Chapter 3

In the case of a quadric equation x² − px + q = 0, we can by the assistance of the sign √( ) or ( )1/2find an expression for x as a 2-valued function of the coefficients p, q such that substituting this value in the equation, the equation is thereby identically satisfied; it has been found that this expression isx = ½ {p ± √(p² − 4q) },and the equation is on this account said to be algebraically solvable, or more accurately solvable by radicals. Or we may by writing x = −½ p + z reduce the equation to z² = ¼ (p² − 4q), viz. to an equation of the form x² = a; and in virtue of its being thus reducible we say that the original equation is solvable by radicals. And the question for an equation of any higher order, say of the order n, is, can we by means of radicals (that is, by aid of the signm√( ) or ( )1/m, using as many as we please of such signs and with any values of m) find an n-valued function (or any function) of the coefficients which substituted for x in the equation shall satisfy it identically?It will be observed that the coefficients p, q ... are not explicitly considered as numbers, but even if they do denote numbers, the question whether a numerical equation admits of solution by radicals is wholly unconnected with the before-mentioned theorem of the existence of the n roots of such an equation. It does not even follow that in the case of a numerical equation solvable by radicals the algebraical solution gives the numerical solution, but this requires explanation. Consider first a numerical quadric equation with imaginary coefficients. In the formula x = ½ {p ± √(p² − 4q) }, substituting for p, q their given numerical values, we obtain for x an expression of the form x = α + βi ± √(γ + δi), where α, β, γ, δ are real numbers. This expression substituted for x in the quadric equation would satisfy it identically, and it is thus an algebraical solution; but there is no obviousa priorireason why √(γ + δi) should have a value = c + di, where c and d are real numbers calculable by the extraction of a root or roots of real numbers; however the case is (what there was noa prioriright to expect) that √(γ + δi) has such a value calculable by means of the radical expressions √{√(γ² + δ²) ± γ}; and hence the algebraical solution of a numerical quadric equation does in every case give the numerical solution. The case of a numerical cubic equation will be considered presently.

x = ½ {p ± √(p² − 4q) },

and the equation is on this account said to be algebraically solvable, or more accurately solvable by radicals. Or we may by writing x = −½ p + z reduce the equation to z² = ¼ (p² − 4q), viz. to an equation of the form x² = a; and in virtue of its being thus reducible we say that the original equation is solvable by radicals. And the question for an equation of any higher order, say of the order n, is, can we by means of radicals (that is, by aid of the signm√( ) or ( )1/m, using as many as we please of such signs and with any values of m) find an n-valued function (or any function) of the coefficients which substituted for x in the equation shall satisfy it identically?

It will be observed that the coefficients p, q ... are not explicitly considered as numbers, but even if they do denote numbers, the question whether a numerical equation admits of solution by radicals is wholly unconnected with the before-mentioned theorem of the existence of the n roots of such an equation. It does not even follow that in the case of a numerical equation solvable by radicals the algebraical solution gives the numerical solution, but this requires explanation. Consider first a numerical quadric equation with imaginary coefficients. In the formula x = ½ {p ± √(p² − 4q) }, substituting for p, q their given numerical values, we obtain for x an expression of the form x = α + βi ± √(γ + δi), where α, β, γ, δ are real numbers. This expression substituted for x in the quadric equation would satisfy it identically, and it is thus an algebraical solution; but there is no obviousa priorireason why √(γ + δi) should have a value = c + di, where c and d are real numbers calculable by the extraction of a root or roots of real numbers; however the case is (what there was noa prioriright to expect) that √(γ + δi) has such a value calculable by means of the radical expressions √{√(γ² + δ²) ± γ}; and hence the algebraical solution of a numerical quadric equation does in every case give the numerical solution. The case of a numerical cubic equation will be considered presently.

