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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            <div xml:id="echoid-div183" type="math:theorem" level="3" n="93">
              <p>
                <s xml:id="echoid-s824" xml:space="preserve">
                  <pb o="62" rhead="IO. BAPT. BENED." n="74" file="0074" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0074"/>
                dendus, alter ex eodem detrahendus ſit, ex quo proferri debeant bina qua-
                  <lb/>
                drata. </s>
                <s xml:id="echoid-s825" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                numeri illi in ſummam collecti dabunt .17. differentiam minoris quadra
                  <lb/>
                ti & maioris. </s>
                <s xml:id="echoid-s826" xml:space="preserve">I am ſi ex hoc .17. binas partes fecerimus, altera erit .8. altera .9. qui
                  <lb/>
                bus in ſeipſis multiplicatis alterum quadratum erit .64. alterum .81. addito
                  <reg norm="itaque" type="simple">itaq;</reg>
                ipſi.
                  <lb/>
                64. 11. aut .6. pro libito, propoſitum numerum conſequemur. </s>
                <s xml:id="echoid-s827" xml:space="preserve">cui addito .6. vel .11.
                  <lb/>
                dabit nobis .81. vel ex ipſo detracto .11. vel .6. relinquet nobis 64. in pręſenti autem
                  <lb/>
                exemplo talis numerus erit, aut .70. vel .75. </s>
                <s xml:id="echoid-s828" xml:space="preserve">Huius autem theorematis ſpeculatio
                  <lb/>
                ex .90. dependet, quo demonſtratum fuit gnomonem proximè quadratum ſequen
                  <lb/>
                tem, vnitate duplo radicis minorem eſſe.</s>
              </p>
            </div>
            <div xml:id="echoid-div184" type="math:theorem" level="3" n="94">
              <head xml:id="echoid-head111" xml:space="preserve">THEOREMA
                <num value="94">XCIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s829" xml:space="preserve">CVR ſi quis cupiat ſummam progreſſionis arithmeticæ quam citiſſimè cogno
                  <lb/>
                ſcere. </s>
                <s xml:id="echoid-s830" xml:space="preserve">Rectè coniunget vltimo termino vnitatem primum terminum, huius
                  <lb/>
                poſtea vltimi termini dimidium cum numero terminorum multiplicabit, ex
                  <lb/>
                quo multiplicationis productum, erit omnium propoſitorum terminorum ſumma,
                  <lb/>
                aut eundem vltimum terminum iunctum primo, per dimidium numeri terminorum
                  <lb/>
                multiplicabit. </s>
                <s xml:id="echoid-s831" xml:space="preserve">Nam idipſum eueniet.</s>
              </p>
              <p>
                <s xml:id="echoid-s832" xml:space="preserve">Exempli gratia, ſi proponerentur .17. termini in prima progreſſione arithmeti-
                  <lb/>
                ca naturali, vltimus eſſet .17. cui coniuncta vnitate primo termino ſumma erit .18.
                  <lb/>
                cuius dimidium cum numero terminorum, nempe .17. multiplicatum cum fuerit,
                  <lb/>
                oritur productum .153. </s>
                <s xml:id="echoid-s833" xml:space="preserve">Idpſum eueniet, multiplicato dimidio numeri
                  <reg norm="terminorum" type="context">terminorũ</reg>
                  <lb/>
                per vltimum coniunctum vnitati primo termino.</s>
              </p>
              <p>
                <s xml:id="echoid-s834" xml:space="preserve">Quod vt ſciamus, cogitemus terminos progreſſionis collocari, vt in figura ſub-
                  <lb/>
                ſcripta
                  <var>.a.o.n.</var>
                collocantur,
                  <reg norm="tanquam" type="context">tanquã</reg>
                per gradus, ſumpto principio ab vnitate
                  <var>.n.</var>
                tum
                  <var>.
                    <lb/>
                  u.t.</var>
                atque ita gradatim. </s>
                <s xml:id="echoid-s835" xml:space="preserve">Sic cogitato abſoluto parallelogrammo
                  <var>.q.o.</var>
                ſciemus aper-
                  <lb/>
                tè ſummam progreſſionis tanto maiorem eſſe dimidio totius
                  <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>
                , quan
                  <lb/>
                tum dimidium numeri diametri
                  <var>.a.e.i.c.u.n.</var>
                requirit. </s>
                <s xml:id="echoid-s836" xml:space="preserve">Nam cum parallelogram-
                  <lb/>
                mum diuidatur à dl
                  <unsure/>
                ametro in tres partes, diameter vnam occupat, reliquæ verò duę
                  <lb/>
                ambientes diametrum inter ſe ſunt æquales. </s>
                <s xml:id="echoid-s837" xml:space="preserve">Sumpto
                  <reg norm="itaque" type="simple">itaq;</reg>
                diametro cum altera di
                  <lb/>
                ctarum duarum partium, patet ſumi pluſquam
                  <reg norm="dimidium" type="context">dimidiũ</reg>
                totius
                  <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>
                . </s>
                <s xml:id="echoid-s838" xml:space="preserve">pro
                  <lb/>
                tanta portione, quantum eſt dimidiam occupatam à diametro, qui
                  <reg norm="cam" type="context">cã</reg>
                  <unsure/>
                ex diſcretis
                  <lb/>
                reſpondentibus numero terminorum componatur, conſtat numero æquali eſſe di-
                  <lb/>
                cto numero terminorum
                  <var>.o.n</var>
                . </s>
                <s xml:id="echoid-s839" xml:space="preserve">Iam ſi quis multiplicet
                  <var>.a.o.</var>
                per dimidium
                  <var>.o.n.</var>
                procul
                  <lb/>
                dubio, ex prima ſexti aut .18. ſeptimi, orietur
                  <reg norm="dimidium" type="context">dimidiũ</reg>
                numeri
                  <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>
                  <var>.q.o.</var>
                  <lb/>
                quod minus erit ſumma progreſſionis dimidio numeri diametri, aut quod idem eſt
                  <lb/>
                dimidio
                  <var>.o.n.</var>
                ſed hoc dimidium
                  <var>.o.n.</var>
                æquale eſt producto dimidij vnitatis
                  <var>.n.</var>
                in
                  <var>.o.n.</var>
                  <lb/>
                ex .20. ſeptimi, cum dimidium
                  <var>.o.n.</var>
                ſit eius productum in
                  <reg norm="vnitatem" type="context">vnitatẽ</reg>
                . </s>
                <s xml:id="echoid-s840" xml:space="preserve">Itaque multipli-
                  <lb/>
                cato
                  <var>.n.o.</var>
                per dimidium
                  <var>.o.a.</var>
                coniunctum dimidio vnitatis
                  <var>.n.</var>
                oritur ſumma quæſita
                  <lb/>
                propoſitæ progreſſionis. </s>
                <s xml:id="echoid-s841" xml:space="preserve">Idipſum accidet multiplicata ſumma
                  <var>.o.a.</var>
                & vnitate
                  <var>.n.</var>
                  <reg norm="per" type="simple">ꝑ</reg>
                  <lb/>
                dimidium
                  <var>.o.n.</var>
                ex .20. ſeptimi, cum proportio totius ad totum eadem ſit, quæ dimi
                  <lb/>
                dijad dimidium, ex cauſa permutationalitatis. </s>
                <s xml:id="echoid-s842" xml:space="preserve">Patet etiam in progreſſionibus,
                  <lb/>
                quæ ab vnitate initium ducunt, ſi fiat aſcenſus per binarium ſumma vltimi termini
                  <lb/>
                cum primo ſemper duplam futuram eſſe numero terminorum, quod ſequentes figu­ </s>
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