[Footnote 11: As the calculation of the power of an engine depends on the number of square inches in the section of the piston, it may be useful to give a rule for computing the number of square inches in a circle. The following rule will always give the dimensions with sufficient accuracy:--_Multiply the number of inches in the diameter by itself; divide the product by 14, and multiply the quotient thus obtained by 11, and the result will be the number of square inches in the circle._ Thus if there be 12 inches in the diameter, this multiplied by itself gives 144, which divided by 14 gives 10-4/14, which multiplied by 11 gives 115, neglecting fractions. There are, therefore, 115 square inches in a circle whose diameter is 12 inches.] (38.) In order to accomplish this design, two things were necessary: 1. To make a speedy and effectual vacuum below the piston in the descent; and 2. To contrive a counterpoise for the atmosphere in the ascent. The condensation of steam immediately presented itself as the most effectual means of accomplishing the former; and the elastic force of the same steam previous to condensation an obvious method of effecting the latter. Nothing now remained to carry the design into execution, but the contrivance of means for the alternate introduction and condensation of the steam; and Newcomen and Cawley were accordingly granted a patent in 1707, in which Savery was united, in consequence of the principle of condensation for which he had previously received a patent being necessary to the projected machine. We shall now describe the _atmospheric engine_, as first constructed by Newcomen:-- The boiler K is placed over a furnace I, the flue of which winds round it, so as to communicate heat to every part of the bottom of it. In the top, which is hemispherical, two gauge-pipes G G´ are placed, as in Savery's engine, and a _puppet valve_ V, which opens upward, and is loaded at one pound per square inch; so that when the steam produced in the boiler exceeds the pressure of the atmosphere by more than one pound on the square inch, the valve V is lifted, and the steam escapes through it, and continues to escape until its pressure is sufficiently diminished, when the valve V again falls into its seat. The great steam-tube is represented at S, which conducts steam from the boiler to the cylinder; and a feeding pipe T furnished with a cock, which is opened and closed at pleasure, proceeds from a cistern L to the boiler. By this pipe the boiler may be replenished from the cistern, when the gauge cock G´ indicates that the level has fallen below it. The cistern L is supplied with hot water by means which we shall presently explain.