Archimedes
,
Natation of bodies
,
1662
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make equall Angles; and that in the Triangles N F S and G
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C
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the Angles at F and
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are equall; as alſo, that S B and B C, and
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S R and C R are equall to one another: And, therefore, N X and
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G Y are alſo equall; and X T and Y I. </
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<
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>And ſince G H is double
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to H I, N X ſhall be leſſer than double of X T. </
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<
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>Let N M therefore
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be double to M T; and drawing a Line from M to K, prolong it
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unto E. </
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<
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>Now the Centre of Gravity of the whole ſhall be the
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Point K; of the part which is in the Liquid the Point M; and
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that of the part which is above the Liquid in the Line prolonged
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as ſuppoſe in E. Therefore, by what was even now demonſtrated
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it is manifeſt that the Portion ſhall not ſtay thus, but ſhall incline, ſo
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as that its Baſe do in no wiſe touch the Surface of the Liquid
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And that the Portion will ſtand, ſo, as to make an Angle with the
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Surface of the Liquid leſſer than
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the Angle
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ſhall thus be demon
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ſtrated. </
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>Let it, if poſſible, ſtand,
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ſo, as that it do not make an Angle
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leſſer than the Angle
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and diſpoſe
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all things elſe in the ſame manner a
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before; as is done in the preſet
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Figure. </
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>We are to demonſtrat
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in the ſame method, that N T is e
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quall to
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and by the ſame reaſor
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equall alſo to G I. </
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<
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>And ſince that in
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the Triangles P
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C and N F S, the Angle F is not leſſer than the
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Angle
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B F ſhall not be greater than B C: And, therefore, neither
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ſhall S R be leſſer than C R; nor N X than P Y: But ſince P F is
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greater than N T, let P F be Seſquialter of P Y: N T ſhall be leſſer
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than Seſquialter of N X: And, therefore, N X ſhall be greate
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than double of X T. </
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<
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>Let N M be double of M T; and drawing
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Line from M to K prolong it. </
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>It is manifeſt, now, by what hath
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been ſaid, that the Portion ſhall not continue in this poſition, but ſhall
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turn about, ſo, as that its Axis do make an Angle with the Surface
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of the Liquid, leſſer than the Angle
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