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The Extension Theory of Hermann Grassmann and
the Creation of Vector Algebra
Juho Vuorikoski

Tiedekunta – Fakultet – Faculty
Matematiikan ja Tilastotieteen Laitos

Koulutusohjelma – Utbildningsprogram – Degree programme
Algebran ja topologian linja

Tekijä – Författare – Author
Juho Vuorikoski
Työn nimi – Arbetets titel – Title
The Extension Theory of Hermann Grassmann and the Creation of Vector Algebra
Työn laji – Arbetets art –Level
Pro Gradu

Aika – Datum – Month and year
05.09.2019

Sivumäärä – Sidoantal – Number of pages
49

Tiivistelmä – Referat – Abstract

Yleisesti Sir William Rowan Hamiltonia pidetään vektorialgebran luojana. Hän oli 1800- luvun alussa
syntynyt arvostettu matemaatikko. Yritettyään vuosia löytää tapaa yleistää kompleksilaskenta kolmeen
ulottuvuuteen, hän lopulta vuonna vuonna 1843 keksi kvaterniot, kompleksilukujen laajennuksen
neliulotteiseksi luvuiksi. Hamiltonin kvaternihin pohjautuva järjestelmä on ensimmäinen julkaistu
vektorialgebra. Hamilton oli jo aikanaan kuuluisa ja vaikutusvaltainen, mutta siitä huolimatta kului yli
kaksikymmentä vuotta, ennen kuin hänen järjestelmäänsä alettiin käyttää laajemmin. Vuosisadan
vaiheessa tämän järjestelmän pohjalta luotiin moderni vektorialgebra.
Mutta modernin järjestelmän pohjaksi harkittiin myös toista järjestelmää. Tämän oli luonut
saksalainen kouluopettaja Hermann Grassmann (1809-1877). Päinvastoin kuin Hamilton, Grassmann ei
ollut aikalaisilleen tunnettu saavutuksistaan matematiikan alalla. Itseasiassa, hän teki ensimmäinen
matemaattinen julkaisunsa yli 30 vuotiaana. Grassmannia oli jo vuosia kiehtonut idea suorien
yhteenlaskusta ja vuonna 1844 hän vihdoin julkaisi kirjan, joka lähti tuosta yksinkertaisesta ideasta ja
päätyi yhdeksi ensimmäisistä oikeista vektorialgebrojen esityksestä ja ensimmäiseksi lineaarialgebran
esitykseksi.
Valitettavasti hänen tuntemattomuutensa ja se että hänen aikalaisensa pitivät kirjaa pahamaineisen
vaikealukuisena, varmisti, että kirja jäi ilman suurempaa huomiota. 15 vuotta myöhemmin hän päätti
yrittää uudelleen ja alkoi kirjoittaa järjestelmästään parannettua versiota. Tämän hän lopulta julkaisi
vuonna 1863 nimellä “Ausdehnungslehre” tai englanniksi “Extension Theory”. Tämäkin kirja jäi vaillesuurempaa huomiota hänen elinaikanansa, mutta noin sata vuotta myöhemmin Grassmannin työt
nousivat uuteen arvoon. Tämä johtui siitä, että “Extension Theory” on paljon muita aikalaisia
järjestelmiä abstraktimpi ja määrittelee kaiken n-ulotteisesti, sen sijaan että asiat määriteltäisiin
lähinnä kolmessa ulottuvuudessa (tai neljässä kvaternioiden tapauksessa). Tässä järjestelmässä
ristitulon sijasta, meillä on ulkotulo ja sisätulo määritellään tämän ulkotulon kautta ja on paljon
monimutkaisempi kuin normaalissa vektorialgebrassa. Tieteen kehittyessä tarpeet käsitellä asioita nulotteisesti kasvoivat ja asiat joidenka takia Grassmannin aikailaiset eivät hänen töitään ymmärtäneet,
toivat hänen työnsä takaisin parrasvaloihin.
Avainsanat – Nyckelord – Keywords
Hermann Grassmann, Vektorialgebra, Lineaarialgebra, Matematiikan historia, William Rowan Hamilton, Extension Theory
Säilytyspaikka – Förvaringställe – Where deposited
Muita tietoja – Övriga uppgifter – Additional information

Contents
1 Introduction

22 Sir William Rowan Hamilton

3

2.1

Before Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.2

Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.3

Hamilton's Legacy

6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Hermann Grassmann

4

7

3.1

Early Life

3.2

Theorie der Ebbe und Flut

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .

7
8

3.3

Ausdehnungslehre of 1844 . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.3.1

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.3.2

Reception of A1. . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.3.3

Priority Claim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.4

Ausdehnungslehre of 1862 . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.5

Later Years

15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ausdehnugslehre of 1862 or Extension Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

4.1

Basics4.2

Product Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

4.3

Combinatorial and Outer Product . . . . . . . . . . . . . . . . . . . . . . .

32

4.4

Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

5 Conclusions

19

45

1

Chapter 1
Introduction
Sir William Rowan Hamilton is widely considered to be the father of vector calculus due tohis discovery of quaternions in 1843. But in 1840 a German school teacher named Hermann
Grassmann completed a 200-page essay called  Theorie der Ebbe und Flut (Theoryof Ebb and Flow), which is the rst known instance of the notion of vector space andalso the rst use of linear algebra. Despite repeated attempts, Grassmann's ideas did not
truly catch on during his lifetime. In the 1890s, there was a brief struggle for what vector
algebra would be used and his work was mostly forgotten in favour of the modern systemof vector algebra and to a lesser extent the system of quaternions it was based on. Yet in
the last 60 years there has been a resurgence of interest in Grassmann and his work andarguably, he is now more relevant than ever before.
Part I of this essay will describe rst the life of Sir William Rowan Hamilton as hisstory illustrates well the diculties and challenges of creating and spreading a brand new
branch of mathematics.

His life and story are also an interesting parallel to the main

topic of this essay, the life and work of Hermann Grassmann. Details are then given on
Grassmann's life and especially the two of the three major works of vector (and linear)calculus written by him,  Theorie der Ebbe und Flut and  Ausdehnungslehre
of 1843 (or A1). In Part II we will cover the basics of Grassmann's third and nal majorwork,  Ausdehnungslehre of 1862 (or A2) in greater detail. .

2

Chapter 2
Sir William Rowan Hamilton
At the turn of the 19th century, many mathematicians were slowly approaching the development of a system of vector calculus. Some like Caspar Wessel (1745-1818) in 1799,
approached this by dening geometric ways to represent complex numbers. Others like
August Ferdinand Möbius (1790-1868) in 1827 with  Der Barycentrische Calkul
approached it by the way of analytical geometry. But it was to be Sir William Rowan
Hamilton and Hermann Grassmann, who would go on to write the rst known major
works of vector calculus.
By most accounts, Hamilton was one of the great mathematicians of the 19th century.
By his own admission, he was not a physicist, but his work was immensely inuential inthe eld. From his development of Hamiltonian mechanics (as it is now known) to his
work on optics, his historical importance should not be underestimated. But in the eldof mathematics, it is the quaternions that he is best known for and it is what will bemainly focused on here.
(O'Connor & Robertson 2015)

2.1

Before Quaternions

Between 3th and 4th of August 1805 at midnight, William Rowan Hamilton was bornin Dublin, Ireland.

From an early age he proved himself to be a child prodigy, under

the tutelage of his uncle, a clergyman he had lived with since age three.

By the time

he was thirteen, he was at least partially versed in Arabic, French, German, Greek,
Hebrew, Hindostanee, Italian, Latin, Malay, Persian, Sanskrit, Spanish and Syriac. He
also studied astronomy, geography, literature, mathematics and religion and found anerror in  Mécanique Céleste by Laplace when he was sixteen.
In 1823 Hamilton received the highest marks on the entrance exam of Trinity College3

of Dublin. Then, during his rst year in college, he was awarded an optime in Classics,
a grade greater than the best grade which was only usually awarded once in 20 years.
In 1826 he presented his now famous  Theory of Systems of Rays, in which hegave the characteristic function for optics. The same year, he also received an optime inboth Classics and science. This was unheard of and Hamilton soon became a celebrity inthe intellectual circles of Dublin. In 1827 he was hired as Andrew's Professor of Astronomy at the University of Dublin and the Royal Astronomer of Ireland. He was still anundergraduate at college at the time.
All in all Hamilton was exceedingly accomplished in multiple elds.

His name waswell known in scientic circles and he was a man from whom a lot was expected. Carl
Gustav Jacob Jacobi (1804-1851) even called him the Lagrange of England and as amathematician, he seemed to do no wrong.
But there was one problem that plagued Hamilton and gave him pause. From at leastas far back as 1827 he tried to nd three dimensional complex numbers or triplets as they
were called. Wessel had discovered the geometrical representation of complex numbers ina plane at the end of the 18th century. This was also discovered/studied at least by veother mathematicians, Jean-Robert Argand (1768-1822), Abbé Buée (1746-1826), CarlFriedrich Gauss (1777-1855), C. V. Mourey (1791-1830) and John Warren (1796-1852).
But Hamilton (and others) wanted to extend this concept into three dimensions.
thought that this would be very useful in geometrical calculations.

He

He also wanted tond a way to calculate space and time in geometry. Negative numbers were somewhat
frowned upon in many elds.

Doubly so in geometry, which was deeply rooted in thereal. How can one measure a negative length? Hamilton's reasoning for tor the use ofnegative numbers was time.

One could not indeed measure a negative length, but one

could measure negative time (i.e. the past). But regardless of how much he thought on
it, a three dimensional complex calculus seemed to elude him.
(Crowe, p. 17-26; O'Connor & Robertson 1998; Wilkins n.d.)

