This book presents methods from the history of mathematics that have not become standard methods or even made it into classic textbooks. These alternative approaches were often popular at the time of their creation, but then lost attention compared to today's standard methods and eventually fell into oblivion. Often these methods still contain unused potential: It is worth developing them further and discovering where exactly their limits lie. The book aims to provide prospective teachers with a view beyond the usual teaching content and provide content suggestions for working with interested and talented students. For example, Archimedes developed a method for calculating the area of a parabolic segment, which in a way anticipates a piece of integral calculus. However, Archimedes' method did not develop into the standard method found in today's textbooks. Instead, you will find the methods designed by Newton and Leibniz. This book develops the Archimedean method further and shows its "residual potential": Other curves, not just parabolas, can be similarly approached and it is interesting and instructive to see how far Archimedes' method can be developed and where it ultimately reaches its limits.This book is a translation of the original German edition Seitenwege in der Mathematikgeschichte (2024). The translation was done with the help of an artificial intelligence machine translation tool. In the subsequent editing, the author and his colleague Helmer Aslaksen (Uni Oslo) carefully reviewed the translation. Still, the book may read stylistically differently from a conventional translation.