17. A cubic equation can be solved by radicals.

Taking for greater simplicity the cubic in the reduced form x³ + qx − r = 0, and assuming x = a + b, this will be a solution if only 3ab = q and a³ + b³ = r, equations which give (a³ − b³)² = r² −4⁄27q³, a quadric equation solvable by radicals, and giving a³ − b³ = √(r² −4⁄27q³), a 2-valued function of the coefficients: combining this with a³ + b³ = r, we have a³ = ½ {r + √(r² −4⁄27q³) }, a 2-valued function: we then have a by means of a cube root, viz.a =3√[½ {r + √(r² −4⁄27q³) }],a 6-valued function of the coefficients; but then, writing q = b/3a, we have, as may be shown, a + b a 3-valued function of the coefficients; and x = a + b is the required solution by radicals. It would have been wrong to complete the solution by writingb =3√[½ {r − √(r² −4⁄27q³) } ],for then a + b would have been given as a 9-valued function having only 3 of its values roots, and the other 6 values being irrelevant. Observe that in this last process we make no use of the equation 3ab = q, in its original form, but use only the derived equation 27a³b³ = q³, implied in, but not implying, the original form.An interesting variation of the solution is to write x = ab(a + b), giving a³b³ (a³ + b³) = r and 3a³b³ = q, or say a³ + b³ = 3r/q, a³b³ =1⁄3q; and consequentlya³ =3⁄2{r + √(r² −4⁄27q³) }, b³ =3⁄2{r − √(r² −4⁄27q³) },qqi.e.here a³, b³ are each of them a 2-valued function, but as the only effect of altering the sign of the quadric radical is to interchange a³, b³, they may be regarded as each of them 1-valued; a and b are each of them 3-valued (for observe that here only a³b³, not ab, is given); and ab(a + b) thus is in appearance a 9-valued function; but it can easily be shown that it is (as it ought to be) only 3-valued.In the case of a numerical cubic, even when the coefficients are real, substituting their values in the expressionx =3√[½ {r + √(r² −4⁄27q³) }] +1⁄3q ÷3√[½ {r + √(r² −4⁄27q³) }],this may depend on an expression of the form3√(γ + δi) where γ and δ are real numbers (it will do so if r² −4⁄27q³ is a negative number), and then wecannotby the extraction of any root or roots of real positive numbers reduce3√(γ + δi) to the form c + di, c and d real numbers; hence here the algebraical solution does not give the numerical solution, and we have here the so-called “irreducible case” of a cubic equation. By what precedes there is nothing in this that might not have been expected; the algebraical solution makes the solution depend on the extraction of the cube root of a number, and there was no reason for expecting this to be a real number. It is well known that the case in question is that wherein the three roots of the numerical cubic equation are all real; if the roots are two imaginary, one real, then contrariwise the quantity under the cube root is real; and the algebraical solution gives the numerical one.The irreducible case is solvable by a trigonometrical formula, but this is not a solution by radicals: it consists in effect in reducing the given numerical cubic (not to a cubic of the form z³ = a, solvable by the extraction of a cube root, but) to a cubic of the form 4x³ − 3x = a, corresponding to the equation 4 cos³ θ − 3 cos θ = cos 3θ which serves to determine cosθ when cos 3θ is known. The theory is applicable to an algebraical cubic equation; say that such an equation, if it can be reduced to the form 4x³ − 3x = a, is solvable by “trisection”—then the general cubic equation is solvable by trisection.

a =3√[½ {r + √(r² −4⁄27q³) }],

a 6-valued function of the coefficients; but then, writing q = b/3a, we have, as may be shown, a + b a 3-valued function of the coefficients; and x = a + b is the required solution by radicals. It would have been wrong to complete the solution by writing

b =3√[½ {r − √(r² −4⁄27q³) } ],

for then a + b would have been given as a 9-valued function having only 3 of its values roots, and the other 6 values being irrelevant. Observe that in this last process we make no use of the equation 3ab = q, in its original form, but use only the derived equation 27a³b³ = q³, implied in, but not implying, the original form.

An interesting variation of the solution is to write x = ab(a + b), giving a³b³ (a³ + b³) = r and 3a³b³ = q, or say a³ + b³ = 3r/q, a³b³ =1⁄3q; and consequently

i.e.here a³, b³ are each of them a 2-valued function, but as the only effect of altering the sign of the quadric radical is to interchange a³, b³, they may be regarded as each of them 1-valued; a and b are each of them 3-valued (for observe that here only a³b³, not ab, is given); and ab(a + b) thus is in appearance a 9-valued function; but it can easily be shown that it is (as it ought to be) only 3-valued.

