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-div207" type="math:theorem" level="3" n="110">
              <p>
                <s xml:id="echoid-s952" xml:space="preserve">
                  <pb o="71" rhead="THEOREM. ARIT." n="83" file="0083" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0083"/>
                te verò à ſecundo milliario vno primus antecedatur, ex communi ſcientia neceſſe
                  <lb/>
                eſt ſecundum tot diebus
                  <reg norm="cum" type="context">cũ</reg>
                primo iter agere quot ſunt
                  <var>.o.n.</var>
                qui ſimul æquales erunt
                  <var>.
                    <lb/>
                  u.n.</var>
                ſed
                  <var>.u.n.</var>
                minor eft numero milliarium diurnorum primi vnitate
                  <var>.e</var>
                . </s>
                <s xml:id="echoid-s953" xml:space="preserve">Itaque rectè
                  <lb/>
                ſequemur regulam, quæ iubet ex numero milliarium vnitatem demere, quo nu
                  <lb/>
                merum dierum habere poſſimus.</s>
              </p>
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                <figure xlink:label="fig-0082-03" xlink:href="fig-0082-03a">
                  <image file="0082-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0082-03"/>
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            <div xml:id="echoid-div209" type="math:theorem" level="3" n="111">
              <head xml:id="echoid-head128" xml:space="preserve">THEOREMA
                <num value="111">CXI</num>
              .</head>
              <p>
                <s xml:id="echoid-s954" xml:space="preserve">SI verò ſecundi viatoris progreſſio per ternarium aſcenderet, ſumpto initio ab
                  <lb/>
                ipſo ternario, animaduertendum eſt an numerus milliarium diurnorum primi,
                  <lb/>
                ternario menſuretur necne, etenim ſi menſuretur, tandem aliquando paria millia-
                  <lb/>
                ria conficient, quæ dies ſit
                  <var>.u.n.</var>
                </s>
                <s xml:id="echoid-s955" xml:space="preserve">quare ſub
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0083-01a" xlink:href="fig-0083-01"/>
                  <var>u.n.</var>
                totidem quot ſupra termini
                  <reg norm="erunt" type="context">erũt</reg>
                ,
                  <lb/>
                &
                  <reg norm="cum" type="context">cũ</reg>
                  <var>.o.n.</var>
                tertia ſit pars
                  <var>.u.n.</var>
                ex .95. theo-
                  <lb/>
                remate. </s>
                <s xml:id="echoid-s956" xml:space="preserve">Itaque tota
                  <var>.o.f.</var>
                minor erit
                  <lb/>
                duabus tertijs
                  <var>.u.n.</var>
                vnitate, vtiam re-
                  <lb/>
                ctè ſumendæ ſint duæ tertiæ partes
                  <var>.u.n.</var>
                  <lb/>
                ex quibus vnitas detrahatur ſuperſitq́ue
                  <lb/>
                numerus
                  <var>.o.f.</var>
                dierum quæſitorum.</s>
              </p>
              <div xml:id="echoid-div209" type="float" level="4" n="1">
                <figure xlink:label="fig-0083-01" xlink:href="fig-0083-01a">
                  <image file="0083-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0083-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div211" type="math:theorem" level="3" n="112">
              <head xml:id="echoid-head129" xml:space="preserve">THEOREMA
                <num value="112">CXII</num>
              .</head>
              <p>
                <s xml:id="echoid-s957" xml:space="preserve">CVM verò milliarium numerus p rimi viatoris metirinon poterit à numero
                  <lb/>
                aſcendente ſecundi, patet nullam futuram diem qua pari milliaria conficient,
                  <lb/>
                </s>
                <s xml:id="echoid-s958" xml:space="preserve">quare illa vltima qua primus ſecundum antecedet, vno aut duobus milliaribus an-
                  <lb/>
                tecedet in præſenti caſu. </s>
                <s xml:id="echoid-s959" xml:space="preserve">Antecedat itaque duobus milliaribus,
                  <reg norm="ſitque" type="simple">ſitq́;</reg>
                dies
                  <var>.u.n.</var>
                & alte
                  <lb/>
                ra
                  <var>.t.i.</var>
                ſecundus primum vno milliari ſuperabit, ita quod ſub
                  <var>.t.i.</var>
                non poterunt plu-
                  <lb/>
                res integros dies iter agere, quam ambulauerunt ante diem
                  <var>.u.n.</var>
                hoc eſt vſquequo
                  <lb/>
                ſecundtis iunctus ſit primo, qui numerus dierum, tertia parte
                  <var>.o.n.</var>
                ipſius
                  <var>.u.n.</var>
                vnitate
                  <lb/>
                minor erit, cum ex .95. theoremate
                  <var>.o.n.</var>
                ſit tertia pars
                  <var>.u.n.</var>
                ex quo numerus
                  <var>.o.f.</var>
                ter-
                  <lb/>
                minorum aut dierum intergrorum cognitus erit, qui ſi cum numero alcendente
                  <lb/>
                cognoſcetur, ſtatim ex .99. theoremate deueniemus in cognitionem vltimi diei in
                  <lb/>
                tegri
                  <var>.s.f.</var>
                atque ita etiam totius ſummæ progreſſionis ex .95. theoremate. </s>
                <s xml:id="echoid-s960" xml:space="preserve">Iam verò
                  <lb/>
                cognito numero milliarium diurnorum primi, ſimul cum numero terminorum, aut
                  <lb/>
                dierum conſequenter nouerimus rectanguli ſummam, hoc eſt productum à primo
                  <lb/>
                viatore formatum, quarum duarum ſummarum in præſenti caſu ſemper ea, quæ
                  <lb/>
                huiuſmodi producti eſt, maior erit, cum conſtitutum fuerit ſecundum viatorem à
                  <lb/>
                primo ſuperari ipſa die
                  <var>.u.n.</var>
                vno milliari amplius quam ſequente die
                  <var>.t.i.</var>
                primus à ſe
                  <lb/>
                cundo ſuperatur, tum pari gradu iter egerunt ſub
                  <var>.t.i.</var>
                quo ſupra
                  <var>.u.n.</var>
                ambulauerant.
                  <lb/>
                </s>
                <s xml:id="echoid-s961" xml:space="preserve">Hoc animaduertendo, quòd ſi ſumma progreſſionis maior eſſet rectangulo, ex ea
                  <lb/>
                ſumma neceſſe eſſet
                  <reg norm="numerum" type="context">numerũ</reg>
                mil
                  <lb/>
                liarium vltimi termini in ſumma
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0083-02a" xlink:href="fig-0083-02"/>
                incluſi detrahere, & reſiduo ope-
                  <lb/>
                rari. </s>
                <s xml:id="echoid-s962" xml:space="preserve">Nunc verò ſummam pro-
                  <lb/>
                greſſionis exſumma rectanguli à
                  <lb/>
                primo viatore facti ſubtrahi de-
                  <lb/>
                bet,
                  <reg norm="reſiduumque" type="simple">reſiduumq́;</reg>
                ſeruari
                  <reg norm="voceturque" type="simple">voceturq́;</reg>
                </s>
              </p>
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