The base measure chosen was the “quinary,” so called (according to the most natural derivation) because its diameter was five quarter-digits. And to this basis are reduced some twenty different _moduli_, to which the names were given of senary, septenary, octonary, denary, duodenary, and so on up to centenary. The reduction of these seems to be tolerably accurate according to the system which he pursues, and making allowance for transcribers’ errors. But either the relation of the circumference to the diameter of a circle had not been worked out to the same extent to which it has been in later times, or Frontinus considered the ratio adopted sufficient for his purpose. One of his calculations stands thus: he is measuring the Appian at a point where it is joined by a later branch; it had been assigned in the register as giving 841 quinaries: he says— “I found the depth of the water five feet, and its breadth one foot and three quarters, which gives an area of eight feet and three quarters. This is equal to twenty-two _centenaries_ and a _quadragenary_, which make 1,825 _quinaries_, or 984 quinaries more than is given in the Commentaries[157].” Each centenary is shewn elsewhere to contain 81 quinaries and a fraction, and therefore 22 would give 1,792 in quinaries and a fraction. The quadragenary is marked to contain 32 quinaries and a fraction. Together, therefore, the 1,825. Now the area of each quinary, according to the system of Frontinus, would be the square of its diameter (i.e. 1¼ digit), multiplied by ¹¹⁄₁₄ (the ratio already ascertained). This equals ²⁷⁵⁄₂₂₄, which, multiplied by 1,825, produces 2,240½ digits (nearly). Finally, the area of 8¾ square feet, reduced to square digits, gives 2,240[158]. This calculation illustrates his mode of computation from absolute measure of the water; but the ordinary way was to affix the _moduli_, which were of brass, or rather of bronze, to certain openings. Provided the _modulus_ was placed perfectly level, the water flowed at its ordinary rate; and when the _moduli_ were sufficient to prevent the water rising in the reservoir or channel, and, on the other hand, not too numerous or too large to reduce the mean level of the water, the sum of them represented the amount of water.