In the case of a numerical cubic, even when the coefficients are real, substituting their values in the expression

x =3√[½ {r + √(r² −4⁄27q³) }] +1⁄3q ÷3√[½ {r + √(r² −4⁄27q³) }],

this may depend on an expression of the form3√(γ + δi) where γ and δ are real numbers (it will do so if r² −4⁄27q³ is a negative number), and then wecannotby the extraction of any root or roots of real positive numbers reduce3√(γ + δi) to the form c + di, c and d real numbers; hence here the algebraical solution does not give the numerical solution, and we have here the so-called “irreducible case” of a cubic equation. By what precedes there is nothing in this that might not have been expected; the algebraical solution makes the solution depend on the extraction of the cube root of a number, and there was no reason for expecting this to be a real number. It is well known that the case in question is that wherein the three roots of the numerical cubic equation are all real; if the roots are two imaginary, one real, then contrariwise the quantity under the cube root is real; and the algebraical solution gives the numerical one.

The irreducible case is solvable by a trigonometrical formula, but this is not a solution by radicals: it consists in effect in reducing the given numerical cubic (not to a cubic of the form z³ = a, solvable by the extraction of a cube root, but) to a cubic of the form 4x³ − 3x = a, corresponding to the equation 4 cos³ θ − 3 cos θ = cos 3θ which serves to determine cosθ when cos 3θ is known. The theory is applicable to an algebraical cubic equation; say that such an equation, if it can be reduced to the form 4x³ − 3x = a, is solvable by “trisection”—then the general cubic equation is solvable by trisection.

18. A quartic equation is solvable by radicals, and it is to be remarked that the existence of such a solution depends on the existence of 3-valued functions such as ab + cd of the four roots (a, b, c, d): by what precedes ab + cd is the root of a cubic equation, which equation is solvable by radicals: hence ab + cd can be found by radicals; and since abcd is a given function, ab and cd can then be found by radicals. But by what precedes, if ab be known then any similar function, say a + b, is obtainable rationally; and then from the values of a + b and ab we may by radicals obtain the value of a or b, that is, an expression for the root of the given quartic equation: the expression ultimately obtained is 4-valued, corresponding to the different values of the several radicals which enter therein, and we have thus the expression by radicals of each of the four roots of the quartic equation. But when the quartic is numerical the same thing happens as in the cubic, and the algebraical solution does not in every case give the numerical one.

It will be understood from the foregoing explanation as to the quartic how in the next following case, that of the quintic, the question of the solvability by radicals depends on the existence or non-existence of k-valued functions of the five roots (a, b, c, d, e); the fundamental theorem is the one already stated, a rational function of five letters, if it has less than 5, cannot have more than 2 values, that is, there are no 3-valued or 4-valued functions of 5 letters: and by reasoning depending in part upon this theorem, N.H. Abel (1824) showed that a general quintic equation is not solvable by radicals; anda fortiorithe general equation of any order higher than 5 is not solvable by radicals.19. The general theory of the solvability of an equation by radicals depends fundamentally on A.T. Vandermonde’s remark (1770) that, supposing an equation is solvable by radicals, and that we have therefore an algebraical expression of x in terms of the coefficients, then substituting for the coefficients their values in terms of the roots, the resulting expression must reduce itself to any one at pleasure of the roots a, b, c ...; thus in the case of the quadric equation, in the expression x = ½ {p + √(p² − 4q) }, substituting for p and q their values, and observing that (a + b)² − 4ab = (a − b)², this becomes x = ½ {a + b + √(a − b)²}, the value being a or b according as the radical is taken to be +(a − b) or −(a − b).So in the cubic equation x³ − px² + qx − r = 0, if the roots are a, b, c, and if ω is used to denote an imaginary cube root of unity, ω² + ω + 1 = 0, then writing for shortness p = a + b + c, L = a + ωb + ω²c, M = a + ω²b + ωc, it is at once seen that LM, L³ + M³, and therefore also(L³ − M³)² are symmetrical functions of the roots, and consequently rational functions of the coefficients; hence½ {L³ + M³ + √(L³ − M³)²}is a rational function of the coefficients, which when these are replaced by their values as functions of the roots becomes, according to the sign given to the quadric radical, = L³ or M³; taking it = L³, the cube root of the expression has the three values L, ωL, ω²L; and LM divided by the same cube root has therefore the values M, ω²M, ωM; whence finally the expression1⁄3[p +3√{½ (L³ + M³ + √(L³ − M³)²) } + LM ÷3√{½ L³ + M³ + √(L³ − M³)²) }]has the three values1⁄3(p + L + M),1⁄3(p + ωL + ω²M),1⁄3(p + ω²L + ωM);that is, these are = a, b, c respectively. If the value M³ had been taken instead of L³, then the expression would have had the same three values a, b, c. Comparing the solution given for the cubic x³ + qx − r = 0, it will readily be seen that the two solutions are identical, and that the function r² −4⁄27q³ under the radical sign must (by aid of the relation p = 0 which subsists in this case) reduce itself to (L³ − M³)²; it is only by each radical being equal to a rational function of the roots that the final expressioncanbecome equal to the roots a, b, c respectively.

