Salusbury, Thomas, Mathematical collections and translations (Tome I), 1667

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

lar C, and let a
Maſs of ſome
Metal be ſu­
ſpended at B,
counterpoiſedby
the Weight D: putting the Weight B into the Water, the
Weight D in A would weigh more: therefore that they may