# When were vectors invented?

In their modern form, vectors appeared late in the 19th century when Josiah Willard Gibbs and Oliver Heaviside (...) independently developed vector analysis to express the new laws of electromagnetism discovered by ... James Clerk Maxwell.

The phrase "in their modern form" suggests vectors were perhaps used earlier in other forms. There seems to be no deep mathematical idea needed for vectors to be "invented" and used much earlier. Can anyone suggest some earlier uses of what we now call vectors? E.g., did Newton use vectors?

This question was actually discussed on this site several times, for example here: When was the vector notation in physics and other sciences first introduced? It indeed looks strange to modern people that this simple idea came so late. Maxwell never uses vectors in his Treatise on electricity and magnetism, which makes his notation somewhat clumsy.

In fact the idea had a predecessor: quaternions. Yes, quaternions were invented before vectors:-) And Newton did not use vectors in the explicit form.

Instead people thought in very "roundabout", complicated ways about subjects which we routinely treat with vectors nowadays. A striking example is the famous theorem of Apollonius, about "equivalence of excentric and epicycle". Motion on epicycle means that a point moves on a circle around the center (E), while the planet moves on another circle (of smaller radius) around this point. Excentric means that the planet moves on a circle of large radius whose center is different from E, and this center rotates about E (on a circle of small radius). In complex notation, Apollonius theorem says that $$Re^{it(\alpha+\alpha_0)}+re^{it(\beta+\beta_0)}=re^{it(\beta+\beta_0)}+Re^{it(\alpha+\alpha_0)}.$$ To us it seems completely evident that these two things are the same. If you look into Apollonius' proof of this, you see that he really proves commutativity of vector addition in the plane. And Ptolemy praises Apollonius for this.

This frequently happens with simple ideas: after they spread, people start thinking of them as "evident" and wonder how could it happen that their predecessors did not see such simple things. Same thing happened with introduction of matrices to write systems of linear differential equations.

• Not only were quaternions invented before vectors, but also the name "vector" comes from quaternions: Hamilton's "vector part" of a quaternion (1844). As does the notation $\mathbf i, \mathbf j, \mathbf k$ for the standard basis vectors in $\mathbf R^3$. – KCd Apr 17 at 8:04
• Probably your "prises" is "praises". – KCd Apr 17 at 8:04
• Quaternions are a particular instance of vectors (over $\mathbb{R}$), as are complex numbers. So I don't find it surprising that they were "invented" before the general concept of vectors. And what does it mean for someone like Newton to not use vectors in explicit form? That he did not use the same terminology as we? That he did not use the same notation? Or the same visualisation (arrows)? – Michael Bächtold Apr 18 at 6:44
• It may be worth pointing out that Maxwell uses vectors extensively, it's only the notation that is different, as the "vectors" came about in quaternions (and quaternion calculus is where we get most of our vector calculus). But he does call them vectors, and uses them as such. – Sam Gallagher May 24 at 11:56
• @Sam Gallagher: Maxwell clearly understood many things which were formalized only much later. For example, in his introductory chapter he clearly explains the difference between vectors and 1-forms, etc. – Alexandre Eremenko May 24 at 12:19

On the non modern form of "vector"... from mathword

The word VECTOR (which, like the word vehicle, derives ultimately from the Latin vehĕre to carry) was first a technical term in astronomical geometry. The OED’s earliest entry is from a technical dictionary of 1704: J. Harris Lexicon Technicum I. s.v., "A Line supposed to be drawn from any Planet moving round a Center, or the Focus of an Ellipsis, to that Center or Focus, is by some Writers of the New Astronomy, called the Vector; because 'tis that Line by which the Planet seems to be carried round its Center."

Vector usually appeared in the phrase radius vector. The French term was rayon vecteur and can be found in e.g. Laplace’s Traité de mécanique céleste (1799-1825).

The term rayon vecteur is used in a non-astronomical context by Monge "Application de l'Analyse à la Géométrie" (1807). On p. 24 he writes, "on nomme la droite r le RAYON VECTEUR du point, et l'origine des coordonnées devient un pôle, d'où partent les rayons vecteurs des différens points de l'espace." Monge does not seem to use the terminology later in the text, however. Cauchy in his Leçons sur les Applications du Calcul Infinitésimal à la Géométrie of 1826 uses the term freely after introducing it in the initial "Preliminaries" chapter: "Une droite AB, menée d'un point A supposé fixe à un point B suppose mobile, sera généralement désignée sous le nom de rayon vecteur." (p.14) (Information from François Ziegler.) Radius vector appears n English in 1831 in Elements of the Integral Calculus (1839) by J. R. Young: "...when the angle Ω between the radius vector and fixed axis is taken for the independent variable, the formula is...."

Hamilton would create a new meaning for vector (see VECTOR & SCALAR) but he used radius vector in the conventional way in On a General Method in Dynamics Philosophical Transactions Royal Society (part II for 1834, pp. 247-308); see article 14.