Trial mixtures may be made by carefully weighing each of the ingredients and then combining them to give a workable concrete. Using a given amount of cement, the proportion of ingredients, of the same total weight, which will give the least volume and therefore the densest concrete is adopted. When making the comparison the consistency of the mixes must be maintained constant. Proportioning may be based on an ideal sieve analysis curve of the mixed cement and aggregates. The sieve analysis of the aggregates is made by screening a predetermined weight of the sample through a series of 5 to 8 sieves graded in size from slightly below the size of the largest particle to slightly above the smallest particle of the aggregate. The analysis is then expressed in the form of a curve. The ideal curve, according to Fuller,[56] is shown in Fig. 77. [Illustration: FIG. 77.—Gravel Analysis. The dotted line indicates the ideal combination of the coarse and fine portions. The heavy full line indicates the combination attained. ] The method of proportioning concrete by surface areas is based on the theory that the strength of a concrete depends on the amount of cement used in proportion to the surface area of the aggregates.[57] The proportioning of concrete on the basis of a water-cement ratio and a fineness modulus was introduced by Prof. D. A. Abrams.[58] It is based on the theory that with fixed conditions of aggregate, moisture, etc., the ratio of water to cement determines the strength of the concrete. A method of proportioning concrete by determining experimentally the voids in mortars made up with a given amount of sand and definite proportions of cement, and then calculating the voids in the concrete made up by adding a definite amount of coarse aggregate to the mixture, has been developed.[59] The method is based on the theory that the strength of the concrete is a known function of the ratio of the volume of cement to the volume of the voids in the concrete. The effect of varying the proportion of the ingredients, including an increase in the amount of mixing water beyond that required to give the densest mixture, may be found by the method, and a comparison may be made of results obtainable with different classes of fine and coarse aggregates. Arbitrarily selected proportions, proportions based on voids, and proportions based on trial mixtures are usually satisfactory for small jobs where the amount of materials involved is not large. Where the saving in materials will permit, more accurate methods should be used. The methods can be studied more fully by reference to the original articles quoted in the footnotes, or to the following texts: