but, if not, still another set of calculations would be necessary. The computation work involved in this investigation is not as great as a perusal of the preceding directions might lead one to surmise, especially after the computer has become accustomed to utilizing the diagrams of the paper and to estimating quickly the approximate quantities of masonry in arch piers.

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Seventh. In the cantilever arch the reduction of the horizontal thrusts at the tops of the piers, as compared with simple or non-cantilevered arches, effects a decided saving in the cost of the said piers; and the greater the proportionate lengths of the flanking spans to the arch span, the larger the economy in pier material.

Eighth. Some engineers who adhere to old-fashioned notions contend that both masonry and steel arch-bridges are applicable to bed-rock foun-