The Introduction of the book says:
The pupils often fail to understand why truths should be proved that seem quite evident without proof, the proofs often appearing to be excessively complicated and cumbersome. It sometimes happens, too, that a seemingly clear and convincing proof turns out, upon closer scrutiny, to be incorrect.
This booklet was written with the aim of helping pupils clear up the following points:
1. What is proof?
2. What purpose does a proof serve?
3. What form should a proof take?
4. What may be accepted without proof in geometry?
The book was translated from the Russian by Mark Samokhvalov and was first publised by Mir in 1978. This books was also publishedin the Topics in Mathematics series by Heath in 1963 and this edition was translated by Theodore M. Switz and Luise Lange. Dover [in 2006] has republished this book along with Mistakes in Geometric Proofs by Y. S. Dubnov as one single volume. You can get the LML edition of the book here.
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The table of contents is as follows:
l. What Is Proof? 3
2. Why Is Proof a Necessity? 12
3. What Should Be Meant by a Proof? 19
4. What Propositions May Be Accepted Without Proof? 44
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