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

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            <p type="main">
              <s>
                <pb xlink:href="040/01/197.jpg" pagenum="179"/>
              of reſt, are alwayes determinate, and anſwer in proportion to the
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              parallels comprehended between two right lines that concur in
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              an angle, like to the angle B A E, or B A D, or any other
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              infinitely more acute, alwayes provided it be
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              But the diminution of the ſpaces thorow which the moveable is
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              to be conducted along the circumference of the wheel, is
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              tionate to another kind of diminution, comprehended between
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              lines that contain an angle infinitely more narrow and acute, than
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              any rectilineal angle, how acute ſoever, which is that in our
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              ſent caſe. </s>
              <s>Let any point be taken in the perpendicular A C, and
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              making it the centre, deſcribe at the diſtance C A, an arch A M P,
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              the which ſhall interſect the parallels that determine the degrees of
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              velocity, though they be very minute, and comprehended within
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              a moſt acute rectilineal angle; of which parallels the parts that
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              lie between the arch and the tangent A B, are the quantities of
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              the ſpaces, and of the returns upon the wheel, alwayes leſſer (and
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              with greater proportion leſſer, by how much neerer they approach
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              to the contact) than the ſaid parallels of which they are parts.
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              </s>
              <s>The parallels comprehended between the right lines in retiring
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              wards the angle diminiſh alwayes at the ſame rate, as
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              v.g.
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              A H
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              ing divided in two equal parts in F, the parallel H I ſhall be
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              ble to F G, and ſub-dividing F A, in two equal parts, the
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              lel produced from the point of the diviſion ſhall be the half of
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              F G; and continuing the ſub-diviſion
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              in infinitum,
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              the ſubſequent
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              parallels ſhall be alwayes half of the next preceding; but it doth
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              not ſo fall out in the lines intercepted between the tangent and
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              the circumference of the circle: For if the ſame ſub-diviſion be
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              made in F A; and ſuppoſing for example, that the parallel which
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              cometh from the point H, were double unto that which commeth
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              from F, this ſhall be more then double to the next following, and
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              continually the neerer we come towards the contact A, we ſhall
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              find the precedent lines contein the next following three, four,
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              ten, an hundred, a thouſand, an hundred thouſand, an hundred
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              millions of times, and more
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              in infinitum.
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              The brevity therefore of
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              ſuch lines is ſo reduced, that it far exceeds what is requiſite to make
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              the project, though never ſo light, return, nay more, continue
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              unremoveable upon the circumference.</s>
            </p>
            <p type="main">
              <s>SAGR. </s>
              <s>I very well comprehend the whole diſcourſe, and upon
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              what it layeth all its ſtreſſe, yet nevertheleſſe methinks that he
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              that would take pains to purſue it, might yet ſtart ſome further
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              queſtions, by ſaying, that of thoſe two cauſes which render the
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              deſcent of the moveable ſlower and ſlower
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              in infinitum,
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              it is
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              feſt, that that which dependeth on the vicinity to the firſt term of
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              the deſcent, increaſeth alwayes in the ſame proportion, like as the
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              parallels alwayes retain the ſame proportion to each other, &c. </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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