The foregoing calculation is based on the assumption that the organism divides in one plane only. If it divides in 2 or 3 planes, the rate is much faster, as is shown by the following formulæ, which indicate the theoretical rate of division: S = number of bacteria after a given number of divisions. a = number at the beginning, and n = number of divisions. 1 plane division S = 2_ⁿ_a 2 „ „ S = 2²_ⁿ_a 3 „ „ S = 2³_ⁿ_a With two-plane or three-plane division, assuming that each organism attains full size, as was assumed in the first calculation, the “mass as large as the earth” would be attained in about thirty-two and twenty-two hours respectively. This extraordinary rate of increase explains in large measure why bacteria are able to bring about such great chemical changes in so short a time as is seen in the rapid “spoiling” of food materials, especially liquids. The reactions brought about by bacteria on substances which are soluble and diffusible are essentially “surface reactions.” The material diffuses into the cell over its entire surface with little hindrance. The bacteria are usually distributed throughout the medium, so that there is very intimate contact in all parts of the mass which favors rapid chemical action. The following calculation illustrates this: The volume of a coccus 1µ in diameter is 0.5236 × 10⁻¹³ cc. The surface of a coccus 1µ in diameter is π × 10⁻⁸ sq. cm. It is not uncommon to find in milk on the point of souring 1,000,000,000 bacteria per cc. Assuming these to be cocci of 1µ diameter the volume of these bacteria in a liter is only 0.05 cc. or in the liter there would be 19999 parts of milk and only 1 part bacteria. The surface area of these bacteria is 3141.6 sq. cm. With this large surface exposed, it is not strange that the change from “on the point of souring” to “sour” occurs within an hour or less. Although large numbers of bacteria can and do cause great chemical changes the amount of material actually utilized for maintenance of the cell is very slight, infinitesimal almost, and yet is fairly comparable to that required for man, as is illustrated by the following computations: E. Kohn has shown that certain water bacteria grew well in water to which there was added per liter 0.000002 mg. dextrose, 0.00000007 mg. (NH₄)₂SO₄ and 0.0000000007 mg. (NH₄)₂HPO₄. The bacteria numbered about 1000 per cc. Taking the specific gravity at 1 (a little too low) the mass of the bacteria in the liter was about 0.001 mg. Hence the bacteria used 0.002 of their weight of carbohydrate and 0.00007 of ammonium sulphate. A 150-pound (75-kilo) man can live on 375 g. of sugar (0.005 of his weight) and 52.5 g. of protein (0.0007 of his weight). From these figures it can be calculated that the man utilizes about two and a half times as much carbohydrate and about seven times as much nitrogen as the bacterium, relatively speaking. CHAPTER IX.