19. The general theory of the solvability of an equation by radicals depends fundamentally on A.T. Vandermonde’s remark (1770) that, supposing an equation is solvable by radicals, and that we have therefore an algebraical expression of x in terms of the coefficients, then substituting for the coefficients their values in terms of the roots, the resulting expression must reduce itself to any one at pleasure of the roots a, b, c ...; thus in the case of the quadric equation, in the expression x = ½ {p + √(p² − 4q) }, substituting for p and q their values, and observing that (a + b)² − 4ab = (a − b)², this becomes x = ½ {a + b + √(a − b)²}, the value being a or b according as the radical is taken to be +(a − b) or −(a − b).

So in the cubic equation x³ − px² + qx − r = 0, if the roots are a, b, c, and if ω is used to denote an imaginary cube root of unity, ω² + ω + 1 = 0, then writing for shortness p = a + b + c, L = a + ωb + ω²c, M = a + ω²b + ωc, it is at once seen that LM, L³ + M³, and therefore also(L³ − M³)² are symmetrical functions of the roots, and consequently rational functions of the coefficients; hence

½ {L³ + M³ + √(L³ − M³)²}

is a rational function of the coefficients, which when these are replaced by their values as functions of the roots becomes, according to the sign given to the quadric radical, = L³ or M³; taking it = L³, the cube root of the expression has the three values L, ωL, ω²L; and LM divided by the same cube root has therefore the values M, ω²M, ωM; whence finally the expression

1⁄3[p +3√{½ (L³ + M³ + √(L³ − M³)²) } + LM ÷3√{½ L³ + M³ + √(L³ − M³)²) }]

has the three values

1⁄3(p + L + M),1⁄3(p + ωL + ω²M),1⁄3(p + ω²L + ωM);

that is, these are = a, b, c respectively. If the value M³ had been taken instead of L³, then the expression would have had the same three values a, b, c. Comparing the solution given for the cubic x³ + qx − r = 0, it will readily be seen that the two solutions are identical, and that the function r² −4⁄27q³ under the radical sign must (by aid of the relation p = 0 which subsists in this case) reduce itself to (L³ − M³)²; it is only by each radical being equal to a rational function of the roots that the final expressioncanbecome equal to the roots a, b, c respectively.

20. The formulae for the cubic were obtained by J.L. Lagrange (1770-1771) from a different point of view. Upon examining and comparing the principal known methods for the solution of algebraical equations, he found that they all ultimately depended upon finding a “resolvent” equation of which the root is a + ωb + ω²c + ω³d + ..., ω being an imaginary root of unity, of the same order as the equation;e.g.for the cubic the root is a + ωb + ω²c, ω an imaginary cube root of unity. Evidently the method gives for L³ a quadric equation, which is the “resolvent” equation in this particular case.

For a quartic the formulae present themselves in a somewhat different form, by reason that 4 is not a prime number. Attempting to apply it to a quintic, we seek for the equation of which the root is (a + ωb + ω²c + ω³d + ω4e), ω an imaginary fifth root of unity, or rather the fifth power thereof (a + ωb + ω²c + ω³d + ω4e)5; this is a 24-valued function, but if we consider the four values corresponding to the roots of unity ω, ω², ω³, ω4, viz. the values

(a + ω b + ω²c + ω³d + ω4e)5,(a + ω²b + ω4c + ω d + ω³e)5,(a + ω³b + ω c + ω4d + ω²e)5,(a + ω4b + ω³c + ω²d + ω e)5,

(a + ω b + ω²c + ω³d + ω4e)5,

(a + ω²b + ω4c + ω d + ω³e)5,

(a + ω³b + ω c + ω4d + ω²e)5,

(a + ω4b + ω³c + ω²d + ω e)5,

any symmetrical function of these, for instance their sum, is a 6-valued function of the roots, and may therefore be determined by means of a sextic equation, the coefficients whereof are rational functions of the coefficients of the original quintic equation; the conclusion being that the solution of an equation of the fifth order is made to depend upon that of an equation of the sixth order. This is, of course, useless for the solution of the quintic equation, which, as already mentioned, does not admit of solution by radicals; but the equation of the sixth order, Lagrange’s resolvent sextic, is very important, and is intimately connected with all the later investigations in the theory.

