Chapter III: Front Matter (3)

23. The process of solution due to Karl Friedrich Gauss (1801) depends essentially on the arrangement of the roots in a certain order, viz. not as above, with the indices of r in arithmetical progression, but with their indices in geometrical progression; the prime number n has a certain number of prime roots g, which are such that g^(n - 1) is the lowest power of g, which is [equivalent to] 1 to the modulus n; or, what is the same thing, that the series of powers 1, g, g^2, ... g^(n - 2), each divided by n, leave (in a different order) the remainders 1, 2, 3, ... n - 1; hence giving to r in succession the indices 1, g, g^2, ... g^(n - 2), we have, in a different order, the whole series of roots r, r^2, r^3, ... r^(n - 1).

In the most simple case, n = 5, the equation to be solved is x^4 + x^3
+ x^2 + x + 1 = 0; here 2 is a prime root of 5, and the order of the
roots is r, r^2, r^4, r^3. The Gaussian process consists in forming an
equation for determining the periods P1, P2, = r + r^4 and r^2 + r^3
respectively;--these being such that the symmetrical functions P1 +
P2, P1P2 are rationally determinable: in fact P1 + P2 = -1, P1P2 = (r
+ r^4)(r^2 + r^3), = r^3 + r^4 + r^6 + r^7, = r^3 + r^4 + r + r^2, =
-1. P1, P2 are thus the roots of u^2 + u - 1 = 0; and taking them to
be known, they are themselves broken up into subperiods, in the
present case single terms, r and r^4 for P1, r^2 and r^3 for P2; the
symmetrical functions of these are then rationally determined in terms
of P1 and P2; thus r + r^4 = P1, r.r^4 = 1, or r, r^4 are the roots of
u^2 - P1u + 1 = 0. The mode of division is more clearly seen for a
larger value of n; thus, for n = 7 a prime root is = 3, and the
arrangement of the roots is r, r^3, r^2, r^6, r^4, r^5. We may form
either 3 periods each of 2 terms, P1, P2, P3 = r + r^6, r^3 + r^4, r^2
+ r^5 respectively; or else 2 periods each of 3 terms, P1, P2 = r +
r^2 + r^4, r^3 + r^6 + r^5 respectively; in each ease the symmetrical
functions of the periods are rationally determinable: thus in the case
of the two periods P1 + P2 = -1, P1P2 = 3 + r + r^2 + r^3 + r^4 + r^5
+ r^6, = 2; and the periods being known the symmetrical functions of
the several terms of each period are rationally determined in terms of
the periods, thus r + r^2 + r^4 = P1, r.r^2 + r.r^4 + r^2.r^4 = P2,
r.r^2.r^4 = 1.

The theory was further developed by Lagrange (1808), who, applying his general process to the equation in question, x^(n - 1) + x^(n - 2) + ... + x + 1 = 0 (the roots a, b, c ... being the several powers of r, the indices in geometrical progression as above), showed that the function (a + [omega]b + [omega]^2 c + ...)^(n - 1) was in this case a given function of [omega] with integer coefficients.

Reverting to the before-mentioned particular equation x^4 + x^3 + x^2
+ x + 1 = 0, it is very interesting to compare the process of solution
with that for the solution of the general quartic the roots whereof
are a, b, c, d.

Take [omega], a root of the equation [omega]^4 - 1 = 0 (whence [omega]
is = 1, -1, i, or -i, at pleasure), and consider the expression

(a + [omega]b + [omega]^2 c + [omega]^3 d)^4,

the developed value of this is

= a^4 + b^4 + c^4 + d^4 + 6(a^2 c^2 + b^2 d^2) + 12(a^2 bd + b^2 ca + c^2 db + d^2ac)
+[omega] {4(a^3 b + b^3 c + c^3 + d^3 a) + 12(a^2 cd + b^2 da + c^2 ab + d^2 bc)}
+[omega]^2{6(a^2 b^2 + b^2 c^2 + c^2 d^2 + d^2 a^2) + 4(a^3 c + b^3 d + c^3 a + d^3 b) + 24abcd}
+[omega]^3{4(a^3 d + b^3 a + c^3 b + d^3 c) + 12(a^2 bc + b^2 cd + c^2 da + d^2 ab)}

that is, this is a 6-valued function of a, b, c, d, the root of a
sextic (which is, in fact, solvable by radicals; but this is not here
material).

If, however, a, b, c, d denote the roots r, r^2, r^4, r^3 of the
special equation, then the expression becomes

r^4 + r^3 + r + r^2 + 6(1 + 1)+12(r^2 + r^4 + r^3 + r)
+ [omega] {4(1 + 1 + 1 + 1) + 12(r^4 + r^3 + r + r^2)}
+ [omega]^2{6(r + r^2 + r^4 + r^3) + 4(r^2 + r^4 + r^3 + r)}
+ [omega]^3{4(r + r^2 + r^4 + r^3) + 12(r^3 + r + r^2 + r^4)}

viz. this is

= -1 + 4[omega] + 14[omega]^2 - 16[omega]^3,

a completely determined value. That is, we have

(r + [omega]r^2 + [omega]^2 r^4 + [omega]^3 r^3) = -1 + 4[omega] +
14[omega]^2 - 16[omega]^3,

which result contains the solution of the equation. If [omega] = 1, we
have (r + r^2 + r^4 + r^3)^4 = 1, which is right; if [omega] = -1,
then (r + r^4 - r^2 - r^3)^4 = 25; if [omega] = i, then we have {r -
r^4 + i(r^2 - r^3)}^4 = -15 + 20i; and if [omega] = -i, then {r - r^4
- i(r^2 - r^3)}^4 = -15 - 20i; the solution may be completed without
difficulty.

