Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

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        <div xml:id="echoid-div169" type="section" level="1" n="79">
          <pb o="434" file="0152" n="161" rhead="VERA CIRCULI"/>
        </div>
        <div xml:id="echoid-div170" type="section" level="1" n="80">
          <head xml:id="echoid-head116" xml:space="preserve">PROP. XII. THEOREMA.</head>
          <p>
            <s xml:id="echoid-s3269" xml:space="preserve">Sit trapezium A B I P, A; </s>
            <s xml:id="echoid-s3270" xml:space="preserve">polygonum
              <lb/>
              <note position="left" xlink:label="note-0152-01" xlink:href="note-0152-01a" xml:space="preserve">TAB. XLIII.
                <lb/>
              Fig. 1. 2. 3.</note>
              <note position="right" xlink:label="note-0152-02" xlink:href="note-0152-02a" xml:space="preserve">
                <lb/>
              A # C # D # B
                <lb/>
              </note>
            A B E I O P, C; </s>
            <s xml:id="echoid-s3271" xml:space="preserve">polygonum A B C G
              <lb/>
            K N P, D; </s>
            <s xml:id="echoid-s3272" xml:space="preserve">& </s>
            <s xml:id="echoid-s3273" xml:space="preserve">polygonum A B D L P, B. </s>
            <s xml:id="echoid-s3274" xml:space="preserve">di-
              <lb/>
            co D eſſe medium harmonicum inter C & </s>
            <s xml:id="echoid-s3275" xml:space="preserve">B. </s>
            <s xml:id="echoid-s3276" xml:space="preserve">ex hujus 4,
              <lb/>
            A: </s>
            <s xml:id="echoid-s3277" xml:space="preserve">C:</s>
            <s xml:id="echoid-s3278" xml:space="preserve">: C: </s>
            <s xml:id="echoid-s3279" xml:space="preserve">B, & </s>
            <s xml:id="echoid-s3280" xml:space="preserve">componendo A + C: </s>
            <s xml:id="echoid-s3281" xml:space="preserve">C:</s>
            <s xml:id="echoid-s3282" xml:space="preserve">: C + B: </s>
            <s xml:id="echoid-s3283" xml:space="preserve">B, ſed ex
              <lb/>
            hujus 5, A + C: </s>
            <s xml:id="echoid-s3284" xml:space="preserve">C:</s>
            <s xml:id="echoid-s3285" xml:space="preserve">: 2 C: </s>
            <s xml:id="echoid-s3286" xml:space="preserve">D; </s>
            <s xml:id="echoid-s3287" xml:space="preserve">& </s>
            <s xml:id="echoid-s3288" xml:space="preserve">ideo C + B: </s>
            <s xml:id="echoid-s3289" xml:space="preserve">B:</s>
            <s xml:id="echoid-s3290" xml:space="preserve">: 2 C: </s>
            <s xml:id="echoid-s3291" xml:space="preserve">D, & </s>
            <s xml:id="echoid-s3292" xml:space="preserve">
              <lb/>
            permutando B + C: </s>
            <s xml:id="echoid-s3293" xml:space="preserve">2 C:</s>
            <s xml:id="echoid-s3294" xml:space="preserve">: B: </s>
            <s xml:id="echoid-s3295" xml:space="preserve">D, & </s>
            <s xml:id="echoid-s3296" xml:space="preserve">dividendo, differentia in-
              <lb/>
            ter B & </s>
            <s xml:id="echoid-s3297" xml:space="preserve">C eſt ad 2 C, ut differentia inter B & </s>
            <s xml:id="echoid-s3298" xml:space="preserve">D ad D, & </s>
            <s xml:id="echoid-s3299" xml:space="preserve">per-
              <lb/>
            mutando differentia inter B & </s>
            <s xml:id="echoid-s3300" xml:space="preserve">C eſt ad differentiam inter
              <lb/>
            B & </s>
            <s xml:id="echoid-s3301" xml:space="preserve">D ut 2 C ad D, hoc eſt, ut C + B ad B, & </s>
            <s xml:id="echoid-s3302" xml:space="preserve">dividendo,
              <lb/>
            differentia inter D & </s>
            <s xml:id="echoid-s3303" xml:space="preserve">C eſt ad differentiam inter B & </s>
            <s xml:id="echoid-s3304" xml:space="preserve">D ut
              <lb/>
