presents in his place (1 Will. and Mary, sess. 1, c. 26). By 13 Anne c. 13 during the pendency of a quare impedit to which either of the universities is a party in right of the patron being a Roman Catholic the court has power to administer an oath for the discovery of any secret trust, and to order the cestui que trust to repeat and subscribe a declaration against transubstantiation.

QUARLES, FRANCIS (1592–1644), a sacred poet of the 17th century, enjoyed considerable celebrity in his own day, and some of his works have shared in the recent revival of interest in our older literature. The work by which he is best known, his Emblemes, was originally published in 1635, with grotesque illustrations, engraved by Marshall, and borrowed from the Pia Desideria of Hermann Hugo. The poems, which are diffuse meditations upon Scriptural texts, seem, in modern phrase, to have been "written up" to the illustrations, and are quite in keeping with their quaint mixture of sublime and familiar thoughts. An edition of the Emblems, with new illustrations by C. H. Bennett, was published in Edinburgh in 1857, and these illustrations are reproduced in Mr Grosart's complete edition in the Chertsey Worthies Library. The incongruous oddities of Quarles's verse are more obvious than the higher qualities that recent admirers claim for him. The following stanza, in a paraphrase of Job xiv. 13, has often been referred to as the supreme example of his occasional sublimity of thought :

""Tis vain to flee; till gentle Mercy show Her better eye, the farther off we go The swing of Justice deals the mightier blow." Quarles would seem to have been a man of good family, and he boasts of his "long-lived genealogy." He was born at Romford in Essex in 1592, and, after a regular education at school, Christ's College, Cambridge, and Lincoln's Inn, had influence enough at court to get the office of cupbearer to the queen of Bohemia. He was afterwards (about 1621) appointed secretary to Ussher, the primate of Ireland, and later on, returning at an uncertain date to England, obtained (in 1639) the post of city chronologer, which had been held before him by Middleton and Ben Jonson. Upon the outbreak of the Civil War he wrote on the Royalist side, and died in 1644, in consequence, his widow suggests, of his harsh treatment by the king's enemies. Quarles's first publication, with the suggestive title of The Feast of Wormes, appeared in 1620, and from that date till his death he was a busy and prolific writer of verse and prose. His Divine Poems, collected in 1630, were published separately at intervals in the course of the preceding ten years. Divine Fancies followed in 1632; then the Emblems, which might well come under that general designation. The earlier poem of Argalus and Parthenia (1622) was in a different vein: the substance was imitated from Sidney's Arcadia, and the verse from Marlowe's Hero and Leander. The speeches are spun out to a most tedious length, but the poem contains more fine lines and fewer incongruous fancies than any other of the author's productions. Quarles also published many Elegies, in the fashion of memorial verses of which Milton's Lycidas was the contemporary masterpiece. His Hieroglyphics, in the same vein as his Emblems, appeared first in 1638. His principal prose work was the Enchyridion (1640), a collection of four "centuries" of miscellaneous aphorisms. This was followed two years later by more of the same kind "concerning princes and states." Lovers of commonplace wisdom dressed in a garb always studiously quaint and sometimes happily epigrammatic, may turn to Quarles with every prospect of enjoyment. QUARTER SESSIONS (in full, GENERAL QUARTER SESSIONS OF THE PEACE) is the name given to a local court with civil and criminal jurisdiction. In England the