The result is perfectly general, thus:--n being a prime number, r a root of the equation x^(n - 1) + x^(n - 2) + ... + x + 1 = 0, [omega] a root of [omega]^(n - 1) - 1 = 0, and g a prime root of g^(n - 1) [equivalent] 1 (mod. n), then

{r + [omega]r^g + ... + [omega]^(n - 2) r^g^(n - 2)}^(n - 1)

is a given function M0 + M1[omega] ... + M_(n - 2)[omega]^(n - 2) with integer coefficients, and by the extraction of (n - 1)th roots of this and similar expressions we ultimately obtain r in terms of [omega], which is taken to be known; the equation x^n - 1 = 0, n a prime number, is thus solvable by radicals. In particular, if n - 1 be a power of 2, the solution (by either process) requires the extraction of square roots only; and it was thus that Gauss discovered that it was possible to construct geometrically the regular polygons of 17 sides and 257 sides respectively. Some interesting developments in regard to the theory were obtained by C.G.J. Jacobi (1837); see the memoir "Ueber die Kreistheilung, u.s.w.," _Crelle_, t. xxx. (1846).

The equation x^(n - 1) + ... + x + 1 = 0 has been considered for its own sake, but it also serves as a specimen of a class of equations solvable by radicals, considered by N.H. Abel (1828), and since called Abelian equations, viz. for the Abelian equation of the order n, if x be any root, the roots are x, [theta]x, [theta]^2 x, ... [theta]^(n - 1)x ([theta]x being a rational function of x, and [theta]^nx = x); the theory is, in fact, very analogous to that of the above particular case.

A more general theorem obtained by Abel is as follows:--If the roots
of an equation of any order are connected together in such wise that
_all_ the roots can be expressed rationally in terms of any one of
them, say x; if, moreover, [theta]x, [theta]1x being any two of the
roots, we have [theta][theta]1x = [theta]1[theta]x, the equation will
be solvable algebraically. It is proper to refer also to Abel's
definition of an _irreducible_ equation:--an equation [phi]x = 0, the
coefficients of which are rational functions of a certain number of
known quantities a, b, c ..., is called irreducible when it is
impossible to express its roots by an equation of an inferior degree,
the coefficients of which are also rational functions of a, b, c ...
(or, what is the same thing, when [phi]x does not break up into
factors which are rational functions of a, b, c ...). Abel applied his
theory to the equations which present themselves in the division of
the elliptic functions, but not to the modular equations.

24. But the theory of the algebraical solution of equations in its most complete form was established by Evariste Galois (born October 1811, killed in a duel May 1832; see his collected works, _Liouville_, t. xl., 1846). The definition of an irreducible equation resembles Abel's,--an equation is reducible when it admits of a rational divisor, irreducible in the contrary case; only the word _rational_ is used in this extended sense that, in connexion with the coefficients of the given equation, or with the irrational quantities (if any) whereof these are composed, he considers any number of other irrational quantities called "adjoint radicals," and he terms rational any rational function of the coefficients (or the irrationals whereof they are composed) and of these adjoint radicals; the epithet irreducible is thus taken either absolutely or in a relative sense, according to the system of adjoint radicals which are taken into account. For instance, the equation x^4 + x^3 + x^2 + x + 1 = 0; the left hand side has here no rational divisor, and the equation is irreducible; but this function is = (x^2 + 1/2 x + 1)^2 -(5/4)x^2, and it has thus the irrational divisors x^2 + 1/2(1 + [root]5)x + 1, x^2 + 1/2(1 - [root]5)x + 1; and these, if we _adjoin_ the radical [root]5, are rational, and the equation is no longer irreducible. In the case of a given equation, assumed to be irreducible, the problem to solve the equation is, in fact, that of finding radicals by the adjunction of which the equation becomes reducible; for instance, the general quadric equation x^2 + px + q = 0 is irreducible, but it becomes reducible, breaking up into rational linear factors, when we adjoin the radical [root](1/4 p^2 - q).

The fundamental theorem is the Proposition I. of the "Memoire sur les
conditions de resolubilite des equations par radicaux"; viz. given an
equation of which a, b, c ... are the m roots, there is always a group
of permutations of the letters a, b, c ... possessed of the following
properties:--

1. Every function of the roots invariable by the substitutions of the
group is rationally known.

2. Reciprocally every rationally determinable function of the roots is
invariable by the substitutions of the group.

Here by an invariable function is meant not only a function of which
the form is invariable by the substitutions of the group, but further,
one of which the value is invariable by these substitutions: for
instance, if the equation be [phi](x) = 0, then [phi](x) is a function
of the roots invariable by any substitution whatever. And in saying
that a function is rationally known, it is meant that its value is
expressible rationally in terms of the coefficients and of the adjoint
quantities.

For instance in the case of a general equation, the group is simply
the system of the 1.2.3 ... n permutations of all the roots, since, in
this case, the only rationally determinable functions are the
symmetric functions of the roots.

In the case of the equation x^(n - 1) ... + x + 1 = 0, n a prime
number, a, b, c ... k = r, r^g, r^g^2 ... r^g^(n - 2), where g is a
prime root of n, then the group is the cyclical group abc ... k, bc
... ka, ... kab ... j, that is, in this particular case the number of
the permutations of the group is equal to the order of the equation.

This notion of the group of the original equation, or of the group of
the equation as varied by the adjunction of a series of radicals,
seems to be the fundamental one in Galois's theory. But the problem of
solution by radicals, instead of being the sole object of the theory,
appears as the first link of a long chain of questions relating to the
transformation and classification of irrationals.

