Welsh Journals

Mathematics. 405 MATHEMATICS. Mary's Euclid.—Chap. VII. In order to understand the first seven axioms thoroughly, it is, above all, necessary to observe how many " things" are taken into account in each statement ; you should first try to make this out for yourself: for, if you get wrong and are corrected, it will make a much stronger impression on your mind than if you were told at once. Now, supposing you have read the seven attentively, and done as I told you, (which, until I found you were an ill bairn, I should always take for granted,) and supposing you have given seven answers, right or wrong, I may as well say that, in the first axiom, as you are comparing " things " with one other thing, you can not possibly have less than three ; though, as all things that are equal to the same are equal to each other, you might have millions, or any greater number, to compare with that one. The same remark applies to the sixth and seventh. But, in the second, third, fourth, and fifth, there can not possibly be less than six; because, when equals are added to equals, there can not be less than two of each, which makes four: then, the two new things produced by the addition or subtraction (the "wholes" or "remainders") make up six. But, here again, millions upon millions of equals may be taken to start with as well as two : and then we shall have three times that number of" things" brought be¬ fore us in the course of the process: for, observe, that in these four axioms there is a process : and this gives them