2.2

Quaternions

Years later, on October 16 1843, Hamilton was walking to the Royal Irish Academy withhis wife. While crossing the Brougham bridge, he had an epiphany and the equation

i2 = j 2 = k2 = ijk = −1
suddenly came to him.

He couldn't make the system work for three variables, but hecould with four. He was so exited by this idea, that he actually stopped at the bridge tocarve the equation on it. The original carving has disappeared a long time ago, but tothis day, the equation is written down on a commemorative plaque on the bridge. For

4this was the discovery of the quaternions, of hypercomplex numbers of the form

w + ix + jy + kz,
where w , x, y and z are real numbers and i, j and k are unit vectors that follow the laws

ij = kji = −kjk = iki = jkj = −iik = −jii = jj = kk = −1

What Hamilton had found and later created in more formal manner was the rst wellknown signicant number system that did not obey the normal arithmetic laws (Grassmann's  Theorie der Ebbe und Flut only getting published in 1911, well after his
death).

This geometric interpretation of quaternions had signicant dierences to thegeometric interpretation of normal complex numbers. While the real and imaginary partof a complex number could be thought of as their x- and y -coordinate in a plane, thesame did not work for quaternions in three dimensions thanks to their four dimensional
nature. Hamilton's interpretation was that the number w represented time and x, y andz represent the forces aecting an object in time. Thus Hamilton avoided thinking about
quaternions as truly four dimensional. This was important for in geometry, just as there
couldn't be negative length, there also couldn't be more than three dimensions.
Although he'd had a breakthrough, Hamilton knew that it would take a lot of work todevelop his idea. According to him, he felt that it would be something worth spending ten
or fteen years on. He read his rst paper on the matter four weeks after his epiphany.
And four years later, he had published 34 papers on the subject.
And so Hamilton, a rising star in the scientic circles made his grand discovery. His
status was such, that he did not originally have many detractors, even though his system
was very revolutionary. In actuality, most simply did not understand it, and just assumed
that Hamilton's genius was above them. One of the most telling examples of the reaction
quaternions had at the time comes from an article published in the North American
Review in 1857. The article in question is a review of  Lectures on Quaternions of1853. The review states that quaternions will be regarded as the great discovery of 19th
century, but yet in America, less than 50 men have seen it and less than ve have read it.
It goes on to describe Hamilton in more and more grandiose manner with lines like:And if the world should stand for twenty-three hundred years longer, thename Hamilton will be found, like that of Pythagoras, made immortal by its
connection with the eternal truth rst revealed through him.
Yet only moments later, the writer states that he has not in fact read the 800-page book inquestion. And so, even receiving rave reviews, Hamilton's system failed to initially catch5

on. Hamilton had failed to write his system in a manner that was understandable to his
peers. And so his system did not see widespread use during his lifetime. He continued towork on quaternions until his death in 1865.
(Crowe, p. 27-42; O'Connor & Robertson 1998; Wilkins n.d.)

2.3

Hamilton's Legacy

Even though Hamilton failed in spreading his system far, there was one specic group ofpeople that he did manage to pass it on to, his students. As he was a college professor
for most of his life, Hamilton taught many aspiring young mathematicians and physicists.
Some of them were able to continue his work. And as many of them, like Peter GuthrieTait, were physicists, they saw more the practical applications of the system. Physics, inelds like thermodynamics, had a growing need for more advanced mathematical systems.
And in 1867 Tait wrote the textbook  Elementary Treatise on Quaternions, therst of many publications on the matter. In addition to being a sort of beginners guide toquaternions (something that was sorely needed), the work also gave practical applications
to physics.

This nally saw the more widespread use of quaternions as it also gave

people a relatively easily explained practical use for the system. The use of the system
would continue to grow until Josiah Willard Gibbs and Oliver Heaviside helped create themodern vector calculus in the 1880s and 1890s by selectively altering and removing parts
from the quaternion system .
(Crowe, p. 27-42; O'Connor & Robertson 1998; Wilkins n.d.)6

Chapter 3
Hermann Grassmann
Unlike Hamilton, the inuence of Grassmann, was not truly felt in the mathematicalcircles of the 19th century. While he did manage to win some acclaim during his life, thiswas mostly due to his work in other elds like linguistics and botany. Truly, if one wereto write about the history of vector algebra in the 1950s, it would not be dicult to notmention Hermann Grassmann at all. This is because even though he was decades aheadof his time in many aspects, it was not until nearly a century after his death with authors
like David Hestenes, that Grassmann's work in vector calculus became valued.

3.1

Early LifeOn 15th of April 1809 in the town of Stettin, Prussia (now Szczecin, Poland), Johanne
Luise Friederike Medenwald gave birth to her third child, Hermann Günther Grassmann.
Hermann's father was Justus Günther Grassmann, who previously worked as an ordained
minister, but had later become a mathematics and physics teacher at Stettin Gymnasium (eectively the high school of Stettin). He was also the author of many books onmathematics and physics.

Johanne, who would go on to have nine more children, wasa daughter of a minister and well educated and so taught Hermann from a young age.
But unlike Hamilton, young Hermann did not distinguish himself in any particular way
and even did badly enough that at one point his father told him that he might want tobecome a craftsman or a gardener instead of a mathematician. But through hard work,
he managed to become the second ranked student at the Stettin Gymnasium by his thirdyear.
Somewhat surprisingly, Grassmann did not focus his studies on mathematics.

In

1827, he went to study theology at the University of Berlin. His courses included classicallanguages, literature, philosophy and theology, but no courses on either mathematics or7physics. This was doubly strange as after he had completed his studies in 1830, he became
determined to become a teacher of mathematics, presumably because of his father. For
the next year, he immersed himself in mathematical studies, so that he could prepare
himself for the examinations for a teaching license. In keeping with the fact that he had
only studied mathematics seriously for a year, the exam did not go as one might havewanted.

In December 1831, he took the exam but received only a level one teaching

license, a license to teach at the lower levels (below university level), and was told that
he would need a much better grasp of mathematics to receive a level two license (whichgave the right to teach at all levels). As one of his goals in life was to ultimately becomea professor at a university, this was not great news.
And so, in 1832 Grassmann went to work at the Stettin Gymnasium as an assistantteacher. He states in the foreword of  Ausdehnungslehre of 1844 that this was theperiod, when he had his rst mathematical discoveries that made him write the book inquestion. Following in his fathers footsteps, in 1834 Grassmann passed theology examinations at level one that allowed him to follow a career as a minister in the Lutheran
Church. Despite this he instead continued his work as a teacher. He mostly worked in
Stettin, but also in a few dierent schools in Berlin. He had not given up on teaching on auniversity level and after passing the level two test for theology by 1839, he set his sights
on an examination that would give him the similar right to teach chemistry, mathematics,
mineralogy and physics. The assignment for that particular examination was that he had
to write an essay on the theory of tides.
(Crowe, p. 44-60; O'Connor & Robertson 2005; Petsche 2011, p. 3-6)

3.2

Theorie der Ebbe und FlutFor his essay Grassmann wrote Theorie der Ebbe und Flut, one of (if not the) rstworks of vector algebra. As stated earlier, Grassmann's father Justus, was a mathematician who had published multiple books. Two of them,  Raumlehre of 1811 (Theory ofSpace) and  Trigonometrie (Trigonometry) contained an idea that a rectangle is a geometric product of its base and height and that this geometric product behaved similarly
to normal arithmetic multiplication. This had led to Hermann thinking about conceptsof geometric product and directed line. Interestingly the basics of his theory is based on
Pierre-Simon Laplace's (1749-1827)  Mécanique Céleste, the same book Hamilton
had famously found an error in as a teenager. In his later writings he explains how hecame up with the basic idea for Theorie der Ebbe und Flut.
Grassmann started by thinking about negatives in geometry. He was accustomed (mostlikely because of his father) to thinking that, if A and B are points, then displacements

AB and BA were opposite magnitudes and so AB + BA = 0. Now if C is a point on the8same line as A and B , thenAB + BC = AC,even if C is situated between A and B . And, if that is the case, the AB and BC aren't
just lengths, but are instead magnitudes with direction and length. This means that thesum of lengths is dierent from the sum of these magnitudes and that there was a needto clearly establish the rules by which such magnitudes were summed. And the easiest
way to do this was to set that AB + BC = AC was always true, even if the points A, B
and C are not on the same straight line.
As his father had thought of the triangle as a product, Hermann realized that thesame could be thought of parallelograms. They could be thought of as a product of twoof their sides, if those sides were thought of as directed magnitudes. By joining the twoideas together, he came to the equation

A [B + C] = AB + AC
where A, B and C are now directed lines (i.e. vectors). Grassmann named this type of
product thegeometric product. This product was similar to the cross product of modern

vector algebra, but it dened a directed area in the plane of the directed lines (perpendicular to the cross product). After this he also dened a linear product identical to themodern dot product.
To prepare himself for the examination, Grassmann applied these new methods to
Lagrange's  Mécanique Analytique and to his delight found that they made most
calculations in the book considerably simpler. His new calculations were actually often
more than ten times shorter than Lagrange's. Naturally he was greatly encouraged by thisand so wrote Theorie der Ebbe und Flut. The work was extremely revolutionary
for its time, containing a large part of vector analysis. It included vector addition andsubtraction, the two major vector products, vector dierentiation and linear functions oflinear algebra.

