Salusbury, Thomas
,
Mathematical collections and translations (Tome I)
,
1667
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331 - 331
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that is, H G to N C: and as
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(d)
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O H is to H P, ſo is G B to C K; For O H is double
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to G B, and H P alſo double to G F; that is, to C K; Therefore H G hath the ſame propor
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tion to N C, that G B hath to C K: And
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Permutando,
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N C hath to C K the ſame proportion
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that H G hath to G B.
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(a)
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By
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2. Lemma.</
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(b)
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By
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4. Lemma.</
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(b)
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By 19. of the
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fifth.
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(d)
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By 15. of the
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fifth.
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>Then take ſome other Point at pleaſure in the Section, which
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let be S: and thorow S draw two Lines, the one S T paral
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lel to D B, and cutting the Diameter in the Point T; the
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other S V parallel to A C, and cutting C E in V. </
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>I ſay
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that V C hath greater proportion to C K, than T G hath
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to G B.</
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For prolong V S unto the Line Q M in X; and from the Point X draw X Y unto the
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Diameter parallel to B D: G T ſhall be leſſe than G Y, in regard that V S is leße than V X:
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And, by the firſt Lemma, Y G ſhall be to V C, as H G to N C; that is, as G B to C K, which
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was demonſtrated but now: And,
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Permutando,
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Y G ſhall be to G B, as V C to C K: But
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T G, for that it is leſſe than Y G, hath leſſe proportion to G B, than Y G hath to the ſame;
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Therefore V C hath greater proportion to C K. than T G hath to G B: Which was to be de
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monſtrated. </
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>Therefore a Poſition given G K, there ſhall be in the Section one only Point, to
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wit M, from which two Lines M E H and M N O being drawn, N C ſhall have the ſame pro
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portion to C K, that H G hath to G B; For if they be drawn from any other, that which fall
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eth betwixt A C, and the Line parallel unto it ſhall alwayes have greater proportion to C K,
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than that which falleth betwixt G K and the Line parallel unto it hath to G B. That, there
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fore, is manifeſt which was affirmed by
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Archimedes,
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to wit, that the Line P I hath unto P H,
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either the ſame proportion that N
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<
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hath to
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O, or greater.
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D</
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>Wherefore P H is to H I either double, or leſſe than double.]
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If leſſe than double, let P T be double to T I: The Centre of Gravity of that part of the
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Portion that is within the Liquid ſhall be the
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Point T: But if P H be double to H I, H ſhall
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be the Centre of Gravity; And draw H F, and
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prolong it unto the Centre of that part of the Por
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tion which is above the Liquid, namely, unto G,
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and the reſt is demonſtrated as before. </
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<
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>And the
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ſame is to be underſtood in the Propoſition that
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followeth.
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>The Solid A P O L, therefore,
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ſhall turn about, and its Baſe ſhall
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not in the leaſt touch the Surface
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of the Liquid.]
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In
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Tartaglia's
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Tranſlation it is rendered
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ut Baſis ipſius non tangent
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ſuperficiem humidi ſecundum unum ſignum;
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but we have choſen to read
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ut Baſis ipſius
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nullo modo humidi ſuperficiem contingent,
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both here, and in the following Propoſitions,
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becauſe the Greekes frequently uſe
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pro
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& οὐδὶν</
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:
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ſo that
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nullus
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eſt;
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ὑπ̓ἑρὸς</
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à nullo,
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and ſo of others of the like nature.
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