Fractions for $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, quasi pythagorean triples and the appearance of the royal cubit, foot and inch in Egypt during the Ancient Kingdom

Based on the analysis of the sizes of Egyptian architectural structures and art objects of the first half of the III millennium BC, a reconstruction of the history of finding fractions for $\sqrt2, \sqrt3, \sqrt5$ is proposed, starting with $7/5, 17/10$ and $11/5$, respectively. Analysis of architectural proportions of the times of pharaohs Djoser and Snefru (XXVII century. BC) speaks of the most likely use, up to the construction of the last Polyline Pyramid, of dimensional modules other than the royal cubit of $7$ palms.