builder's mind, his choice could hardly be directed otherwise, and the adoption of the built and tower-like shaft appears to have been the result of a loss of this sense of symmetry consequent on the employment of intractable materials. § XII. But farther: we have up to this point spoken of shafts as always set in ranges, and at equal intervals from each other. But there is no necessity for this; and material differences may be made in their diameters if two or more be grouped so as to do together the work of one large one, and that within, or nearly within, the space which the larger one would have occupied. § XIII. Let A, B, C, Fig. XIV., be three surfaces, of which B and C contain equal areas, and each of them double that of A: then supposing them all loaded to the same height, B or C would receive twice as much weight as A; therefore, to carry B or C loaded, we should need a shaft of twice the strength needed to carry A. Let S be the shaft required to carry A, and S_2 the shaft required to carry B or C; then S_3 may be divided into two shafts, or S_2 into four shafts, as at S_3, all equal in area or solid contents;[40] and the mass A might be carried safely by two of them, and the masses B and C, each by four of them. [Illustration: Fig. XIV.] Now if we put the single shafts each under the centre of the mass they have to bear, as represented by the shaded circles at _a_, _a2_, _a3_, the masses A and C are both of them very ill supported, and even B insufficiently; but apply the four and the two shafts as at _b_, _b2_, _b3_, and they are supported satisfactorily. Let the weight on each of the masses be doubled, and the shafts doubled in area, then we shall have such arrangements as those at _c_, _c2_, _c3_; and if again the shafts and weight be doubled, we shall have _d_, _d2_, _d3_. § XIV. Now it will at once be observed that the arrangement of the shafts in the series of B and C is always exactly the same in their relations to each other; only the group of B is set evenly, and the group of C is set obliquely,--the one carrying a square, the other a cross. [Illustration: Fig. XV.]