History notes on vectors

Coordinates describe the position of any location on a map-
this is the early use of vectors and even without calculating
you can measure differences of direction and position.

1572 book-keeping was highly 
developed in northern Italy,
but even "simple" negative numbers were just introduced 
(and the + and - signs unknown). So the hydraulic ingenieur 
Bombelli  wrote a poem about "piu" and "meno" to teach 
calculating these.  In his book "L Algebra" he didn't try to 
solve x≤+1=0 any longer; instead he recognized the "necessarissimi"
existence of squareroots of negative numbers and introduced
Just insert numbers and you can calculate every combination.
So he introduces them as new members to the family of numbers, 
or, more precise, of quantities, lets say of a different branch (we 
express this by the word adjungate or adjoin). In modern words 
he is calculating vectors. For him, it was not an abstract construction.
Solving equations were done with geometric constructions and
Bombelli used L-shaped rulers for this:

By the sign-rules you have a generating set of the first vectorspace 
ever  and - i repeat - you can calculate inside, whatever you want,
like 3+4* piu di meno 36 (=3+24*i).When you calculate
a lot, the sign-rules -with the insert of the number 1 -will lead you 
to a smaller generating set {1,-1,sqrt(-1),-sqrt(-1)}.
The basis is {1,sqrt(-1)}, but this you can't expect,when signs 
are unknown. So you have a 2D-vector-space over R, the (R2,+,*).
You can replace sqrt(-1) by anything, letís say the letter
i or euro-sign, it will stay the same vector-space.
Like any real number is 1 a directed quantity, not necessarily
of space (3D) or time, just directed by counting and from
this and the addition you can get minus, the opposite direction.
There is no difference between vector 1 and number 1. Real
numbers are scalars and vectors in one. R or R1 is also a
Vector-space, with a basiselement 1.
Every field is a vectorspace over any sub-field, which is regarded
as the scalar-range, and by identifying the operations respectively.
You call the sub-field embedded in the field.

15.. Cardan called the positive squareroot of minus one
the impossible or fictious number.
1... Descartes called it the imaginary number.
1679 Leibniz opens the quest for a geometrie of
position/situation. He first used the word coordinate axis.
The earth-axis from north- to southpole is
sometimes called "imaginary" (eine gedachte Linie).
1687 Newton Parallelogram of forces

1714 Cotes i*@ = ln (cos @ +i*sin @ )

1728 de Moivre (cos a + sqrt(-1)*sin a) ^n =
(cos n*a+sqrt(-1)*sin n*a)
1747ff Euler knows the polar coordinates and how to
calculate to the power of e and with the logarithm and
the power-series for e and sin.
1777 Euler writes " i " for squareroot minus one.

When you know, that the multiplication of vectors is
geometrically a rotation, then an associative question
arises: Why Leibniz began using the word axis in
connection with coordinates?
Cardan invented the Cardan suspension with three axes
and Descartes developed the cartesian coordinates.
Learning starts with imitating, so arises the question:
How could they do it ? Did they do it all abstract, or
did they had a visualisation of the vectors? An angle is
very geometric.

1797 Caspar Wessel writes "On the analytical representation
of direction", i like this short text.
A length gets orientation by assigning letters to its start- and
endpoints in alphabetical order - we use an arrow-head for
this. The addition of lenghths is done by connecting the starting 
point of the second to the endpoint of the first, without changing
their directions. And he represents his directed lengths analytically 
by a+euro*b, where euro-sign is perpendicular left-orientated to a given
reference-direction. Representing a moving point like this we call 
now translation (A vector-arrow is like a photo with the shutter 
open a time: a moving object is unsharp against the sharp background). 
"Changes of direction" are done by multiplication.
Everything - be aware - is on a simple and ordinary plane.
The problem of 2D and a plane solved, Wessel attempts
in the second part of his treatise 3D. As a surveyor is he
interested in analytical calculations of the surface of earth,
a sphere.
Caspar Wessel's text is still out of reach of most students, let 
alone by price. Are the "keeper of knowledge" afraid ?
1897 first translation into french, 1929 into english.
Martin A.Nordgaard, "A source book in Mathematics",
ed.D.E.Smith (New York:McGraw Hill, 1929;Havard University
Press). A selection was published in
"The Treasury of Mathematics" by Henrietta Midonick,
1965 ff.
First complete english translation 1999 (!):
ISBN:87-7876-158-1 (4$ per pagina)
First part in german