21. It is to be remarked, in regard to the question of solvability by radicals, that not only the coefficients are taken to be arbitrary, but it is assumed that they are represented each by a single letter, or say rather that they are not so expressed in terms of other arbitrary quantities as to make a solution possible. If the coefficients are not all arbitrary, for instance, if some of them are zero, a sextic equation might be of the form x6+ bx4+ cx² + d = 0, and so be solvable as a cubic; or if the coefficients of the sextic are given functions of the six arbitrary quantities a, b, c, d, e, f, such that the sextic is really of the form (x² + ax + b)(x4+ cx³ + dx² + ex + f) = 0, then it breaks up into the equations x² + ax + b = 0, x4+ cx³ + dx² + ex + f = 0, and is consequently solvable by radicals; so also if the form is (x − a) (x − b) (x − c) (x − d) (x − e) (x − f) = 0, then the equation is solvable by radicals,—in this extreme case rationally. Such cases of solvability are self-evident; but they are enough to show that the general theorem of the non-solvability by radicals of an equation of the fifth or any higher order does not in any wise exclude for such orders the existence of particular equations solvable by radicals, and there are, in fact, extensive classes of equations which are thus solvable; the binomial equations xn− 1 = 0 present an instance.

22. It has already been shown how the several roots of the equation xn− 1 = 0 can be expressed in the form cos 2sπ/n + i sin 2sπ/n, but the question is now that of the algebraical solution (or solution by radicals) of this equation. There is always a root = 1; if ω be any other root, then obviously ω, ω², ... ωn−1are all of them roots; xn− 1 contains the factor x − 1, and it thus appears that ω, ω², ... ωn−1are the n-1 roots of the equationxn−1+ xn−2+ ... x + 1 = 0;we have, of course, ωn−1+ ωn−2+ ... + ω + 1 = 0.It is proper to distinguish the cases n prime and n composite; and in the latter case there is a distinction according as the prime factors of n are simple or multiple. By way of illustration, suppose successively n = 15 and n = 9; in the former case, if α be an imaginary root of x³ − 1 = 0 (or root of x² + x + 1 = 0), and β an imaginary root of x5− 1 = 0 (or root of x4+ x³ + x² + x + 1 = 0), then ω may be taken = αβ; the successive powers thereof, αβ, α²β², β³, αβ4, α², β, αβ², α²β³, β4, α, α²β, β², αβ³, α²β4, are the roots of x14+ x13+ ... + x + 1 = 0; the solution thus depends on the solution of the equations x³ − 1 = 0 and x5− 1 = 0. In the latter case, if α be an imaginary root of x³ − 1 = 0 (or root of x² + x + 1 = 0), then the equation x9− 1 = 0 gives x³ = 1, α, or α²; x³ = 1 gives x = 1, α, or α²; and the solution thus depends on the solution of the equations x³ − 1 = 0, x³ − α = 0, x³ − α² = 0. The first equation has the roots 1, α, α²; if β be a root of either of the others, say if β³ = α, then assuming ω = β, the successive powers are β, β², α, αβ, αβ², α², α²β, α²β², which are the roots of the equation x8+ x7+ ... + x + 1 = 0.It thus appears that the only case which need be considered is that of n a prime number, and writing (as is more usual) r in place of ω, we have r, r², r³,...rn−1as the (n − 1) roots of the reduced equationxn−1+ xn−2+ ... + x + 1 = 0;then not only rn− 1 = 0, but also rn−1+ rn−2+ ... + r + 1 = 0.

xn−1+ xn−2+ ... x + 1 = 0;

we have, of course, ωn−1+ ωn−2+ ... + ω + 1 = 0.

