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.
This afternoon, I've arranged for this ceremony to be illuminated by solar power.
When we seed millions of acres of land with these plants, what happens to foraging birds, to insects, to microbes, to the other animals, when they come in contact and digest plants that are producing materials ranging from plastics to vaccines to pharmaceutical products?
From a certain temperature on, the molecules 'condense' without attractive forces; that is, they accumulate at zero velocity. The theory is pretty, but is there some truth in it.
You know what they call alternative medicine that's been proved to work? - Medicine.
Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.
Science does not promise absolute truth, nor does it consider that such a thing necessarily exists. Science does not even promise that everything in the Universe is amenable to the scientific process.
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