The parallelogram parallelogram law for for the addition of vectors is so intuitive that its origin origin is unknown. It may have appeared in a now lost work of Aristotle Aristotle (384--3 (384--3 !.".#$ and it is in the %echanics of &eron ('rst century century A..# of Ale)andria Ale)andria.. It was also the the 'rst corollary in Isaac *ewton+s (,4--,# /rincipia %athematica (,8#. In the /rincipia$ *ewton dealt e)tensively with what are now considered vectorial entities (e.g.$ velocity$ force#$ 0ut never the concept of a vector. The systematic study and use of vectors were a ,1th and early 2th century phenomenon. ectors were 0orn in the 'rst two decades of the ,1th century with the geometric representations of comple) num0ers. "aspar essel (,45--,8,8#$ 6ean (,45--,8,8#$ 6ean 7o0ert 7o0ert Argand (,8--,8#$ "arl riedrich 9auss (,--,855#$ (,--,855#$ and at least one or two others conceived of comple) num0ers as points in the two-dimensional plane$ i.e.$ as two-dimensional vectors. %athematicians %athematician s and scientists worked with with and applied these new num0ers in various ways: for e)ample$ 9auss made crucial use of comple) num0ers to prove the undamental undamental Theorem of Alge0ra (,11#. In ,83$ illiam 7owan &amilton (,825-,85# showed that the comple) num0ers could 0e considered a0stractly as ordered pairs (a$ 0# of real num0ers. This idea was a part of the campaign of many mathematicians$ including &amilton himself$ to search for a way to e)tend the two-dimensional ;num0ers; to three dimensions : 0ut no one was a0le to accomplish this$ while preserving the 0asic alge0raic properties of real and comple) num0ers. In ,8$ August erdinand %<0ius pu0lished a short 0ook$ The !arycentric "alculus$ in which he introduced directed line segments that he denoted 0y letters of the alpha0et$ vectors in all 0ut the name. In his study of centers of gravity and pro=ective geometry$ geometry$ %<0ius developed an arithmetic of these directed line segments: he added them and he showed how to multiply them 0y a real num0er. &is interests were elsewhere$ elsewhere$ however$ and no one else 0othered to notice the importance of these computations. After a good deal of frustration$ &amilton &amilton was was 'nally inspired to give up the search for such a three-dimensional ;num0er; system and instead he invented a fourdimensional system that he called >uaternions. In his own words? @cto0er ,$ ,843 $ which happened to 0e a %onday$ and a "ouncil day of the 7oyal Irish Academy I was walking in to attend and preside$ B$ along the 7oyal "anal$ B an under-current of thought was going on in my mind$ which at last gave a result$ whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close: and a spark Cashed forth$ B I could not resist the impulse B to cut with a knife on a stone of !rougham !ridge$ as we passed it$ the fundamental formulaeB. &amilton+s >uaternions were written$ > D w E i) E =y E kF $ where w$ )$ y$ and F were real num0ers. &amilton >uickly realiFed that his >uaternions consisted of two distinct parts. The 'rst term$ which he called the scalar and ;)$ y$ F for its three rectangular components$ or pro=ections on three rectangular a)es$ he Greferring to
himselfH has 0een induced to call the trinomial e)pression itself$ as well as the line which it represents$ a "T@7.; &amilton used his ;fundamental formulas$; i D = D k D -i=k D -,$ to multiply >uaternions$ and he immediately discovered that the product$ >,> D - >>,$ was not commutative. &amilton had 0een knighted in ,835$ and he was a well-known scientist who had done fundamental work in optics and theoretical physics 0y the time he invented >uaternions$ so they were given immediate recognition. In turn$ he devoted the remaining years of his life to their development and promotion. &e wrote two e)haustive 0ooks$ Jectures on Kuaternions (,853# and lements of Kuaternions (,8#$ detailing not =ust the alge0ra of >uaternions 0ut also how they could 0e used in geometry. At one point$ &amilton wrote$ ;I still must assert that this discovery appears