observations

|

court consists in counties of two or more justices of the peace, one of whom must be of the quorum (see JUSTICE OF THE PEACE), in cities and boroughs of the recorder alone. The quarter sessions are a court of record. The records in a county are nominally in the custody of the custos rotulorum, the highest civil officer in the county, practically in that of the clerk of the peace, who is nominated by the custos and removable by the quarter sessions. In a city or borough he is appointed by the council and removable by the recorder. The original jurisdiction of quarter sessions seems to have been confined to cognizance of breaches of the peace. By a series of statutes passed in the reign of Edward III. the time of the holding of the sessions was fixed, and their jurisdiction extended to the trial of felonies and trespasses. The jurisdiction now depends upon a mass of legislation reaching from 1344 to the present time. The dates at which the county sessions must meet are fixed, by 2 Geo. IV. and 1 Will. IV. c. 70, to be the first weeks after the 11th of October, the 28th of December, the 31st of March, and the 24th of June. Quarter sessions in a city or borough depend upon the Municipal Corporations Act, 1882, 45 & 46 Vict., c. 50. A grant of quarter sessions to a city or borough is made by the crown in council on petition of the town council. The main points in which borough differ from county quarter sessions are these: (1) the recorder, a barrister of five years' standing, is sole judge in place of a body of laymen; (2) the recorder has a discretion to fix his own dates for the holding of a court, as long as he holds it once every quarter of a year; it may be held more frequently if the recorder think fit or a secretary of state so direct; (3) the recorder has no power to levy a borough rate or to grant a licence for the sale of exciseable liquors by retail. In some few boroughs the recorder is judge of the borough civil court. Quarter sessions in the counties of Middlesex, Kent, and Lancaster, as also in London and the Cinque Ports, are governed by special legislation.

The jurisdiction of quarter sessions is either original or appellate. Original Jurisdiction.-Civil.-This includes the levying of a county rate and its application, the appointment of a county licensing committee, and of public officers, such as the county treasurer, the public analyst, and the inspectors of weights and measures, the confirmation of bye-laws made by local authorities, the increase or alteration of polling places and petty sessional of Parliament, such as the Highway Acts, and the Contagious divisions, the regulation of police, and powers under various Acts Diseases (Animals) Act. The quarter sessions of the metropolitan counties have by 25 Geo. II. c. 36 the power of licensing music-halls, &c., within 20 miles of London.

Criminal.-Apart from statute, the commissions of justices of trial at assizes. They are now forbidden by several Acts of Parliathe peace provide that they shall reserve the graver felonies for ment to try a prisoner for treason or murder, or for any felony punishable without a previous conviction by penal servitude for life (such as burglary and rape), or for any of the offences enumerated in 5 & 6 Vict. c. 38, the most practically important of which are perjury, forgery, bigamy, abduction, concealment of birth, libel, bribery, and conspiracy. The procedure is by indictment, as at assizes, and the trial of offences by jury. In the case of incorrigible rogues and of sureties of the peace the quarter sessions exercise a recognizances entered into for appearance in a court of summary quasi-criminal jurisdiction without a jury. They may also estreat jurisdiction, but this part of their jurisdiction may be considered practically obsolete, as the Summary Jurisdiction Act, 1879, gives the court of summary jurisdiction power to estreat the recognizances itself.

Appellate Jurisdiction.-Civil.-The principal cases in which this jurisdiction is exercised are in appeals from orders of a court of summary jurisdiction as to the assessment of the poor-rate and the removal and settlement of paupers, and orders made under the Highway, Licensing, and Bastardy Acts.

Criminal.-An appeal lies to quarter sessions from a court of summary jurisdiction only where such an appeal is expressly given by statute. The appellate jurisdiction has been considerably increased by the Summary Jurisdiction Act, 1879 (42 & 43 Vict. conviction or order of a court of summary jurisdiction inflicting c. 49), which allows an appeal (with certain exceptions) from every imprisonment without the option of a fine. The appeal may be

brought in accordance with either the Act giving the appeal or | mycelium of a fungus. The wood has a pure bitter taste, the Summary Jurisdiction Act.

There is no appeal from quarter sessions on the facts, but their decision may be reviewed by the High Court of Justice by means of certiorari, mandamus, prohibition, or a case stated under 12 & 13 Vict. c. 45, § 11. A case may be stated for the opinion of the court of criminal appeal under 11 & 12 Vict. c. 78, § 1. Ireland.--In Ireland the chairman of quarter sessions is a salaried professional lawyer, and has important civil jurisdiction corresponding very much to that of a county court judge in England. His jurisdiction depends chiefly upon 14 & 15 Vict. c. 57. The recorders of Dublin and Cork are judges of the civil bill courts in those cities.

