*THE POPULAR SCIENCE MONTHLY*

entropy diagram, which represents the efficiency of a Carnot cycle as a simple rectangular figure and is, he points out, "nothing more nor less than a geometrical representation of the second law of thermodynamics." The area of the Watts diagram represents the work done by the engine; the area of the Gibbs diagram represents the heat it has received and, either upon separate blackboards or upon "quadrant diagrams," the two taken together have proved invaluable in teaching thermodynamics to engineers. As the indicator diagram tells the engineer what he wants to know about the work done upon the piston, the efficiency of the valves and passages and the total horse power of the engine, the entropy diagram gives him the heat taken in or given out and shows directly the losses of efficiency from such heat wastes as wire-drawing of steam, incomplete expansion, etc. Professor John Perry says that the thermodynamics of heat engines is revealed by the entropy diagram "as it can be revealed in no other way," and he describes how "a man almost illiterate, innocent of algebra, can use his diagram of water steam or air or ammonium anhydride, obtaining in a few minutes answers to problems which the mathematical engineers of years ago spent days in solving."^{[1]} In England the temperature-entropy diagram has been found very useful in "engine testing laboratories," and its ultimate adoption is due to the persistent crusade of Mr. Macfarlane Gray, late chief engineer of the Royal Navy, who introduced it independently in 1880 as the "theta-phi" () diagram. American engineers should not forget that this diagram was first described in scientific literature by Professor Willard Gibbs,^{[2]} who clearly pointed out its advantages, in visualizing the second law, for teaching purposes and its use and significance when attached to heat-engines. In his second memoir^{[3]} Gibbs extends his graphical methods to three-dimensional space, the first example of which was the volume-pressure-temperature diagram employed by James Thomson^{[4]} in 1871. The first solid diagram described by Gibbs had for its coordinates, volume, entropy and energy and is now generally known as the "thermodynamic surface." It is a solid model or relief-map, affording a bird's-eye view of the chemico-physical changes of a system at constant temperature and pressure as it passes through the coexistent states of solid, liquid, vapor or gas. Maxwell, who had himself written learn-

- ↑
*Nature,*London, 1902-3, LXVII., 604. - ↑ Gibbs,
*Tr. Connect. Acad.,*April, 1873, II., 317-25. The equivalent of an entropy diagram was laid down and described by the Belgian physicist M. Belpaire in 1872 (*Bull. Acad. roy. d. sc, Brux.,*1872, 2. s., XXXIV., 520-6), but his treatment of the matter is so sketchy and slight in comparison with the exhaustive and illuminative handling of Gibbs that it seems negligible. The mere plotting of the diagram itself is nothing, for it was for years implicit in Rankine's algebraic use of the "thermodynamic function" () as a coordinate (1854), and to this day the British unit of entropy is called a "Rank." - ↑
*Tr. Connect. Acad.,*1873, II., 382-404. - ↑ J. Thomson,
*Proc. Roy. Soc. Lond.,*1871, XX., 1.