# Quaternions Sentence Examples

His earliest work dealt mainly with mathematical subjects, and especially with

**quaternions**(q.v.), of which he may be regarded as the leading exponent after their originator, Hamilton.He was the author of two text-books on them - one an Elementary Treatise on

**Quaternions**(1867), written with the advice of Hamilton, though not published till after his death, and the other an Introduction to**Quaternions**(1873), in which he was aided by Professor Philip Kelland (1808-1879), who had been one of his teachers at Edinburgh.In addition,

**quaternions**was one of the themes of his address as president of the mathematical section of the British Association in 1871.Among his articles may be mentioned those which he wrote for the ninth edition of this Encyclopaedia on Light, Mechanics,

**Quaternions**, Radiation and Thermodynamics, besides the biographical notices of Hamilton and Clerk Maxwell.For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER;

**QUATERNIONS**; VECTOR ANALYSIS.AdvertisementThe sum and product of two

**quaternions**are defined by the formulae mi ase + F+lases = (a s + 133) es 2arer X ZO,es = Fiarfseres, where the products e,e, are further reduced according to the following multiplication table, in which, for example, the eo e1 e2 e3 second line is to be read eieo = e1, e 1 2 = - eo, e i e 2 = es, eie3 = - e2.Thus e 1 e 2 = - e2ei, and if q, q are any two

**quaternions**, qq is generally different from q'q.Thus every quaternion may be written in the form q = Sq+Vq, where either Sq or Vq may separately vanish; so that ordinary algebraic quantities (or scalars, as we shall call them) and pure vectors may each be regarded as special cases of

**quaternions**.As in

**quaternions**, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units.**Quaternions**afford an example of a quadruple algebra of this kind; ordinary algebra is a special case of a duplex linear algebra.AdvertisementVarious special algebras (for example,

**quaternions**) may be expressed in the notation of the algebra of matrices.This applies also to

**quaternions**, but not to extensive quantities, nor is it true for linear algebras in general.**Quaternions**(as a mathematical method) is an extension, or improvement, of Cartesian geometry, in which the artifices of co-ordinate axes, &c., are got rid of, all directions in space being treated on precisely the same terms. It is therefore, except in some of its degraded forms, possessed of the perfect isotropy of Euclidian space.Something far more closely analogous to

**quaternions**than anything in Argand's work ought to have been suggested by De Moivre's theorem (1730).The course of his investigations is minutely described in the preface to his first great work (Lectures on

**Quaternions**, 1833) on the subject.AdvertisementBut in 1877, in the M athematische Annalen, xii., he gave a paper " On the Place of

**Quaternions**in the Ausdehnungslehre," in which he condemns, as far as he can, the nomenclature and methods of Hamilton.More general systems, having close analogies to

**quaternions**, have been given since Hamilton's discovery was published.Had

**quaternions**effected nothing more than this, they would still have inaugurated one of the most necessary, and apparently impracticable, of reforms.The 2 Lectures on

**Quaternions**, ยง 513.Here the symmetry points at once to the selection of the three principal axes as the directions for i, j, k; and it would appear at first sight as if

**quaternions**could not simplify, though they might improve in elegance, the solution of questions of this kind.AdvertisementEven in Hamilton's earlier work it was shown that all such questions were reducible to the solution of linear equations in

**quaternions**; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol.Sufficient has already been said to show the close connexion between

**quaternions**and the theory of numbers.Neither of these men professed to employ the calculus itself, but they recognized fully the extraordinary clearness of insight which is gained even by merely translating the unwieldy Cartesian expressions met with in hydrokinetics and in electrodynamics into the pregnant language of

**quaternions**.We select for description stage (3) above, as the most characteristic development of

**quaternions**in recent years.For (3) (a) we are constrained to refer the reader to Joly's own Manual of

**Quaternions**(1905).AdvertisementClifford makes use of a quasi-scalar w, commutative with

**quaternions**, and such that if p, q, &c., are**quaternions**, when p-I-wq= p'+wq', then necessarily p= p', q = q'.In 1904 Alexander Macfarlane published a Bibliography of

**Quaternions**and allied systems of Mathematics for the International Association for promoting the study of**Quaternions**and allied systems of Mathematics (Dublin University Press); the pamphlet contains 86 pages.Tait's Elementary Treatise on

**Quaternions**appeared (Cambridge).Joly published his Manual of

**Quaternions**(London); the valuable contents of this are doubled by copious so-called examples; every earnest student should take these as part of the main treatise.The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamiltons

**Quaternions**,the tensor expressing the velocity ratio, and the versor the directional relation.He retained his wonderful faculties unimpaired to the very last, and steadily continued till within a day or two of his death, which occurred on the 2nd of September 1865, the task (his Elements of

**Quaternions**) which had occupied the last six years of his life.The other great contribution made by Hamilton to mathematical science, the invention of

**Quaternions**, is treated under that heading.His first great work, Lectures on

**Quaternions**(Dublin, 1852), is almost painful to read in consequence of the frequent use of italics and capitals.Rowan Hamilton, in the preface to his Lectures on

**Quaternions**, refers more than once to those papers as having led and encouraged him in the working out of the new system of**quaternions**.Under the general heading "Fundamental Notions" occur the subheadings "Foundations of Arithmetic," with the topics rational, irrational and transcendental numbers, and aggregates; "Universal Algebra," with the topics complex numbers,

**quaternions**, ausdehnungslehre, vector analysis, matrices, and algebra of logic; and "Theory of Groups," with the topics finite and continuous groups.The effect of these definitions is that the sum and the product of two

**quaternions**are also**quaternions**; that addition is associative and commutative; and that multiplication is associative and distributive, but not commutative.With this notation the values of x and y may be expressed in the forms x q q /N q ', gg /Nq', which are free from ambiguity, since scalars are commutative with

**quaternions**.In England, multiple algebra was developed by j ames Joseph Sylvester, who, in company with Arthur Cayley, expanded the theory of matrices, the germs of which are to be found in the writings of Hamilton (see above, under (B); and

**Quaternions**).