lon on incomparable Archimedes' Sphere and Cylinder, Book II {1}: for these are cases of two lines (and numerous others could also be imagined), which, being prolonged without end, always come nearer, but it is impossible that they finally meet; hence the distance between them is always diminished, yet it is never consumed. </s>
<s id="id.1.2.8.03.04">And if a line is drawn at right angles to the straight line that lies under the conchoid, or at right angles to the asymptote, and we assume that this line, always remaining at right angles, is moved without end in the direction in which the lines that do not meet are extended without end; on this perpendicular line, the point at which it is intersected by the hyperbola or the conchoid will always be moved towards the other extremity by coming near it, but will never reach its end point. </s>
<s id="id.1.2.8.03.05">What happens concerning the swiftness is something similar: indeed the slowness of motion can always be diminished and, therefore, the swiftness increased, and yet never be finally consumed. </s>
<s id="id.1.2.8.03.06">As, for example, let ab be the slowness, so that if the mobile consumed it entirely, the motion would happen in an instant: I say that, even if it is always diminished without end, it is not necessary that in the end it is consumed. </s>
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<s id="id.1.2.8.04.00.fig"/>
<s id="id.1.2.8.04.01">For let a motion begin which can be intensified without end: but let it be such that in the first distance of a mile it is accelerated to such an extent that it diminishes the slowness ab by an eighth part, let us say, ac; but at the end of the second mile let it diminish it by the eighth part of the remainder cb; and at the end of another mile let it diminish it by an eighth part of the next remainder.</s>
<s id="id.1.2.8.04.02">And thus this diminution will be able to be produced without end, since the seven-eighths that remain can always be divided into eight equal parts; and the mobile will be able to be moved over miles without end, by consuming at each mile some of its slowness: and yet it is not necessary that this slowness be completely consumed.</s>
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<s id="id.1.2.8.05.00"/>
<s id="id.1.2.8.05.01">What is more: those who with Aristotle have believed that if slowness is always diminished, the mobile must finally reach a swiftness without end -- what will they say if it is shown to them that, not only is it not necessary for that swiftness to come to be without end, but it is even demonstrated that a mobile can be always accelerated, yet without its swiftness being intensified to the extent that it is equal to, much less in excess of, a certain finite swiftness? </s>
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<s id="id.1.2.8.06.00.fig"/>
<s id="id.1.2.8.06.01">And, to express it more clearly, let something be moved whose swiftness at the beginning of the motion is ab; now let there be another swiftness cd greater than ab; I say that the mobile, moving without end, can increase without end its swiftness ab, which nevertheless, increased </s>