Returning to the question of solution by radicals, it will be readily
understood that by the adjunction of a radical the group may be
diminished; for instance, in the case of the general cubic, where the
group is that of the six permutations, by the adjunction of the square
root which enters into the solution, the group is reduced to abc, bca,
cab; that is, it becomes possible to express rationally, in terms of
the coefficients and of the adjoint square root, any function such as
a^2 b + b^2 c + c^2 a which is not altered by the cyclical
substitution a into b, b into c, c into a. And hence, to determine
whether an equation of a given form is solvable by radicals, the
course of investigation is to inquire whether, by the successive
adjunction of radicals, it is possible to reduce the original group of
the equation so as to make it ultimately consist of a single
permutation.

The condition in order that an equation of a given prime order n may
be solvable by radicals was in this way obtained--in the first
instance in the form (scarcely intelligible without further
explanation) that every function of the roots x1, x2 ... x_n,
invariable by the substitutions x_(ak + b) for x_k, must be rationally
known; and then in the equivalent form that the resolvent equation of
the order 1.2 ... (n - 2) must have a rational root. In particular,
the condition in order that a quintic equation may be solvable is that
Lagrange's resolvent of the order 6 may have a rational factor, a
result obtained from a direct investigation in a valuable memoir by E.
Luther, _Crelle_, t. xxxiv. (1847).

Among other results demonstrated or announced by Galois may be
mentioned those relating to the modular equations in the theory of
elliptic functions; for the transformations of the orders 5, 7, 11,
the modular equations of the orders 6, 8, 12 are depressible to the
orders 5, 7, 11 respectively; but for the transformation, n a prime
number greater than 11, the depression is impossible.

The general theory of Galois in regard to the solution of equations
was completed, and some of the demonstrations supplied by E. Betti
(1852). See also J.A. Serret's _Cours d'algebre superieure_, 2nd ed.
(1854); 4th ed. (1877-1878).

25. Returning to quintic equations, George Birch Jerrard (1835) established the theorem that the general quintic equation is by the extraction of only square and cubic roots reducible to the form x^5 + ax + b = 0, or what is the same thing, to x^5 + x + b = 0. The actual reduction by means of Tschirnhausen's theorem was effected by Charles Hermite in connexion with his elliptic-function solution of the quintic equation (1858) in a very elegant manner. It was shown by Sir James Cockle and Robert Harley (1858-1859) in connexion with the Jerrardian form, and by Arthur Cayley (1861), that Lagrange's resolvent equation of the sixth order can be replaced by a more simple sextic equation occupying a like place in the theory.

The theory of the modular equations, more particularly for the case n = 5, has been studied by C. Hermite, L. Kronecker and F. Brioschi. In the case n = 5, the modular equation of the order 6 depends, as already mentioned, on an equation of the order 5; and conversely the general quintic equation may be made to depend upon this modular equation of the order 6; that is, assuming the solution of this modular equation, we can solve (not by radicals) the general quintic equation; this is Hermite's solution of the general quintic equation by elliptic functions (1858); it is analogous to the before-mentioned trigonometrical solution of the cubic equation. The theory is reproduced and developed in Brioschi's memoir, "Uber die Auflosung der Gleichungen vom funften Grade," _Math. Annalen_, t. xiii. (1877-1878).

26. The modern work, reproducing the theories of Galois, and
exhibiting the theory of algebraic equations as a whole, is C.
Jordan's _Traite des substitutions et des equations algebriques_
(Paris, 1870). The work is divided into four books--book i.,
preliminary, relating to the theory of congruences; book ii. is in two
chapters, the first relating to substitutions in general, the second
to substitutions defined analytically, and chiefly to linear
substitutions; book iii. has four chapters, the first discussing the
principles of the general theory, the other three containing
applications to algebra, geometry, and the theory of transcendents;
lastly, book iv., divided into seven chapters, contains a
determination of the general types of equations solvable by radicals,
and a complete system of classification of these types. A glance
through the index will show the vast extent which the theory has
assumed, and the form of general conclusions arrived at; thus, in book
iii., the algebraical applications comprise Abelian equations,
equations of Galois; the geometrical ones comprise Q. Hesse's
equation, R.F.A. Clebsch's equations, lines on a quartic surface
having a nodal line, singular points of E.E. Kummer's surface, lines
on a cubic surface, problems of contact; the applications to the
theory of transcendents comprise circular functions, elliptic
functions (including division and the modular equation), hyperelliptic
functions, solution of equations by transcendents. And on this last
subject, solution of equations by transcendents, we may quote the
result--"the solution of the general equation of an order superior to
five cannot be made to depend upon that of the equations for the
division of the circular or elliptic functions"; and again (but with a
reference to a possible case of exception), "the general equation
cannot be solved by aid of the equations which give the division of
the hyperelliptic functions into an odd number of parts." (See also
GROUPS, THEORY OF.) (A. Ca.)

BIBLIOGRAPHY.--For the general theory see W.S. Burnside and A.W.
Panton, _The Theory of Equations_ (4th ed., 1899-1901); the Galoisian
theory is treated in G.B. Matthews, _Algebraic Equations_ (1907). See
also the _Ency. d. math. Wiss._ vol. ii.

FOOTNOTES:

[1] The coefficients were selected so that the roots might be nearly
1, 2, 3.

[2] The third edition (1826) is a reproduction of that of 1808; the
first edition has the date 1798, but a large part of the contents is
taken from memoirs of 1767-1768 and 1770-1771.

[3] The earlier demonstrations by Euler, Lagrange, &c, relate to the
case of a numerical equation with real coefficients; and they consist
in showing that such equation has always a real quadratic divisor,
furnishing two roots, which are either real or else conjugate
imaginaries [alpha] + [beta]i (see Lagrange's _Equations
numeriques_).

