itself more directly to the intellect, and through the intellect to the soul.
The reform in the teaching of mathematics, now in agitation, depends essentially on getting teachers to understand that the chalk in the lecturer's hand becomes, at a given moment in the lesson, a Revealer, independent of (and, for the moment, superior to) the man who holds it; a Teacher of Teachers, King of Kings, and Lord of Lords. Yet how easily this essential doctrine of mathematics slips over into the slavish dogmas which ignorant people connect with the so-called doctrine of Transubstantiation! And how wisely did the English Church decree that the "transubstantiated" bread shall be eaten at once, not preserved as sacred! We do not wish the children to attach superstitious ideas to the chalk (or bread) when the demonstration is over; but if there is to be any vital reform in method we must make young teachers realize that, for a few moments in each lesson, he and the chalk change places; that for those moments the chalk, not he, is the true intermediary (or mediator) between the Unseen Revealer and the class. We cannot continue to boycott in England all vital mathematical teaching, just because stupid people have talked grovelling nonsense about the doctrine which is its vital essence.
The manner in which a problem that baffles us when treated on its own level can often be solved by bringing to bear on the solution truths of a higher order than that contemplated when the question was first propounded, is well illustrated by the famous 47th Proposition of Euclid. The question proposed for solution is this:— Is there any constant relation between the length of