The

Scotland.-In Scotland quarter sessions were established by the Act 1661, c. 338, under which justices were to meet on the first Tuesday of March, May, and August, and the last Tuesday of October to "administrate justice to the people in things that are within their jurisdiction and punish the guilty for faults and crimes done and committed in the preceding quarter.' jurisdiction of quarter sessions in Scotland is more limited than in England, much of what would be quarter-sessions work in England being done by the sheriff or the commissioners of supply. Quarter sessions have appellate jurisdiction in poaching, revenue, and licensing cases, and under the Pawnbrokers and other Acts. All appeals from proceedings under the Summary Jurisdiction Acts are taken to the High Court of Justiciary at Edinburgh or on circuit (44 & 45 Vict. c. 33). The original jurisdiction of quarter sessions is very limited, and almost entirely civil. Thus they have power to divide a county and to make rules for carrying into effect the provisions of the Small Debts Act, 6 Geo. IV. c. 48. The decision of quarter sessions may be reviewed by advocation, suspension, or appeal.

United States. In the United States courts of quarter sessions exist in many of the States; their jurisdiction is determined by State legislation, and extends as a rule only to the less grave crimes. They are in some States constituted of professional judges. QUARTZ, the name of a mineralogical species which includes nearly all the native forms of silica. It thus embraces a great number of distinct minerals, several of

which are cut as ornamental stones or otherwise used in

and is without odour or aroma. It is usually to be met with in the form of turnings or raspings, the former being obtained in the manufacture of the "bitter cups" which are made of this wood. The medicinal properties are due to the presence of quassiin (first obtained by Winckler in 1835), which exists in the wood to the extent of 10th per cent. It is a neutral crystalline substance, soluble in hot dilute alcohol and chloroform and in 200 parts of water. It is also readily soluble in alkalies, and is reprecipitated by acids. It is almost insoluble in ether, and forms an insoluble compound with tannin.

Quassia is used in medicine in the form of infusion and tincture as a pure bitter tonic and febrifuge, and in consequence of containing no tannin is often prescribed in combination with iron. An infusion of the wood sweetened with sugar is also used as a fly poison, and forms an effectual injection for destroying thread worms. Quassia also forms a principal ingredient of several "hop substitutes," for which use it was employed as long ago as 1791, when John Lindsay, a medical practitioner in Jamaica, wrote that the bark was exported to England "in considerable quantities for the purposes of brewers of ale and porter." The word quaternion properly means "a set of four." In employing such a word to denote a new mathematical method, Sir W. R. HAMILTON (q.v.) was probably influenced by the recollection of its Greek equivalent, the Pythagorean Tetractys, the mystic source of all things.

QUATERNIONS.

Quaternions (as a mathematical method) is an extension, or improvement, of Cartesian geometry, in which the artifices of coordinate 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.

From the purely geometrical point of view, a quaternion may be regarded as the quotient of two directed lines in space-or, what comes to the same thing, as the factor, or operator, which changes one directed line into another. Its analytical definition cannot be given for the moment; it will appear in the course of the article.

History of the Method. The 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. Sir W. R. Hamilton was led to his great invention by keeping geometrical applications constantly before him while he endeavoured to give a real significance to √1. We will therefore confine ourselves, so far as his predecessors are concerned, to attempts at interpretation which had geometrical appli

