The first and typical example of the application of mathematics to the indirect investigation of truth, is within the limits of the pure science itself; the application of algebra to geometry, the introduction of which, far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences. Its rationale is simple. It is grounded on the general truth, that the position of every point, the direction of every line, and consequently the shape and magnitude of every enclosed space, may be fixed by the length of perpendiculars thrown down upon two straight lines, or (when the third dimension of space is taken into account) upon three plane surfaces, meeting one another at right angles in the same point. A consequence or rather a part of this general truth is that, curve lines and surfaces may be determined by their equations.

What It Means

This is a placeholder explanation generated by the low-overhead model. It interprets "The first and typical example of the application of mathematics to the indirect investigation of truth, is within the limits of the pure science itself; the application of algebra to geometry, the introduction of which, far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences. Its rationale is simple. It is grounded on the general truth, that the position of every point, the direction of every line, and consequently the shape and magnitude of every enclosed space, may be fixed by the length of perpendiculars thrown down upon two straight lines, or (when the third dimension of space is taken into account) upon three plane surfaces, meeting one another at right angles in the same point. A consequence or rather a part of this general truth is that, curve lines and surfaces may be determined by their equations." in the context of "John Stuart Mill, An Examination of Sir William Hamilton's Philosophy (1865) as quoted in 5th ed. (1878) pp. 617-618." to mean that wisdom is timeless.