Theorie der Ebbe und Flut was the rst important system of vector analysis
and also the rst major work in linear algebra. According to Michael J. Crowe it could
be compared to non-Euclidean geometry in how revolutionary it was. Interestingly Crowe
also states that Grassmann's later insistence on proving everything in n-dimensions could
be thought of as a discovery of non-Euclidean geometry.
After submitting the essay it was given to its chief reader on the 26th of April 1840.
The chief reader was mathematician Carl Ludwig Conrad and he returned the essay vedays later with barely a comment. Grassmann did receive his level two teachers license
allowing him to teach his chosen subjects at all levels. But it is safe to say that Conrad
did not realize the importance of the work he had reviewed. Theorie der Ebbe undFlut was only published in 1911, 72 years after its writing.
(Crowe, p. 60-63; O'Connor & Robertson 2005)

9

3.3

Ausdehnungslehreof 1844

Grassmann was kept very busy by his career as a teacher, a career that he took very
seriously.

During the years he wrote many textbooks to be used at schools and twoof them were published in 1842.

But he did not forget about the ideas he had ex-pressed in Theorie der Ebbe und Flut and he became increasingly intrigued bya system that incorporate the ideas of that essay, but in a more general form, not restricted to three dimensions. And so, in the spring of 1842 Grassmann started working
on  Ausdehnungslehre (or A1 as it is sometimes known). The work was completed
by fall of 1843 under the full title of  Die lineale Ausdehnungslehre, ein neuer

Zweig der Mathematik dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehrevom Magnetismus und die Krystallonomie erläutert (which translates roughly
to The Linear Extension Theory, a new branch of mathematics presented and explained
through application to other branches of mathematics, which include statics, mechanics,
magnetics and crystallography) and was published in 1844.
(Crowe, p. 63-77; O'Connor & Robertson 2005)

3.3.1

Contents

In the foreword, in addition to explaining how he came to write Theorie der Ebbe

und Flut he also regales on his reasons and goals in writing  Ausdehnungslehre.
Grassmann had for a long time thought that geometry should not be viewed as a branchof mathematics in the same way as arithmetics. This was because geometry was denedby real world limitations, in this case the limitation of three dimensions. He thought that
there had to be a completely abstract system of calculus that's laws coincided with thoseof geometry. The system he had created for Theorie der Ebbe und Flut was hisrst glimpse of this new region of science. And A1 was his attempt at formulating thatsystem on a purely abstract level, without any assumptions from the real world.
So in A1 Grassmann created a purely formal system that was above and independentof geometry and also of all mathematics known to the time. Included here, rewritten andin an abbreviated form, is the rst example in A1.

connections between
synthetic connection is symbolized by a and it is

Let us say that a and b are contentless forms. We shall now denethese forms.

A connection called adened by

aab=baaand10

(a a b) a c = a a (b a c) = a a b a c.
So, in a synthetic connection the parentheses can always be omitted without changing theresult. On the other hand, ananalytical connection is symbolized by ` and it is denedby(a ` b) a b = a,a ` b ` c = a ` c ` b = a ` (b a c)anda ` (b ` c) = a ` b a c.
It should also be noted that these analytical connections are unique, so ifa ` b = c,then there does not exist a d such thata ` b = d 6= c.
At this point in the book Grassmann explained that synthetic connections could be called
addition and analytical connections subtraction. He then went on to dene an indierent
form (i.e. zero or null) and an analytic form ` b (i.e.

−b). And nally two additional

types of connection. The rst was multiplication, a type of synthetic connection and thesecond was division, an analytical connection.
The work also contained many writings that were mainly philosophical in their content.
As Grassmann was also highly interested in philosophy, which was also an extremely
respected eld of science at the time, this was meant to better explain the reasons for thisabstract system to readers.

Even though A1 contained many incredibly revolutionary

results and denitions worthy of mention, we will only shortly go over some of them, as
many will be gone over in greater detail as we go through A2 (as   Ausdehnungslehre
of 1862 is sometimes known) later in this essay. The linear and geometric product had theirnames changed to inner and outer product respectively.

Later in the book Grassmannalso in practice dened the free linear space generated by units e1 , e2 , · · · and all of theirlinear combinations a1 e1 + a2 + e2 + · · · , where a1 , a2 , · · · are numbers. He then went on
to dene the addition of such linear combinations as well as multiplication with numbers.
These were dened similarly as in modern vector algebra.
of the operations was also formally proven.

The linear space properties

A1 also had denitions for (with dierent

names) dimension, independence, subspace, join and meet of subspaces and span and11projections of elements onto subspaces. And notably, in the book Grassmann also invents
exterior algebra (as it is now known). It also had very many other proofs and identities,
for example the proof of the Steinitz Exchange Theorem, which received its name from
German mathematician Ernst Steinitz (1871-1928) who proved it in 1913.
(Crowe, p. 63-77; O'Connor & Robertson 2005)

3.3.2

Reception of

A1

It might not be terribly surprising to learn that readers at the time did not really understand what Grassmann was trying to achieve. The extreme abstractness added to thephilosophical writings made the book infamously dicult for people to read, let alone
understand.

Even though philosophy was prevalent at the time, most mathematicians

were still not actually philosophers and vice versa. As an eort to both promote his workand to get feedback, Grassmann sent copies of A1 to numerous eminent mathematicians
to little eect.

One was sent to Gauss, who sent a reply letter dated the 14th of De-

cember 1844. Gauss was cordial, but also said that he had worked on similar concepts
50 years earlier. He had also concluded that to truly understand Grassmann's book, hewould have to spend a considerable amount of time acclimating himself to his peculiar
terminology, and unfortunately he was far too busy to do so at the time. Gauss was most
likely referring to his work on representing complex numbers geographically, when talking
about having worked on similar concepts.

Gauss apparently never did get the time tofamiliarize himself to A1, which was very unfortunate for Grassmann as Gauss was morequalied than most qualied to understand it.
Another mathematician who had A1 sent to the and also had the potential to understand it was Möbius, thanks of his work on Barycentric Calculus. Barycentric Calculus
was similar to vector calculus, but it did not deal with lines, only with points. Yet having
attempted to read it, even Möbius stated that A1 was simply unreadable. Grassmann
actually still asked Möbius to write a review of A1, but Möbius, unsurprisingly, declined.
He did go on to have a correspondence with Grassmann to help out his countryman. First
Möbius tried to get Moritz Wilhelm Drobisch (1802-1896) to review the book, as Drobisch
was a philosopher as well as a mathematician. Unfortunately he declined and so Möbius
suggested to Grassmann that he himself should write a review of A1. This Grassmannwent on to do and it was the only review that A1 ever received.
As a continued attempt at helping Grassmann, Möbius told him about a contest heldby the Fürstliche Jablonowski'schen Gesellschaft der Wissenschaft (The First Jablonowski
Society of Science). The entries had to solve a problem originally presented by Gottfried
Wilhelm Leibniz (1646-1716), to establish geometric characteristic without using metricproperties. As Grassmann had already done as much, this seemed like a good way to get
his name and system known. And so Grassmann wrote  Die Geometrische Analyse12geknüpft und die von Leibniz Characteristik (Geometric analysis and the Leibniz's characteristic, Die Geometrische Analyse for short), which did win the prize in1846. This was less impressive than it sounds as his was the only entry. His work was alsoheavily criticized for being too abstract and unintuitive by one of the judges, Möbius.
By 1852, Hamilton had heard about a system similar to his own and so decided
to read it.

As did others, he also found the work extremely hard to understand, but

he also thought that Grassmann was original and brilliant.

He thought that, if only

Grassmann had thought to combine his inner and outer products, then he too could havediscovered quaternions.

And so Hamilton became perhaps the rst mathematician toactually appreciate Grassmann's work, but he also naturally thought that quaternions
were superior and so concentrated on them.
Almost ten years after its publication in 1853, Möbius wrote in a letter that Carl
Anton Bretschnider (1808-1878) was the only mathematician he knew that had read A1
in its entirety. Heinrich Richard Baltzer (1818-1887) said that trying to read A1 madehim dizzy and Hamilton, wrote to Augustus De Morgan (1806-1871) that one should learn
to smoke if one wanted to be able to read Grassmann.
During the 1850's an Italian mathematician named Giusto Bellavitis (1803-1880)
learned of Grassmann. As his work on the Calculus of Equipollence (a more limited studyof lines and points) was actually one of Grassmann's inuences for writing A1, Bellavitis had more appreciation for his system than most.

But Luigi Cremona (1830-1903)

seemed to have been the only mathematician actually inspired by A1, during this period. He actually used methods from A1 to solve mathematical problems in an journal
called  Nouvelle Annales de mathématiques, published in 1860 and credited him
accordingly. He would go on to use Grassmann's ideas in at least one of his books.
To sum up the books reception, here is a quote from Grassmann's biography, written
by Friedrich Engel (1861-1941) :Thus Grassmann experienced what must be the most painful experience forthe author of a new work: his book nowhere received attention; the public wascompletely silent about it; there was no one who discussed it or even publicly
found fault with it.And even more telling is the letter Grassmann received from his books publisher in 1876:
Your book Die Ausdehnungslehre has been out of print for some time.
Since your work hardly sold at all, roughly 600 copies were in 1864 used aswaste paper and the remainder, a few odd copies, have now been sold withthe exception of one copy that remains in our library.
(Crowe, p. 77-89; O'Connor & Robertson 2005)

13

3.3.3

Priority Claim

In 1845 Adhémar Jean Claude Barré de Saint-Venant (1797-1886), better known as Comtede Saint-Venant, wrote a paper in which he dened, among other things discovered by
Grassmann, the produit géométrique (the geometric product). After rst hearing about
this and then traveling 150 km to Berlin just to be able to read the paper, Grassmanndecided to send a letter and a copy of A1 to Saint-Venant.

