IN MATH: 1. adj. refering to rectangular coordinate system created by Rene Descartes, mathematician and philosopher. EX. The Cartesian coordinates of point A are (2,4).

IN ENGLISH: 1. as defined above.

APPLICATION: See list 230.

 See: A Graph is A Portrait - an essay on the concepts of graph Point-Plotting Animation - see step-by-step how to create a graph

 Some History       Above, the picture of Descarte was first published in 1995. The web version appeared in 2005. At the right, Descarte is depicted in 2021. The picture has changed, but, so has the "history."  The former appears in a dictionary for those just beginning to study algebra, analytic geometry, graphing, functions -- it is simple.   The latter appears in a Geometer's® Sketch Pad for those studying precalc, and calc, and any other interested party -- it is more complicated.       Years ago, stories of the fly on Descarte's ceiling and Newton's falling apple were told. Now, some support the stories but others do not. Information availability and volume has increased (in evidence, compare the two Descartes).       The history presented is elaborations on the content of each headshot w/thumbnail info. Sources are provided, but with links rather than academic citations.       Ancient Computing Devises Used Digitally (text format) and 50+ Centuries of Computation in a 21st-Century Format (lecture format) are other math history pages unrelated to this material. Disclaimer       This is the "Azzolino Version" of math history.       It is left for you to determine the "truth."             -- Agnes (A2) Azzolino

 Analytic Geometry & Graphing Functions       Years ago we credited Descartes with the coordinate plane.  Descartes in Latin is Cartesius [Descartes 1].   Now we elaborate a bit more, witness the players shown above.       Though Galileo [analytic 1] was working on the same thing, while Fermat and Descartes worked independently, credit is given to Fermat and Descartes -- based first publication date and evidence of private, unpublished work. Each also developed the idea of and computed derivatives before calculus was "invented," but Newton and Leibniz are given credit for calculus.       Note in the image that these men also worked on other things. Fermat was better known for Fermat's Last Theorem [Fermat 1] and Descartes for "I think, therefore I am" [Descartes 2].       Leibniz and van Schooten also worked on other things. Leibniz discovered calculus!       van Schooten taught and translated and edited Descartes' La Geometrie [Descartes 1] when he introduced the use of two axes. It has a horizontal base line AB, but no axes. The translation into Latin made it available to more people. The axes made it easier for all.       2graphs.gsp -- graphs and functions sketch pad

Precalc Functions in Sketch Pad Format
(if you do not wish to use 2graphs.gsp or fX.f'X.f''X.intX.gsp

 · Functions, Lines · Parabols, Quadratics · Exponentials, Logs, then scroll down · Trig, then scroll down

 Reimann Sums       ReimannSums.gsp animates partitions, boxes, Reimann sums, then area under (or between) curve(s), and plotting an antiderivative by using the sums.

 Limit       limit -- history, symbols, by definition, by epsilon-delta, by approach

 Derivatives       fX.f'X.f''X.intX.gsp -- derivatives, be definition (limit of the difference quotient), br tangent, by trace, antiderivative by plotting sums of sums, history

 Sources geometry 1 - ms.uky.edu - earlier Ptolemy, Oresme. then Galileo. geometry - 2 - download.tuxfamily.org - pdf copy of paft of Descartes' Geometry. It has a horizontal base line AB,but no axes.   Descartes 1 - famousscientists.org - name in Latin. Descartes 2 - britannica.com - "Je pense, donc je suis."   "Cogito, ergo sum."   "I think, therefore I am." & Cartesian coordinate system   e - 1 - wired.com - "'Discover' by either Napier in the early 1600s or Bernoulli in the late 1600s" e- 2 - en.wikipedia.org - Jacob Bernoulli discovered the fundamental mathematical constant e. Compound interest   Euler - 1 - brilliant,org 1707-1783 Euler - 2 - google - Introduced many current notations, such as Σ for the sum; the symbol e for the base of natural logarithms; the letter f and parentheses for a function; and i for square root of −1, a, b and c for the sides of a triangle and A, B, and C for the opposite angles Euler - 3 - storyofmathematics.com -- "The use of a, b and c as constants and x, y and z as unknowns – was either created, popularized or standardized by Euler. His efforts to standardize these and other symbols (including π and the trigonometric functions).   Fermat 1 - britannica.com - Fermat's Last Theorem - xn + yn n for n > 2, proveda few years ago by Andrew Wiles   Leibniz - 1 - wired.com - Introduced integration symbol in 1675. Liebnitz- 2 - abebooks.com - picture   limit - 1 - en.wikipedia.org - Bernard Bolzano in 1817, introduced (unpublished) the basics of the epsilon-delta technique to define continuous functions. - Cauchy in 1821, introduced the basic idea behind Delta-epsilon proofs. Weierstrass introduced the epsilon-delta definition of limit in today's form and the notations lim and limx→x0.   Weierstrass - 1 - wikipedia Weierstrass - 2 - math.libretexts.org - modern limit form

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