My answer to the question as to these fundamental mathematical truths is, that in any absolute sense there are none. There is no unique small body of independent primitive unproved propositions which are the necessary starting points of all mathematical reasoning on these subjects. In mathematical reasoning the only absolute necessary pre-suppositions are those which make logical deduction possible. Between these absolute logical truths and so-called fundamental truths concerning geometry, quantity and number, there is a whole new world of mathematical subjects concerning the logic of propositions, of classes, and of relations. But this subject is too abstract to form an elementary training ground in the difficult art of abstract thought. It is for this reason that we have to make a compromise and start from such obvious general ideas as naturally occur to all men when they perceive objects with their senses. In geometry, the ideas elaborated by the Greeks and presented by Euclid are, roughly speaking, those adapted for our purpose, namely, ideas of volumes, surfaces, lines, of straightness and of curvature, of intersection and of congruence, of greater and less, of similarity, shape, and scale. In fact, we use in education those general ideas of spatial properties which must be habitually present in the mind of any one who is to observe the world of phenomena with understanding. Thus we come back to Plato's opinion that for a liberal education, geometry, as he knew it, is the queen of sciences. In addition to geometry, there remain the ideas of quantity, ratio, and of number. This in practice means, elementary algebra. Here the prominent ideas are those of "any number," in other words, the use of the familiar _x_, _y_, _z_, and of the dependence of variables on each other, or otherwise, the idea of functionality. All this is to be gradually acquired by the continual use of the very simplest functions which we can devise: of linear functions, graphically represented by straight lines; of quadratic functions, graphically represented by parabolas; and of those simple implicit functions, graphically represented by conic sections. Thence, with good fortune and a willing class, we can advance to the ideas of rates of increase, still confining ourselves to the simplest possible cases. I wish here emphatically to remind you that both in geometry and in algebra a clear grasp of these general ideas is not what the pupil starts from, it is the goal at which he is to arrive. The method of progression is continual practice in the consideration of the simplest particular cases, and the goal is not philosophical analysis but the power of use. But how is he to practise himself in their use? He cannot simply sit down and think of the relation _y_ = _x_ + 1, he must employ it in some simple obvious way.