the arts. For a general description of the species, see
MINERALOGY, vol. xvi. p. 389; and for its chief varieties,
see AGATE, vol. i. p. 277; AMETHYST, vol. i. p. 736;
FLINT, vol. ix. p. 325; and JASPER, vol. xiii. p. 596.
The crystallography of quartz has been fully investigated
by Des Cloizeaux in his classical Mémoire sur la cristal-
lisation et la structure intérieure du Quartz, Paris, 1855.
QUASSIA, the generic name given by Linnæus to a
small tree of Surinam in honour of the negro Quassi or
Coissi, who employed the intensely bitter bark of the tree
as a remedy for fever. This bark was introduced into
European medicine about the middle of the last century,
and was officially recognized in the London Pharmacopoeia
of 1788. In 1809 it was replaced by the bitter wood or
bitter ash of Jamaica, Picræna excelsa, Lindl., which was
found to possess similar properties and could be obtained
in pieces of much larger size. Since that date this wood
has continued in use in Britain under the name of quassia
to the exclusion of the Surinam quassia, which, however,
is still employed in France and Germany. Picræna ex-
celsa, Lindl. (Quassia excelsa, Swartz) is a tree 50 to 60
feet in height, and resembles the common ash in appear-
ance. It has imparipinnate leaves composed of four or
five pairs of short-stalked, oblong, blunt, leathery leaflets,
and inconspicuous green flowers. The fruit consists of
shining drupes about the size of a pea. It is found also
in Antigua and St Vincent. Quassia amara, L., is a
shrub or small tree belonging to the same natural order as
Picrana, viz., Simarubacex, but is readily distinguished
by its large handsome red flowers arranged in terminal
clusters. It is a native of Panama, Venezuela, Guiana,
and northern Brazil. Jamaica quassia is imported into
England in logs several feet in length and often nearly
one foot in thickness, consisting of pieces of the trunk
and larger branches. The thin greyish bark is usually
removed. The wood is nearly white, or of a yellowish
tint, but sometimes exhibits blackish markings due to the

cations in view.

One geometrical interpretation of the negative sign of algebra was early seen to be mere reversal of direction along a line. Thus, when an image is formed by a plane mirror, the distance of any point in it from the mirror is simply the negative of that of the corresponding point of the object. Or if motion in one direction along a line be treated as positive, motion in the opposite direction along the same line is negative. In the case of time, measured from the Christian era, this distinction is at once given And to by the letters A.D. or B.C., prefixed to the date. find the position, in time, of one event relatively to another, we have only to subtract the date of the second (taking account of its sign) from that of the first. Thus to find the interval between the battles of Marathon (490 B.C.) and Waterloo (1815 A.D.) we have

is essential to notice that this is by no means necessarily | Cauchy, Gauss, and others, the properties of the expression true of operators. To turn a line through a certain angle in a given plane, a certain operator is required; but when we wish to turn it through an equal negative angle we must not, in general, employ the negative of the former operator. For the negative of the operator which turns a line through a given angle in a given plane will in all cases produce the negative of the original result, which is not the result of the reverse operator, unless the angle involved be an odd multiple of a right angle. This is, of course, on the usual assumption that the sign of a product is changed when that of any one of its factors is changed, -which merely means that - 1 is commutative with all other quantities.

a+b√1 were developed into the immense and most important subject now called the theory of complex numbers (see NUMBERS, THEORY OF). From the more purely symbolical view it was developed by Peacock, De Morgan, &c., as double algebra.

The celebrated Wallis seems to have been the first to push this idea further. In his Treatise of Algebra (1685) he distinctly proposes to construct the imaginary roots of a quadratic equation by going out of the line on which the roots, if real, would have been constructed.

In 1804 the Abbé Buée,1 apparently without any knowledge of Wallis's work, developed this idea so far as to make it useful in geometrical applications. He gave, in fact, the theory of what in Hamilton's system is called Composition of Vectors in one plane-i.e., the combination, by + and of complanar directed lines. His constructions are based on the idea that the imaginaries ±√-1 represent a unit line, and its reverse, perpendicular to the line on which the real units ±1 are measured. In this sense the imaginary expression a+b-1 is constructed by measuring a length a along the fundamental line (for real quantities), and from its extremity a line of length 6 in some direction perpendicular to the fundamental line. But he did not attack the question of the representation of products or quotients of directed lines. The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but expressed in terms of the algebraic imaginary.