[4] The square root of [alpha] + [beta]i can be determined by the
extraction of square roots of positive real numbers, without the
trigonometrical tables.

EQUATION OF THE CENTRE, in astronomy, the angular distance, measured around the centre of motion, by which a planet moving in an ellipse deviates from the mean position which it would occupy if it moved uniformly. Its amount is the correction which must be applied positively or negatively to the mean anomaly in order to obtain the true anomaly. It arises from the ellipticity of the orbit, is zero at pericentre and apocentre, and reaches its greatest amount nearly midway between these points. (See ANOMALY and ORBIT.)

EQUATION OF TIME, the difference between apparent time, determined by the meridian passage of the real sun, and mean time, determined by the passage of the mean sun. It goes through a double period in the course of a year. Its amount varies a fraction of a minute for the same date, from year to year and from one longitude to another, on the same day. The following table shows an average value for any date and for the Greenwich meridian for a number of years, from which the actual value will seldom deviate more than 20 seconds until after 1950. The + sign indicates that the real sun reaches the meridian _after_ mean noon; the - sign _before_ mean noon.

_Table of the Equation of Time._

m. s. m. s. m. s.
Jan. 1 +3 26 Mar. 1 +12 39 May 1 -2 55
6 5 45 6 11 35 6 -3 27
11 7 51 11 10 20 11 -3 46
16 9 43 16 8 58 16 -3 51
21 11 19 21 7 30 21 -3 40
26 12 36 26 5 59 26 -3 16

Feb. 1 +13 42 Apr. 1 +4 9 June 1 -2 32
6 14 14 6 2 40 6 -1 44
11 14 25 11 +1 15 11 -0 48
16 14 17 16 -0 3 16 +0 14
21 13 52 21 -1 12 21 1 19
26 13 11 26 -2 10 26 2 24

July 1 +3 26 Sept. 1 +0 9 Nov. 1 -16 18
6 4 21 6 -1 28 6 -16 19
11 5 8 11 -3 10 11 -15 58
16 5 44 16 -4 55 16 -15 15
21 6 8 21 -6 41 21 -14 12
26 6 18 26 -8 25 26 -12 49

Aug. 1 +6 10 Oct. 1 -10 5 Dec. 1 -11 7
6 5 47 6 -11 38 6 - 9 9
11 5 9 11 -13 2 11 - 6 57
16 4 17 16 -14 14 16 - 4 35
21 3 12 21 -15 11 21 - 2 7
26 1 55 26 -15 52 26 + 0 23

EQUATOR (Late Lat. _aequator_, from _aequare_, to make equal), in geography, that great circle of the earth, equidistant from the two poles, which divides the northern from the southern hemisphere and lies in a plane perpendicular to the axis of the earth; this is termed the "geographical" or "terrestrial equator." In astronomy, the "celestial equator" is the name given to the great circle in which the plane of the terrestrial equator intersects the celestial sphere; it is consequently equidistant from the celestial poles. The "magnetic equator" is an imaginary line encircling the earth, along which the vertical component of the earth's magnetic force is zero; it nearly coincides with the terrestrial equator.

EQUERRY (from the Fr. _ecurie_, a stable, through its older form _escurie_, from the Med. Lat. _scuria_, a word of Teutonic origin for a stable or shed, cf. Ger. _Scheuer_; the modern spelling has confused the word with the Lat. _equus_, a horse), a contracted form of "gentleman of the equerry," an officer in charge of the stables of a royal household. At the British court, equerries are officers attached to the department of the master of the horse, the first of whom is called chief equerry (see HOUSEHOLD, ROYAL).

EQUIDAE, the family of perissodactyle ungulate mammals typified by the horse (_Equus caballus_); see HORSE. According to the older classification this family was taken to include only the forms with tall-crowned teeth, more or less closely allied to the typical genus _Equus_. There is, however, such an almost complete graduation from the former to earlier and more primitive mammals with short-crowned cheek-teeth, at one time included in the family _Lophiodontidae_ (see PERISSODACTYLA), that it has now become a very general practice to include the whole "phylum" in the family _Equidae_. The _Equidae_, in this extended sense, together with the extinct _Palaeotheriidae_, are indeed now regarded as forming one of four main groups into which the Perissodactyla are divided, the other groups being the Tapiroidea, Rhinocerotoidea and Titanotheriide. For the horse-group the name Hippoidea is employed. All four groups were closely connected in the Lower Eocene, so that exact definition is almost impossible.

In the Hippoidea there is generally the full series of 44 teeth, but the first premolar is often deciduous or wanting in the lower or in both jaws. The incisors are chisel-shaped, and the canines tend to become isolated so as in the now specialized forms to occupy nearly the middle of a longer or shorter gap between the incisors and premolars. In the upper molars the two outer columns of the primitive tubercular molar coalesce to form an outer wall, from which proceed two crescentic transverse crests; the connexion between the crests and the wall being imperfect or slight, and the crests themselves sometimes tubercular. Each of the lower molars carries two crescentic ridges. The number of toes ranges from four to one in the fore-foot, and from three to one in the hind-foot. The paroccipital, postglenoid and post-tympanic processes of the skull are large, and the latter always distinct. Normally there are no traces of horn-cores. The calcaneum lacks the facet for the fibula found in the Titanotheroidea.

In the earlier _Equidae_ the teeth were short-crowned, with the premolars simpler than the molars; but there is a gradual tendency to an increase in the height of the crowns of the teeth, accompanied by increasing complexity of structure and the filling up of the hollows with cement. Similarly the gap on each side of the canine tooth in each jaw continues to increase in length; while in all the later forms the orbit is surrounded by a ring of bone. A third modification is the increasing length of limb (as well as in general bodily size), accompanied by a gradual reduction in the number of toes from three or four to one.

