The application of algebra to geometry ... has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences.
Interpretation
John Stuart Mill praises the impact of algebra on geometry and its significance in the development of exact sciences.
In this quote, John Stuart Mill highlights how the integration of algebraic principles into geometry fundamentally transformed the study of mathematics and paved the way for advancements in the sciences. He attributes immense importance to this intellectual leap made by RenΓ© Descartes, suggesting that it has fundamentally altered our understanding of the exact sciences, representing a monumental step in scientific progress.
In practice
In a lecture about the history of mathematics, one might use this quote to illustrate the importance of Descartes' contributions.
The peculiar evil of silencing the expression of an opinion is, that it is robbing the human race; posterity as well as the existing generation; those who dissent from the opinion, still more than those who hold it. If the opinion is right, they are deprived of the opportunity of exchanging error for truth: if wrong, they lose, what is almost as great a benefit, the clearer perception and livelier impression of truth, produced by its collision with error.
As for charity, it is a matter in which the immediate effect on the persons directly concerned, and the ultimate consequence to the general good, are apt to be at complete war with one another.
To think that because those who wield power in society wield in the end that of government, therefore it is of no use to attempt to influence the constitution of the government by acting on opinion, is to forget that opinion is itself one of the greatest active social forces. One person with a belief is a social power equal to ninety-nine who have only interests.
There should be perfect freedom, legal and social, to do the action and stand the consequences. It would be a great misunderstanding of this doctrine to suppose that it is one of selfish indifference, which pretends that human beings have no business with each other's conduct in life, and that they should not concern themselves about the well-doing or well-being of one another, unless their own interest is involved.
Political Economy, in truth, has never pretended to give advice to mankind with no lights but its own; though people who knew nothing but political economy (and therefore knew it ill) have taken upon themselves to advise, and could only do so by such lights as they had.
Marriage is the only actual bondage known to our law. There remain no legal slaves, except the mistress of every house.
When an economist says the evidence is "mixed," he or she means that theory says one thing and data says the opposite.
I have been struck again and again by how important measurement is to improving the human condition.
Nothing is so dangerous to the progress of the human mind than to assume that our views of science are ultimate, that there are no mysteries in nature, that our triumphs are complete and that there are no new worlds to conquer.
Science has to be understood in its broadest sense, as a method for apprehending all observable reality, and not merely as an instrument for acquiring specialized knowledge.
If there is no solace in the fruits of our research, there is at least some consolation in the research itself. Men and women are not content to comfort themselves with tales of gods and giants, or to confine their thoughts to the daily affairs of life; they also build telescopes and satellites and accelerators and sit at their desks for endless hours working out the meaning of the data they gather.
This preservation of favourable variations and the destruction of injurious variations, I call Natural Selection, or the Survival of the Fittest. Variations neither useful nor injurious would not be affected by natural selection and would be left a fluctuating element.
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