Benedetti, Giovanni Battista de, Io. Baptistae Benedicti ... Diversarvm specvlationvm mathematicarum, et physicarum liber : quarum seriem sequens pagina indicabit ; [annotated and critiqued by Guidobaldo Del Monte]

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THEOREM. ARIT.
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            <div xml:id="echoid-div184" type="math:theorem" level="3" n="94">
              <p>
                <s xml:id="echoid-s842" xml:space="preserve">
                  <pb o="63" rhead="THEOREM. ARIT." n="75" file="0075" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0075"/>
                ras confideranti ſpeculari licebit, Diametros harum figurarum notaui literis ſiue
                  <lb/>
                characteribus
                  <var>.a.e.i.c.u.n</var>
                .</s>
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              <figure position="here">
                <image file="0075-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0075-01"/>
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              <figure position="here">
                <image file="0075-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0075-02"/>
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            <div xml:id="echoid-div185" type="math:theorem" level="3" n="95">
              <head xml:id="echoid-head112" xml:space="preserve">THEOREMA
                <num value="95">XCV</num>
              .</head>
              <p>
                <s xml:id="echoid-s843" xml:space="preserve">IN progreſſionibus, quæ ab alio termino quam vnitate incohantur, idipſum vt
                  <lb/>
                monuimus accidit, hoc tamen notato, quòd ex conſequenti quælibet pars dia-
                  <lb/>
                metri
                  <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>
                , minimo termino æqualis erit, prout in progreſſionibus quæ
                  <lb/>
                ab vnitate originem ducunt, ſingulæ partes diametri, vnitati ſui primi termini æ-
                  <lb/>
                quales ſunt. </s>
                <s xml:id="echoid-s844" xml:space="preserve">At in reliquis progreſſionibus, vt in figura patet, eadem eſt propor-
                  <lb/>
                tio totius diametri ad
                  <var>.o.n.</var>
                quæ minimi termini ad vnitatem ex .13. quinti, nempe
                  <var>.
                    <lb/>
                  a.o.</var>
                ad
                  <var>.o.n.</var>
                vt
                  <var>.n.n.n.n.</var>
                ad
                  <var>.n</var>
                . </s>
                <s xml:id="echoid-s845" xml:space="preserve">In eiuſmodi progreſſionibus accidit quoque
                  <reg norm="parallelo- grammum" type="context">parallelo-
                    <lb/>
                  grãmum</reg>
                à diametro in tres partes diuidi, quarum vnam ipſe occupat, reliquæ ve-
                  <lb/>
                ro inter ſe æquales ipſum ambiunt. </s>
                <s xml:id="echoid-s846" xml:space="preserve">Ex quo illud etiam ſequitur, productum
                  <var>.a.o.</var>
                in
                  <lb/>
                dimidium
                  <var>.o.n.</var>
                æquale eſſe dimidio
                  <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>
                , quod minus eſt ſumma progreſ-
                  <lb/>
                ſionis dimidio diametri, quod dimidum ſi inuenire voluerimus, minimum
                  <reg norm="terminum" type="context">terminũ</reg>
                  <var>.
                    <lb/>
                  n.n.n.n.</var>
                per dimidium
                  <var>.o.n.</var>
                multiplicabimus, & ex .18. aut .19. ſeptimi ipſum habe-
                  <lb/>
                bimus,
                  <reg norm="quandoquidem" type="context">quandoquidẽ</reg>
                minimo termino per totum
                  <var>.o.n.</var>
                multiplicato profertur integer
                  <lb/>
                diameter ex .20. prædicti. </s>
                <s xml:id="echoid-s847" xml:space="preserve">Etenim vt diximus, eadem eſt proportio totius diame-
                  <lb/>
                tri ad
                  <var>.o.n.</var>
                quæ minimi termini ad vnitatem. </s>
                <s xml:id="echoid-s848" xml:space="preserve">Ita etiam dico ex dicta .20. ſeptimi.
                  <lb/>
                </s>
                <s xml:id="echoid-s849" xml:space="preserve">idem dimidium diametri oriri, ſi quis dimidium minimi termini nempè
                  <var>.n.n.</var>
                per to
                  <lb/>
                tum
                  <var>.o.n.</var>
                multiplicauerit. </s>
                <s xml:id="echoid-s850" xml:space="preserve">Quamobrem qui ſtatim ſummam propoſitæ progreſſionis
                  <lb/>
                cognoſcere voluerit,
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0075-03a" xlink:href="fig-0075-03"/>
                ſemper primum termi
                  <lb/>
                num
                  <var>.n.n.n.n.</var>
                cum
                  <var>.a.o.</var>
                  <lb/>
                coniunget, qua ſumma
                  <lb/>
                per
                  <reg norm="dimidium" type="context">dimidiũ</reg>
                  <var>.o.n.</var>
                mul-
                  <lb/>
                tiplicata, aut
                  <var>.o.n.</var>
                per
                  <lb/>
                dimidium dictæ ſum-
                  <lb/>
                mæ, ex prædictis rationibus propofitum conſequemur.</s>
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                <figure xlink:label="fig-0075-03" xlink:href="fig-0075-03a">
                  <image file="0075-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0075-03"/>
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            <div xml:id="echoid-div187" type="math:theorem" level="3" n="96">
              <head xml:id="echoid-head113" xml:space="preserve">THEOREMA
                <num value="96">XCVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s851" xml:space="preserve">CVR ſi quis numerum terminorum inuenire velit, cognitis tantummodo pri
                  <lb/>
                mo atque vltimo, rectè vltimum per primum diuidet, ex quo proueniens </s>
              </p>
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