Theodosius <Bithynius>; Clavius, Christoph, Theodosii Tripolitae Sphaericorum libri tres
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            KS, ſinus verſi ſunt arcuum ſimilium DL, KC; </s>
            <s xml:id="echoid-s15827" xml:space="preserve">erit, vt DX, ad KY, ſinus
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            totus ad ſinum totum, ita DZ, ad KS, per lem ma propoſ. </s>
            <s xml:id="echoid-s15828" xml:space="preserve">1. </s>
            <s xml:id="echoid-s15829" xml:space="preserve">noſtræ Gnomo-
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            nices. </s>
            <s xml:id="echoid-s15830" xml:space="preserve">Quia vero proportio rectanguli ſub DX, XA, ad rectangulum ſub KY,
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            AV, componitur ex proportione DX, ad KY, hoc eſt, DZ, ad KS, & </s>
            <s xml:id="echoid-s15831" xml:space="preserve">ex
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              <note position="right" xlink:label="note-459-01" xlink:href="note-459-01a" xml:space="preserve">23. ſexti.</note>
            proportione XA, ad AV: </s>
            <s xml:id="echoid-s15832" xml:space="preserve">Et proportio DZ, ad KT, componitur ex eiſdẽ
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            proportionibus, nempe (poſita media recta KS) ex proportione DZ, ad KS,
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            & </s>
            <s xml:id="echoid-s15833" xml:space="preserve">ex proportione KS, ad KT, hoc eſt, ex proportione XA, ad AV; </s>
            <s xml:id="echoid-s15834" xml:space="preserve">erit, vt
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            rectangulum ſub DX, XA, id eſt, quadratum ſinus totius, ad rectangulum
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            ſub KY, AV, ſinubus rectis arcuum inæqualium AC, AB, ita DZ, ſinus ver
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            ſus anguli A, ad KT, differentiam inter BR, ſinum verſum arcus BC, angu-
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            lo A, oppoſiti, & </s>
            <s xml:id="echoid-s15835" xml:space="preserve">BQ, ſinum verſum differentiæ arcuum inæqualium AC,
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            AB. </s>
            <s xml:id="echoid-s15836" xml:space="preserve">Quod eſt propoſitum.</s>
            <s xml:id="echoid-s15837" xml:space="preserve"/>
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            <s xml:id="echoid-s15838" xml:space="preserve">2. </s>
            <s xml:id="echoid-s15839" xml:space="preserve">SINT duo arcus inæquales AB, AC, quadrante quidem minores, at
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              <note position="right" xlink:label="note-459-02" xlink:href="note-459-02a" xml:space="preserve">2. caſus.</note>
            BC, quadrans. </s>
            <s xml:id="echoid-s15840" xml:space="preserve">Compleatur minoris arcus AB, circulus ABDGH, & </s>
            <s xml:id="echoid-s15841" xml:space="preserve">pro-
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            ducto arcu AC, vt fiat quadrans AL, deſcribantur ex polis A, B, ad inter-
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            ualla quadrantum AL, BC, cir-
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            culi maximi DELF, GECH:
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            </s>
            <s xml:id="echoid-s15842" xml:space="preserve">
              <figure xlink:label="fig-459-01" xlink:href="fig-459-01a" number="319">
                <image file="459-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/459-01"/>
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            Item ex polo A, ad interuallum
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            AC, circulus non maximus
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            KCN, qui ipſi DELF, paral-
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              <note position="right" xlink:label="note-459-03" xlink:href="note-459-03a" xml:space="preserve">2.2. Theo.</note>
            lelus erit, ſecabitq́ue circulus
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            ABDGH, circulos DELF,
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              <note position="right" xlink:label="note-459-04" xlink:href="note-459-04a" xml:space="preserve">15.1. Theod.</note>
            GECH, KCN, ad angulos re-
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            ctos, & </s>
            <s xml:id="echoid-s15843" xml:space="preserve">bifariam: </s>
            <s xml:id="echoid-s15844" xml:space="preserve">ac proinde ho-
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            rum cum illo communes ſectio-
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            nes DF, GH, KN, diametri eo-
