_e_ by a less difference than any quantity that can be assigned. Again, let H I be drawn parallel and equal to G E, cutting C B in P, the crooked line in F, and O E in I, and the proportion of C _g_ to C H will be triplicate to the proportion of H F to H P, and I F will be greater than the third part of P I. But again, setting H in _h_ so near to _g_, as that the difference between C _h_ and C _g_ may be but as a point, the point P will also in _p_ be so near to _l_, as that the difference between C _p_ and C _l_ will be but as a point; and drawing _h p_ till it meet with B D in _i_, cutting the crooked line in _f_ and having drawn _e o_ parallel to B D, cutting D C in _o_, the parallelogram _f o_ will differ less from the third part of the parallelogram _g f_, than by any quantity that can be given. And so it will be in all other spaces generated in the same manner. Wherefore the differences of the arithmetical and geometrical means, which are but as so many points B, _e_, _f_, &c. (seeing the whole figure is made up of so many indivisible spaces) will constitute a certain line, such as is the line B E F C, which will divide the complete figure A D into two parts, whereof one, namely, A B E F C, which I call a deficient figure, is triple to the other, namely, B D C F E, which I call the complement thereof. And whereas the proportion of the altitudes to one another is in this case everywhere triplicate to that of the decreasing quantities to one another; in the same manner, if the proportion of the altitudes had been everywhere quadruplicate to that of the decreasing quantities, it might have been demonstrated that the deficient figure had been quadruple to its complement; and so in any other proportion. Wherefore, a deficient figure, which is made, &c. which was to be demonstrated. The same rule holdeth also in the diminution of the bases of cylinders, as is demonstrated in the second article of chapter XV. [Sidenote: The proportion of deficient figures to the parallelograms in which they are described, set forth in a table.]