In 1806 (the year of publication of Buée's paper) Argand published a pamphlet 2 in which precisely the same ideas are developed, but to a considerably greater extent. For an interpretation is assigned to the product of two directed lines in one plane, when each is expressed as the sum of a real and an imaginary part. This product is interpreted as another directed line, forming the fourth term of a proportion, of which the first term is the real (positive) unit-line, and the other two are the factor-lines. Argand's work remained unnoticed until the question was again raised in Gergonne's Annales, 1813, by Français. This writer stated that he had found the germ of his remarks among the papers of his deceased brother, and that they had come from Legendre, who had himself received them from some one unnamed. This led to a letter from Argand, in which he stated his communications with Legendre, and gave a résumé of the contents of his pamphlet. In a further communication to the Annales, Argand pushed on the applications of his theory. He has given by means of it a simple proof of the existence of n roots, and no more, in every rational algebraic equation of the nth order with real coefficients. About 1828 Warren in England, and Mourey in France, independently of one another and of Argand, reinvented these modes of interpretation; and still later, in the writings of

Argand's method may be put, for reference, in the following
form. The directed line whose length is a, and which makes au
angle with the real (positive) unit line, is expressed by
a(cose +isine),

where i is regarded as +-1. The sum of two such lines (formed
by adding together the real and the imaginary parts of two such
expressions) can, of course, be expressed as a third directed line-
the diagonal of the parallelogram of which they are conterminous
sides. The product, P, of two such lines is, as we have seen, given
by
1: a(cos +isin0):: a'(cose'+isine'): P,
P=aa'{cos(0+0')+isin (0+0')}.

or

Its length is, therefore, the product of the lengths of the factors, and its inclination to the real unit is the sum of those of the factors. If we write the expressions for the two lines in the form A+ Bi, A'+B'i, the product is AA′ – BB′+¿(AB′+BA'); and the fact that the length of the product line is the product of those of

the factors is seen in the form

(A2 + B2)(A22+B′2) = (AA′ – BB′)2+(AB′+BA')2. In the modern theory of complex numbers this is expressed by saying that the Norm of a product is equal to the product of the norms of the factors.

Argand's attempts to extend his method to space generally were fruitless. The reasons will be obvious later; but we mention them just now because they called forth from Servois (Gergonne's Annales, 1813) a very remarkable comment, in which was contained the only yet discovered trace of an anticipation of the method of Hamilton. Argand had been led to deny that such an expression as i could be expressed in the form A + Bi,— although, as is well known, Euler showed that one of its values is a real quantity, the exponential function of — π/2. Servois says, with reference to the general representation of a directed line in space :—

"

L'analogie semblerait exiger que le trinôme fût de la forme pcosa+qcos B+rcos y; a, B, y étant les angles d'une droite avec trois axes rectangulaires; et qu'on eût

(pcosa+qcos8+rcos y)(p'cosa+q'cos B+r'cosy)

= cos 2a + cos 2B+cos2y-1. Les valeurs de p, q, r, p, q, qui
satisferaient à cette condition seraient absurdes; mais seraient-elles
imaginaires, reductibles à la forme générale A+B√-1? Voilà
une question d'analyse fort singulière que je soumets a vos lumières.
La simple proposition que je vous en fais suffit pour vous faire voir
que je ne crois point que toute fonction analytique non réelle soit
vraiment reductible à la forme A+ B-1."
As will be seen later, the fundamental i, j, k of quater-
nions, with their reciprocals, furnish a set of six quantities
which satisfy the conditions imposed by Servois. And it
is quite certain that they cannot be represented by ordin-
ary imaginaries.

Something far more closely analogous to quaternions than anything in Argand's work ought to have been suggested by De Moivre's theorem (1730). Instead of regarding, as Buée and Argand had done, the expression a (cos + i sin 0) as a directed line, let us suppose it to represent the operator which, when applied to any line in the plane in which is measured, turns it in that plane through the angle 0, and at the same time increases its length in the ratio a: 1. From the new point of view we see at once, as it were, why it is true that

(cos + isin 0)-cosme+isin mo. For this equation merely states that m turnings of a line through successive equal angles, in one plane, give the same result as a single turning through m times the common angle. To make this process applicable to any plane in space, it is clear that we must have a special value of i for each such plane. In other words, a unit line, drawn in any direction whatever, must have - 1 for its square. In such XX.

21

星辰

康管

xxyyzz+i(yx+xy')+j(xz' + zx')+ij(yz+zy').

