term would therefore be unlimited. In the case of explicit definitions a similar error is called _circulus in definiendo_. When your reviewer goes on to attack my definition of “direction” why does he change his front all at once, and disregard all the considerations he has just been discussing? Why does he not apply my, or his own, requirements for a definition to the case in point? The criticism he actually does put forward will not bear a moment’s investigation. If my definition is “circular,” the assertion must be equally true whatever meaning is ascribed to the term. Well, then, let us try the effect of giving to it the meaning we ordinarily ascribe to “cheese.” Is it equally true that “a cheese may be conceived to be indicated by naming two points, as the cheese from one to the other”? Clearly not. But not only does this one assertion out of my definition exclude the import of “cheese” from the meaning of “direction,” but, more particularly, it distinguishes between the “three distinct but closely associated notions” which your reviewer quite rightly says “become confused in thought and expression unless the most solicitous care is taken to distinguish them.” This is exactly the care which I _have_ taken, by framing my definition. I need not say much about the rest of the criticism. Your reviewer’s remarks on my definition of “angle” are simply due to the fact that he has not read the definition carefully, and probably has not read the note on the top of page 36 at all. It may make it clearer to him if I point out that if “we imagine a northeast-southwest line cutting an east-west line,” we imagine _four_ different directions and therefore (4⋅3)/(1⋅2) = 6 angles. Two of these are the straight angles between the opposite directions of each of the two lines. The other four are what Euclid calls “the angles between the lines.” As an angle, according to my definition, has no local habitation in space, it is, _prima facie_, meaningless to talk of the “right hand upper angle.” But if this is only an abbreviation for “the angle between the directions upwards and to the right,” then “the right hand upper angle” means the same as it would in Euclid. With the remarks about the nature of the challenge I have thrown down I heartily agree. May I however suggest that I have a right to expect that criticism should be, not only “competent and candid,” but careful? It is a difficult subject, and _I_ at least am not always able to express myself in such a way that my meaning cannot be misunderstood by any one. I think if your reviewer looks at what I have said again, with the aid of what I say further in my _Essay on Reasoning_, he will see that his criticisms have really originated in misunderstandings, and perhaps he will alter his judgment that I have “come short of the high result to which I aspired.”[18]