to me to 0e as important for the middle of the nineteenth century as the discovery of Cu)ions was for the close of the seventeenth.; &e ac>uired a disciple$ /eter 9uthrie Tait (,83,--,12,#$ who in the ,852s 0egan applying >uaternions to pro0lems in electricity and magnetism and to other pro0lems in physics. In the second half of the ,1th century$ Tait+s advocacy of >uaternions produced strong reactions$ 0oth positive and negative$ in the scienti'c community. At a0out the same time that &amilton discovered >uaternions$ &ermann 9rassmann (,821--,8# was composing The "alculus of )tension (,844#$ now well known 0y its 9erman title$ Ausdehnungslehre. In ,83$ 9rassmann 0egan development of ;a new geometric calculus; as part of his study of the theory of tides$ and he su0se>uently used these tools to simplify portions of two classical works$ the Analytical %echanics of 6oseph Jouis Jagrange (,3-,8,3# and the "elestial %echanics of /ierre Limon Japlace (,41-,8#. In his Ausdehnungslehre$ 'rst$ 9rassmann e)panded the conception of vectors from the familiar two or three dimensions to an ar0itrary num0er$ n$ of dimensions: this greatly e)tended the ideas of space. Lecond$ and even more generally$ 9rassmann anticipated a good deal of modern matri) and linear alge0ra and vector and tensor analysis. Mnfortunately$ the Ausdehnungslehre had two strikes against it. irst$ it was highly a0stract$ lacking in e)planatory e)amples and written in an o0scure style with an overly complicated notation. ven after he had given it serious study$ %<0ius was not a0le to understand it fully. Lecond$ 9rassmann was a secondary school teacher without a ma=or scienti'c reputation (compared to &amilton#. ven though his work was largely ignored$ 9rassmann promoted its message in the ,842s and ,852s with applications to electrodynamics and to the geometry of curves and surfaces$ 0ut without much general success. In ,8$ 9rassmann pu0lished a second and much revised edition of his Ausdehnungslehre$ 0ut it too was o0scurely written and too a0stract for the mathematicians of the time$ and it met essentially the same fate as his 'rst edition. In the later years of his life$ 9rassmann turned away from mathematics and launched a second and very successful research career in phonetics and comparative linguistics. inally$ in the late ,82s and ,82s$ the Ausdehnungslehre slowly 0egan to 0e understood and appreciated$ and 9rassmann
0egan receiving some favora0le recognition for his visionary mathematics. A third edition of the Ausdehnungslehre was pu0lished in ,88$ the year after 9rassmann died. uring the middle of the nineteenth century$ !en=amin /eirce (,821--,882# was far and away the most prominent mathematician in the Mnited Ltates$ and he referred to &amilton as$ ;the monumental author of >uaternions.; /eirce was a professor of mathematics and astronomy at &arvard from ,833 to ,882$ and he wrote a massive Lystem of Analytical %echanics (,855: second edition ,8#$ in which$ surprisingly$ he did not include >uaternions. 7ather$ /eirce e)panded on what he called ;this wonderful alge0ra of space; in composing his Jinear Associative Alge0ra (,82#$ a work of totally a0stract alge0ra. 7eportedly$ >uaternions had 0een /eirce+s favorite su0=ect$ and he had several students who went on to 0ecome mathematicians and who wrote a good num0er of 0ooks and papers on the su0=ect. 6ames "lerk %a)well (,83,--,81# was a discerning and critical proponent of >uaternions. %a)well and Tait were Lcottish and had studied together in din0urgh and at "am0ridge Mniversity$ and they shared interests in mathematical physics. In what he called ;the mathematical classi'cation of physical >uantities$; %a)well divided the varia0les of physics into two categories$ scalars and vectors . Then$ in terms of this strati'cation$ he pointed out that using >uaternions made transparent the mathematical analogies in physics that had 0een discovered 0y Jord Nelvin (Lir illiam Thomson$ ,84--,12# 0etween the Cow of heat and the distri0ution of electrostatic forces. &owever$ in his papers$ and especially in his very inCuential Treatise on lectricity and %agnetism (,83#$ %a)well emphasiFed the importance of what he descri0ed as ;>uaternion ideas B or the doctrine of ectors; as a ;mathematical method B a method of thinking.; At the same time$ he pointed out the inhomogeneous nature of the product of >uaternions$ and he warned scientists away from using ;>uaternion methods; with its details involving the three vector components. ssentially$ %a)well was suggesting a purely vectorial analysis. illiam Ningdon "liOord (,845--,81# e)pressed ;profound admiration; for 9rassmann+s Ausdehnungslehre and clearly favored vectors$ which he often called steps$ over >uaternions. In his lements of ynamic (,88#$ "liOord 0roke down the product of two >uaternions into two very diOerent vector products$ which he called the scalar product (now known as the dot product# and the vector product (today we call it the cross product#. or vector analysis$ he asserted ;G%Hy conviction GisH that its principles will e)ert a vast inCuence upon the future of mathematical science.; Though the lements of ynamic was supposed to have 0een the 'rst of a se>uence of te)t0ooks$ "liOord never had the opportunity to pursue these ideas 0ecause he died >uite young. The development of the alge0ra of vectors and of vector analysis as we know it today was 'rst revealed in sets of remarka0le notes made 0y 6. illard 9i00s (,831--,123# for his students at Pale Mniversity. 9i00s was a native of *ew &aven$
"onnecticut (his father had also 0een a professor at Pale#$ and his main scienti'c accomplishments were in physics$ namely thermodynamics. %a)well strongly supported 9i00s+s work in thermodynamics$ especially the geometric presentations of 9i00s+s results. 9i00s was introduced to >uaternions when he read %a)well+s Treatise on lectricity and %agnetism$ and 9i00s also studied 9rassmann+s Ausdehnungslehre. &e concluded that vectors would provide a more eQcient tool for his work in physics. Lo$ 0eginning in ,88,$ 9i00s privately printed notes on vector analysis for his students$ which were widely distri0uted to scholars in the Mnited Ltates$ !ritain$ and urope. The 'rst 0ook on modern vector analysis in nglish was ector Analysis (,12,#$ 9i00s+s notes as assem0led 0y one of his last graduate students$ dwin !. ilson (,81--,14#. Ironically$ ilson received his undergraduate education at &arvard (!.A. ,811# where he had learned a0out >uaternions from his professor$ 6ames %ills /eirce (,834--,12#$ one of !en=amin /eirce+s sons. The 9i00sRilson 0ook was reprinted in a paper0ack edition in ,12. Another contri0ution to the modern understanding and use of vectors was made 0y 6ean renet (,8,--,112#. renet entered Scole normale suprieure in ,842$ then studied at Toulouse where he wrote his doctoral thesis in ,84. renetUs thesis contains the theory of space curves and contains the formulas known as the renetLerret formulas (the T*! frame#. renet gave only si) formulas while Lerret gave nine. renet pu0lished this information in the 6ournal de mathemati>ue pures et appli>ues in ,85. In the ,812s and the 'rst decade of the twentieth century$ Tait and a few others derided vectors and defended >uaternions while numerous other scientists and mathematicians designed their own vector methods. @liver &eaviside (,852-,15#$ a self-educated physicist who was greatly inCuenced 0y %a)well$ pu0lished papers and his lectromagnetic Theory (three volumes$ ,813$ ,811$ ,1,# in which he attacked >uaternions and developed his own vector analysis. &eaviside had received copies of 9i00s+s notes and he spoke very highly of them. In introducing %a)well+s theories of electricity and magnetism into 9ermany (,814#$ vector methods were advocated and several 0ooks on vector analysis in 9erman followed. ector methods were introduced into Italy (,88$ ,888$ ,81#$ 7ussia (,12#$ and the *etherlands (,123#. ectors are now the modern language of a great deal of physics and applied mathematics and they continue to hold their own intrinsic mathematical interest.
