The book is addressed to high school students interested in mathematics. It popularly and interestingly outlines some issues related to the foundations of geometry, elements of Lobachevsky geometry. The entire presentation relies entirely on school math programs. This knowledge is enough to navigate the text well and read the book without difficulty. Historical excursions enliven the text, help to better understand the development of mathematics. Book for extracurricular reading of students in grades 9-10 of high school.