Organic Numbers.
The forms and properties of brute nature having been sufficiently defined by the previous classes of numbers, the organic world, both vegetable and animal, remains outstanding, and offers a higher series of phenomena for our investigation. All exact knowledge relating to the forms and sizes of living things, their numbers, the quantities of various compounds which they consume, contain, or excrete, their muscular or nervous energy, &c. must be placed apart in a class by themselves. All such numbers are doubtless more or less subject to variation, and but in a minor degree capable of exact determination. Man, so far as he is an animal, and as regards his physical form, must also be treated in this class.
Social Numbers.
Little allusion need be made in this work to the fact that man in his economic, sanitary, intellectual, æsthetic, or moral relations may become the subject of sciences, the highest and most useful of all sciences. Every one who is engaged in statistical inquiry must acknowledge the possibility of natural laws governing such statistical facts. Hence we must allot a distinct place to numerical information relating to the numbers, ages, physical and sanitary condition, mortality, &c., of different peoples, in short, to vital statistics. Economic statistics, comprehending the quantities of commodities produced, existing, exchanged and consumed, constitute another extensive body of science. In the progress of time exact investigation may possibly subdue regions of phenomena which at present defy all scientific treatment. That scientific method can ever exhaust the phenomena of the human mind is incredible.
CHAPTER XV.
ANALYSIS OF QUANTITATIVE PHENOMENA.
In the two preceding chapters we have been engaged in considering how a phenomenon may be accurately measured and expressed. So delicate and complex an operation is a measurement which pretends to any considerable degree of exactness, that no small part of the skill and patience of physicists is usually spent upon this work. Much of this difficulty arises from the fact that it is scarcely ever possible to measure a single effect at a time. The ultimate object must be to discover the mathematical equation or law connecting a quantitative cause with its quantitative effect; this purpose usually involves, as we shall see, the varying of one condition at a time, the other conditions being maintained constant. The labours of the experimentalist would be comparatively light if he could carry out this rule of varying one circumstance at a time. He would then obtain a series of corresponding values of the variable quantities concerned, from which he might by proper hypothetical treatment obtain the required law of connection. But in reality it is seldom possible to carry out this direction except in an approximate manner. Before then we proceed to the consideration of the actual process of quantitative induction, it is necessary to review the several devices by which a complicated series of effects can be disentangled. Every phenomenon measured will usually be the sum, difference, or it may be the product or quotient, of two or more different effects, and these must be in some way analysed and separately measured before we possess the materials for inductive treatment.
Illustrations of the Complication of Effects.
It is easy to bring forward a multitude of instances to show that a phenomenon is seldom to be observed simple and alone. A more or less elaborate process of analysis is almost always necessary. Thus if an experimentalist wishes to observe and measure the expansion of a liquid by heat, he places it in a thermometer tube and registers the rise of the column of liquid in the narrow tube. But he cannot heat the liquid without also heating the glass, so that the change observed is really the difference between the expansions of the liquid and the glass. More minute investigation will show the necessity perhaps of allowing for further minute effects, namely the compression of the liquid and the expansion of the bulb due to the increased pressure of the column as it becomes lengthened.
In a great many cases an observed effect will be apparently at least the simple sum of two separate and independent effects. The heat evolved in the combustion of oil is partly due to the carbon and partly to the hydrogen. A measurement of the heat yielded by the two jointly, cannot inform us how much proceeds from the one and how much from the other. If by some separate determination we can ascertain how much the hydrogen yields, then by mere subtraction we learn what is due to the carbon; and vice versâ. The heat conveyed by a liquid, may be partly conveyed by true conduction, partly by convection. The light dispersed in the interior of a liquid consists both of what is reflected by floating particles and what is due to true fluorescence;[233] and we must find some mode of determining one portion before we can learn the other. The apparent motion of the spots on the sun, is the algebraic sum of the sun’s axial rotation, and of the proper motion of the spots upon the sun’s surface; hence the difficulty of ascertaining by direct observations the period of the sun’s rotation.
We cannot obtain the weight of a portion of liquid in a chemical balance without weighing it with the containing vessel. Hence to have the real weight of the liquid operated upon in an experiment, we must make a separate weighing of the vessel, with or without the adhering film of liquid according to circumstances. This is likewise the mode in which a cart and its load are weighed together, the tare of the cart previously ascertained being deducted. The variation in the height of the barometer is a joint effect, partly due to the real variation of the atmospheric pressure, partly to the expansion of the mercurial column by heat. The effects may be discriminated, if, instead of one barometer tube we have two tubes containing mercury placed closely side by side, so as to have the same temperature. If one of them be closed at the bottom so as to be unaffected by the atmospheric pressure, it will show the changes due to temperature only, and, by subtracting these changes from those shown in the other tube, employed as a barometer, we get the real oscillations of atmospheric pressure. But this correction, as it is called, of the barometric reading, is better effected by calculation from the readings of an ordinary thermometer.