinus com. secundæ. Secans 2
              <emph style="super">æ</emph>
            . Secans primæ.
              <lb/>
            6. Tangens primæ. Tangens 2
              <emph style="super">æ</emph>
            . Tangens comp. 2
              <emph style="super">æ</emph>
            . Tangens compl.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Therefore, when there are given two arcs:
              <lb/>
            4. Sine of the first arc : Sine of the second = Secant of the complement of the second : Secant of the complement of the first.
              <lb/>
            5. Sine of the complement of the first : SIne of the complement of the second = Secant of the second : Secant of the first.
              <lb/>
            6. Tangent of the first : Tangent of the second = Tangent of the complemnt of the second : Tangent of the complement of the first.</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Menda in Vieta.
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Wrong in Viète.
              <lb/>
            Correction. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> sinus
              <lb/>
            tangens
              <lb/>
            secans
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
Impressum