in time allow considerable election of high school studies by applicants for admission. The high school course can no longer be called a preparatory course in the sense that it is prerequisite to work in the university. Work in the university no longer fits on to the high school course in any definite way. The university offers rather a course of general culture, and in order that the student may pursue such a course intelligently he should have four years' training in the high school. But that this training should be of any specific kind is not necessary. There is every reason to believe that in the near future the universities will see that it is the amount and quality of high school training rather than the specific character of such training that is needed in their applicants. Silence and Serenity Silence and serenity Are precious words to me, And so I chant them as a nun ELIZABETH PORTER GOULD. What is Multiplication? A. LATHAM BAKER, PH.D., BROOKLYN, N. Y. TTENTION has been called to this question by the recent appearance of Judd's Genetic Psychology for Teachers, in which is reproduced the old misapprehension that multiplication is a special phase of addition. The prominence of the writer and the unusual merit of the book compel attention to an error that otherwise might be passed over in silence as another instance of the conservatism of pedagogy and of the more than feline multiplicity of lives possessed by this ancient inaccuracy. Its reappearance in such authoritative form is unfortunate for those who still adhere to the ancient error, who thus are strengthened in their orthodoxy. He says, "When we come to multiplication and division we have modifications of the process of grouping [addition and subtraction]. . . Division is therefore a special case of subtraction." "Division and multiplication thus come to be processes dealing with units of higher order" [produced by addition and subtraction]. These quotations show that the writer has confused the simple and universally applicable operation of addition with the far more complicated and, in its applicability, limited operation of multiplication. It is curiously surprising that the acumen so conspicuous throughout the book did not discover that the definition breaks down the moment it is applied to fractions. Particularly noticeable is the failure when applied to stroke and vector multiplication, where no amount of addition of the original elements will produce the quotient. Likewise in his definition of division he has confused the abstract operation of division with the concrete semi-mechanical operation of finding the quotient. To define division as a special case of subtraction because in the mechanico-mental operation of getting the result we do perform subtractions is no more reasonable than to define division as a special phase of crank-turning because we can arrive at the result by turning the crank of a calculating machine. It should be carefully distinguished that division is a mental operation the result of which is arrived at, except in the case of simple and familiar quantities, by a series of partial divisions and subtractions, each part of the quotient being subtracted as soon as ascertained. But these subtractions are not division, nor have they properly anything to do with the division, being merely operations to clear away the débris for a new partial division. They have nothing to do with the division per se, being merely a house-cleaning process to prepare for a new partial division. Addition is merely accumulative combination of elements, and is applicable not only to mathematical concepts but to concrete objects, fingers, pebbles, etc., and to this extent is not a purely mathematical operation; it is concrete as well as abstract, in that it can be applied to concrete objects. concreteness. Subtraction is the decumulative combination of elements, and is likewise applicable to concrete objects, within the limits of When applied to abstract concepts, debt, direction, etc., its continued application results in a new number concept, the negative number. These two operations are operations which do not change the original elements but merely adjoin them unchanged into a cumulative (or decumulative) result. So long as the elements. are unchanged these are the only possible operations. To arrive at an idea of multiplication we must ascertain the permissible operations performable upon number. To do this we must find out the properties of number. Primary number has only one property, its differentness from unity. This property can be used as a directory mandate for the possible operations. If we use the evolutory differentness from unity, the dif ferentness which produces the number from unity, as the mandate we get multiplication, the doing to the operand (multiplicand) what was done to unity to produce the operator (multiplier). If we use the involutory differentness, the differentness which converts the number into unity, we get division, the doing to the operand (dividend) what was done to the operator (divisor) to convert it into unity. These are transformational operations; operations which transform the operand into an entirely new and different number. They are purely algebraic, being limited, unlike addition and subtraction, to operations upon discrete number. A not entirely satisfactory illustration of the difference between the two operations of addition and multiplication is the putting of two canes together for addition; for multiplication we would be compelled, because the first was triangular, black, smooth and uniform in size, to transform the second cane, which was round, white, knotted, tapering and with a knob on the end, into a triangular, black, smooth, prismatic stick. It is transformed into an entirely new and distinct form, and its original features have disappeared. So in numbers, addition and subtraction are merely adjunction without change of the original elements. Multiplication and division are destructive transformations, the destruction of one element and the production of an entirely new and different one, like the growth of a plant from the seed. From primary number these are all the operations permissible until we produce by subtraction the new concept, negative numbers, introducing reversion; and by division (evolution) the additional concept, complex numbers, introducing mean reversion. These definitions of the operations are general, holding for all branches of discrete mathematics, algebra, complex functions, quaternions. Involution is merely that special case of multiplication in which the first operand and all operators are the same, working from unity to the result. Evolution is that special case of division in which all the operators and the last operand are the same, working from the first operand toward unity. Incidentally, as a corollary to this, comes up the similarly faulty definition of exponent, so persistent in the schoolroom, the number which shows how many times the base is taken as a factor. This is absurd, of course, when the exponent is a fraction, and is apologized for by asking for an extension of the idea to cover fractions. Why not start right at the beginning, and define exponent as the number which shows how many equal multiplicative and divisive operations are to be performed upon the base? Whether in the case of fractional exponents we should speak of power or not, about which many make so much ado, is merely a question of definition; whether we define power as the result of equal integral multiplicative operations, or of equal operations, multiplicative or divisive. The question. is largely one of convenience and convention, narrowness or breadth, and not one of right or wrong. In the Piazza San Marco Mosque-like and many-domed in the sky it shone, With rich mosaic, vari-colored stone, The triple pedestals their mastheads raise, I look to see the fair Venetian girls, Like Caliari's proud Saint Catharines, Their blond hair bound with loops of pendant pearls. ISABEL HUNTER. |