The purpose of this paper is to consider some examples of what happens in major turning-points in scientific thinking. It is also aimed to show how the critical study of the history of science can throw light both on how science works as an intellectual discipline and on how scientists think and what they do. In critical history the analysis of history involves at the same time an analysis of science. The critical history of the scientific revolution of the seventeenth century raises the larger question of why and how the West became a scientific civilization at all. This is suggested as an ideal topic for the treatment of science as a general subject in the fifth form. It leads to several different kinds of question concerning the internal content and structure of science, the influence of general ideas, and the influence of social and economic conditions, which can be discussed from different backgrounds in the arts and sciences. But the central problem of the scientific revolution is what happened in science. There seems to have been a fundamental mutation in ideas and in the questions asked in science. These led to new techniques for answering the questions, especially by means of experiment and mathematics. These changes are illustrated by means of Galileo's contribution to the invention of the concept of inertia, Descartes' model of the animal automaton, and Darwin's theory of evolution by natural selection. In every case the turning-point appears as a result of active speculation and the manipulation and dissection of nature and critical re-assessment of basic theoretical assumptions. Of special importance is the explicit use of designed analysis by the manipulation of mathematical concepts, designed experiments, and hypothetical models.