32 ON MAGNITUDE [ch. 



there be a similar tendency among animals for the rate of supply 

 o^ kinetic energy to vary with the surface of the lung, that is to say 

 (other things being equal) with the square of the linear dimensions 

 of the animal ; which means that, caeteris paribus, the small animal 

 is stronger (having more power per unit weight) than a large one. 

 We may of course (departing from the condition of similarity) increase 

 the heating-surface of the boiler, by means of an internal system of 

 tubes, without increasing its outward dimensions, and in this very 

 way Nature increases the respiratory surface of a lung by a complex 

 system of branching tubes and minute air-cells ; but nevertheless in 

 two similar and closely related animals, as also in two steam-engines 

 of the same make, the law is bound to hold that the rate of working 

 tends to vary with the square of the linear dimensions, according to 

 Froude's law of steamship comparison. In the case of a very large 

 ship, built for speed, the difficulty is got over by increasing the size 

 and number of the boilers, till the ratio between boiler-room and 

 engine-room is far beyond what is required in an ordinary small 

 vessel*; but though we find lung-space increased among animals 

 where greater rate of working is required, as in general among birds, 

 I do not know that it can be shewn to increase, as in the "over- 

 boilered" ship, with the size of the animal, and in a ratio which 

 outstrips that of the other bodily dimensions. If it be the case then, 

 that the working mechanism of the muscles should be able to exert 

 a force proportionate to the cube of the linear bodily dimensions, 



* Let L be the length, S the (wetted) surface, T the tonnage, D the displacement 

 (or volume) of a ship; and let it cross the Atlantic at a speed V. Then, in com- 

 paring two ships, similarly constructed but of different magnitudes, we know that 

 L=V\ S=L^ = V\ D = T = L^=V^; also B (resistance) =/Sf. F^^ F«; H (horse- 

 power) = i2 . F = F' ; and the coal (C) necessary for the voy a,ge— HjV = V^. That 

 is to say, in ordinary engineering language, to increase the speed across the Atlantic 

 by 1 per cent, the ship's length must be increased 2 per cent., her tonnage or 

 displacement 6 per cent., her coal- consumption also 6 per cent., her horse-power, 

 and therefore her boiler-capacity, 7 per cent. Her bunkers, accordingly, keep 

 pace with the enlargement of the ship, but her boilers tend to increase out of 

 proportion to the space available. Suppose a steamer 400 ft. long, of 2000 tons, 

 2000 H.P., and a speed of 14 knots. The corresponding vessel of 800 ft. long should 

 develop a speed of 20 knots (I : 2 :: 14^ : 20^), her tonnage would be 16,000, her 

 H.p. 25,000 or thereby. Such a vessel would probably be driven by four propellers 

 instead of one, each carrying 8000 h.p. See (int. al.) W. J. Millar, On the most 

 economical speed to drive a steamer, Proc. Edin. Math. Soc. vii, pp. 27-29, 1889; 

 Sir James R. Napier, On the most profitable speed for a fully laden cargo steamer 

 for a given voyage, Proc. Phil. Soc, Glasgow, vi, pp. 33-38, 1865. 