As he did not know his

address, he instead sent them to Augustin Cauchy (1789-1857), and asked him to forward
them to Saint-Venant. Unfortunately Cauchy only forwarded the letter and not the book.
Saint-Venant attempted to nd a copy A1, but was unable to nd a copy as the FrenchInstitute had the book, but mistakenly classied.
In 1853 Cauchy and Saint-Venant wrote papers on subjects Grassmann had alreadydiscovered.

The following year, after Grassmann had had time to actually read them,Grassmann decided to make a priority claim on the paper Cauchy wrote, the  Comptes

Rendus. The French Academy formed a committee of three mathematicians to deliberate on the matter.

One of the members was Cauchy.

The committee never made adecision before (or after) Cauchy died in 1857.
(Crowe, p. 77-89; O'Connor & Robertson 2005)

3.4

Ausdehnungslehreof 1862

In 1847 Grassmann became an Oberlehrer (senior teacher) at his school and decided
to nally apply for a position at a university.

He was not happy about still being a

gymnasium teacher, even though he had written a mathematical work that he felt, wasof great import. This was partly because, as a gymnasium teacher, he could not teachhis his system to his students as it was too advanced. So he wrote an application to thePrussian Ministry of Education.

They felt that they were unable to accurately gauge

if Grassmann was worthy of a post and so asked the mathematician Eduard Kummer(1810-1893) to give his input on the matter. Kummer read through Grassmann's essay

Die Geometrische Analyse, but felt that it had good material, but just presented
in a decient form.

This ended up permanently ending Grassmann's dreams of ever

becoming a university professor.
As Grassmann became increasingly frustrated with people ignoring his work, he decided to rewrite A1. Originally he intended to write a sequel to it, but because of A1's
bad reception thought that reworking the title would yield better results.
was completed under the full title of  Ausdehnungslehre:

In 1862, it

Vollständing und instenger Form bearbeitet (Linear extension theory: Fully processed in strict form),
known as  Extension Theory in English. Grassmann had 300 copies of A2 printed
and the book very dierent from A1. Gone were all the philosophical musings and the14book (which was a third longer than A1) was structured more like a modern textbook.
This was Grassmann's attempt to make the book more readable, and this partially worked.
But unfortunately A2 also had almost no physical applications or practical examples. The
book had many new and exiting results, like the solution to the Pfaan Problem fteen
years ahead of its time, but it was also, once again, exceedingly dicult for the people ofthe time to read.
As much the contents of the book will be gone over in length later, it suces to go overits reception. A2 actually received even less attention than A1. Less copies of Grassmann

A2 were printed and not even Grassmann wrote a review of the book this time. And theonly reaction he received for the books he sent to other mathematicians was one letter of
thanks. Engel describes its reception thus:As in the rst Ausdehnungslehre so in the second: matters which Grassmann had published in it were later independently rediscovered by others, andonly much later was it realized that Grassmann had discovered them earlier.
(Crowe, p. 80-96; O'Connor & Robertson 2005)

3.5

Later Years

For the rest of his life Hermann Grassmann stayed very active.

He published multiple

language and mathematics textbooks as well as scientic papers. He was an editor fora political magazine and published materials for the evangelization of China. He was anactive teacher, married and had 11 children. He made many contributions to linguistics,
foremost of which was his translation of  Rig-Veda (Sacred text of the Hindu), forwhich he was given a honorary doctorate from the University of Tübingen.
published writings on botany, music and religion.

He alsoGrassmann did not completely stop

writing mathematical works, but it was not his focus.
Gradually, after 1865, a few mathematicians started to pay attention to Grassmann.
Victor Schlegel (1843-1905) was a fellow teacher at Grassmann's school who became interested in his work. He was very enthusiastic about Grassmann's calculus and even wrote
papers using his methods.

Unfortunately, he was not a gifted mathematician and his

papers had practically no eect.

The biography he later went on to write in 1878 forGrassmann was more inuential, thanks to its extreme praising of the man.
In a cruel twist of fate, four men who might have had more success all died within a
few years of becoming interested in Grassmann. They were Herman Henkel (1839-1873),
Alfred Clebsch (1833-1872), William Klingdon Cliord (1845-1879) and Hermann Noth(1840-1882).

Clebsch had actually learned of Grassmann's work while studying withhis son Justus. All of them were inspired by Grassmann's work and wrote papers on the15

matter, though Cliord is the most inuential of them. This slowly started to make Grassmann's name more known and actually this was enough for A1 to be called for a second
printing shortly before Grassmann's death. On September 26, 1877 Hermann Grassmanndied. Here is an excerpt from his biography, written by his son Justus Grassmann:Only a few days before his death, Prof. Burmeister in Dresden informed
him that he was thinking of giving lec[tures] on the Ausdehnugslehre in theupcoming winter semester; and with joy he could now leave to others thatwhich an unfavourable fate had denied to himself.
(Crowe, p. 80-96; O'Connor & Robertson 2005)

16

Chapter 4

Ausdehnugslehre of 1862 or
Extension Theory
The 1862 Ausdehnungslehre is the most complete form of Grassmann's vector algebra, extension theory. It has many similarities with the modern vector algebra, but
it's closest spiritual successor is geometric algebra popularized by David Hestenes in the
1960s, which was based on work done by Cliord in the 1880's, which in turn was based on
the work of Grassmann and Hamilton. This essay aims to go over the basics of extension
theory, hopefully allowing people to better understand this (for its time) revolutionary
system and at the same time shed some light into why the system was and is so hard forpeople to understand.
For the most part, the symbols and terms used by Grassmann might not be familiar
for the modern reader and so an attempt will be made to translate terms to their vector
calculus equivalents when appropriate. When possible, terms and symbols have been kept
the same as to avoid confusion as one of the aims of this essay is to aid in reading thebook.
One of the key diculties in learning the system, is the abstractness of it. Grassmann
was motivated by a desire to completely dene an abstract algebra that encompassed
geometry. Unfortunately because of this, his book is very sparse on practical examples
on the applications of the system. Geometric applications are investigated only well after
the beginning, over a hundred pages in.

Coordinates are never used.

One can only

assume that Grassmann was worried about limiting the potential readers understandingof the system and thus stayed purposefully abstract.

This might have been a relevant

concern for the time, considering the entire concept of a vector algebra was foreign toalmost all mathematicians of the time. Luckily modern readers are more cognizant to thepossibilities of vectors and so some basic parallels to vector calculus will be included here
and there.

But it is good to remember that one should not fully equate such parallels

17because things in extension theory tend to be more general in nature than those in basic
vector calculus.
An added diculty is that the system is generated from a few denitions which areiteratively built upon, layer by layer. Because Grassmann had to do the book practicallyby himself, it does not have the sort of editing as one is accustomed to these days.
Meaning, among other things, that it can be dicult to gauge the importance of any one
theorem for the understanding of the system as a whole..
We will now go over the basics of extension theory hopefully giving readers an elementary understanding of it and its similarities and dierences to modern vector algebra.
As such, most proofs will be skipped, as most readers have already read similar proofs,
and there will be periodically inserted clarifying texts. After explaining the basics of thesystem, we will mostly be looking into the product structure in Extension Theory as thatis perhaps the most striking dierence of the system compared to modern vector calculus.

18

4.1

BasicsExtensive Magnitude
Extension theory is based on

extensive magnitudes (also called magnitudes for short). It

is to be taken as a primitive concept that can represent anything that can be given a
numerical amount.

If a vector has a direction and magnitude, an extensive magnitude

only needs to have the magnitude. They can represent anything that can have a numericrepresentation of magnitude. So for example 5a could be ve apples, a cube with a volume
3of 5m or a vector a multiplied by the scalar 5. A
simply means anumerical magnitude

number.

Numerical Relation
A magnitude a isderived from magnitudes b, c, ... ifa = βb + γc + ...

where β, γ, ... are real or irrational numbers.

An assembly of magnitudes (i.e.

a set)a, b, c, · · · is subject to numerical relation if one of them can be derived from the rest. If,

an assembly only has two magnitudes neither of which is zero, it is symbolized bya≡band say that a iscongruent to b. From this we get that two real numbers are always

congruent and that any assembly where one magnitude is 0 is subject to numerical relation.

Unit
unit is any magnitude we use to derive other magnitudes from (similar
to a basis vector). The unit of numbers is one and it is called the absolute unit. A systemof units is an assembly of magnitudes that are not in any numerical relation to each other.
We then say that a(Grassmann 1862, p. 3)
Side comments:
It is notable that, as is proved later in A2, there is really no dierence between a non-zero
magnitude and a unit.

This is because (similarly as with basis vectors) any non-zero

magnitude may be taken as a unit. To a modern reader, accustomed to vector and linear
algebra, these concepts would most likely feel very familiar. One could easily nd similar
theorems from modern vector calculus, if one were to change the term magnitude with
vector and numerical relation with linear dependence. Or replace a system of units withbasis. And so one would have some idea of the reason for why Grassmann describes these19things.

These were not luxuries that any of Grassmann's peers had in the middle of

19th-century. And so, calling the system obtuse, would be an understatement for them.
But one noteworthy thing here is that one should not just equate magnitudes with
vectors (at least, not in the Euclidean sense). All vectors are magnitudes but not viceversa. They are closer to the concept of the multivector in geometric algebra. In geometric algebra a 1-vector is a normal vector (a line segment), but a 2-vector (or bivector) isa plane segment and so forth. Even this is not really an accurate comparison because,
for one, points can also be magnitudes. The sheer abstractness and generalness of thedenitions in the system makes many things relatively simple in vector calculus, like theinner product, considerably more dicult. Of course, this should not be taken to meanthat everything is harder to do in Extension Theory.

