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

Page concordance

< >
Scan Original
111 390
112 391
113 392
114 393
115 394
116 395
117 396
118 397
119 398
120
121
122
123 399
124 400
125 401
126 402
127 403
128 404
129
130
131
132
133
134
135
136
137
138
139
140 413
< >
page |< < (434) of 568 > >|
    <echo version="1.0RC">
      <text xml:lang="la" type="free">
        <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"/>
          </p>
        </div>
      </text>
    </echo>
Impressum