All the existing members of the family, such as the domesticated horse (_Equus caballus_) and its wild or half-wild relatives, the asses and the zebras, are included in the typical genus. In all these the crowns of the cheek-teeth are very tall (fig. 1, b) and only develop roots late in life; while their grinding-surfaces (fig. 2, b and c) are very complicated and have all the hollows filled with cement. The summits of the incisors are infolded, producing, when partially worn, the "mark." In the skull the orbit is surrounded by bone, and there is no distinct depression in front of the same. Each limb terminates in one large toe; the lateral digits being represented by the splint-bones, corresponding to the lateral metacarpals and metatarsals of _Hipparion_. Not unfrequently, however, the lower ends of the splint-bones carry a small expansion, representing the phalanges.

Remains of horses indistinguishable from _E. caballus_ occur in the Pleistocene deposits of Europe and Asia; and it is from them that the dun-coloured small horses of northern Europe and Asia are probably derived. The ancestor of these Pleistocene horses is probably _E. stenonis_, of the Upper Pliocene of Europe, which has a small depression in front of the orbit, while the skull is relatively larger, the feet are rather shorter, and the splint-bones somewhat more developed. In India a nearly allied species (_E. sivalensis_), occurs in the Lower Pliocene, and may have been the ancestor of the Arab stock, which shows traces of the depression in front of the orbit characteristic of the earlier forms. In North America species of _Equus_ occur in the Pleistocene and from that continent others reached South America during the same epoch. In the latter country occurs _Hippidium_, in which the cheek-teeth are shorter and simpler, and the nasal bones very long and slender, with elongated slits at the side. The limbs, especially the cannon-bones, are relatively short, and the splint-bones large. The allied Argentine _Onohippidium_, which is also Pleistocene, has still longer nasal bones and slits, and a deep double cavity in front of the orbit, part of which probably contained a gland. _Onohippidium_ is certainly off the direct line of descent of the modern horses, and, on account of the length of the nasals and their slits, the same probably holds good for _Hippidium_.

Species from the Pliocene of Texas and the Upper Miocene (Loup Fork) of Oregon were at one time assigned to _Hippidium_, but this is incorrect, that genus being exclusively South American. The name _Pliohippus_ has been applied to species from the same two formations on the supposition that the foot-structure was similar to that of _Hippidium_, but Mr J.W. Gidley is of opinion that the lateral digits may have been fully developed.

Apparently there is here some gap in the line of descent of the horse, and it may be suggested that the evolution took place, not as commonly supposed, in North America, but in eastern central Asia, of which the palaeontology is practically unknown; some support is given to this theory by the fact that the earliest species with which we are acquainted occur in northern India.

a, _Hyracotherium_ (Eocene).
b, _Mesohippus_ (Oligocene).
c, _Anchitherium_ (Miocene).
d, _Hipparion_ (Pliocene).
e, _Equus_ (Pleistocene).]

Be this as it may, the next North American representatives of the family constitute the genera _Protohippus_ and _Merychippus_ of the Miocene, in both of which the lateral digits are fully developed and terminate in small though perfect hoofs. In both the cheek-teeth have moderately tall crowns, and in the first named of the two those of the milk-series are nearly similar to their permanent successors. In _Merychippus_, on the other hand, the milk-molars have short crowns, without any cement in the hollows, thus resembling the permanent molars of the under-mentioned genus _Anchitherium_. From the well-known _Hipparion_, or _Hippotherium_, typically from the Lower Pliocene of Europe, but also occurring in the corresponding formation in North Africa, Persia, India and China, and represented in the Upper Miocene Loup Fork beds of the United States by species which it has been proposed to separate generically as _Neohipparion_, we reach small horses which are now generally regarded as a lateral offshoot from the _Merychippus_ type. The cheek-teeth, which have crowns of moderate height, differ from those of all the foregoing in that the postero-internal pillar (the projection on the right-hand top corner of c in fig. 2) is isolated in place of being attached by a narrow neck to the adjacent crescent. The skull, which is relatively short, has a large depression in front of the orbit, commonly supposed to have contained a gland, but this may be doubtful. In the typical, and also in the North American forms these were complete, although small, lateral toes in both feet (fig. 3, d), but it is possible that in _H. antilopinum_ of India the lateral toes had disappeared. If this be so, we have the development of a monodactyle foot in this genus independently of _Equus_.

The foregoing genera constitute the subfamily _Equinae_, or the _Equidae_ as restricted by the older writers. In all the dentition is of the hypsodont type, with the hollows of the cheek-teeth filled by cement, the premolars molariform, and the first small and generally deciduous. The orbit is surrounded by a bony ring; the ulna and radius in the fore, and the tibia and fibula in the hind-limb are united, and the feet are of the types described above. Between this subfamily and the second subfamily, _Hyracotheriinae_, a partial connexion is formed by the North American Upper Miocene genera _Desmatippus_ and _Anchippus_ or _Parahippus_. The characteristics of the group will be gathered from the remarks on the leading genera; but it may be mentioned that the orbit is open behind, the cheek-teeth are short-crowned and without cement (fig. 1, a), the gap between the canine and the outermost incisor is short, the bones of the middle part of the leg are separate, and there are at least three toes to each foot.