Derivation Numbers
Numbers used to derive a magnitude from a system of units are called thenumbers of that magnitude. For example, if

derivationa = α1 e1 + α2 e2 + · · ·
where

α1 , α2 , · · · are real numbers and e1 , e2 , · · · form a system of units, then a is amagnitude and α1 , α2 , · · · its derivation numbers. The magnitude can also be marked as

X
X

αe,

αr er

or

[

r= 1]nSide Comment:Xαr er .

It should be noted thatP

αe is the most common notation used by

Grassmann in his book as shorthand notations for sums were the norm at the time. In
PP[
this text, mainly theαr er form will be used, as it is more explicit. r= 1]n αr er will
be used when required for clarity.

Basic Operations
If two extensive magnitudes are derived from the same system of units, theirsimply adding together the derivation numbers with the same unit, meaning

20

addition is

X

(4.1)

αr e r +Xβr er =

X

(αr + βr ) er .

Subtraction is dened in the same way. To multiply an extensive magnitude by a number,you multiply all of its derivation numbers by the number in question, meaning

X

(4.2)

αr er  β =X(βαr ) er .

Division is similarly dened, as long as β 6= 0.

Basic Properties
From (1) we get the following if a, b and c are extensive magnitudes:
1.

a+b=b+a2.

(a + b) + c = a + (b + c)3.

a+b−b=a4.

a+b−a=b(Grassmann 1862, p. 4-5)

DomainFor any given assembly of magnitudes a1 , a2 , . . . , an , their

domain is the collection of all

magnitudes derivable from them. So if A is the domain of the magnitudes ar (1 ≤ r ≤ n),then a ∈ A ifa=X

α r arfor some αr . If the domain cannot be derived from less than n such magnitudes, it is calleda domain ofn th order. A domain is of zeroth order if it only includes the magnitude zero.

Domain Relations
identical if a ∈ A⇐⇒ a ∈ B
If A ⊂ B , then A is subordinate to B and B superordinate to A.
A and B are incident, if A ⊂ B or B ⊂ A.
A ∩ B is the common domain of A and B .
A + B is the covering domain of A and B .
DomainsA and B are

21

Example
If domain A is derivable from units e1 , e2 , e3 and domain B is derivable from units e2 , e3 , e4 ,then their common domain is derivable from e2 , e3 and their covering domain is derivablefrom e1 , e2 , e3 , e4 .
(Grassmann 1862, p. 8)
Side comments:
At this point one would likely wonder if a domain is a vector space. So we shall check fora domain A, (here αr , βr , β, γ are numbers):
1. Numbers (real and irrational) are a eld, and the sum of two magnitudes in A is
still in A (by denition). Same for scalar multiplication.
2. Associativity and commutativity were already noted.
3. If magnitude a is in A then so is 0a = 0, thus 0 is in A.
4. If

P

αr er is in A, then so is −P

αr er =P(−αr ) er , so the additive inverse of all

magnitudes are included in their domains.

PPPP
β (γ αr er ) = β ( (γαr ) er ) =(βγαr ) er = βγ ( αr er ).
PP
6. 1 (
αr er ) =αr er .
PPPPP
7. γ (
αr er + βr eP(γαr + γβr ) er =(γαr ) er + (γβr ) err) =P
= γ ( αr er ) + γ ( βr er ) .
PPP
8. (β + γ)
α
e
=
((β
+
γ)
α
)
e
=
(βαr + γαr ) er
r
r
r
r
P
P
= β ( αr er ) + γ ( αr er ) .
5.

And so, domains are vector spaces and a subordinate domain is a vector subspace.

Properties of Magnitudes and Domains
Next we will go through several theorems about the properties of assemblies of magnitudes,
most of which should be very familiar to modern readers.

• Numerical relation between magnitudes a1 , · · · , an can also be given in a more familiar form,

α1 a1 + · · · + αn an = 0
if α1 , · · · , αn are not all simultaneously zero.

22• In the previous case, it is always possible to separate from the n magnitudes, anassembly of magnitudes less than n, from which the rest can be derived.

• If we have a magnitude a1 , that can be derivable from an assembly of magnitudes
b1 , b2 , · · · , bn , where b1 6= 0, then the domain of b1 , b2 , · · · , bn is identical to that of
a1 , b 2 · · · , b n .
• If magnitudes a1 , · · · , an stand in no numerical relation and can be derived frommagnitudes b1 , · · · , bn , then their domains are identical.
• If the domain of an assembly of magnitudes a1 , · · · , an is derivable from less than nmagnitudes b1 , · · · , bm , then a1 , · · · , an always stand in numerical relation.
• All domains of n th order can be derived from any n of its magnitudes that stand inno numerical relation to one another.

• If m and n are orders of the domains M and N , r the order of their common domain,
and v of their covering domain, thenm + n = r + v.
In vector algebra, this would be in the form ofdim (M ) + dim (N ) = dim (M ∩ N ) + dim (M + N ) .
• If domains A and B , of orders α and β respectively, are in a domain N , with anorder n, they must have a domain of (α + β − n)th order in common if α + β > n.
Or in vector algebra, if

dim (A) + dim (B) > dim (N )and A, B ⊂ N , thendim (A ∩ B) = dim (A) + dim (B) − dim (N ) .
PP• If a =αr er and b =βr er are magnitudes in a domain of n th order a = b only
if αr = βr for all 1 ≤ r ≤ n.
(Grassmann 1862, p. 8-13)

23

Shadow
Let a1 , a2 , · · · , an be magnitudes from which you can derive a domain ofn th order A. If

a ∈ A, then there are n real numbers α1 , α2 , · · · , αn so thata = α 1 a1 + α 2 a2 + · · · + α n an .
Now if m < n then we call the magnitude

α 1 a1 + α 2 a2 + · · · + α m am ,shadow of the magnitude a on the domain of a1 , a2 , · · · , am , excluding the domainof am+1 , am+2 , + · · · + an . We say that shadows of several magnitudes are taken in thesame sense if the magnitudes are shadowed on the same domain and exclude the samethe 

3domain. So, for example, if a, c ∈ R and a = (1, 1, 1) = e1 + e2 + e3 , c = (2, 2, 2). Now if

b is the shadow of a on to the domain of e1 , e2 (i.e. the xy-plane), excluding the domainof the e3 (the z-axis), then b = e1 + e2 . And if d is c:s shadow in the same sense thend = 2e1 + 2e2 .
(Grassmann 1862, p. 15-16)

Side comments:

We have now gone over the rst chapter or 18 pages of A2.

In

those pages Grassmann has goes through the basics of his system, and although dierent
in many of the terms, they are also strikingly similar to many parts of modern vector
calculus. It is noteworthy that nowhere in those 18 pages was there references to three
dimensions or to geometry and the system is generated almost wholly in the abstract.
There will also never be any real use of coordinates. This is in some ways very laudable,
as overt focus on real applications has often stymied mathematical progress (as in thecases of 0, negative numbers, i or π ). But it was hardly conducive to motivating people
to keep reading, as they would not have been able to understand the point of it all.
From the rst chapter one could easily think that extension theory is mostly similar tomodern vector calculus and so far it has been. But that all changes rather drastically when
we start investigating the part of A2 that is perhaps the most inuential, the productstructure.

24

4.2

Product Structure

ProductPP[
αr er and b =s= 1]m βs es be extensive magnitudes. Now choose aPP[[domain of n th degree A, such that a, b ∈ A. Then a =r= 1]nαr er and b =s= 1]n βs es .
Then their product [ab], which is always a magnitude, is dened as
Let a

[

=r= 1]n(4.3)[ab] =



X
X
X [
X
[
[
r= 1]n αr ers= 1]n βs ess= 1]n αr βs [er es ] .
=r= 1]n[

It is also said that a and b are

factors of the product [ab].

ExampleIf

P = [ab] = [(α1 e1 + α2 e2 ) (β1 e1 + β2 e2 )] ,then this would equal

α1 β1 [e1 e1 ] + α1 β2 [e1 e2 ] + α2 β1 [e2 e1 ] + α2 β2 [e2 e2 ] .
As of yet, it is not dened what type of magnitudes the four products[er es ] are.

Here are some examples of dierent ways they could be dened, forming dierent productstructures.
1. We set all of the four products as dierent units that one can derive P from. Thenα1 β1 , α1 β2 , α2 β1 and α2 β2 are its derivation numbers and then no establishingequations are necessary for the product structure.
2. By setting

e1 e2 = e2 e1 ,you would only have three units [e1 e1 ], [e1 e2 ] and [e2 e2 ] and their respective derivation numbers α1 β1 , α1 β2 + α2 β1 and α2 β2 . This product structure would be denedby its commutative property.
3. If[e1 e1 ] = [e2 e2 ] = 0
and[e1 e2 ] = − [e2 e1 ] ,25then, there would be only one unit for the product P and there would be only thederivation number α1 β2 − α2 β1 .This product structure we call

combinatorial.

4. If one were to set[e1 e1 ] = [e2 e2 ] = 1
and[e1 e2 ] = [e2 e1 ] = 0,then there would also be only one unit for the product P and it is a number. From
this we get thatP = α1 β1 + α2 β2 .
This product structure we callinner.

(Grassmann 1862, p. 19-20)

General Product Identities
For now we will only look at the product structure in its most general form. From equation(3) we get a number of identities for product structures, in these a, b, · · · , ar , bs and p
are magnitudes and α,β ,· · · are numbers:
1.