For the square of j, like that of i, was assumed to be
negative unity. But the interpretation of presented a
difficulty, in fact the main difficulty of the whole investi-
gation, and it is specially interesting to see how Hamilton
simpler case, already thoroughly discussed, provided the
attacked it. He saw that he could get a hint from the
two factor lines were in one plane through the real unit
line. This requires merely that
line. This requires merely that

y: z:: y': z'; or yz' — zy' =0;

a system there will be no line in space specially distin- | the coordinate axes are x, y, z. The composition of two
guished as the real unit line: all will be alike imaginary, such lines by the algebraic addition of their several pro-
or rather alike real. We may state, in passing, that every jections agreed with the assumption of Buée and Argand
quaternion can be represented as a(cose + sine),-where for the case of coplanar lines. But, assuming the dis-
a is a real number, a real angle, and a directed unit tributive principle, the product of two lines appeared to
line whose square is -1. Hamilton took this grand step, give the expression
but, as we have already said, without any help from the
previous work of De Moivre. The course of his investiga-
tions is minutely described in the preface to his first great
work on the subject. Hamilton, like most of the many
inquirers who endeavoured to give a real interpretation to
the imaginary of common algebra, found that at least two
kinds, orders, or ranks of quantities were necessary for
the purpose. But, instead of dealing with points on a
line, and then wandering out at right angles to it, as Buée
and Argand had done, he chose to look on algebra as the
science of pure time,2 and to investigate the properties of
"sets
of time-steps. In its essential nature a set is a
linear function of any number of distinct units of the
same species. Hence the simplest form of a set is a
couple; and it was to the possible laws of combination of
couples that Hamilton first directed his attention. It is
obvious that the way in which the two separate time-steps
are involved in the couple will determine these laws of
But Hamilton's special object required that
these laws should be such as to lead to certain assumed
results; and he therefore commenced by assuming these,
and from the assumption determined how the separate
time-steps must be involved in the couple. If we use
Roman letters for mere numbers, capitals for instants of
time, Greek letters for time-steps, and a parenthesis to
denote a couple, the laws assumed by Hamilton as the
basis of a system were as follows:-

a

For

but then the product should be of the same form as the
separate factors. Thus, in this special case, the term in
j ought to vanish. But the numerical factor appears to
be yz+zy', while it is the quantity yz-zy' which really
vanishes. Hence Hamilton was at first inclined to think
that ij must be treated as nil. But he soon saw that "
less harsh supposition " would suit the simple case.
his speculations on sets had already familiarized him with
the idea that multiplication might in certain cases not be
commutative; so that, as the last term in the above pro-
duct is made up of the two separate terms ijyz' and jizy',
the term would vanish of itself when the factor-lines are
coplanar provided ij-ji, for it would then assume the
form ij (yz' —zy'). He had now the following expression
for the product of any two directed lines

Thus the number-couple (0,1), when twice applied to a step-couple, simply changes its sign. That we have here a perfectly real and intelligible interpretation of the ordinary algebraic imaginary is easily seen by an illustration, even if it be a somewhat extravagant one. Some Eastern potentate, possessed of absolute power, covets the vast possessions of his vizier and of his barber. He determines to rob them both (an operation which may be very satisfactorily expressed by -1); but, being a wag, he chooses his own way of doing it. He degrades his vizier to the office of barber, taking all his goods in the process; and makes the barber his vizier. Next day he repeats the tion. Each of the victims has been restored to his former rank, but the operator 1 has been applied to both.

opera

Hamilton, still keeping prominently before him as his great object the invention of a method applicable to space of three dimensions, proceeded to study the properties of triplets of the form x+iy+jz, by which he proposed to represent the directed line in space whose projections on

1 Lectures on Quaternions, Dublin, 1853.

2 Theory of Conjugate Functions, or Algebraic Couples, with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time, read in 1833 and 1835, and published in Trans. R. I. A., xvii. ii. (1835).