The longest-known genus and the one containing the largest species is _Anchitherium_, typically from the Middle Miocene of Europe, but also represented by one species from the Upper Miocene of North America. The European _A. aurelianense_ was of the size of an ordinary donkey. The cheek-teeth are of the type shown in a of figs. 1 and 2; the premolars, with the exception of the small first one, being molar-like; and the lateral toes (fig. 3, c) were to some extent functional. The summits of the incisors were infolded to a small extent. Nearly allied is the American _Mesohippus_, ranging from the Lower Miocene to the Lower Oligocene of the United States, of which the earliest species stood only about 18 in. at the shoulder. The incisors were scarcely, if at all, infolded, and there is a rudiment of the fifth metacarpal (fig. 3, b). By some writers all the species of _Mesohippus_ are included in the genus _Miohippus_, but others consider that the two genera are distinct.

_Mesohippus_ and _Miohippus_ are connected with the earliest and most primitive mammal which it is possible to include in the family _Equidae_ by means of _Epihippus_ of the Uinta or Upper Eocene of North America, and _Pachynolophus_, or _Orohippus_, of the Middle and Lower Eocene of both halves of the northern hemisphere. The final stage, or rather the initial stage, in the series is presented by _Hyracotherium_ (_Protorohippus_), a mammal no larger than a fox, common to the Lower Eocene of Europe and North America. The general characteristics of this progenitor of the horses are those given above as distinctive of the group. The cheek-teeth are, however, much simpler than those of _Anchitherium_; the transverse crests of the upper molars not being fully connected with the outer wall, while the premolars in the upper jaw are triangular, and thus unlike the molars. The incisors are small and the canines scarcely enlarged; the latter having a gap on each side in the lower, but only one on their hinder aspect in the upper jaw. The fore-feet have four complete toes (fig. 3, a), but there are only three hind-toes, with a rudiment of the fifth metatarsal. The vertebrae are simpler in structure than in _Equus_. From _Hyracotherium_, which is closely related to the Eocene representatives of the ancestral stocks of the other three branches of the Perissodactyla, the transition is easy to _Phenacodus_, the representative of the common ancestor of all the Ungulata.

See also H.F. Osborn, "New Oligocene Horses," _Bull. Amer. Mus._ vol.
xx. p. 167 (1904); J.W. Gidley, _Proper Generic Names of Miocene
Horses_, p. 191; and the article PALAEONTOLOGY. (R. L.*)

EQUILIBRIUM (from the Lat. _aequus_, equal, and _libra_, a balance), a condition of equal balance between opposite or counteracting forces. By the "sense of equilibrium" is meant the sense, or sensations, by which we have a feeling of security in standing, walking, and indeed in all the movements by which the body is carried through space. Such a feeling of security is necessary both for maintaining any posture, such as standing, or for performing any movement. If this feeling is absent or uncertain, or if there are contradictory sensations, then definite muscular movements are inefficiently or irregularly performed, and the body may stagger or fall. When we stand erect on a firm surface, like a floor, there is a feeling of resistance, due to nervous impulses reaching the brain from the soles of the feet and from the muscles of the limbs and trunk. In walking or running, these feelings of resistance seem to precede and guide the muscular movements necessary for the next step. If these are absent or perverted or deficient, as is the case in the disease known as locomotor ataxia, then, although there is no loss of the power of voluntary movement, the patient staggers in walking, especially if he is not allowed to look at his feet, or if he is blind-folded. He misses the guiding sensations that come from the limbs; and with a feeling that he is walking on a soft substance, offering little or no resistance, he staggers, and his muscular movements become irregular. Such a condition maybe artificially brought about by washing the soles of the feet with chloroform or ether. And it has been observed to exist partially after extensive destruction of the skin of the soles of the feet by burns or scalds. This shows that tactile impulses from the skin take a share in generating the guiding sensation. In the disease above mentioned, however, tactile impressions may be nearly normal, but the guiding sensation is weak and inefficient, owing to the absence of impulses from the muscles. The disease is known to depend on morbid changes in the posterior columns of the spinal cord, by which impulses are not freely transmitted upwards to the brain. These facts point to the existence of impulses coming from the muscles and tendons. It is now known that there exist peculiar spindles, in muscle, and rosettes or coils or loops of nerve fibres in close proximity to tendons. These are the end organs of the sense. The transmission of impulses gives rise to the _muscular sense_, and the guiding sensation which precedes co-ordinated muscular movements depends on these impulses. Thus from the limbs streams of nervous impulses pass to the sensorium from the skin and from muscles and tendons; these may or may not arouse consciousness, but they guide or evoke muscular movements of a co-ordinated character, more especially of the limbs.

In animals whose limbs are not adapted for delicate touch nor for the performance of complicated movements, such as some mammals and birds and fishes, the guiding sensations depend largely on the sense of vision. This sense in man, instead of assisting, sometimes disturbs the guiding sensation. It is true that in locomotor ataxia visual sensations may take the place of the tactile and muscular sensations that are inefficient, and the man can walk without staggering if he is allowed to look at the floor, and especially if he is guided by transverse straight lines. On the other hand, the acrobat on the wire-rope dare not trust his visual sensations in the maintenance of his equilibrium. He keeps his eyes fixed on one point instead of allowing them to wander to objects below him, and his muscular movements are regulated by the impulses that come from the skin and muscles of his limbs. The feeling of insecurity probably arises from a conception of height, and also from the knowledge that by no muscular movements can a man avoid a catastrophe if he should fall. A bird, on the other hand, depends largely on visual impressions, and it knows by experience that if launched into the air from a height it can fly. Here, probably, is an explanation of the large size of the eyes of birds. Cover the head, as in hooding a falcon, and the bird seems to be deprived of the power of voluntary movement. Little effect will be produced if we attempt to restrain the movements of a cat by covering its eyes. A fish also is deprived of the power of motion if its eyes are covered. But both in the bird and in the fish tactile and muscular impressions, especially the latter, come into play in the mechanism of equilibrium. In flight the large-winged birds, especially in soaring, can feel the most delicate wind-pressures, both as regards direction and force, and they adapt the position of their body so as to catch the pressure at the most efficient angle. The same is true of the fish, especially of the flat-fishes. In mammals the sense of equilibrium depends, then, on streams of tactile, muscular and visual impressions pouring in on the sensorium, and calling forth appropriate muscular movements. It has also been suggested that impulses coming from the abdominal viscera may take part in the mechanism. The presence in the mesentery of felines (cats, &c.) of large numbers of Pacinian corpuscles, which are believed to be modified tactile bodies, favours this supposition. Such animals are remarkable for the delicacy of such muscular movements, as balancing and leaping.

