A second difficulty of the pupil is seen in his attitude of mind towards proofs in general. He does not see why vertical angles should be proved equal when he knows that they are so by looking at the figure. This difficulty should also be anticipated by giving him some opportunity to know the weakness of his judgment, and for this purpose figures like the following should be placed before him. He should be asked which of these lines is longer, _AB_ or _XY_. Two equal lines should then be arranged in the form of a letter T, as here shown, and he should be asked which is the longer, _AB_ or _CD_. A figure that is very deceptive, particularly if drawn larger and with heavy cross lines, is this one in which _AB_ and _CD_ are really parallel, but do not seem to be so. Other interesting deceptions have to do with producing lines, as in these figures, where it is quite difficult in advance to tell whether _AB_ and _CD_ are in the same line, and similarly for _WX_ and _YZ_. Equally deceptive is this figure, in which it is difficult to tell which line _AB_ will lie along when produced. In the next figure _AB_ appears to be curved when in reality it is straight, and _CD_ appears straight when in reality it is curved. The first of the following circles seems to be slightly flattened at the points _P_, _Q_, _R_, _S_, and in the second one the distance _BD_ seems greater than the distance _AC_. There are many equally deceptive figures, and a few of them will convince the beginner that the proofs are necessary features of geometry. It is interesting, in connection with the tendency to feel that a statement is apparent without proof, to recall an anecdote related by the French mathematician, Biot, concerning the great scientist, Laplace: Once Laplace, having been asked about a certain point in his "Celestial Mechanics," spent nearly an hour in trying to recall the chain of reasoning which he had carelessly concealed by the words "It is easy to see." A third difficulty lies in the necessity for putting a considerable number of definitions at the beginning of geometry, in order to get a working vocabulary. Although practically all writers scatter the definitions as much as possible, there must necessarily be some vocabulary at the beginning. In order to minimize the difficulty of remembering so many new terms, it is helpful to mingle with them a considerable number of exercises in which these terms are employed, so that they may become fixed in mind through actual use. Thus it is of value to have a class find the complements of 27 deg., 32 deg. 20', 41 deg. 32' 48", 26.75 deg., 33 1/3 deg., and 0 deg.. It is true that into the pure geometry of Euclid the measuring of angles in degrees does not enter, but it has place in the practical applications, and it serves at this juncture to fix the meaning of a new term like "complement."