At present, a great amount of scientific papers, monographs, and reference books dealing with analytical and differential geometry of surfaces have been published. They contain materials for following geometric investigations, for implementation of received earlier geometrical results into architecture, building, and machinery manufacturing. In the paper it has been shown on the specific examples that sometimes the results of geometric investigations for shells’ middle surfaces taken in published references for the following application without check could lead to serious errors because of ones in the surfaces equations or inexactitudes in a surfaces class definition. At present, 38 classes of surfaces, uniting more than 600 ones that have their own names and are described in scientific publications, are known. The author has worked up a great number of researches and found errors, inaccuracies, and alternative versions in monographs and scientific papers, related to questions on geometry of developable surfaces (conic and torse surfaces), surfaces of rev olution (paraboloid and ellipsoid of revolution, nodoid), minimal surfaces (catenoids), conoids, and cyclic surfaces including the canal ones. In actual practice there are much more geometric errors, but in this paper are discussed only well-known geometricians and architects’ works, as well as in this paper there is no information on surfaces that are presented at specialized sites in Internet. Here are encountered misreckoned coefficients for surfaces’ fundamental quadratic forms, there are errors in the formulae for the quadratic forms’ coefficients determination, as well as in the formulae for the calculation a surface element’s area, surface’s principle curvatures, and so on. All of encountered errors have been divided into four groups. The fourth group’s errors named as “typographical errors and authors’ slips of the pen” have been considered fragmentarily because they are encountered the most frequently, and can be corrected by the authors themselves in the following papers.
On the differential geometry of ruled surfaces in $4$-space and cyclic surfaces in $3$-space
