A Short History

of the Cartesian

Coordinates

By Ernesto Pérez

The first intent

of formalizing

the geometry

concepts and

knowledge came

from Euclid’s

Elements (c. 300

BC).

Elements is a

collection of

definitions,

postulates, theorems

and mathematical

proofs.

Elements is probably a

collection of theorems

previously proved by

other mathematicians

that predated Euclid

with some original

work; among them

Pythagoras.

However, geometry

became analytic

with the

publication of

René Descartes'

(1596–1650)

masterpiece in

1637: Discourse on

the Method.

In an appendix to the

Discourse, titled La

Gèomètrie, he

included applications

of algebra to

geometry, giving a

push to the use of

geometry to solve some

algebraic problems.

But some authors claim

that the discovery of

the coordinate system

should be attributed

to Pierre de Fermat

(1601–1665), because

Fermat had more

geometrical insight

than Descartes.

E. T. Bell, a

historian of

mathematics

writes: “There

is no doubt

that he

preceded

Descartes”.

“But as his work of about

1629 was not

communicated to others

until 1636, and was

published posthumously

only in 1679, it could

not be possibly have

influenced Descartes in

his own invention, and

Fermat never hinted that

it had.”

James Newman,

expressing his

thought about to

whom should be

attributed the

'modern' use of the

coordinate system

states:

“Fermat may have preceded

Descartes in stating

problems of maxima and

minima; but Descartes went

far past Fermat in the use

of symbols, in

"arithmetizing" analytic

geometry., in extending it

to equations of higher

degree.”

But… in order to

develop a useful

geometric coordinate

system to solve

mathematical problems

related to geometry and

physics two important

steps are needed:

1 - the recognition

of the zero as a

number, and

2 - the

introduction of

the negative

numbers.

The number zero has no

zero-history, on the

contrary, its uses,

manipulations and

rejections has been

traced as far as the

beginning of

civilization itself.

Some form of acceptance

can be found in Greece,

India, Babylon.

With the acceptance

of the negative

numbers the story is

similar: some

mathematicians

argued against the

existence of numbers

"below zero". Other

considered that

subtraction from zero

was "nonsense".

Descartes used only an

X-axis and did not

refers to a Y-axis. For

each value of x he

computed the

corresponding y from

the equation, thus

getting the coordinates

x and y.

Bell again: “The use of two

axes obviously is not a

necessity but a

convenience. In our

terminology, he used the

equivalent of both

rectangular and oblique

axes”. “In Descartes' work

every thing was measured

with positive distances.”

The earliest use of

negative coordinates

is attributed to Isaac

Newton (1642–1727) in a

collection of figures

and graphs of

polynomials of the

third degree of his

book ...

Enumeratio

linearum tertii

ordinis, or

Enumeration of

Curves of Third

Degree.

In this

publication

Newton used

perpendicular

axes and

included both

positive and

negative

numbers.

In fact, in some of the

figures he used the

capital letter X to label

the horizontal axis, the

capital letter Y for the

vertical axis, and even

the capital letter O to

label the point of

intersection of both

axes.

References:

[1] Bell E. T. (1945). Development of Mathematics.

2nd. ed. McGraw-Hill Book Company. New York.

[2] Descartes, Rene. (1956). The Geometry. In James R.

Newman (Ed.) The World of Mathematics. Vol. 1. (pp.

235-253). Simon and Schuster. New York.

[3] Newton, Sir Isaac. (c. 1760). Enumeration Of

Lines Of The Third Order, Generation Of Curves By

Shadows, Organic Description Of Curves, And

Construction, Of Equations By Curves. Retrieved

2007 from the digital version at Google Book

Search at http://books.google.com/books?

id=6I97byFB3v0C&dq=newton+enumeration+curves.

Visit my Website at

http://4dlab.info

for the full article

and other historical

curiosities in

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