[(a + b + · · · ) p] = [ap] + [bp] +· · ·[p (a + b + · · · )] = [pa] + [pb]+· · ·PP
3. [ab] = [(αr er ) b] =αr [er b]2.

4.

[(αa) b] = [a (αb)] = α [ab]5.

[(αa + βb + · · · ) p] = α [ap] + β [bp] + · · ·6. If a =P

αr ar and b =P[

βs bs , then [ab] =r= 1]nP [P
s= 1]m αr βs [ar bs ](Grassmann 1862, p. 20-21)
These formulas also apply if the product has more than two magnitudes, for example,
if a, b,· · · are magnitudes that all are in the same domain ofn th order, then

(4.4)





X
X
X [
X
[
[
[
r= 1]n αr ers= 1]n βs es · · · =r= 1]ns= 1]n[ab · · · ] =· · · (αr βs · · · [er es · · · ]) .

26

We use Pa as a way to mark a product which includes a as one of its factors. If a, b, · · ·are factors and α, β, · · · numbers, thenPαa+βb+··· = αPa + βPb + · · · .

Side comment:

From these properties we can rst prove that the product is multilinear.

Let a, b, p ∈ U be magnitudes, α a number andU a domain, thenB : U × U 7−→ V
is the map of the product (V is also a domain). By 1., 2. and 4. we get that

B (a + b, p) = B (a, p) + B (b, p) = [ap] + [bp] ,B (a, b + p) = B (a, b) + B (a, p) = [ab] + [ap]andB (αa, b) = B (a, βb) = αB (a, b) = α [ab] .
Thus the product is bilinear. And then because of (4.4) it is actually multilinear.
(Grassmann 1862, p. 22-23)

Dening Equations
In a product structure,dening equations are established numerical relations between

the products of the units. If an assembly of dening equations includes all the numerical
relations of the products of the units and none of the assembly can be derived fromthe others, we call it asystem of dening equations. If we have a system of m dening

equations between n unit products E1 , E2 , · · · , En , then n − m of those unit products
form a system of units from which all products of this product structure are derivable.
The dening equations also tell how to derive the remaining m unit products from then − m unit products.

ExampleIf in a domain of 2nd order, if we have a system of three dening equations,

E1 = [e1 e1 ] = 0,
E2 = [e2 e2 ] = 0
27

and

E3 = [e1 e2 ] = − [e2 e1 ] = E4 ,then n = 4, m = 3 and all products E1 , E2 , E3 and E4 can be derived from E3 .

Linear Product Structure
If all dening equations of a product structure remain true, even if you replace the units
in the equation with arbitrary magnitudes, we call it aSide comment:

linear product structure.

This can be very misleading name for modern readers as the linear

product property is stronger than bilinearity or multilinearity. For example, if you havea product with three dening equations,[e1 e1 ] = [e2 e1 ] = [e2 e2 ] = 0,
then, if a = α1 e1 + α2 e2 and b = β1 e1 + β2 e2[ab] = [(α1 e1 + α2 e2 ) (β1 e1 + β2 e2 )] = α1 β2 [e1 e2 ] ,which is bilinear (as shown before), but it is not a linear product structure. This is because[e1 e1 ] = 0,but[aa] = α1 α2 [e1 e2 ] 6= 0,
when α1 6= 0 6= α2 .
(Grassmann 1862, p. 24)

Theorem 51
Because of the importance of the next theorem, we will also go through its proof.
In addition to a product structure with no dening equations and one with all products
being zero, there are only two dierent types of linear product structure for two factors:(4.5)[er es ] + [es er ] = 0

and

28

[er es ] = [es er ] ,(4.6)

for all unitser and es from which the two factors are derived.

Proof: Any dening equation of two magnitudes [ab] can be written in the form of[

r= 1]n

(4.7)

X [
X
s= 1]n αr,s [er es ] = 0,

where αr,s = αr βs cannot all be zero. Now if we assume the product structure to be linear,then we can replace the units by arbitrary magnitudes derived from them. So we shall
replace all er withX
γr,u euu
and then all es withX

γs,v ev .

v
Now the derivation numbers γr,u and γs,v can be any chosen numbers. We get

"
X

!
X
γr,u eu

αr,s

r,s

!#
X

u

=

X

γs,v evv

αr,s γr,u γs,v [eu ev ] = 0.

r,s,u,v
Because all the indexes have been chosen arbitrarily, we can exchange r with s and u withv and we getX

αs,r γs,v γr,u [ev eu ] = 0

s,r,v,uand if we add these two equations together we getXαr,s γr,u γs,v [eu ev ] +r,s,u,v

(4.8)

=

X

X

αs,r γs,v γr,u [ev eu ]s,r,v,u

γr,u γs,v (αr,s [eu ev ] + αs,r [ev eu ]) = 0,r,s,u,vwhich must hold for all values γr,u and γs,v . So if we construct an equation from (4.8)
where we set one of the values γa,c = 1 and then another one where its value is −1, but
29keep all other values of γr,u the same in both. Both of these equations equal zero and areequal in their terms except for terms with γa,c once (if both γr,u and γs,v equal γa,c , then
12 = (−1)2 ). So we subtract the second from the rst and divide the result by two and
get

(4.9)

X
γs,v (αa,s [ec ev ] + αs,a [ev ec ]) − 2αa,a [ec ec ] = 0.
s,vThen we do the same for (4.9) for values γb,d and get that

(4.10)

αa,b [ec ed ] + αb,a [ed ec ] = 0,

with a 6= b if c = d and c 6= d if a = b. Applying this to (4.8) we haveXγr,u γr,u αr,r [eu eu ] = 0.
r,uNow if we set γr,u = 0 for all values except γa,c = 1,then

αa,a [ec ec ] = 0
and so (4.10) applies for all a, b, c, d and thus follows from (4.7), if (4.7) is a dening
equation.
If for (4.10) we set c = d it is changed into

(4.11)(αa,b + αb,a ) [ec ec ] = 0

and if instead we set a = b, it is changed to(4.12)

αa,a ([ec ed ] + [ed ec ]) = 0.

In either case, it must be that either the number or magnitude is zero.
Let us rst investigate (4.11) and assume that [ec ec ] 6= 0, then

αa,b + αb,a = 0,
meaning

αa,b = −αb,a ,for all a, b. And by this (4.10) transforms into30

αa,b ([ec ed ] − [ed ec ]) = 0,
where either αa,b = 0 for all a, b (which is contrary to our assumption) or[ec ed ] = [ed ec ] ,which is the same dening equation as (4.6).
Now let us assume that [ec ec ] = 0 or equivalently that [ea ea ] = 0. This can now betaken as a dening equation where the derivation number αa,a = 1 and by applying it to(4.12) you have[ec ed ] + [ed ec ] = 0,
which is the same as equation (4.5).
Finally, if equations (4.5) and (4.6) were to both apply, then [ec ed ] = 0 for all c, d,which cannot be true by assumption. And thus they are the only possible linear productstructures.

Side comment:

There are a few important things to note here. Firstly, both of theselinear product structures have important modern analogues.

(4.5) is the outer product(similar to the cross product) and (4.6) is the inner product. Thus their formulation is, atthe very least, interesting as well as a part of the justication of why Grassmann devotes
so much time on dening both later in the book.
Also of import is to note that Grassmann only thought about the product of two
factors and did not try to generalize the system for additional factors.

For example,

the multiplication of quaternions has a dening equation with a product of three factors

ijk = −1.
(Grassmann 1862, p. 24-27)

31

4.3

Combinatorial and Outer Product

Combinatorial ProductFirstly, let us study products of the type[ec ed ] + [ed ec ] = 0.
Let P be a product for which:
1. All the factors are derived from a system of units.
2. For all a =P

αr er and b =P

βs es that are factors ofP , if
αr 6= 0 =⇒ βr = 0

then P 6= 0.
3. For an arbitrary series of units A and factors b, c

[Abc] + [Acb] = 0.
And thus also[bc] + [cb] = 0.
Then we call P a

combinatorial product and its factors its elementary factors. One can

now iteratively prove increasingly strong statements about the combinatorial product.
The generalizations will be listed here in order:• [Abc] + [Acb] = 0 continues to hold if A is an arbitrary series of factors.
• [AbcD] + [AcbD] = 0 if D is also an arbitrary series of factors.
• Pa,b + Pb,a = 0, if Pa,b is an arbitrary combinatorial product which includes elementary factors a and b and Pb,a is an otherwise identical combinatorial product where
you have exchanged a and b.

Combinatorial Product Properties• If a1 , a2 , · · · , am are elementary factors, then[a1 a2 · · · am ] = ± [aj · · · ak ] ,where [aj · · · ak ] is an arbitrary rearrangement of the elementary factors of the rstproduct. The latter product is positive if you can get it from the rst one by doingan even amount of exchanges and negative if the amount is odd.

32• It also follows that
Pa,a = 0,
meaning that a combinatorial product is zero if two of its elementary factors are thesame. And then, if a1 , a2 , · · · , am are numerically related, then[a1 a2 · · · am ] = 0.
• The converse also holds, meaning that if a combinatorial product is zero, then there
is a numerical relation between its elementary factors. And

Pa,b+αa = Pa,b + αPa,a = Pa,b ,(4.13)

meaning that you can add an arbitrary multiple one of its elementary factors to one
of its other elementary factors without changing the product.

Side comment:

It should be noted that the modern version of this type of product,

the outer product, is marked with a ∧ symbol. So for example, if magnitudes a and b are
linearly independent vectors, then[ab] = a ∧ band this is a bivector (i.e.

a plane segment).

