Galilei, Galileo, Mechanics, 1665

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              De æquiponder antium,
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              there came into my thoughts a Rule which
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              exquiſitely reſolveth our Queſtion; which Rule I believe to be
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              the ſame that
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              Archimedes
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              made uſe of, ſeeing that beſides the
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              uſe that is to be made of the Water, the exactneſs of the Work
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              dependeth alſo upon certain Demonſtrations found by the ſaid
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                <emph type="italics"/>
              Archimedes.
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              <s>The way is by help of a Ballance, whoſe Conſtruction and Uſe
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              ſhall be ſhewn by and by, after we ſhall have declared what is
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              neceſſary for the knowledge thereof. </s>
              <s>You muſt know there­
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              fore, that the Solid Bodies that ſink in the Water weigh ſo much
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              leſs in the Water than in the Air, as a Maſs of Water equal to
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              the ſaid Solid doth weigh in the Air: which hath been demon­
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              ſtrated by
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              Archimedes.
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              But, in regard his Demonſtration is very
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              mediate, becauſe I would not be over long, laying it aſide, I ſhall
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              declare the ſame another way. </s>
              <s>Let us conſider, therefore, that
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              putting into the Water
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              v. </s>
              <s>g.
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              a Maſs of Gold, if that Maſs were
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              of Water it would have no weight at all: For the Water moveth
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              neither upwards, nor downwards in the Water: It remains,
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              therefore, that the Maſs of Gold weigheth in the Water only ſo
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              much as the Gravity of the Gold exceeds the Gravity of the Wa­
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              ter. </s>
              <s>And the like is to be underſtood of other Metals. </s>
              <s>And be­
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              cauſe the Metals are different from each other in Gravity, their
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              Gravity in the Water ſhall diminiſh according to ſeveral proporti­
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              ons. </s>
              <s>As for example: Let us ſuppoſe that Gold weigheth twenty
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              times more than Water, it is manifeſt by that which hath been
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              ſpoken, that the Gold will weigh leſs in the Water than in the
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              Air by a twentieth part of its whole weight. </s>
              <s>Now, let us ſuppoſe
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              that Silver, as being leſs Grave than Gold, weigheth 12 times more
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              than Water: this then, being weighed in the Water, ſhall di­
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              miniſh in Gravity the twelfth part of its whole weight. </s>
              <s>Therefore
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              the Gravity of Gold in the Water decreaſeth leſs than that of
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              Silver; for that diminiſheth a twentieth part, and this a twelfth.
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              </s>
              <s>If therefore in an exquiſite Ballance we ſhall hang a Metal at the
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              one Arm, and at the other a Counterpoiſe that weigheth equally
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              with the ſaid Metal in the Water, leaving the Counterpoiſe in the
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              Air, to the end that it may equivalate and compenſate the Me­
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              tal, it will be neceſſary to hang it nearer the Perpendicular or
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              Cook. </s>
              <s>As for example, Let the Ballance be A B, its Perpendicu­
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                <figure id="id.070.01.034.1.jpg" xlink:href="070/01/034/1.jpg" number="28"/>
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              lar C, and let a
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              Maſs of ſome
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              Metal be ſu­
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              ſpended at B,
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              counterpoiſedby
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              the Weight D: putting the Weight B into the Water, the
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              Weight D in A would weigh more: therefore that they may </s>
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