1806 Argand and others
1814 Cauchy called certain functions from R2 to R2
"monotypique", Briot and Bouquet later called these

1835 Hamilton , a new kind of notation, ordered
pairs of numbers. The addition is by components and the
multiplication looks a bit more complicated:
(a,b)*(c,d)=(a*c-b*d,a*d+b*c), than the simple: i*i= - 1.
They have both have their advantages. Hamiltons is easy
expandable to many mathematical dimensions.
Hamilton "Theory of Algebraic Couples"

From the time of Wessel on you can talk of the representation
on an ordinary or real plane, and from Hamilton on, you can
talk of calculating with these numbers inside the R2,
the set of ordered pairs of real numbers.
And if you look for a strict definition of C, you only can find,
that it is (R2,+,*), a vector-space and commutative field - i stick
to this! Why C still lingers on (with imaginary axis, complex
plane, Gauss-Ebene or Argand diagramm)?
i only have an emotional answer.

1840 Grassmann created the dot-product and an "outer
product", somehow similar to the cross -product.
1843 Hamilton solved the 3D-problem, itís similar to a
work of art, like an irish Aaran-pullover. The translation
is done by addition ,i.e addition of components. To do
3D-rotations, he quested for a multiplication. First he
developed the division, than he got the solution for the
product: i*j*k= - 1 , where i,j,k represent the unit-arrows
of the three perpendicular directions. That sounds easy, but
a rotation is done by a transformation of a vector v (he 
coined this word 1846 ) by a quaternion q, a 4D-vector, 
which has a scalar-part (scalar is another word from 
Hamilton) and a vector-part.
When v=a*i+b*j+c*k and q= r+s*i+t*j+u*k
with n= (sqrt(r≤+s≤+t≤+u≤) ,then q is too:
q=n*(cos @ +sin @*(s*i+t*j+u*k)/(sqrt(s≤+t≤+u≤))).
A rotation is done by this transformation :
v after rotation =q*v/q,
but - be aware - the angle of rotation is 2*@.
Multiply the vector-parts of two quaternions v and w with
v*w = - v dot w + v cross w, a scalar-part +a vector-part
All papers of Hamilton edited by Dr. D.R.Wilkins

He started drawing vectors with an arrow alongside

 1843 Cayley writes about Algebra in n-dimensions
1844 Grassmann writes about vector-spaces of n dimensions.
1851 Riemann still uses the words x-axis and y-axis in his "Dissertation"
The words "imaginary axis" "Gauss Ebene"or "complex plane"
still not in use.
1858 Arthur Cayley,"A Memoir on the Theory of Matrices"
Matrix-multiplication alike the dot-product of a row of the first by a
column of the second matrix. 
3D-Rotation of any
vector around an axis of the direction (a,b,c) by an angle @"
and how to "Extract axis and angle out of a rotation -matrix"
3D-matrices are quite comfortable, but numerically instable for
angles around 90 degrees. Here are Quaternions better.
18.. oder 19.. with the norm | a+i*b| you can get a metrical
vector-space (R2,+,r.s.m.)
18.. oder 19.. with the dot-multiplication you get the
euklidian vector-space (R2,+,r.s.m,dot)
1873 (or 1881) Maxwell gives a short impression of the
state of the art of calculating vectors.J.C.Maxwell, 
"A Treatise on Electricity and Magnetism",
Vol.I, Preliminary, "On the Relation of Physical Quatities
To Directions in Space" (just 5 pages)

2003 Is there another representation of (R2,+,*) , letís say in
bookkeeping or chemistry -or is the real plane the only one ?