There is another channel by which nervous impulses reach the sensorium and play their part in the sense of equilibrium, namely, from the semicircular canals, a portion of the internal ear. It is pointed out in the article HEARING that the appreciation of sound is in reality an appreciation of variations of pressure. The labyrinth consists of the vestibule, the cochlea and the semicircular canals. The cochlea receives the sound-waves (variations of pressure) that constitute musical tones. This it accomplishes by the structures in the ductus cochlearis. In the vestibule we find two sacs, the saccule next to and communicating with the ductus cochlearis, and the utricle communicating with the semicircular canals. The base of the stapes communicates pressures to the utricle. The membranous portion of the semicircular canals consists of a tube, dilated at one end into a swelling or pouch, termed the ampulla, and each end communicates freely with the utricle. On the posterior wall of both the saccule and of the utricle there is a ridge, termed in each case the macula acustica, bearing a highly specialized epithelium. A similar structure exists in each ampulla. This would suggest that all three structures have to do with hearing; but, on the other hand, there is experimental evidence that the utricle and the canals may transmit impressions that have to do with equilibrium. Pressure of the base of the stapes is exerted on the utricle. This will compress the fluid in that cavity, and tend to drive the fluid into the semicircular canals that communicate with that cavity by five openings. Each canal is surrounded by a thin layer of perilymph, so that it may yield a little to this pressure, and exert a pull or pressure on the nerve-endings in each ampulla. Thus impulses may be generated in the nerves of the ampullae.

The three semicircular canals lie in the three directions in space, and it has been suggested that they have to do with our appreciation of the direction of sound. But our appreciation of sound is very inaccurate: we look with the eyes for the source of a sound, and instinctively direct the ears or the head, or both, in the direction from which the sound appears to proceed. But the relationship of the canals on the two sides must have a physiological significance. Thus (1) the six canals are parallel, two and two; or (2) the two horizontal canals are in the same plane, while the superior canal on one side is nearly parallel with the posterior canal of the other. These facts point to the two sets of canals and ampullae acting as one organ, in a manner analogous to the action of two retinae for single vision.

We have next to consider how the canals may possibly act in connexion with the sense of equilibrium. In 1820 J. Purkinje studied the vertigo that follows rapid rotation of the body in the erect position on a vertical axis. On stopping the rotation there is a sense of rotation in the opposite direction, and this may occur even when the eyes are closed. Purkinje noticed that the position of the imaginary axis of rotation depends on the axis around which the head revolves. In 1828 M.J.P. Flourens discovered that injury to the canals causes disturbance to the equilibrium and loss of co-ordination, and that sections of the canals produce a rotatory movement of a kind corresponding to the canal that had been divided. Thus division of a membranous canal causes rotatory movements round an axis at right angles to the plane of the divided canal. The body of the animal always moves in the direction of the cut canal. Many other observers have corroborated these experiments. F. Goltz was the first who formulated the conditions necessary for equilibration. He put the matter thus:--(1) A central co-ordinating organ--in the brain; (2) centripetal fibres, with their peripheral terminations--in the ampullae; and (3) centrifugal fibres, with their terminal organs--in the muscular mechanisms. A lesion of any one of these portions of the mechanism causes loss or impairment of balancing. Cyon also investigated the subject, and concluded:--(1) To maintain equilibrium, we must have an accurate notion of the position of the head in space; (2) the function of the semicircular canals is to communicate impressions that give a representation of this position--each canal having a relation to one of the dimensions of space; (3) disturbance of equilibrium follows section; (4) involuntary movements following section are due to abnormal excitations; (5) abnormal movements occurring a few days after the operation are caused by irritation of the cerebellum.

On theoretical considerations of a physical character, E. Mach, Crum-Brown and Breuer have advanced theories based on the idea of the canals being organs for sensations of acceleration of movement, or for the sense of rotation. Mach first pointed out that Purkinje's phenomena, already alluded to, were in all probability related to the semicircular canals. "He showed that when the body is moved in space, in a straight line, we are not conscious of the velocity of motion, but of variations in this velocity. Similarly, if a body is rotated round a vertical axis, we perceive only angular acceleration and not angular velocity. The sensations produced by angular acceleration last longer than the acceleration itself, and the position of the head during the movements enables us to determine direction." Both Mach and Goltz state that varying pressures of the fluid in the canals produced by angular rotation produce sensations of movement (always in a direction opposite to the rotation of the body), and that these, in turn, cause the vertigo of Purkinje and the phenomena of Flourens. Mach, Crum-Brown and Breuer advance hydrodynamical theories in which they assume that the fluids move in the canals. Goltz, on the other hand, supports a hydrostatical theory in which he assumes that the phenomena can be accounted for by varying pressures. Crum-Brown differs from Mach and Breuer as follows:--(1) In attributing movement or variation of pressure not merely to the endolymph, but also to the walls of the membranous canals and to the surrounding perilymph; and (2) in regarding the two labyrinths as one organ, all the six canals being required to form a true conception of the rotating motion of the head. He sums up the matter thus: "We have two ways in which a relative motion can occur between the endolymph and the walls of the cavity containing it--(1) When the head begins to move, here the walls leave the fluid behind; (2) when the head stops, here the fluid flows on. In both cases the sensation of rotation is felt. In the first this sensation corresponds to a real rotation, in the second it does not, but in both it corresponds to a real acceleration (positive or negative) of rotation, using the word acceleration in its technical kinematical sense."

