Welsh Journals

496 Old Prices Remains. something of the nature of problems, whereas the first, sixth, and seventh, are mere theorems, where you take things as you find them, and do nothing to them, but just observe and compare them. I should, therefore, class them differently, in two sets, only that it is a great evil to dis¬ turb the order, where it is so desirable that, when two people are speaking of the first, seventh, or tenth axiom, they should both mean the same one. And now I advise you to try the axioms in every possible way (at least) with pieces of string and cups of water to be measured, groups of beans to be counted, and small quantities of "tea, coffee, tobacco, or snuff," to be weighed in scales—faute de mieux—try sand. The eighth axiom requires particular attention and thought. 1st—Two points, A and B, may coincide in position ; as when the ends of two lines A B, C D are brought together to form an angle. 2ndly—Two lines, however unequal in length, may coincide in direction, as far as the shorter one reaches when laid on a?ty part of the longer. Notice the minute hand and hour hand of a clock, twenty-four times a day. 3rdly and lastly—Both lines and surfaces may coincide in their whole extent. This is the only kind of coincidence noticed by Euclid in the eighth axiom ; and it proves not only equality, but Geometric identity. Take notice of this, very particu¬ larly. Two equal circles, two equal squares, or any two equal and similar figures, are only two different pictures of the same thing: just as all the impressions of a certain portrait or map, are only different pictures of the same person or country. The first and second kind of coinci¬ dence, viz., in position and direction, though not named by Euclid, are absolutely necessary to a clear view of the proof of his fourth proposition, " as I shall tell you by- and-by;" and so is this identity, as distinguished from