3 Compare these with the long-subsequent ideas of Grassmann, presently to be described.

xx' -yy' — zz' +i(yx'+xy')+j(xx′+zx')+ij(yz −zy).
But his result had to be submitted to another test, the
Law of the Norms. As soon as he found, by trial, that
this law was satisfied, he took the final step. "This led
me," he says, "to conceive that perhaps, instead of seeking
to confine ourselves to triplets, . we ought to regard
these as only imperfect forms of Quaternions, .
and that thus my old conception of sets might receive a
new and useful application." In a very short time he
settled his fundamental assumptions. He had now three
distinct space-units i, j, k; and the following conditions
regulated their combination by multiplication:-

ï2=j2=k2= −1, ij— —ji—k, jk— — kj—i, ki— −ik−j.a
And now the product of two quaternions could be at
once expressed as a third quaternion, thus-

-(xx+yy+zz)+i(yz'′ − zy')+j (zx' — xz′)+k(xy' — yx').
To any one acquainted, even to a slight extent, with the
elements of Cartesian geometry of three dimensions, a
glance at the extremely suggestive constituents of this
expression shows how justly Hamilton was entitled to say
"When the conception
had been so far unfolded
and fixed in my mind, I felt that the new instrument for
applying calculation to geometry, for which I had so long

It will be easy to see that, instead of the last three of these, we may write the single one ijk – 1.

sought, was now, at least in part, attained." The date of this memorable discovery is October 16, 1843.

kinds of multiplication correspond respectively to the numerical and the directed parts of Hamilton's quaternion product. But We can devote but a few lines to the consideration of leisure to extend his method to angles in space. Hamilton and GrassGrassmann distinctly states in his preface that he had not had the expression above. Suppose, for simplicity, the factor- mann, while their earlier work had much in common, had very diflines to be each of unit length. Then x, y, z, x, y, z' express ferent objects in view. Hamilton, as we have seen, had geometrical their direction-cosines. Also, if be the angle between application as his main object; when he realized the quaternion them, and x", ", " the direction-cosines of a line perpendi-system, he felt that his object was gained, and thenceforth confined himself to the development of his method. Grassmann's cular to each of them, we have xx+yy'+zz' = cose, object seems to have been, all along, of a much more ambitious yz-zy=x'sine, &c., so that the product of two unit lines character, viz., to discover, if possible, a system or systems in is now expressed as - cose + (ix" + jy''+kz") sine. Thus, which every conceivable mode of dealing with sets should bo included. when the factors are parallel, or 0=0, the product, which That he made very great advances towards the attainment of this object all will allow; that his method, even as comis now the square of any (unit) line, is-1. And when pleted in 1862, fully attains it is not so certain. But his claims, the two factor lines are at right angles to one another, however great they may be, can in no way conflict with those of or 0/2, the product is simply ix"+jy'+kz", the unit Hamilton, whose mode of multiplying couples (in which the line perpendicular to both. Hence, and in this lies "inner" and "outer" multiplication are essentially involved) was the main element of the symmetry and simplicity of produced in 1833, and whose quaternion system was completed and published before Grassmann had elaborated for press even the the quaternion calculus, all systems of three mutually rudimentary portions of his own system, in which the veritable diffirectangular unit lines in space have the same properties as culty of the whole subject, the application to angles in space, had not the fundamental system i, j, k. In other words, if the even been attacked. Grassmann made in 1854 a somewhat savage system (considered as rigid) be made to turn about till onslaught on Cauchy and De St Venant, the former of whom had invented, while the latter had exemplified in application, the the first factor coincides with i and the second with j, system of "clefs algébriques," which is almost precisely that of the product will coincide with k. This fundamental Grassmann. But it is to be observed that Grassmann, though he system, therefore, becomes unnecessary; and the quater- virtually accused Cauchy of plagiarism, does not appear to have nion method, in every case, takes its reference lines solely to Hamilton in the second edition of his work. But in 1877, in preferred any such charge against Hamilton. He does not allude from the problem to which it is applied. It has therefore, the Mathematische Annalen, xii., he gave a paper "On the Place of as it were, a unique internal character of its own. Quaternions in the Ausdehnungslehre," in which he condemns, as far as he can, the nomenclature and methods of Hamilton. There been given for application to geometry of directed lines, but those are many other systems, based on various principles, which have which deal with products of lines are all of such complexity as to be practically useless in application. Others, such as the Barycentrische Calcul of Möbius, and the Méthode des Equipollences of Bellavitis, give elegant modes of treating space problems, so long as we confine ourselves to projective geometry and matters of that order; but they are limited in their field, and therefore need not be discussed here. More general systems, having close analogies to quaternions, have been given since Hamilton's discovery was published. As instances we may take Goodwin's and O'Brien's papers in the Cambridge Philosophical Transactions for 1849.