Cyon states that the semicircular canals only indirectly assist in giving a notion of spatial relations. "He holds that knowledge of the position of bodies in space depends on nervous impulses coming from the contracting ocular muscles; that the oculomotor centres are in intimate physiological relationship with the centres receiving impulses from the nerves of the semicircular canals; and that the oculomotor centres, thus excited, produce the movements of the eyeballs, which then determine our notions of spatial relations." These views are supported by experiments of Lee on dog-fish. When the fish is rotated round different axes there are compensating movements of the eyes and fins. "It was observed that if the fish were rotated in the plane of one of the canals, exactly the same movements of the eyes and fins occurred as were produced by experimental operation and stimulation of the ampulla of that canal." Sewall, in 1883, carried out experiments on young sharks and skates with negative results. Lee returned to the subject in 1894, and, after numerous experiments on dog-fish, in which the canals or the auditory nerves were divided, obtained evidence that the ampullae contain sense-organs connected with the sense of equilibrium.

It has been found by physicians and aurists that disease or injury of the canals, occurring rapidly, produces giddiness, staggering, nystagmus (a peculiar twitching movement of the muscles of the eyeballs), vomiting, noises in the ear and more or less deafness. It is said, however, that if pathological changes come on slowly, so that the canals and vestibule are converted into a solid mass, none of these symptoms may occur. On the whole, the evidence is in favour of the view that from the semicircular canals nervous impulses are transmitted, which, co-ordinated with impulses coming from the visual organs, from the muscles and from the skin, form the bases of these guiding sensations on which the sense of equilibrium depends. These impulses may not reach the level of consciousness, but they call into action co-ordinated mechanisms by which complicated muscular movements are effected.

Full bibliographical references are given in the article on "The Ear"
by J.G. McKendrick, in Schafer's _Textbook of Physiology_, vol. ii. p.
1194. (J. G. M.)

EQUINOX (from the Lat. _aequus_, equal, and _nox_, night), a term used to express either the moment at which, or the point at which, the sun apparently crosses the celestial equator. Since the sun moves in the ecliptic, it is in the last-named sense the point of intersection of the ecliptic and the celestial equator. This is the usual meaning of the term in astronomy. There are two such points, opposite each other, at one of which the sun crosses the equator toward the north and at the other toward the south. They are called vernal and autumnal respectively, from the relation of the corresponding times to the seasons of the northern hemisphere. The line of the equinoxes is the imaginary diameter of the celestial sphere which joins them.

The vernal equinox is the initial point from which the right ascensions and the longitudes of the heavenly bodies are measured (see ASTRONOMY: _Spherical_). It is affected by the motions of Precession and Nutation, of which the former has been known since the time of Hipparchus. The actual equinox is defined by first taking the conception of a fictitious point called the Mean Equinox, which moves at a nearly uniform rate, slow varying, however, from century to century. The true equinox then moves around the mean equinox in a period equal to that of the moon's nodes. These two motions are defined with greater detail in the articles PRECESSION OF THE EQUINOXES and NUTATION.

_Equinoctial Gales._--At the time of the equinox it is commonly believed that strong gales may be expected. This popular idea has no foundation in fact, for continued observations have failed to show any unusual prevalence of gales at this season. In one case observations taken for fifty years show that during the five days from the 21st to the 25th of March and September, there were fewer gales and storms than during the preceding and succeeding five days.

EQUITES ("horsemen" or "knights," from _equus_, "horse"), in Roman history, originally a division of the army, but subsequently a distinct political order, which under the empire resumed its military character. According to the traditional account, Romulus instituted a cavalry corps, consisting of three _centuriae_ ("hundreds"), called after the three tribes from which they were taken (Ramnes, Tities, Luceres), divided into ten _turmae_ ("squadrons") of thirty men each. The collective name for the corps was _celeres_ ("the swift," or possibly from [Greek: keles], "a riding horse"); Livy, however, restricts the term to a special body-guard of Romulus. The statements in ancient authorities as to the changes in the number of the equites during the regal period are very confusing; but it is regarded as certain that Servius Tuillus found six centuries in existence, to which he added twelve, making eighteen in all, a number which remained unchanged throughout the republican period. A proposal by M. Porcius Cato the elder to supplement the deficiency in the cavalry by the creation of four additional centuries was not adopted. The earlier centuries were called _sex suffragia_ ("the six votes"), and at first consisted exclusively of patricians, while those of Servius Tullius were entirely or for the most part plebeian. Until the reform of the comitia centuriata (probably during the censorship of Gaius Flaminius in 220 B.C.; see COMITIA), the equites had voted first, but after that time this privilege was transferred to one century selected by lot from the centuries of the equites and the first class. The equites then voted with the first class, the distinction between the _sex suffragia_ and the other centuries being abolished.

Although the equites were selected from the wealthiest citizens, service in the cavalry was so expensive that the state gave financial assistance. A sum of money (_aes equestre_) was given to each eques for the purchase of two horses (one for himself and one for his groom), and a further sum for their keep (_aes hordearium_); hence the name _equites equo publico_. In later times, pay was substituted for the _aes hordearium_, three times as much as that of the infantry. If competent, an eques could retain his horse and vote after the expiration of his ten years' service, and (till 129 B.C.) even after entry into the senate.