Bion, Nicolas, Traité de la construction et principaux usages des instruments de mathématique, 1723

Table of Notes

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            une fois le compas commun ainſi ouvert vers l'extremité de ladite
              <lb/>
            ligne, la pointe tombera ſur un nombre de plan neuf fois plus grand.
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            </s>
            <s xml:id="echoid-s1598" xml:space="preserve">Ainſi, par exemple, ſi l'on a pris la diſtance depuis le centre juſqu'au
              <lb/>
            plan marqué 2, arrêtant une pointe du compas ſur ledit point 2,
              <lb/>
            l'autre pointe doittomber ſur le point 8, & </s>
            <s xml:id="echoid-s1599" xml:space="preserve">en tournant encore une
              <lb/>
            fois le compas, ſans changer l'ouverture, en arrêtant une de ſes
              <lb/>
            pointes ſur ledit point 8, l'autre pointe doit rencontrer le dix-hui-
              <lb/>
            tiéme plan, qui contient neuf fois le ſecond plan; </s>
            <s xml:id="echoid-s1600" xml:space="preserve">tournant encore
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            une fois le compas, on rencontrera le trente deuxiéme plan, qui
              <lb/>
            contient ſeize fois le ſecond plan. </s>
            <s xml:id="echoid-s1601" xml:space="preserve">Si enfin on tourne encore une au-
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            tre fois, on doit rencontrer le cinquantiéme plan, qui contient ce-
              <lb/>
            lui de deux 25 fois, & </s>
            <s xml:id="echoid-s1602" xml:space="preserve">ainſi des autres plans ſemblables, parce qu'ils
              <lb/>
            ſont entr'eux, comme les quarrez de leurs côtez homologues. </s>
            <s xml:id="echoid-s1603" xml:space="preserve">
              <lb/>
            C'eſt ce qui facilite la diviſion de cette ligne des plans, puiſqu'ayant
              <lb/>
            le premier, on a le quatriéme, le neuviéme, le ſeiziéme, le vingt-
              <lb/>
            cinquiéme, le trente-ſixiéme, le quarante-neuviéme, & </s>
            <s xml:id="echoid-s1604" xml:space="preserve">le ſoixan-
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            te-quatriéme; </s>
            <s xml:id="echoid-s1605" xml:space="preserve">ayant trouvé le ſecond, on a le huitiéme, le dix-
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            huitiéme, le trente-deuxiéme, & </s>
            <s xml:id="echoid-s1606" xml:space="preserve">cinquantiéme; </s>
            <s xml:id="echoid-s1607" xml:space="preserve">ayant pareille-
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            ment trouvé le troiſiéme, on a le douziéme, le vingt-ſeptiéme, & </s>
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            le quarante-huitiéme; </s>
            <s xml:id="echoid-s1609" xml:space="preserve">& </s>
            <s xml:id="echoid-s1610" xml:space="preserve">ainſi du reſte.</s>
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          <head xml:id="echoid-head107" style="it" xml:space="preserve">Preuve de la ligne des Solides.</head>
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            <s xml:id="echoid-s1612" xml:space="preserve">ON connoît ſi cette ligne eſt bien diviſée par la methode ſuivan-
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            te. </s>
            <s xml:id="echoid-s1613" xml:space="preserve">Prenez avec un compas ordinaire la diſtance de quelque
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            point que ce ſoit de cette ligne jnſqu'au centre du compas de pro-
              <lb/>
            portion; </s>
            <s xml:id="echoid-s1614" xml:space="preserve">arrêtez une pointe du compas ainſi ouvert ſur le même
              <lb/>
            point de diviſion, & </s>
            <s xml:id="echoid-s1615" xml:space="preserve">tournez l'autre pointe vers l'extremité de ladite
              <lb/>
            ligne, elle doit rencontrerun nombre de ſolides huit fois plus grand
              <lb/>
            que celui que vous aurez choiſi. </s>
            <s xml:id="echoid-s1616" xml:space="preserve">Si vous tournez encore une fois le
              <lb/>
            compas, une de ſes pointes tombera ſur un ſolide vingt-ſeptfois plus
              <lb/>
            grand que le nombre choiſi. </s>
            <s xml:id="echoid-s1617" xml:space="preserve">Ainſi, par exemple, l'ouverture du pre-
              <lb/>
            mier ſolide donnera celle du huitiéme, du vingt-ſeptiéme, & </s>
            <s xml:id="echoid-s1618" xml:space="preserve">du
              <lb/>
            ſoixante-quatriéme; </s>
            <s xml:id="echoid-s1619" xml:space="preserve">l'ouverture du ſecond ſolide donnera celle du
              <lb/>
            ſeiziéme, & </s>
            <s xml:id="echoid-s1620" xml:space="preserve">du cinquante-quatriéme; </s>
            <s xml:id="echoid-s1621" xml:space="preserve">l'ouverture du troiſiéme pri-
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            ſe deux fois donnera celle du vingt quatriéme. </s>
            <s xml:id="echoid-s1622" xml:space="preserve">Par le quatriéme
              <lb/>
            ſolide on aura le trente-deuxiéme, de même que par le cinquiéme
              <lb/>
            on aura le quarantiéme; </s>
            <s xml:id="echoid-s1623" xml:space="preserve">par le ſixiéme on aura le quarante-huitié-
              <lb/>
            me, & </s>
            <s xml:id="echoid-s1624" xml:space="preserve">enfin par le moyen du ſeptiéme on aura le cinquante-ſixié-
              <lb/>
            me ſolide, parce que les ſolides ſemblables ſont entr'eux, comme les
              <lb/>
            cubes de leurs côtez homologues; </s>
            <s xml:id="echoid-s1625" xml:space="preserve">& </s>
            <s xml:id="echoid-s1626" xml:space="preserve">c'eſt ce qui facilite la diviſion
              <lb/>
            de la ligne des ſolides.</s>
            <s xml:id="echoid-s1627" xml:space="preserve"/>
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