            C ad B; </s>
            <s xml:id="echoid-s3305" xml:space="preserve">& </s>
            <s xml:id="echoid-s3306" xml:space="preserve">proinde D eſt medium harmonicum inter C & </s>
            <s xml:id="echoid-s3307" xml:space="preserve">B,
              <lb/>
            quod demonſtrare oportuit.</s>
            <s xml:id="echoid-s3308" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3309" xml:space="preserve">Hæc propoſitio eodem modo locum habet in omnibus po-
              <lb/>
            lygonis complicatis, ut patet ex ſcholio 5 hujus.</s>
            <s xml:id="echoid-s3310" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div172" type="section" level="1" n="81">
          <head xml:id="echoid-head117" xml:space="preserve">PROP. XIII. THEOREMA.</head>
          <p>
            <s xml:id="echoid-s3311" xml:space="preserve">Inter duas quantitates A, B, ſit media a-
              <lb/>
              <note position="right" xlink:label="note-0152-03" xlink:href="note-0152-03a" xml:space="preserve">
                <lb/>
              A ## B
                <lb/>
              C # D # E
                <lb/>
              </note>
            rithmetica C, media geometrica D & </s>
            <s xml:id="echoid-s3312" xml:space="preserve">me-
              <lb/>
            dia harmonica E. </s>
            <s xml:id="echoid-s3313" xml:space="preserve">dico C, D, E, eſſe con-
              <lb/>
            tinuè proportionales. </s>
            <s xml:id="echoid-s3314" xml:space="preserve">quoniam A, E, B,
              <lb/>
            ſunt in ratione harmonica; </s>
            <s xml:id="echoid-s3315" xml:space="preserve">erit differentia inter A & </s>
            <s xml:id="echoid-s3316" xml:space="preserve">E ad
              <lb/>
            differentiam inter E & </s>
            <s xml:id="echoid-s3317" xml:space="preserve">B ut A ad B; </s>
            <s xml:id="echoid-s3318" xml:space="preserve">& </s>
            <s xml:id="echoid-s3319" xml:space="preserve">componendo erit
              <lb/>
            differentia inter A & </s>
            <s xml:id="echoid-s3320" xml:space="preserve">B ad differentiam inter E & </s>
            <s xml:id="echoid-s3321" xml:space="preserve">B, ut
              <lb/>
            A + B ad B; </s>
            <s xml:id="echoid-s3322" xml:space="preserve">deinde permutando & </s>
            <s xml:id="echoid-s3323" xml:space="preserve">componendo 2 A: </s>
            <s xml:id="echoid-s3324" xml:space="preserve">A + B:</s>
            <s xml:id="echoid-s3325" xml:space="preserve">:
              <lb/>
            E: </s>
            <s xml:id="echoid-s3326" xml:space="preserve">B, ſed 2A eſt duplum ipſius A & </s>
            <s xml:id="echoid-s3327" xml:space="preserve">A + B duplum ipſius C;
              <lb/>
            </s>
            <s xml:id="echoid-s3328" xml:space="preserve">& </s>
            <s xml:id="echoid-s3329" xml:space="preserve">ideo A: </s>
            <s xml:id="echoid-s3330" xml:space="preserve">C:</s>
            <s xml:id="echoid-s3331" xml:space="preserve">: E: </s>
            <s xml:id="echoid-s3332" xml:space="preserve">B; </s>
            <s xml:id="echoid-s3333" xml:space="preserve">& </s>
            <s xml:id="echoid-s3334" xml:space="preserve">proinde CE = AB, & </s>
            <s xml:id="echoid-s3335" xml:space="preserve">AB = DD, ideo-
              <lb/>
            que CE = DD; </s>
            <s xml:id="echoid-s3336" xml:space="preserve">& </s>
            <s xml:id="echoid-s3337" xml:space="preserve">igitur C: </s>
            <s xml:id="echoid-s3338" xml:space="preserve">D:</s>
            <s xml:id="echoid-s3339" xml:space="preserve">: D: </s>
            <s xml:id="echoid-s3340" xml:space="preserve">E, quod demonſtrare o-
              <lb/>
            portuit.</s>
            <s xml:id="echoid-s3341" xml:space="preserve"/>
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