Psychologically as well as logically, there is a distinction between arithmetic and mathematics. In both respects the former is but one phase or branch of the latter. By arithmetic is meant those functions of mathematicians which involve numerical calculation. This includes the four fundamental processes, with whole numbers and fractions, enumeration, and the solution of problems requiring choice of process to be employed. Mathematics includes arithmetic, and also the relationships of space, time, proportion, and probability, as subsumed in algebra, geometry, trigonometry, and calculus. Psychologists find a positive intercorrelation among abilities in these various branches of mathematics, which is, however, not sufficiently close to unity so that the possibility of marked specialization in some cases is excluded. Judd has concluded that the abilities demanded by algebra, geometry, and arithmetic represent, respectively, elements not included in the others. Lightning calculators have been recorded, who could accomplish nothing, apparently, in the derivation of formulæ, or abstraction of principles. Rogers decided as a result of experimental tests of mathematical ability, that “a marked degree of the power to analyze a complex and abstract situation, and to seize upon its implications, is the most indispensable element in mathematical proficiency.” This is the power that makes for proficiency in all life’s difficulties, and he who has it has unusual general intelligence—not mathematical proficiency only. There is certainly slight possibility that a generally stupid individual can ever deal with “higher mathematics.” Since the processes other than the arithmetical have been very little studied, the discussion of special aptitude in mathematics will here be restricted largely to aptitude for arithmetic. III. MENTAL FUNCTIONS IN ARITHMETICAL CALCULATION In his recent presentation of the psychology of arithmetic, Thorndike writes as follows: “Achievement in arithmetic depends upon a number of different abilities. For example, accuracy in copying numbers depends upon eyesight, ability to perceive visual details, and short-term memory for these. Long column addition depends chiefly upon great strength of the addition combinations, especially in higher decades, ‘carrying,’ and keeping one’s place in the column. The solution of problems framed in words requires understanding of language, the analysis of the situation described into its elements, the selection of the right elements for use at each step, and their use in the right relations.”