This book is intended to lead students to develop their mathematical ability, to learn the art of mathematics, and to create mathematical ideas. This is not a compendium of mathematical facts and inventions to be read over as a connoisseur of art looks over the paintings in a gallery. It is, instead, a sketchbook in which readers may try their hands at mathematical discovery.

The American painter Winslow Homer is said to have declared that painters should not look at the works of others for fear of damaging their own directness of expression. I believe the same is true of the mathematician. The fresher the approach the better—there is less to unlearn and there are fewer bad thinking habits to overcome. In my teaching experience, some of my best students have been among those who entered my classes with the least previous mathematical course work. On the other hand, I have usually found it very difficult, if not impossible, to get any kind of creative effort from a student who has had many poor courses in mathematics. This has been true in some cases even though, as it developed later on, the student had very unusual mathematical ability.

The development of mathematical ability does not occur quickly. There are no short cuts. This book is written for the person who seeks an intellectual challenge and who can find genuine pleasure in spending hours and even weeks in constructing proofs for the theorems of one chapter or even a portion of one chapter.