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]

Table of handwritten notes

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            <div xml:id="echoid-div191" type="math:theorem" level="3" n="100">
              <p>
                <s xml:id="echoid-s870" xml:space="preserve">
                  <pb o="65" rhead="THEOR. ARITH." n="77" file="0077" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0077"/>
                to dimidio ipſius nempe .3. & dimidio, cum numero ipſum terminum
                  <reg norm="ſequenti" type="context">ſequẽti</reg>
                , nem
                  <lb/>
                pè .8. ſumma dictorum terminorum erit .28.</s>
              </p>
              <p>
                <s xml:id="echoid-s871" xml:space="preserve">Huius autem ſpeculatio ex .94. theoremate dependet, in quo facilè depræhen-
                  <lb/>
                dere licet ex figura continuæ progreſſionis naturalis, numerum terminorum maxi-
                  <lb/>
                mo termino ſemper æqualem eſſe; </s>
                <s xml:id="echoid-s872" xml:space="preserve">ex quo
                  <reg norm="tantum" type="context">tãtum</reg>
                eſt dimidium numeriterminorum,
                  <lb/>
                quantum maximi dimidium,
                  <reg norm="tantusque" type="simple">tantusq́;</reg>
                eſt vltimus terminus vnitati coniunctus, quan
                  <lb/>
                tus numerus is, qui vltimum terminum conſequitur.</s>
              </p>
            </div>
            <div xml:id="echoid-div192" type="math:theorem" level="3" n="101">
              <head xml:id="echoid-head118" xml:space="preserve">THEOREMA
                <num value="101">CI</num>
              .</head>
              <p>
                <s xml:id="echoid-s873" xml:space="preserve">CVR antiqui idip fum, quod iam dictum eft, in ea progreſſione, cuius vltimus ter
                  <lb/>
                minus diſpar eſt ſcire cupientes, numerum integrorum proximè dimidium
                  <lb/>
                maximi ſequentem ſumebant, quem per maximum multiplicabant, ex quo
                  <lb/>
                ſumma quæſita oriebatur.</s>
              </p>
              <p>
                <s xml:id="echoid-s874" xml:space="preserve">Exempli gratia, ſi dimidium maximi fuiſſet .3. cum dimidio, fumebant quatuor,
                  <lb/>
                & per maximum .7. multiplicabant, ex quo pariter proferebatur ſumma .28.</s>
              </p>
              <p>
                <s xml:id="echoid-s875" xml:space="preserve">Cuius ratio ex .20. ſeptimi Euclidis oritur, cum eadem ſit proportio numeri fe-
                  <lb/>
                quentis ma ximum ad numerum dimidium maximi ſequentem; </s>
                <s xml:id="echoid-s876" xml:space="preserve">quæ maximi ad
                  <reg norm="fuum" type="context">fuũ</reg>
                  <lb/>
                dimidium, eſt enim dupla.</s>
              </p>
            </div>
            <div xml:id="echoid-div193" type="math:theorem" level="3" n="102">
              <head xml:id="echoid-head119" xml:space="preserve">THEOREMA
                <num value="102">CII</num>
              .</head>
              <p>
                <s xml:id="echoid-s877" xml:space="preserve">TRaditum eſt à nonnullis, à veteribus obſeruatam fuiſſe hancregulam, qua ſci-
                  <lb/>
                re poſſent ſummam alicuius progreſſionis arithmeticæ diſcontinuæ aut inter
                  <lb/>
                cifæ, quæ numero pari terminetur. </s>
                <s xml:id="echoid-s878" xml:space="preserve">
                  <reg norm="Multiplicabant" type="context">Multiplicabãt</reg>
                enim
                  <reg norm="dimidium" type="context">dimidiũ</reg>
                vltimi termini per
                  <lb/>
                pro ximum numerum dimidio dicto maiorem, ex quo
                  <reg norm="inquiebant" type="context">inquiebãt</reg>
                ſemper productum
                  <lb/>
                ſummæ quæſitæ æquale eſſe,
                  <reg norm="ſubijciuntque" type="simple">ſubijciuntq́;</reg>
                exemplum progreſſionis, quæ à binario in-
                  <lb/>
                choata crefcit per binarium. </s>
                <s xml:id="echoid-s879" xml:space="preserve">In qua quidem progreſſione non per fe, fed per acci-
                  <lb/>
                dens regula vera eft. </s>
                <s xml:id="echoid-s880" xml:space="preserve">Hoc eſt, non quia ex ſe vnus ex producentibus numeris dimi-
                  <lb/>
                dium termini maioris futurus ſit, alter uerò proximè ſequens dimidium, fed quia
                  <lb/>
                vt dictum eſt .95. theoremate, eadem eſt proportio maximi termini ad numerum
                  <lb/>
                terminorum, quæ minimi ad vnitatem. </s>
                <s xml:id="echoid-s881" xml:space="preserve">Cumq́ue in præfenti exemplo minimum
                  <lb/>
                ſit duplum vnitati in eiuſmodi caſu, numerus terminorum, dimidio maximi termini
                  <lb/>
                æqualis eſt, qui terminorum numerus ex ſe, vt patet, vnus eſt ex producentibus, al-
                  <lb/>
                ter verò producens numerus, eſt proximè dimidium ſequens, non exſe, fed quia nu
                  <lb/>
                merus ſequens, dimidium eſt ſummæ maximi, & minimi, quæ per fe alter eſſe de-
                  <lb/>
                bet producens numerus. </s>
                <s xml:id="echoid-s882" xml:space="preserve">In cæteris enim progreſſionibus, quæ binario non creſcút
                  <lb/>
                regulafal
                  <unsure/>
                fa eſt, prout facilè patere poteſt ei, qui ex ſcientiæ legibus ope ſpeculatio-
                  <lb/>
                nis .95. theorematis ſpeculatus fuerit.</s>
              </p>
            </div>
            <div xml:id="echoid-div194" type="math:theorem" level="3" n="103">
              <head xml:id="echoid-head120" xml:space="preserve">THEOREMA
                <num value="103">CIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s883" xml:space="preserve">ALIAM quoque tradunt regulam, qua veteres vſos fuiſſe dicunt, quo ſum-
                  <lb/>
                mam ſcire poſſent progreſſionis diſcontinuæ, quænumero diſpari abſolui-
                  <lb/>
                tur. </s>
                <s xml:id="echoid-s884" xml:space="preserve">Ea autem eſt eiuſmodi. </s>
                <s xml:id="echoid-s885" xml:space="preserve">Vltimum terminum in duas quam maximè poterant ma-
                  <lb/>
                ximas partes diuidebant, quarum vna ſemper altera maior erat, banc autem maio-
                  <lb/>
                rem in ſeipſam multiplicabant, at que quadratum hoc, ſummam progreffionis effe </s>
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