The relation between two variables, and the idea of function
Relations are the fundamental stuff out of which all science is built. To say that a given piece of metal weighs so many pounds is to state a relationship. The weight simply means that there is a certain relationship between the pull of gravity on that piece of metal and the pull on another piece which has been named the "pound." We can tell what our "pound" is only by defining it in terms of still other units, or by comparing it to a master lump of metal carefully sheltered in the Bureau of Standards. If the pull is twice as great on the given piece of metal as it is on the standard pound, then we say that the lump weighs 2 pounds. If, further, we say it weighs 2 pounds per cubic inch, that is stating a composite relationship, involving at the same time the arbitrary units which we use to measure extent or distance in space and the units for measuring the gravitational force or attracting power of the earth.
Relations between Variables. Besides these very simple relationships which are implicit in all our statements of numerical description--weight, length, temperature, size, age, and so on--there are more complicated relationships where two or more variables are concerned. A variable is any measurable characteristic which can assume varying or different values in successive individual cases. The yield of corn on different farms is a variable, since it may differ widely from farm to farm. So is the length of time which a falling body takes to reach the earth, or the quantity of sugar that can be dissolved in a glass of water, or the distance it takes for an automobile to stop after the brakes are applied, or the quantity of milk that one cow will produce in a year, or the tensile strength of a piece of metal, or the length of time it takes a person to memorize a quotation. In contrast to these variables there are other numerical values called constants, because they never change. Thus one foot always