This paper treats important questions at the interface of mathematics and the engineering sciences. It starts off with a quick quotation tour through 2300 years of mathematical history. At the beginning of the 21st century, technology has developed beyond every expectation. But do we also learn and practice an adequately modern form of mathematics? The paper argues that this role is very likely to be played by universal geometric calculus. The fundamental geometric product of vectors is introduced. This gives a quick-and-easy description of rotations as well as the ultimate geometric interpretation of the famous quaternions of Sir W.R. Hamilton. Then follows a one page review of the historical roots of geometric calculus. In order to exemplify the role of geometric calculus for the engineering sciences three representative examples are looked at in some detail: elasticity, image geometry and pose estimation. Next a current snapshot survey of geometric calculus software is provided. Finally the value of geometric calculus for teaching, research and development is commented.