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TREATISE OF PLANE GEOMETRY

THROUGH GEOMETRIC ALGEBRA

Ramon González Calvet

TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA

Ramon González Calvet

The geometric algebra,

initially discovered by Hermann

Grassmann (1809-1877) was

reformulated by William Kingdon

Clifford (1845-1879) through the

synthesis of the Grassmann’s

extension theory and the

quaternions of Sir William Rowan

Hamilton (1805-1865). In this way

the bases of the geometric algebra

were established in the XIX

century. Notwithstanding, due to

the premature death of Clifford, the

vector analysis −a remake of the

quaternions by Josiah Willard

Gibbs (1839-1903) and Oliver

Heaviside (1850-1925)− became,

after a long controversy, the

geometric language of the XX century; the same vector analysis whose beauty attracted the

attention of the author in a course on electromagnetism and led him -being still

undergraduate- to read the Hamilton’s Elements of Quaternions . Maxwell himself already

applied the quaternions to the electromagnetic field. However the equations are not written

so nicely as with vector analysis. In 1986 Ramon contacted Josep Manel Parra i Serra,

teacher of theoretical physics at the Universitat de Barcelona, who acquainted him with the

Clifford algebra. In the framework of the summer courses on geometric algebra which they

have taught for graduates and teachers since 1994, the plan of writing some books on this

subject appeared in a very natural manner, the first sample being the Tractat de geometria

plana mitjançant l’àlgebra geomètrica (1996) now out of print. The good reception of the

readers has encouraged the author to write the Treatise of plane geometry through

geometric algebra (a very enlarged translation of the Tractat ) and publish it at the Internet

site http://campus.uab.es/~PC00018 , writing it not only for mathematics students but also

for any person interested in geometry. The plane geometry is a basic and easy step to enter

into the Clifford-Grassmann geometric algebra, which will become the geometric language

of the XXI century.

Dr. Ramon González Calvet (1964) is high school teacher of mathematics since 1987,

fellow of the Societat Catalana de Matemàtiques ( http://www-ma2.upc.es/~sxd/scma.htm )

and also of the Societat Catalana de Gnomònica ( http://www.gnomonica.org ).

TREATISE OF PLANE GEOMETRY

THROUGH GEOMETRIC ALGEBRA

Dr. Ramon González Calvet

Mathematics Teacher

I.E.S. Pere Calders, Cerdanyola del Vallès

To my son Pere, born with the book.

 Ramon González Calvet ( rgonzal1@teleline.es )

This is an electronic edition by the author at the Internet site

http://campus.uab.es/~PC00018 . All the rights reserved. Any electronic or

paper copy cannot be reproduced without his permission. The readers are

authorised to print the files only for his personal use. Send your comments

or opinion about the book to ramon.gonzalezc@campus.uab.es .

ISBN: 84-699-3197-0

First Catalan edition: June 1996

First English edition: June 2000 to June 2001

PROLOGUE

The book I am so pleased to present represents a true innovation in the field of the

mathematical didactics and, specifically, in the field of geometry. Based on the long

neglected discoveries made by Grassmann, Hamilton and Clifford in the nineteenth

century, it presents the geometry -the elementary geometry of the plane, the space, the

spacetime- using the best algebraic tools designed specifically for this task, thus making

the subject democratically available outside the narrow circle of individuals with the

high visual imagination capabilities and the true mathematical insight which were

required in the abandoned classical Euclidean tradition. The material exposed in the

book offers a wide repertory of geometrical contents on which to base powerful,

reasonable and up-to-date reintroductions of geometry to present-day high school

students. This longed-for reintroductions may (or better should) take advantage of a

combined use of symbolic computer programs and the cross disciplinary relationships

with the physical sciences.

The proposed introduction of the geometric Clifford-Grassmann algebra in high

school (or even before) follows rightly from a pedagogical principle exposed by

William Kingdon Clifford (1845-1879) in his project of teaching geometry, in the

University College of London, as a practical and empirical science as opposed to

Cambridge Euclidean axiomatics: “ ... for geometry, you know, is the gate of science,

and the gate is so low and small that one can only enter it as a little child”. Fellow of the

Royal Society at the age of 29, Clifford also gave a set of Lectures on Geometry to a

Class of Ladies at South Kengsinton and was deeply concerned in developing with

MacMillan Company a series of inexpensive “very good elementary schoolbook of

arithmetic, geometry, animals, plants, physics ...”. Not foreign to this proposal are Felix

Klein lectures to teachers collected in his book Elementary mathematics from an

advanced standpoint 1 and the advice of Alfred North Whitehead saying that “the hardest

task in mathematics is the study of the elements of algebra, and yet this stage must

precede the comparative simplicity of the differential calculus” and that “the

postponement of difficulty mis no safe clue for the maze of educational practice” 2 .

Clearly enough, when the fate of pseudo-democratic educational reforms,

disguised as a back to basic leitmotifs, has been answered by such an acute analysis by

R. Noss and P. Dowling under the title Mathematics in the National Curriculum: The

Empty Set? 3 , the time may be ripen for a reappraisal of true pedagogical reforms based

on a real knowledge, of substantive contents, relevant for each individual worldview

construction. We believe that the introduction of the vital or experiential plane, space

and space-time geometries along with its proper algebraic structures will be a

substantial part of a successful (high) school scientific curricula. Knowing, telling,

learning why the sign rule, or the complex numbers, or matrices are mathematical

structures correlated to the human representation of the real world are worthy objectives

in mass education projects. And this is possible today if we learn to stand upon the

shoulders of giants such as Leibniz, Hamilton, Grassmann, Clifford, Einstein,

Minkowski, etc. To this aim this book, offered and opened to suggestions to the whole

world of concerned people, may be a modest but most valuable step towards these very

good schoolbooks that constituted one of the cheerful Clifford's aims.

1 Felix Klein, Elementary mathematics from an advanced standpoint . Dover (N. Y., 1924).

2 A.N. Whitehead, The aims of education . MacMillan Company (1929), Mentor Books (N.Y.,

1949).

3 P. Dowling, R. Noss, eds., Mathematics versus the National Curriculum: The Empty Set?. The

Falmer Press (London, 1990).

III

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