This plane segment is orthogonal to thecross producta×b
and if a = αe1 and b = βe2 , thena × b = αβ (e1 × e2 )anda ∧ b = αβ (e1 ∧ e2 ) .
(Grassmann 1862, p. 29-35)

Combinatorial Product as a Magnitude
It has been mentioned before that products, and thus combinatorial products as well, aremagnitudes. Let us now investigate more fully what types of magnitudes they are. When
are they numerically related and how?
Let us say that A, B, · · · , are all the dierent combinatorial products of the magnitudes
a1 , a2 , · · · , an that stand in no numerical order. For example, if n = 2, then A = [a1 ],B = [a2 ] and C = [a1 a2 ]. Now if α, β, · · · are numbers, then the equation

33

αA + βB + · · · = 0,
can be replaced by the equation

α = β = · · · = 0.
This is because, if for example A = [a1 ] and A1 = [a2 · · · an ] and we multiply the entire
equation with A1 , we get

αAA1 + βBA1 + · · · = 0.
Now AA1 = [a1 a2 · · · an ] 6= 0, but BA1 = 0 for all possible values of B (if for example

B = [a1 a2 ] , then BA1 = [a1 a2 a2 · · · an ] = 0 and thus we get
αAA1 = 0,
meaning

α = 0.
And by doing this for all the products we get that all the derivation numbers are zero.
An interesting result of this is that if A and B are combinatorial products, thenA≡B
if and only if the elementary factors of A and B dene the same domains.

Side comment:

The modern equivalent to this would be saying thata1 ∧ a2 ∧ · · · ∧ an = α (b1 ∧ b2 ∧ · · · ∧ bn ) ,for some scalar α if and only if

ha1 , a2 , · · · , an i = hb1 , b2 , · · · , bn i .
Where a1 ∧ a2 ∧ · · · ∧ an and b1 ∧ b2 ∧ · · · ∧ bn are m-vectors and ha1 , a2 , · · · , an i and

hb1 , b2 , · · · , bn i are vector subspaces formed of the vectors a1 , a2 , · · · , an and b1 , b2 , · · · , bn
respectively.
For example, if a1 = (1, 0, 0) and a2 = (0, 1, 0), then if

a1 ∧ a2 = β (b1 ∧ b2 ) ,for some number β , then b1 and b2 are both linearly independent vectors in the xy -plane.

34One should note that these results also mean that the combinatorial product is clearlydierent from the cross product (which is its closest parallel in normal vector calculus).
This is because, if a1 and a2 are as previously and a3 = (0, 0, 1) ,thena1 × a2 = a3 ,but

[a1 a2 ] 6= αa3 ,for all numbers α.
(Grassmann 1862, p. 36-37)

Linear evolution
In a series of magnitudes, if you add an arbitrary multiple of a neighboring magnitude
to a magnitude, we call it alinear evolution.

The new series has been arrived by anelementary linear evolution from the original. If you linearly evolve the new series again,
then it has been arrived by a multiple linear evolution from the rst series. So, if a seriesof magnitudes has the magnitudes p and q as neighboring magnitudes,· · · , p, q, · · ·then· · · , p + αq, q, · · ·where α is a any number, would be an (elementary) linear evolution of the rst series.
The same can be done by multiple linear evolution, even if p and q are not neighbors.
Meaning that you can get from· · · , p, · · · , q, · · ·to· · · , p + αq, · · · , q, · · ·by multiple linear evolution.

ExampleOne of the important things about linear evolution is that any combinatorial product isunchanged after a linear evolution, as shown by (4.13). So[a1 a2 · · · an ]35

can be changed to[a1 (a2 + αan ) · · · an ]= [a1 a2 · · · an ] + α [a1 an · · · an ] = [a1 a2 · · · an ] ,where a1 , a2 , · · · , an are magnitudes and α is a number. We have the converse, so that if[a1 a2 · · · an ] = [b1 b2 · · · bn ] ,where a1 , a2 , · · · , an and b1 , b2 , · · · , bn are magnitudes, then b1 , b2 , · · · , bn can be derived
from a1 , a2 , · · · , an by linear evolution.
(Grassmann 1862, p. 38-44)

Order of Magnitudes
Original units and other magnitudes that cannot be numerically derived from combinatorial products of multiple magnitudes are called magnitudes of rst order. A magnitude

a that is only numerically derivable by a combinatorial product of n magnitudes of rstorder is called a magnitude of nth order. Magnitudes and units of rst or more degree
can also be called ofhigher order.

Elementary and Compound Magnitude
The magnitude

A is elementary if it can be represented by a single combinatorial

product, meaning

A = [a1 · · · an ]where a1 · · · an are magnitudes of rst order, is an elementary magnitude of nth order. If
a magnitude A is not elementary andA = α [a1 · · · an ] + β [b1 · · · bn ] + · · · ,where a1 , · · · , an , b1 , · · · , bn , · · · are magnitudes of rst order and α, β, · · · are numbers,
then A is acompound magnitude of nth order. As an example, a vector (and thus asum of vectors) and a point are both elementary magnitudes of rst order and a trivector
would be an elementary magnitude of third order.

Side comment:

We will mostly be looking at products with elementary magnitudes,

as the scope of this essay is rather limited and compound magnitudes can make many of
the theorems very messy.

36Magnitude Relations
If we have a magnitude A = [a1 a2 · · · an ], where a1 , · · · , an are elementary factors, thenbelonging to the magnitude A. The
magnitude A is also superordinate, subordinate or incident to other magnitudes as their
the domain derivable from those factors is the domain

domains would be. So if B = [a1 ] and A == [a1 a2 · · · an ], then A and B are incident, B
is subordinate to A and A is superordinate to B . As shown previously, the combinatorial
product of two incident magnitudes is always zero.

Side comment:
the word

It should also be noted that there is a natural inclination to equate

order with the word dimension, but this should be avoided as they are not

the same and the meaning of order diers according to context (meaning it is dierent
3
for magnitudes than for domains). In modern vector calculus space or R has three3
dimensions. Any vector in R can be dened by a linear combination of the three basevectors ej (j= 1,2 or 3), so if we have a vector a, then a = (x1 , x2 , x3 ) = x1 e1 +x2 e2 + x3 e3 for some real numbers x1 , x2 , x3 . In Extension Theory, you can also dene3
any displacement (as vectors are called in Extension Theory) in space (i.e. R ) withthree unit displacements.

But points are also magnitudes of rst order, and with only

displacements, you can never form a point (When you add a point and a displacement,
you make the displacement starting from that point and get the end point. So point (1, 1)
added to vector (1, 1) gives point (2, 2) ). And so you need four units e1 , e2 , e3 , e4 to form
space, for example, the origo and the three base vectors. And so the domain belonging
3
to them, (i.e. space or R ) is a domain of fourth order and a magnitude a in space would
be a = α1 e1 + α2 e2 + α3 e3 + α4 e4 .
For example, as stated, the collection of all magnitudes of rst order in space is adomain of fourth order, with e1 , e2 , e3 and e4 its units of rst order. And the collectionof all magnitudes of second order in space is a domain of sixth order, with [e1 e2 ], [e1 e3 ],[e1 e4 ], [e2 e3 ], [e2 e4 ] and [e3 e4 ] as its units of second order.
(Grassmann 1862, p. 44-45)

Outer Product
If you have elementary magnitudes A = a1 a2 · · · am and B = am+1 · · · an , then tooutermultiply them is to combinatorially multiply their elementary factors in order withoutchanging their position, meaning[AB] = [(a1 a2 · · · am ) (am+1 · · · an )] = [a1 a2 · · · an ] .

Side comment:

Grassmann's reasoning for calling it outer multiplication was that

[AB] 6= 0 only if A was outside of the domain dened by B .
37This also made it theopposite of inner multiplication.

Outer Product Properties
Here is a short list of some of the properties of the outer product.

• If A and B are nonzero elementary magnitudes, and A is subordinate to B , then B
can be presented as the outer product of A and some elementary magnitude C sothat

B = [AC] .
• Outer multiplication is also associative and so for magnitudes A, B and C[A (BC)] = [ABC] .
• Let elementary magnitudes a1 , a2 , · · · , an , b1 , b2 , · · · , bm stand in no numerical relation. Now if A is formed of a1 , a2 , · · · , an by summation and/or multiplication (asan example, A = [a1 a2 + a3 · · · an ]) and B similarly from b1 , b2 , · · · , bm . Now if theouter product[AB] = 0,then either A = 0 or B

= 0. Also if S = A1 + A2 + · · · is a sum of elementarymagnitudes (which we will call Aj ), and the outer product[aS] = 0,
for some magnitude of rst order a, then a is incident to all the magnitudes Aj andthere exists a magnitude P such thatS = [aP ] .
If instead

0 = [a1 S] = [a2 S] = · · ·for m outer products and magnitudes of rst order not standing in a numerical
relation, a1 , a2 , · · · , am thenS = [a1 a2 · · · am P ] ,for some magnitude P .
(Grassmann 1862, p. 45-48)

38

Principal Domain
We call the domain of all the magnitudes under consideration as theprincipal domain. If

the sum of the orders of two elementary magnitudes A and B is more than the order ofthe principal domain by γ , thenA = [CA1 ]and

B = [CB1 ] ,for some elementary magnitudes A1 , B1 and C and C has order γ .

