“Thus, all unknown quantities can be expressed in terms of a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth.
But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by anyone at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise.”

—  René Descartes , book La Géométrie

First Book
La Géométrie (1637)

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "Thus, all unknown quantities can be expressed in terms of a single quantity, whenever the problem can be constructed by…" by René Descartes?
René Descartes photo
René Descartes 47
French philosopher, mathematician, and scientist 1596–1650

Related quotes

René Descartes photo

“I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and them classify them in order, is by recognizing the fact that all points of those curves which we may call "geometric," that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by a single equation. If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely.”

René Descartes book La Géométrie

Second Book
La Géométrie (1637)

Lance Armstrong photo

“By fight I mean arm yourself with all the available information, get second opinions, third opinions, and fourth opinions.”

Lance Armstrong book It's Not About the Bike: My Journey Back to Life

Source: It's Not About the Bike: My Journey Back to Life (2000), p. 267
Context: Anything is possible. You can be told you have a 90-percent chance or a 50-percent chance or a 1-percent chance, but you have to believe, and you have to fight. By fight I mean arm yourself with all the available information, get second opinions, third opinions, and fourth opinions. Understand what has invaded your body, and what the possible cures are. It's another fact of cancer that the more informed and empowered patient has a better chance of long-term survival. What if I had lost? What if I relapsed and the cancer came back? I still believe I would have gained something in the struggle, because in what time I had left I would have been a more complete, compassionate, and intelligent man, and therefore more alive.

Roger Joseph Boscovich photo

“But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes.
We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line.”

Roger Joseph Boscovich (1711–1787) Croat-Italian physicist

"Boscovich's mathematics", an article by J. F. Scott, in the book Roger Joseph Boscovich (1961) edited by Lancelot Law Whyte.
"Transient pressure analysis in composite reservoirs" (1982) by Raymond W. K. Tang and William E. Brigham.
"Non-Newtonian Calculus" (1972) by Michael Grossman and Robert Katz.

Vladimir Lenin photo

“Human knowledge is not (or does not follow) a straight line, but a curve, which endlessly approximates a series of circles, a spiral. Any fragment, segment, section of this curve can be transformed (transformed one-sidedly) into an independent, complete, straight line, which then (if one does not see the wood for the trees) leads into the quagmire, into clerical obscurantism (where it is anchored by the class interests of the ruling classes).”

Vladimir Lenin (1870–1924) Russian politician, led the October Revolution
John Ramsay McCulloch photo

“The division of labour is a consequence of the previous accumulation of capital… As the accumulation of capital must have preceded the division of labour, so its subsequent division can only be extended as capital is more and more accumulated. Accumulation and division act and react on each other. The quantity of raw materials which the same number of people can work up increases in a great proportion, as labour comes to be more and more subdivided; and acccording as the operations of each workman are reduced to a greater degree of identity and simplicity, he has, as already explained, a greater chance of discovering machines and processes for facilitating and abridging his labour. The quantity of industry, therefore, not only increases in every country with the increase of the stock or capital which sets it in motion; but, in consequence of this increase, the division of labour becomes extended, new and more powerful implements and machines are invented, and the same quantity of labour is thus made to produce an infinitely greater quantity of commodities”

John Ramsay McCulloch (1789–1864) Scottish economist, author and editor

Source: The principles of political economy, 1825, p. 95-96

David Eugene Smith photo

“He used capital vowels for the unknown quantities and capital consonants for the known, thus being able to express several unknowns and several knowns.”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1925) Vol.2, p. 430; footnote

“The great Cartesian invention had its roots in those famous problems of antiquity which originated in the days of Plato. In endeavoring to solve the problems of the trisection of an angle, of the duplication of the cube and of the squaring of the circle, the ruler and compass having failed them, the Greek geometers sought new curves. They stumbled on the conic sections…There we find the nucleus of the method which Descartes later erected into a principle. Thus Apollonius referred the parabola to its axis and principal tangent, and showed that the semichord was the mean propotional between the latus rectum and the height of the segment. Today we express this relation by x2 = Ly, calling the height the ordinate (y) and the semichord the abscissa (x); the latus rectum being… L. …the Greeks named these curves and many others… loci… Thus the ellipse was the locus of a point the sum of the distances of which from two fixed points was constant. Such a description was a rhetorical equation of the curve…”

Tobias Dantzig (1884–1956) American mathematician

Number: The Language of Science (1930)

Isaac Barrow photo

“As a Line, I say, is looked upon to be the Trace of a Point moving forward, being in some sort divisible by a Point, and may be divided by Motion one Way, viz. as to Length; so Time may be conceiv'd as the Trace of a Moment continually flowing, having some Kind of Divisibility from an Instant, and from a successive Flux, inasmuch as it can be divided some how or other. And like as the Quantity of a Line consists of but one Length following the Motion; so the Quantity of Time pursues but one Succession stretched out as it were in Length, which the Length of the Space moved over shews and determines. We therefore shall always express Time by a right Line; first, indeed, taken or laid down at Pleasure, but whose Parts will exactly answer to the proportionable Parts of Time, as its Points do to the respective Instants of Time, and will aptly serve to represent them. Thus much for Time.”

Isaac Barrow (1630–1677) English Christian theologian, and mathematician

p, 125
Geometrical Lectures (1735)

“Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate.”

Gregory Bateson (1904–1980) English anthropologist, social scientist, linguist, visual anthropologist, semiotician and cyberneticist

Bateson (1978) " Number is Different from Quantity http://www.oikos.org/batesnumber.htm". In: CoEvolution Quarterly, Spring 1978, pp. 44-46

“Like drawing a straight line – you draw a straight line and it's crooked and you draw another straight line on top of it and it's crooked a different way and then you draw another one and eventually you have a very rich thing on your hands which is not a straight line. If you can do that the it seems to me you are doing more than most people. The thing is, it is very difficult to know oneself whether one is doing that or not, whether you mean what you do; and there is the other problem of the way you do it and whether sometimes you do more than you mean or you do less than you mean. It's very good if you can establish a language where it's clear that that is what you are doing – that you do what you mean to do.”

Jasper Johns (1930) American artist

interview at John's studio, Billy Klüver, March 1963, as quoted in Jasper Johns, Writings, sketchbook Notes, Interviews, ed. Kirk Varnedoe, Moma New York, 1996, p. 85
1960s

Related topics