It is proper to distinguish the cases n prime and n composite; and in the latter case there is a distinction according as the prime factors of n are simple or multiple. By way of illustration, suppose successively n = 15 and n = 9; in the former case, if α be an imaginary root of x³ − 1 = 0 (or root of x² + x + 1 = 0), and β an imaginary root of x5− 1 = 0 (or root of x4+ x³ + x² + x + 1 = 0), then ω may be taken = αβ; the successive powers thereof, αβ, α²β², β³, αβ4, α², β, αβ², α²β³, β4, α, α²β, β², αβ³, α²β4, are the roots of x14+ x13+ ... + x + 1 = 0; the solution thus depends on the solution of the equations x³ − 1 = 0 and x5− 1 = 0. In the latter case, if α be an imaginary root of x³ − 1 = 0 (or root of x² + x + 1 = 0), then the equation x9− 1 = 0 gives x³ = 1, α, or α²; x³ = 1 gives x = 1, α, or α²; and the solution thus depends on the solution of the equations x³ − 1 = 0, x³ − α = 0, x³ − α² = 0. The first equation has the roots 1, α, α²; if β be a root of either of the others, say if β³ = α, then assuming ω = β, the successive powers are β, β², α, αβ, αβ², α², α²β, α²β², which are the roots of the equation x8+ x7+ ... + x + 1 = 0.

It thus appears that the only case which need be considered is that of n a prime number, and writing (as is more usual) r in place of ω, we have r, r², r³,...rn−1as the (n − 1) roots of the reduced equation

xn−1+ xn−2+ ... + x + 1 = 0;

then not only rn− 1 = 0, but also rn−1+ rn−2+ ... + r + 1 = 0.

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 gn−1is the lowest power of g, which is ≡ 1 to the modulus n; or, what is the same thing, that the series of powers 1, g, g², ... gn−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, g²,...gn−2, we have, in a different order, the whole series of roots r, r², r³,...rn−1.

In the most simple case, n = 5, the equation to be solved is x4+ x³ + x² + x + 1 = 0; here 2 is a prime root of 5, and the order of the roots is r, r², r4, r³. The Gaussian process consists in forming an equation for determining the periods P1, P2, = r + r4and r² + r³ respectively;—these being such that the symmetrical functions P1+ P2, P1P2are rationally determinable: in fact P1+ P2= −1, P1P2= (r + r4) (r² + r³), = r³ + r4+ r6+ r7, = r³ + r4+ r + r², = −1. P1, P2are thus the roots of u² + u − 1 = 0; and taking them to be known, they are themselves broken up into subperiods, in the present case single terms, r and r4for P1, r² and r³ for P2; the symmetrical functions of these are then rationally determined in terms of P1and P2; thus r + r4= P1, r·r4= 1, or r, r4are the roots of u² − 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, r³, r², r6, r4, r5. We may form either 3 periods each of 2 terms, P1, P2, P3= r + r6, r³ + r4, r² + r5respectively; or else 2 periods each of 3 terms, P1, P2= r + r² + r4, r³ + r6+ r5respectively; 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 + r² + r³ + r4+ r5+ r6, = 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 + r² + r4= P1, r·r² + r·r4+ r²·r4= P2, r·r²·r4= 1.

The theory was further developed by Lagrange (1808), who, applying his general process to the equation in question, xn−1+ xn−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 + ωb + ω²c + ...)n−1was in this case a given function of ω with integer coefficients.

Reverting to the before-mentioned particular equation x4+ x³ + x² + 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 ω, a root of the equation ω4− 1 = 0 (whence ω is = 1, −1, i, or −i, at pleasure), and consider the expression(a + ωb + ω²c + ω³d)4,the developed value of this is=a4+ b4+ c4+ d4+ 6 (a²c² + b²d²) + 12 (a²bd + b²ca + c²db + d²ac)+ω{4 (a³b + b³c + c³ + d³a) + 12 (a²cd + b²da + c²ab + d²bc) }+ω²{6 (a²b² + b²c² + c²d² + d²a²) + 4 (a³c + b³d + c³a + d³b) + 24abcd}+ω³{4 (a³d + b³a + c³b + d³c) + 12 (a²bc + b²cd + c²da + d²ab) }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, r², r4, r³ of the special equation, then the expression becomesr4+ r³ + r + r² + 6 (1 + 1)+ 12 (r² + r4+ r³ + r)+ ω {4 (1 + 1 + 1 + 1)+ 12 (r4+ r³ + r + r²) }+ ω²{6 (r + r² + r4+ r³)+ 4 (r² + r4+ r³ + r) }+ ω³{4 (r + r² + r4+ r³)+ 12 (r³ + r + r² + r4) }viz. this is= −1 + 4ω + 14ω² − 16ω³,a completely determined value. That is, we have(r + ωr² + ω²r4+ ω³r³) = −1 + 4ω + 14ω² − 16ω³,which result contains the solution of the equation. If ω = 1, we have (r + r² + r4+ r³)4= 1, which is right; if ω = −1, then (r + r4− r² − r³)4= 25; if ω = i, then we have {r − r4+ i(r² − r³) }4= −15 + 20i; and if ω = −i, then {r − r4− i (r² − r³) }4= −15 − 20i; the solution may be completed without difficulty.

