the following statement: "The matrix has also significance apart from its development into a determinant." In view of the fact that a matrix, as commonly used by mathematicians, has no possible development, it is clear that the given sentence does not convey any information. In fact, the other remarks under this term are almost equally objectionable, and they raise the question whether a philosopher should be selected to define the mathematical terms of a standard work.
It is not implied that such an extensive work as a large dictionary could be expected to be free from defects, but there is always a limit to the number and the type of those which appear excusable. When one reads in such a dictionary that "algebra is that branch of mathematics which treats the relations and properties of quantity by means of letters and symbols," and then turns to page 22 of volume 1 of the large French mathematical encyclopedia and reads that "it is convenient, in arithmetic, to represent any number by a letter, it being understood that this letter denotes a single and the same number whenever one remains in the same subject," it becomes evident that the given definition of algebra is not supported by some of the highest authorities.
In fact, such terms as arithmetic, algebra and geometry are used with such a wide range of meanings by eminent authorities that it seems impossible to give satisfactory definitions of them, and our dictionaries would convey more reliable information about mathematical terms by stating this fact, together with some indication of what broad subjects are generally classed under these terms, than by giving categorical definitions which can be accepted only by those who have a meager knowledge of mathematics.
Without implying that Webster's New International Dictionary is any less reliable with respect to mathematical matters than most others, we shall refer to one more instance of misleading statements in this work. On page 2547 we read as follows: "The cipher was originally a dot, used as a mere arbitrary sign to mark place or local value." Such a definite statement seems strange in view of the fact that the origin of zero is one of the unsettled questions of the history of mathematics. It is of interest to note in this connection that Cantor changed his view with respect to the origin of this concept and this symbol, in the third edition of Volume I. of his classic "Vorlesungen ueber Geschichte der Mathematik," where he states that the symbol for zero and the positional arithmetic are probably due to the Babylonians instead of to the Hindus, as he had stated in the earlier editions of this work, and as is stated in a large number of other works.
Our encyclopedias also frequently exhibit careless editing along the line of mathematical terms, and the choice of editors for such work often seems to indicate that the general editor regarded the choice of the mathematical editors as a matter of little consequence. Possibly
