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Hamilton's Quaternions.

00Hamilton and quaternions.

00His 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.

00He 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.

00In addition, quaternions was one of the themes of his address as president of the mathematical section of the British Association in 1871.

00Among 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.

00For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS.

00The 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.

00Thus e 1 e 2 = - e2ei, and if q, q are any two quaternions, qq is generally different from q'q.

00Thus 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.

00Quaternions).

00All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.

00As 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.

00Quaternions afford an example of a quadruple algebra of this kind; ordinary algebra is a special case of a duplex linear algebra.

00Various special algebras (for example, quaternions) may be expressed in the notation of the algebra of matrices.

00This applies also to quaternions, but not to extensive quantities, nor is it true for linear algebras in general.

00Hamilton, Lectures on Quaternions (Dublin, 1853), Elements of Quaternions (ibid., 1866); H.

00QUATERNIONS, in mathematics.

00Quaternions (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.

00The evolution of quaternions belongs in part to each of two weighty branches of mathematical history - the interpretation of the imaginary (or impossible) quantity of common algebra, and the Cartesian application of algebra to geometry.

00Something far more closely analogous to quaternions than anything in Argand's work ought to have been suggested by De Moivre's theorem (1730).

00The course of his investigations is minutely described in the preface to his first great work (Lectures on Quaternions, 1833) on the subject.

00we ought to regard these as only imperfect forms of Quaternions,.

00He had now three distinct space-units, i, j, k; and the following conditions regulated their combination by multiplication: - I T = 12 '=' 2 = _ 1, ij= - ji=k, jk= - kj=i, ki= - ik =j.3 And now the product of two quaternions could be at once expressed as a third quaternion, thus (a+ib+jc+kd) (a'+ib'+jc'+kd') = A+iB+jC+kD, where A=aa' - bb' - cc' - dd', B = ab'+ba'+cd' - dc', C = ac'+ca'+db' - bd', D =ad' +da'+bc' - cb'.

00But 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.

00More general systems, having close analogies to quaternions, have been given since Hamilton's discovery was published.

00- The above narrative shows how close is the connexion between quaternions and the ordinary Cartesian space-geometry.

00to the resources of ordinary algebra, for the solution of equations in quaternions."

00Had quaternions effected nothing more than this, they would still have inaugurated one of the most necessary, and apparently impracticable, of reforms.

00The 2 Lectures on Quaternions, § 513.

00Here 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.

00Even 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.

00Sufficient has already been said to show the close connexion between quaternions and the theory of numbers.

00Neither 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.

00- There are three fairly wellmarked stages of development in quaternions as a geometrical method.

00method quaternions have from the beginning received much attention from mathematicians.

00We select for description stage (3) above, as the most characteristic development of quaternions in recent years.

00For (3) (a) we are constrained to refer the reader to Joly's own Manual of Quaternions (1905).

00Clifford 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'.

00In 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.

00Hamilton's classical Elements of Quaternions of 1866 was republished under C. J.

00Tait's Elementary Treatise on Quaternions appeared (Cambridge).

00Joly 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.

00The 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.

00He 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.

00The other great contribution made by Hamilton to mathematical science, the invention of Quaternions, is treated under that heading.

00His first great work, Lectures on Quaternions (Dublin, 1852), is almost painful to read in consequence of the frequent use of italics and capitals.

00Rowan 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.

00Hamilton's Quaternions.

00Hamilton and quaternions.

00His 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.

00He 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.

00In addition, quaternions was one of the themes of his address as president of the mathematical section of the British Association in 1871.

00Among 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.

00Under 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.

00For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS.

00The 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.

00The 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.

00Thus e 1 e 2 = - e2ei, and if q, q are any two quaternions, qq is generally different from q'q.

00Thus 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.

00With 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.

00Clifford's biquaternions are quantities Eq+nr, where q, r are quaternions, and E, n are symbols (commutative with quaternions) obeying the laws E 2 = E, n 2 =,g, = 1 j E=0 (cf.

00All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.

00As 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.

00Quaternions afford an example of a quadruple algebra of this kind; ordinary algebra is a special case of a duplex linear algebra.

00Various special algebras (for example, quaternions) may be expressed in the notation of the algebra of matrices.

00This applies also to quaternions, but not to extensive quantities, nor is it true for linear algebras in general.

00Hamilton, Lectures on Quaternions (Dublin, 1853), Elements of Quaternions (ibid., 1866); H.

00In 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).

00QUATERNIONS, in mathematics.

00Quaternions (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.

00The evolution of quaternions belongs in part to each of two weighty branches of mathematical history - the interpretation of the imaginary (or impossible) quantity of common algebra, and the Cartesian application of algebra to geometry.

00Something far more closely analogous to quaternions than anything in Argand's work ought to have been suggested by De Moivre's theorem (1730).

00The course of his investigations is minutely described in the preface to his first great work (Lectures on Quaternions, 1833) on the subject.

00we ought to regard these as only imperfect forms of Quaternions,.

00He had now three distinct space-units, i, j, k; and the following conditions regulated their combination by multiplication: - I T = 12 '=' 2 = _ 1, ij= - ji=k, jk= - kj=i, ki= - ik =j.3 And now the product of two quaternions could be at once expressed as a third quaternion, thus (a+ib+jc+kd) (a'+ib'+jc'+kd') = A+iB+jC+kD, where A=aa' - bb' - cc' - dd', B = ab'+ba'+cd' - dc', C = ac'+ca'+db' - bd', D =ad' +da'+bc' - cb'.

00But 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.

00More general systems, having close analogies to quaternions, have been given since Hamilton's discovery was published.

00- The above narrative shows how close is the connexion between quaternions and the ordinary Cartesian space-geometry.

00to the resources of ordinary algebra, for the solution of equations in quaternions."

00Had quaternions effected nothing more than this, they would still have inaugurated one of the most necessary, and apparently impracticable, of reforms.

00The 2 Lectures on Quaternions, § 513.

00Here 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.

00Even 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.

00Sufficient has already been said to show the close connexion between quaternions and the theory of numbers.

00Neither 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.

00- There are three fairly wellmarked stages of development in quaternions as a geometrical method.

00method quaternions have from the beginning received much attention from mathematicians.

00We select for description stage (3) above, as the most characteristic development of quaternions in recent years.

00For (3) (a) we are constrained to refer the reader to Joly's own Manual of Quaternions (1905).

00Clifford 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'.

00In 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.

00Hamilton's classical Elements of Quaternions of 1866 was republished under C. J.

00Tait's Elementary Treatise on Quaternions appeared (Cambridge).

00Joly 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.

00The 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.

00He 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.

00The other great contribution made by Hamilton to mathematical science, the invention of Quaternions, is treated under that heading.

00His first great work, Lectures on Quaternions (Dublin, 1852), is almost painful to read in consequence of the frequent use of italics and capitals.

00Rowan 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.

00

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