Analytical Geometry Quotes

The female mind is capable of understanding analytic geometry... The difficulty may just be that we have never yet discovered a way to communicate with the female mind. If it is done in the right way, you may be able to get something out of it.

Wow. I'm twenty years old. Rene Descartes invented analytic geometry in his early twenties. Talk about pressure.

The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.

In an ever-more complex world, Mandelbrot argues, scientists need both tools: image as well as number, the geometric view as well as the analytic. The two should work together. Visual geometry is like an experienced doctor's savvy in reading a patient's complexion, charts, and X-rays. Precise analysis is like the medical test results-the raw numbers of blood pressure and chemistry. "A good doctor looks at both, the pictures and the numbers. Science needs to work that way too," he says.

In asking for a relic of Descartes, the chevalier de Terlon was standing at the crossroads of the ancient and modern. He was applying to a modern thinker - the inventor of analytic geometry, no less - a primitive tradition that extends back not only to the institutionalization of Christianity in the fourth century, when Christians first broke into the tombs of saints to gather relics, but farther still, beyond the horizon of recorded history. The request is all the stranger for the fact that the man whose remains were treated in this quasisaintlike way would go down in history as the progenitor of materialism, rationalism, and a whole tradition that looked on such veneration as nonsense.

therefore, did learn a lesson: The female mind is capable of understanding analytic geometry. Those people who have for years been insisting (in the face of all obvious evidence to the contrary) that the male and female are equal and capable of rational thought may have something. The difficulty may just be that we have never yet discovered a way to communicate with the female mind. If it is done in the right way, you may be able to get something out of it.

A maze is a puzzle to be solved, with twists and turns and dead ends. It requires logical, analytical thinking and usually has a different way out than the way in. The maze could be a metaphor of struggling through life, going one way and then another until the exit takes your by surprise. A maze signifies entrapment, while the labyrinth, with its unicursal path leading into the center and out again the same way, provides enlightenment. It's the process, the journey into your deepest self, your soul, the part where God abides. It's a passive path, a surrender even, to an order and design repeated through creation. A sacred geometry.

Descartes was a philosopher, a mathematician, and a man of science. In philosophy and mathematics, his work was of supreme importance; in science, though creditable, it was not so good as that of some of his contemporaries. His great contribution to geometry was the invention of co-ordinate geometry, though not quite in its final form. He used the analytic method, which supposes a problem solved, and examines the consequences of the supposition; and he applied algebra to geometry. In both of these he had had predecessors—as regards the former, even among the ancients. What was original in him was the use of co-ordinates, i.e. the determination of the position of a point in a plane by its distance from two fixed lines. He did not himself discover all the power of this method, but he did enough to make further progress easy.

Kant distinguished between two types of truths: (1) analytic propositions, which derive from logic and “reason itself” rather than from observing the world; for example, all bachelors are unmarried, two plus two equals four, and the angles of a triangle always add up to 180 degrees; and (2) synthetic propositions, which are based on experience and observations; for example, Munich is bigger than Bern, all swans are white. Synthetic propositions could be revised by new empirical evidence, but not analytic ones. We may discover a black swan but not a married bachelor or (at least so Kant thought) a triangle with 181 degrees. As Einstein said of Kant’s first category of truths: “This is held to be the case, for example, in the propositions of geometry and in the principle of causality. These and certain other types of knowledge… do not previously have to be gained from sense data, in other words they are a priori knowledge.” Einstein

Two possibilities present themselves for the analytical treatment of metrical geometry.

And why, after all, may not the world be so complex as to consist of many interpenetrating spheres of reality, which we can thus approach in alternation by using different conceptions and assuming different attitudes, just as mathematicians handle the same numerical and spatial facts by geometry, by analytical geometry, by algebra, by the calculus, or by quaternions, and each time come out right? On this view religion and science, each verified in its own way from hour to hour and from life to life, would be co-eternal.

Americans have all these classes that mean they just know odd things, so engineers know about William Blake and poets know about analytical geometry. She probably took one on Aristotle and the politics of gender.

Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates. Rightly understood, there exists only a difference of gradation between these two kinds of geometry, according as one gives more prominence to the figures or to the formulas. Analytic geometry which dispenses entirely with geometric representation can hardly be called geometry; synthetic geometry does not get very far unless it makes use of a suitable language of formulas to give precise expression to its results.

Descartes was responsible for analytical geometry, a mechanism for translating from geometrical forms to the equivalent algebraic equations and vice versa.

In Kant we find an old form of intuitionism, now almost completely abandoned, in which time and space are taken to be forms of conception inherent in human reason. For Kant the axioms of arithmetic and geometry were synthetic a priori judgments, i.e., judgments independent of experience and not capable of analytical demonstration; and this explained their apodictic [necessarily true] exactness in the world of experience as well as in abstracto. For Kant, therefore, the possibility of disproving arithmetical and geometrical laws experimentally was not only excluded by a firm belief, but it was entirely unthinkable.

Some technological invention is made, like that of a steam engine or a printing press, for example; or some discovery of scientific method, like that of analytical geometry or the infinitesimal calculus; or [pg 020] some discovery of natural law, like that of falling bodies or the Newtonian law of gravitation. What happens? What is the effect upon the progress of knowledge and invention? The effect is stimulation. Each invention leads to new inventions and each discovery to new discoveries; invention breeds invention, science begets science, the children of knowledge produce their kind in larger and larger families; the process goes on from decade to decade, from generation to generation, and the spectacle we behold is that of advancement in scientific knowledge and technological power according to the law and rate of a rapidly increasing geometric progression or logarithmic function.

Analytic truths also take a blow from the fact that dictionaries need to be updated from time to time. The meaning and usage of words is not static and unchanging; there is nothing 'necessary' about it. Hence, trying to base analytic truths on definitions, which are supposedly immune to revision and absolutely certain, would seem to be risky business. This is true even of scientific terms. For example, at one point the statement 'Atoms are indivisible' would have been widely accepted as analytic, but not so today. At one point in history, Euclidean geometry and Newtonian physics appeared to provide analytic truths. However, the rise of Riemannian geometry and Einstein's relativity theory made analytic truths in those areas debatable.

The main interest of Fermat, who shares the credit for inventing calculus with Newton and analytic geometry with Descartes, was number theory —“the higher arithmetic.

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ANALYTIC GEOMETRY (1637)

Among other elements, the test had a vestigial examination in drawing, and Mandelbrot discovered a latent facility for copying the Venus de Milo. On the mathematical sections of the test—exercises in formal algebra and integrated analysis—he managed to hide his lack of training with the help of his geometrical intuition. He had realized that, given an analytic problem, he could almost always think of it in terms of some shape in his mind. Given a shape, he could find ways of transforming it, altering its symmetries, making it more harmonious. Often his transformations led directly to a solution of the analogous problem. In physics and chemistry, where he could not apply geometry, he got poor grades. But in mathematics, questions he could never have answered using proper techniques melted away in the face of his manipulations of shapes.