From these considerations it follows that there are two sciences of number. There is the true science which deals with quantities really seen in objects and imagined in the mind, and an artificial science dealing with figures which have only a historical connection with real quantity. Of the latter, unfortunately, our arithmetical education chiefly consists. We are never taught to distinguish number and size in things by the 'eye,' that is, by reason. The symbolism that was originally intended to assist real arithmetical thought has ended by supplanting it. An ignorant shepherd, bricklayer, or carpenter, who is accustomed to make a rapid estimate of the number of things in a mass, or the area of planking in a log, has a better training in real arithmetical science than some mathematicians. If we are obliged to practise genuine arithmetical thought in engineering, astronomy, and other professions, our scholastic symbolism gets realised to some extent, and is a great assistance in arithmetical estimation. But without this it has no more reference to number and quantity than a musical education, based entirely on the printed or written notation, has to the appreciation of musical sounds. A book arithmetician is in the position of a person thoroughly acquainted with theoretical music, and who can even compose music _according to rule_, but who is unable to distinguish a high note from a low one or harmony from discord in actual sound. It will thus be seen that it is only in the real arithmetic that reasoning can enter. The judgment in free arithmetical observation is the counting of actual groups and the measurement of actual surfaces, and the argument consists in estimating the number of individuals in other groups, and the size of other surfaces, without counting or measurement. But this exercise never enters into symbolic arithmetic. All the apparent conclusions of book arithmetic are tautological; they consist in repeating in one combination of symbols the whole or part of what has been already given in another combination. It is an exercise in expression--nothing more. Arithmetical ratio has a resemblance to the rational parallel. 3:5::9:15 might be arranged thus-- 5 | 15 --+--- 3 | 9 This is not argument, for two reasons. (1) The apparent conclusion is not an effort of rational imagination; it is a figure that can be obtained with infallible certainty by treating the other figures according to a rule, which has only to be recollected and applied. (2) The relation between the left-hand figures and the right-hand figures is not a categorical judgment; it is a form of resemblance, and so it cannot yield a valid conclusion. XXXIV--GEOMETRICAL DEMONSTRATION This exercise is regarded by logicians as one of the purest forms of argument. It is nothing more than an aid to a certain kind of perception.