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 x²+1=0 any longer; instead he recognized the "necessarissimi" existence of squareroots of negative numbers and introduced sign-rules,f.e.: 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: Commentary: 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, let’s say the letter i or , 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. Commentary: 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 , where 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 "holomorphic". 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" Commentary: 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, it’s 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(r²+s²+t²+u²) ,then q is too: q=n*(cos @ +sin @*(s*i+t*j+u*k)/(sqrt(s²+t²+u²))). 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 quaternion-multiplication: 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,+,*) , let’s say in bookkeeping or chemistry -or is the real plane the only one ?-