It was the mistake of the teachers of the Middle Ages to believe that the first step in knowledge was to get a correct set of concepts of all things, and then to deduce or bring out all knowledge from them. Admirable plan if you can get your concepts! But unfortunately concepts do not exist ready made—they must be grown; and as your knowledge increases, so do your concepts change. A concept of a thing is not a mere definition, it is a complete history of it. And you must build up your edifice of scientific knowledge from the earth, brick by brick and stone by stone. There is no magic process by which it can with a word be conjured into existence like a palace in the Arabian Nights. For nothing is more fatal than a juggle with words such as force, weight, attraction, mass, time, space, capacity, or gravity. Words are like purses, they contain only as much money as you put into them. You may jingle your bag of pennies till they sound like sovereigns, but when you come to pay your bills the difference is soon discovered. This fatal practice of learning words without trying to obtain a clear comprehension of their meaning, causes many teachers to use mathematical formulæ not as mere steps in a logical chain, but like magical chaldrons into which they put the premises as the witches put herbs and babies’ thumbs into their pots, and expect the answers to rise like apparitions by some occult process that they cannot explain. This tendency is encouraged by foolish parents who like to see their infant prodigies appear to understand things too hard for themselves, and look on at their children’s lessons in mathematics like rustics gaping at a fair. They forget that for the practical purposes of life one thing well understood is worth a whole book-full of muddled ill-digested formulæ. Unfortunately it is possible to cram boys up and run them through the examination sieves with the appearance of knowledge without its reality. If it were cricket or golf that were being tested how soon would the fraud be discovered. No humbug would be permitted in those interesting and absorbing subjects. And really, when one reflects how easy it is to present the appearance of book knowledge without the reality, one can hardly blame those who select men for service in India and Egypt a good deal for their proficiency in sports and games. Better a good cricketer than a silly pedant stuffed full of learning that “lies like marl upon a barren soil encumbering what is not in its power to fertilize.” Another kindred error is to expect too much of science. For with all our efforts to obtain a further knowledge of the mysteries of nature, we are only like travellers in a forest. The deeper we penetrate it, the darker becomes the shade. For science is “but an exchange of ignorance for that which is another kind of ignorance”[A] and all our analysis of incomprehensible things leads us only to things more incomprehensible still. [A] _Manfred_, Act II., scene iv.