used extensive notations; Lorenz brought out a very compact edition of all books of Euclid’s Elements.
Our data for the eighteenth and nineteenth centuries have been drawn mainly from the field of elementary mathematics. A glance at the higher mathematics indicates that the great mathematicians of the eighteenth century, Euler, Lagrange, Laplace, used symbolism freely, but expressed much of their reasoning in ordinary language. In the nineteenth century, one finds in the field of logic all gradations from no symbolism to nothing but symbolism. The well-known opposition of Steiner to Plücker touches the question of sign language.
The experience of the past certainly points to conservatism in the use of symbols in elementary instruction. In our second volume we indicate more fully that the same conclusion applies to higher fields. Individual workers who in elementary fields proposed to express practically everything in ideographic form have been overruled. It is a question to be settled not by any one individual, but by large groups or by representatives of large groups. The problem requires a consensus of opinion, the wisdom of many minds. That widsom discloses itself in the history of the science. The judgment of the past calls for moderation.
The conclusion reached here may be stated in terms of two school-boy definitions for salt. One definition is, “Salt is what, if you spill a cupful into the soup, spoils the soup.” The other definition is, “Salt is what spoils your soup when you don’t have any in it.”
