The embarrassments that involve the foundations of elementary geometry are mainly, if not wholly, those which involve the general problems of definition. Now a definition is the certification of the purport of a name by means of a statement or a conspiracy of statements necessary and sufficient to that end. But names are constituents absolutely necessary for the formation of any statement, so that the above definition of a definition may be restated thus: A definition is the certification of the purport of one name by means of other names, necessary and sufficient to certify the purport of the one defined. Evidently then, definition can only lead us from name to name in unending process, or to some undefinable name, or to some name that we choose to leave undefined; and the question arises, on what sort of names shall we take our stand as ultimate grounds? Our author answers this question as follows: “The propounder of a scientific theory is not of course expected to teach his readers to speak, it is only necessary for him to define the terms peculiar to his science, or those to which he wishes to attach peculiar meanings. He may therefore assume that the meanings of all other words are known to his readers.” He then propounds that “all that is logically required for a definition is one or more assertions with regard to the word to be defined or, its attributes,” provided “they are not demonstrably incompatible with each other.” Although our author conceives that logical competence requires no more than this for a good definition, he yet goes on to remark, that “if the definition is to form the basis of a deductive science it is further advisable that the assertions should be independent,” and that “where it is required to define a term whose denotation is already known, it is further necessary not only that the assertions should be commonly accepted as true with respect to it, but that they should restrict the meaning of the term exactly to its accepted denotation, neither more nor less, and should do so in the simplest manner that can be devised.” It is upon this theory of definition that our author requests of his critics a “categorical” answer to his first question, “Do you accept the requirements I have laid down for a logical definition? (If not please state which of them you object to, why you object to it, and what you would propose to substitute for it.)” Since it is a “categorical” answer that is requested and since also it is the matter of definition that is put in issue, we wish that our author had been more definite and had made his propositions better issuable, for we must protest that we regard ourselves obliged to answer to what we can best conceive to be the author’s true meanings rather than to what he has explicitly said.