How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)
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A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out--from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft--indeed, brilliant--instructions on stripping away irrelevancies and going straight to the heart of the problem.
understood the problem as a whole, its aim, its main point, we wish to go into detail. Where should we start? In almost all cases, it is reasonable to begin with the consideration of the principal parts of the problem which are the unknown, the data, and the condition. In almost all cases it is advisable to start the detailed examination of the problem with the questions: What is the unknown? What are the data? What is the condition? If we wish to examine further details, what should we do?
Lamb’s book Mechanics is incorrect for him, since he always worked in an armchair with his feet up! Then, after asking how his reader would present the picture of a closed curve lying all on one side of its tangent, he states that there are four main schools (to left or right of vertical tangent, or above or below horizontal one) and that by lecturing without a figure, presuming that the curve was to the right of its vertical tangent, he had unwittingly made nonsense for the other three schools.
general pattern suggested by our example can be exhibited thus: If A is true, then B is also true, as we know. Now, it turns out that B is true. Therefore, A becomes more credible. Still shorter: If A then B B true A more credible In this schematic statement the horizontal line stands for the word “therefore” and expresses the implication, the essential link between the premises and the conclusion.] [7. Nature of plausible reasoning. In this little book we are discussing a philosophical
the smaller container! That’s the idea. See Fig. 26. (The step that we have just completed is not easy at all. Few persons are able to take it without much foregoing hesitation. In fact, recognizing the significance of this step, we foresee an outline of the following solution.) FIG. 26 But how can we reach the situation that we have just found and illustrated by Fig. 26? (Let us inquire again what could be the antecedent of that antecedent.) Since the amount of water in the river is, for our
various parts of the condition. Can you write them down? Let a and b stand for the lengths of the (unknown) lines of vision, α and β for their inclinations to the horizontal plane, respectively. We may distinguish three parts in the condition, concerning (1) the inclination of a (2) the inclination of b (3) the triangle with sides a, b, and c. 17. Do you recognize the denominators 2, 6, 24? Do you know a related problem? An analogous problem? (INDUCTION AND MATHEMATICAL INDUCTION.) 18.