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

Table of figures

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              <s>
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              half of
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              O D,
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              would paſſe in the Times
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              E A
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              and
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              D A.
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              If it be proved
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              therefore that theſe Spaces
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              M H
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              and
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              L H
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              are in duplicate proportion to
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              the Times
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              E A
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              and
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              D A;
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              We ſhall have done that which was intended.
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              <s>But in the fourth Propoſition of the Firſt Book we have demonſtrated:
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              That the Spaces paſt by two Moveables with an Equable Motion are
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              to each other in a proportion compounded of the proportion of the Velo­
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              cities and of the proportion of the Times: But in this caſe the propor­
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              tion of the Velocities and the proportion of the Times is the ſame
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              (
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              for
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              as the half of
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              P E
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              is to the half of
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              O D,
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              or the whole
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              P E
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              to the whole
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              O D,
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              ſo is
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              A E
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              to
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              A D
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              : Therefore the proportion of the Spaces paſ­
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              ſed is double to the proportion of the Times. </s>
              <s>Which was to be demon­
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              ſtrated.
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              Hence likewiſe it is manifeſt, that the proportion of the ſame Spaces
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              is double to the proportions of the greateſt degrees of Velocity: that is,
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              of the Lines
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              P E
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              and
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              O D
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              : becauſe
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              P E
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              is to
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              O D,
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              as
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              E A
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              to
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              D A.</s>
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            <p type="head">
              <s>COROLARY I.</s>
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              <s>
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              Hence it is manifeſt, that if there were many equal Times taken in or­
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              der from the firſt Inſtant or beginniug of the Motion, as ſuppoſe
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              A D, D E, E F, F G,
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              in which the Spaces
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              H L, L M, M N, N I
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              are paſſed, thoſe Spaces ſhall be to one another as the odd numbers
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              from an Vnite:
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              ſcilicet,
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              as 1, 3, 5, 7. For this is the Rate or pro­
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              portion of the exceſſes of the Squares of Lines that equally exceed
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              one another, and the exceſſe of which is equal to the least of them,
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              or, if you will, of Squares that follow one another, beginning
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              ab
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              Unitate.
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              Whilſt therefore the degree of Velocity is encreaſed ac­
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              cording to the ſimple Series of Numbers in equal Times, the Spaces
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              paſt in thoſe Times make their encreaſe according to the Series of
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              odd Numbers from an Vnite.
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              </s>
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            <p type="main">
              <s>SAGR. </s>
              <s>Be pleaſed to ſtay your Reading, whilſt I do paraphraſe
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              touching a certain Conjecture that came into my mind
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              but even now; for the explanation of which, unto your under­
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              ſtanding and my own, I will deſcribe a ſhort Scheme: in which I
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              fanſie by the Line A I the continuation of the Time after the firſt
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              Inſtant, applying the Right Line A F unto A according to any
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              Angle: and joyning together the Terms I F, I divide the Time A I
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              in half at C, and then draw C B parallel to I F.
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              A
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              nd then conſide­
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              ring B C, as the greateſt degree of Velocity which beginning from
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              Reſt in the firſt Inſtant of the Time
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              A
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              goeth augmenting accord­
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              ing to the encreaſe of the Parallels to B C, drawn in the Triangle
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              A
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              B C, (which is all one as to encreaſe according to the encreaſe
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              of the Time) I admit without diſpute, upon what hath been ſaid
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              already, That the Space paſt by the falling Moveable with the </s>
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