Hamilton, having gone thus far, proceeded to evolve these results from a train of a priori or metaphysical reasoning, which is so interesting in itself, and so characteristic of the man, that we briefly sketch its nature. Let it be supposed that the product of two directed lines is something which has quantity; i.e., it may be halved, or doubled, for instance. Also let us assume (a) space to have the same properties in all directions, and make the convention (6) that to change the sign of any one factor changes the sign of a product. Then the product of two lines which have the same direction cannot be, even in part, a directed quantity. For, if the directed part have the same direction as the factors, (b) shows that it will be reversed by reversing either, and therefore will recover its original direction when both are reversed. But this would obviously be inconsistent with (a). If it be perpendicular to the factor lines, (a) shows that it must have simultaneously every such direction. Hence it must be a mere number.

Again, the product of two lines at right angles to one another cannot, even in part, be a number. For the reversal of either factor must, by (b), change its sign.

But, if we look at the two factors in their new position by the light of (a), we see that the sign must not change. But there is nothing to prevent its being represented by a directed line if, as farther applications of (a) and (6) show we must do, we take it perpendicular to each of the factor lines. Hamilton seems never to have been quite satisfied with the apparent heterogeneity of a quaternion, depending as it does on a numerical and a directed part. He indulged in a great deal of speculation as to the exist ence of an extra-spatial unit, which was to furnish the raison d'être of the numerical part, and render the quaternion homogeneous as well as linear. But, for this, we must refer to his own works.

The

Hamilton was not the only worker at the theory of sets. year after the first publication of the quaternion method, there appeared a work of great originality, by Grassmann,' in which results closely analogous to some of those of Hamilton were given. In particular two species of multiplication ("inner" and "outer") of directed lines in one plane were given. The results of these two

1 Die Ausichnungslehre, Leipsic, 1844; 2d ed., “rollständig und in strenger Form bearbeitet," Berlin, 1862. works of Mobius, and those of Clifford, for a general explanation of See also the collected Grassmann's method.

Relations to other Branches of Science.-Even the above brief narrative shows how close is the connexion between quaternions and the ordinary Cartesian space-geometry. Were this all, the gain by their introduction would consist mainly in a clearer insight into the mechanism of coordinate systems, rectangular or not a very important addition to theory, but little advance so far as practical application is concerned. But we have now to consider that, as yet, we have not taken advantage of the perfect symmetry Hamilton's grand step becomes evident, and the gain is of the method. When that is done, the full value of quite as extensive from the practical as from the theoretical point of view. Hamilton, in fact, remarks,2 "I regard it as an inelegance and imperfection in this calculus, or rather in the state to which it has hitherto been unfolded, whenever it becomes, or seems to become, necessary to to the resources of ordinary algebra, for the solution of equations in quaternions." This refers to the use of the x, y, z coordinates-associated, of course, with i, j, k. But when, instead of the highly artificial expression ix+jy+kz, to denote a finite directed line, we employ a single letter, a (Hamilton uses the Greek alphabet for this purpose), and find that we are permitted to deal with it exactly as we should have dealt with the more complex expression, the immense gain is at least in part obvious. Any quaternion may now be expressed in numerous simple forms. regard it as the sum of a number and a line, a+a, or Thus we may as the product, By, or the quotient, de', of two directed lines, &c., while, in many cases, we may represent it, so far as it is required, by a single letter such as q, r, &c.

have recourse

2 Lectures on Quaternions, § 513.