Our primary aim is to illustrate the power and elegance of mathematical reasoning in science; our secondary aim, to show also that science egenders mathematics. We start with Archimedes work on the lever, which, based on our common experience of carrying ladders, confirms our bone borne intuition. We finish with the work of Einstein, which , as we have no experience of "unaccustomed dimensions" such as intergalatic travel at nearly the speed of light, disconcertingly discomforts us with situations beyond our intuitive grasp. Here mathematics comes into its own; to give brain borne counsel where bone borne guidance fails; to give such comforting assurance that scientists justifiably call it a sixth sense. Our ultimate aim is to help develop this sixth sense in as wide a readership as possible; we have restricted technical mathematics to minimum, just algebra and some calculus. therefore let none be deterred!