without end, will never be as great as the swiftness cd. </s>
<s id="id.1.2.8.06.02">As would be the case if a mobile, receding from rest, acquired after the first mile of its motion the swiftness ab, which is two-thirds of swiftness cd; and if after the second mile its swiftness were increased by one-third of swiftness ab; if after the third it were increased by one-third of one-third of swiftness ab; after the fourth by one-third of one-third of one-third of ab; and if in this way without end there came about an increase for each mile of one third of the increase in the preceding mile: and the swiftness will always, surely, be increased, yet it will never be as great as cd, but always one-half {1} of the last increase will be wanting. {2}</s>
<s id="id.1.2.8.06.03">Now here is the demonstration of this. </s>
</p>
<p>
<s id="id.1.2.8.07.00.fig"/>
<s id="id.1.2.8.07.01">Let there be any number of continuous swiftnesses, each one three times its successor, ab, bc, cd; let the greatest be ab, whose sesquialter {1} is ea.</s>
<s id="id.1.2.8.07.02">I say that the sum of all the magnitudes ab, bc, cd, together with one half of cd, is equal to ea itself. </s>
<s id="id.1.2.8.07.03"> For since ea = 3 ab/2,ab + 1/2 ab = ae.And since ab = 3 bc, bc + 1/2 bc = 1/2 ab: but it has been demonstrated that ab + 1/2 ab = ae.Therefore, abc + 1/2 bc = ae.</s>
<s id="id.1.2.8.07.04">Similarly, since bc = 3 cd, cd + 1/2 cd = 1/2 bc. but it has been demonstrated that ac + 1/2 bc = ae:Therefore, ad + 1/2 dc = ae.</s>
<s id="id.1.2.8.07.05">And the same demonstration being always repeated, it will be demonstrated that any number of swiftnesses, each one three times its successor, taken together with one-half the smallest, are equal to a swiftness which is 3/2 that of the greatest swiftness among them. </s>
<s id="id.1.2.8.07.06">And if this is so, it is evident that the sum of the swiftnesses, taken together, each one three times its successor, is less than a swiftness 3/2 that of the greatest, since they always fall short of it by half the smallest swiftness. </s>
<s id="id.1.2.8.07.07">It is thus evident how the swiftness ab could always be increased without end; and yet it would never be equal to swiftness ae. </s>
<s id="id.1.2.8.07.08">Consequently let us conclude that, in the case of a mobile, for the reasons adduced above, the speed is not always increased; but it reaches a certain motion, beyond which its determinate hea</s>