Supplement
Let the principal domain be of nth order and e1 , e2 , · · · , en be its original units and E aunit of arbitrary order (meaning thatE can be one of the original units or a combinatorial
product of some of them). Now, if we set[e1 e2 · · · en ] = 1(4.14)and E 'is the combinatorial product of the units not incident to E , then[EE 0 ] = ±1.
Then thesupplement of E , marked as| E is ±E 0 so that
[E | E] = 1

or said in another way| E = [EE 0 ] E 0 .
This means, naturally, that the supplement can only be dened in a principal domain.

Properties of the Supplement
Here is a short list of some of the properties of the supplement.

• The supplement of a number is set as the number, so | α = α if α is a number.

39• The supplement of an arbitrary magnitude A = α1 E1 + α2 E2 + · · · derived fromunits E1 , E2 , · · · by numbers α1 , α2 , · · · is dened as| A =| (α1 E1 + α2 E2 + · · · ) = α1 | E1 + α2 | E2 + · · · .
This means that in a principal order of nth order, the supplement of a magnitudewith order m, has order n − m.

• If q is the order of a magnitude A and r is the order of its supplement, then|| A = (−1)qr A.
q
If the principal domain is odd, then || A = A, otherwise || A = (−1) A.

Side comment:Denition (4.14) changes how the outer product works considerably.

For example, in a normal outer product[(e1 e2 ) (e2 e3 )] = 0,
where e1 , e2 and e3 are original units. But if the principal domain is of third order and
we set [e1 e2 e3 ] = 1,[(e1 e2 ) (e2 e3 )] = − [e1 e2 e3 e2 ] = −e2 .
And so, when denition (4.14) is enforced, Grassmann's outer product works somewhat
dierently than its modern counterpart.
(Grassmann 1862, p. 49-52)

Progressive and Regressive Product
Let E1 and E2 are units of orders α and β respectively in a principal domain of order n.
If α + β ≤ n, then theirprogressive product is the outer product as long as we take theprogressive product of all original units to be 1. If on the other hand α + β > n, then theirregressive product is the magnitude that is the supplement to the progressive product of
their supplements. Both progressive and regressive products are products relative to aprincipal domain. And so in a relative product we always take the progressive product of
the original units to equal one.

ExampleIf n = 3, E1 = e1 e2 and E2 = e2 e3 , then the regressive product of E1 and E2 is[E1 E2 ] =| [| E1 | E2 ] =| [| (e1 e2 ) | (e2 e3 )] =| [e3 e1 ] = −e240which is the same as[E1 E2 ] = [(e1 e2 ) (e2 e3 )] = − [e1 e2 e3 e2 ] = −e2 .
(Grassmann 1862, p. 52-53)

41

4.4

Inner ProductLastly we will be taking a quick look at the inner product in Extension Theory. In normal
vector calculus, the basic calculations using inner product are relatively simple operations,while calculations using the cross product are more dicult. In Extension Theory, it is
the opposite, at least as far as denitions go. This is partly because in Extension Theory,the inner product is dened using the outer product.

It is also a fact that the innerproduct of two magnitudes is considerably simpler if you assume the magnitudes to beof the same order, otherwise the inner product is not commutative. This section will beseverely condensed with many parts of the original book skipped as it is only meant togive a basic grasp of the basics as anything more would be outside the scope of this essay.

Inner ProductLet A and B be magnitudes of arbitrary order.

Their

inner product is dened as therelative product[A | B] .

Side comment:

It is very interesting, that in Extension Theory, there is no separate

symbol for the inner product.

It is somewhat less general, as it is always relative innature. In his previous work Grassmann had a separate symbol for it, but as he found a
way to represent the product with the supplement, he omitted it from A2. Amusingly, heoriginally chose the modern symbol for cross product (×), which would no doubt confuse
modern readers.

Inner Product Properties
We will now go through some of the basic properties of the inner product as a quick glance
to the similarities and dierences to modern vector calculus.

• Let A be and B be magnitudes of ordersα and β respectively, in a principal domainof order n. Then their inner product has order α − β if α ≥ β and order n + α − βotherwise. This means that if A and B are of the same order, then their innerproduct is of order zero and so a number and also commutative.

• The inner product of a unit with the same unit is one[Er | Er ] = 1.

42• Otherwise the inner product of units of the same order is zero . Meaning that forunits of the same order Er and Es order[Er | Es ] = 0,
when r 6= s.

• If Er and Es are units of arbitrary order and[Er | Es ] 6= 0,
then they are incident.

ExampleLet e1 , · · · , en be the original units of a principal domain of order 3. Now[(e1 e2 ) | (e2 e1 )] = − [(e1 e2 ) (e3 )] = −1
and[(e2 e1 ) | (e1 e2 )] = [(e2 e1 ) (e3 )] == − [(e1 e2 ) (e3 )] − 1But,[(e1 e2 ) | (e2 )] = [(e1 e2 ) (e1 e3 )] = −e1and[(e2 ) | (e1 e2 )] = [e2 e3 ] .

Normal and Completely Normal
normal to one another. Two domains arecompletely normal to one another if the inner product of all magnitudes of the rst order
If their inner product is zero, Er and Es arefrom one of the domains is normal to all similar magnitudes from another.

Inner Square and Numerical Value

We say that

[A | A] = A2
is called theinner square of A and if A = α1 E1 + · · · + αn En , thenA2 = α12 + · · · + αn2 .
43

We also say thatis the√A2 =q
α12 + · · · + αn2numerical value of A.

(Grassmann 1862, p. 93-98)

The Angle

If magnitudes A and B are of the same order, which is not zero, then ∠AB(the angle AB ) is dened so thatcos ∠AB =[A | B],αβwhere 0 ≤ ∠AB ≤ π and α and β are the numerical values of A and B . This gives us themore recognizable form

[A | B] = αβ cos ∠AB.
And it is also true that if a and b are magnitudes of rst order, then[ab]2 = (αβ sin ∠AB)2 ,where α and β are the numerical values of a and b.
(Grassmann 1862, p. 117-119)

44

Chapter 5
Conclusions
The comparison of Grassmann to Hamilton is hard to avoid as they both came from exceedingly dierent backgrounds. As an example, the title page of Hamilton's  Lectures

on Quaternions read:SIR WILLIAM ROWAN HAMILTON, LL.D., M. R. I. A., FELLOW
OF THE AMERICAN SOCIETY OF ARTS FOR SCOTLAND; OF THE
ROYAL ASTRONOMICAL SOCIETY OF LONDON; AND OF THE ROYALNORTHERN SOCIETY OF ANTIQUARIES AT COPENHAGEN; CORRESPONDING MEMBER OF THE INSTITUTE OF FRANCE; HONORARY
OR CORRESPONDING MEMBER OF THE IMPERIAL OR ROYAL ACADEMIES
OF ST. PETERSBURGH, BERLIN, AND TURIN; OF THE ROYAL SOCIETIES OF EDINBURGH AND DUBLIN; OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY; THE NEW YORK HISTORICAL SOCIETY; THE
SOCIETY OF NATURAL SCIENCES AT LAUSANNE; AND OF OTHERSCIENTIFIC SOCIETIES IN BRITISH AND FOREIGN COUNTRIES; ANDREWS PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF DUBLIN;
AND ROYAL ASTRONOMER OF IRELAND.
While the title page of  Ausdehnungslehre of 1844 read:Hermann Graÿmann
Lehrer an der Friedrich-Wilhelms-Schule zu Stettin
And the title page of  Ausdehnungslehre of 1862 read:Hermann Graÿmann
Professor am Gymnasium zu Stettin

45

One was a prodigy, immensely respected who everyone thought was destined for greatness.
And the other was a school teacher who came in to the profession of mathematics relatively
late and who had no fame to speak of. Hamilton was immensely respected when he createdthe quaternions. He was a man people expected to revolutionize mathematics. Yet he wasunable to present his work in a form that others could understand, at least in writing.
It was only after his students started writing about quaternions that the system truly
became used.
In contrast, no one expected a mathematical revolution from Hermann Grassmann.
As Hamilton, Grassmann failed to present his work in a way understandable to his peers,
but even more unfortunately, he lacked the position and fame necessary to inspire others.
Strangely the dening dierence may be that Grassmann never could teach his system tostudents. As he never received a post at a university he could never propagate his system
personally. But it is also likely that many of his peers, like Gauss, might have tried tounderstand his writing more, had he more fame.
Fortunately in the last 50 years, there has been a growing appreciation of Grassmannand his work as uses for a more general form of vector calculus have become apparent.
And so I will end with a quote from the foreword of A2, written by Grassmann himself:I remain completely condent that the labour I have expended on thescience presented here and which has demanded a signicant part of my life
as well as the most strenuous application of my powers, will not be lost. It istrue that I am aware that the form which I have given the science is imperfectand must be imperfect. But I know and feel obliged to state (though I run therisk of seeming arrogant) that even if this work should again remain unused
for another seventeen years or even longer, without entering into the actual
development of science, still that time will come when it will be brought forthfrom the dust of oblivion and when ideas now dormant will bring forth fruit.
I know that if I also fail to gather around me (as I have until now desired invain) a circle of scholars, whom I could fructify with these ideas, and whom
I could stimulate to develop and enrich them further, yet there will come atime when these ideas, perhaps in a new form, will arise anew and will enter
into a living communication with contemporary developments.

For truth iseternal and divine and no phase of it ... can pass without a trace; it remains
in existence even if the cloth in which weak mortals dress it disintegrates into
dust.
(O'Connor & Robertson 2005)

46

Bibliography[1] Crowe, M. 1967. A History of Vector Analysis. Notre Dame. University of Notre Dame
Press.
[2] Grassmann, H. 1862. Extension Theory. United States of America. American Mathematical Society & London Mathematical Society.
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