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            rum erunt, & </s>
            <s xml:id="echoid-s15845" xml:space="preserve">DF, GH, ſe in X, centro ſphæræ interſeca bunt, parallelæq́ue
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            erunt DF, KN. </s>
            <s xml:id="echoid-s15846" xml:space="preserve">Reliqua fiant, vt in præcedenti caſu, niſi quòd hic punctum
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              <note position="right" xlink:label="note-459-05" xlink:href="note-459-05a" xml:space="preserve">16. vndec.</note>
            R, idem eſt, quod X, propterea quòd circulus OCP, à circulo GEH, atq;
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            </s>
            <s xml:id="echoid-s15847" xml:space="preserve">adeo recta ORP, à recta GH, non differt. </s>
            <s xml:id="echoid-s15848" xml:space="preserve">Erit, vt prius, AV, ſinus rectus ar-
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            cus AB; </s>
            <s xml:id="echoid-s15849" xml:space="preserve">& </s>
            <s xml:id="echoid-s15850" xml:space="preserve">KY, ſinus rectus arcus AK, hoc eſt, arcus AC, ipſi AK, ex de-
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            fin. </s>
            <s xml:id="echoid-s15851" xml:space="preserve">poli, æqualis. </s>
            <s xml:id="echoid-s15852" xml:space="preserve">Item BR, ſinus verſus erit arcus BG, id eſt, arcus BC,
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            ipſi BG, æqualis. </s>
            <s xml:id="echoid-s15853" xml:space="preserve">At BQ, ſinus erit verſus arcus BK, differentiæ arcuum
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            AB, AC; </s>
            <s xml:id="echoid-s15854" xml:space="preserve">ideoq; </s>
            <s xml:id="echoid-s15855" xml:space="preserve">QR, vel KT, differentia erit inter ſinus verſos BR, BQ,
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            arcuum BC, BK. </s>
            <s xml:id="echoid-s15856" xml:space="preserve">Deniq; </s>
            <s xml:id="echoid-s15857" xml:space="preserve">KS, erit ſinus verſus arcus KC. </s>
            <s xml:id="echoid-s15858" xml:space="preserve">Sumpto ergo DZ,
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            ſinu verſo arcus DL, hoc eſt, anguli A, demonſtrandum eſt, ita eſſe quadra-
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            tum ſinus totias, id eſt, rectangulum ſub DX, XA, ad rectangulum ſub AV,
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            KY, ſinubus rectis arcuum AB, AC, vt eſt ſinus verſus DZ, anguli A, ad
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            KT, differentiam ſinuum verſorum BR, BQ, arcuum BC, BK. </s>
            <s xml:id="echoid-s15859" xml:space="preserve">quod quidem
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            demonſtrabitur, vt in præcedenti caſu, niſi quod triangulum XAV, oſtende-
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            tur hic æquiangulum eſſe triangulo SKT, ex eo quòd angulus IXY, angu-
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            lo YSX, æqualis eſt, propterea quòd triangula IXY, YSX, ſimilia ſunt inter
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              <note position="right" xlink:label="note-459-06" xlink:href="note-459-06a" xml:space="preserve">8. ſexti.</note>
            ſe. </s>
            <s xml:id="echoid-s15860" xml:space="preserve">Hinc enim fit, rectangula triangula XAV, SKT, inter ſe omnino æquian-
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            gula eſſe.</s>
            <s xml:id="echoid-s15861" xml:space="preserve"/>
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            <s xml:id="echoid-s15862" xml:space="preserve">3. </s>
            <s xml:id="echoid-s15863" xml:space="preserve">SINT rurſus AB, AC, quadrante minores, at BC, maior. </s>
            <s xml:id="echoid-s15864" xml:space="preserve">Com-
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              <note position="right" xlink:label="note-459-07" xlink:href="note-459-07a" xml:space="preserve">3. caſus.</note>
            pleatur minoris arcus AB, circulus, & </s>
            <s xml:id="echoid-s15865" xml:space="preserve">ex BC, abſcindatur quadrans BM,
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            producaturq́; </s>
            <s xml:id="echoid-s15866" xml:space="preserve">AC, vt fiat quadrans AL. </s>
            <s xml:id="echoid-s15867" xml:space="preserve">Reliqua conſtruantur, vt in prmo
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            caſu. </s>
            <s xml:id="echoid-s15868" xml:space="preserve">Erunt hic ſinus, vt ibi. </s>
            <s xml:id="echoid-s15869" xml:space="preserve">Demonſtrandum ergo eſt, ita eſſe quadratũ </s>
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