Take ω, a root of the equation ω4− 1 = 0 (whence ω is = 1, −1, i, or −i, at pleasure), and consider the expression

(a + ωb + ω²c + ω³d)4,

the developed value of this is

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, r², r4, r³ of the special equation, then the expression becomes

viz. this is

= −1 + 4ω + 14ω² − 16ω³,

a completely determined value. That is, we have

(r + ωr² + ω²r4+ ω³r³) = −1 + 4ω + 14ω² − 16ω³,

which result contains the solution of the equation. If ω = 1, we have (r + r² + r4+ r³)4= 1, which is right; if ω = −1, then (r + r4− r² − r³)4= 25; if ω = i, then we have {r − r4+ i(r² − r³) }4= −15 + 20i; and if ω = −i, then {r − r4− i (r² − r³) }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 xn−1+ xn−2+ ... + x + 1 = 0, ω a root of ωn−1− 1 = 0, and g a prime root of gn−1≡ 1 (mod. n), then

(r + ωrg+ ... + ωn − 2rgn−2)n−1

is a given function M0+ M1ω ... + Mn−2ωn−2with integer coefficients, and by the extraction of (n − 1)th roots of this and similar expressions we ultimately obtain r in terms of ω, which is taken to be known; the equation xn− 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 xn−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, θx, θ²x, ... θn−1x (θx being a rational function of x, and θnx = 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 thatallthe roots can be expressed rationally in terms of any one of them, say x; if, moreover, θx, θ1x being any two of the roots, we have θθ1x = θ1θx, the equation will be solvable algebraically. It is proper to refer also to Abel’s definition of anirreducibleequation:—an equation φx = 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 φx 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 wordrationalis 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 x4+ x³ + x² + x + 1 = 0; the left hand side has here no rational divisor, and the equation is irreducible; but this function is = (x² + ½ x + 1)² −5⁄4x², and it has thus the irrational divisors x² + ½ (1 + √5)x + 1, x² + ½ (1 − √5)x + 1; and these, if weadjointhe radical √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 x² + px + q = 0 is irreducible, but it becomes reducible, breaking up into rational linear factors, when we adjoin the radical √(¼ p² − q).

The fundamental theorem is the Proposition I. of the “Mémoire sur les conditions de résolubilité des équations 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 φ(x) = 0, then φ(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 xn−1... + x + 1 = 0, n a prime number, a, b, c ... k = r, rg, rg²... rgn−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 a²b + b²c + c²a 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... xn, invariable by the substitutions xak + bfor xk, 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’sCours d’algèbre supérieure, 2nd ed. (1854); 4th ed. (1877-1878).

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 φ(x) = 0, then φ(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 xn−1... + x + 1 = 0, n a prime number, a, b, c ... k = r, rg, rg²... rgn−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 a²b + b²c + c²a 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... xn, invariable by the substitutions xak + bfor xk, 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’sCours d’algèbre supérieure, 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 x5+ ax + b = 0, or what is the same thing, to x5+ 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 6depends, 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, “Über die Auflösung der Gleichungen vom fünften 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’sTraité des substitutions et des équations algébriques(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 alsoGroups, 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 theEncy. d. math. Wiss.vol. ii.

(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 theEncy. d. math. Wiss.vol. ii.

1The coefficients were selected so that the roots might be nearly 1, 2, 3.2The 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 1770-1771.3The 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 α + βi (see Lagrange’sÉquations numériques).4The square root of α + βi can be determined by the extraction of square roots of positive real numbers, without the trigonometrical tables.

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

2The 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.

3The 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 α + βi (see Lagrange’sÉquations numériques).

4The square root of α + βi 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. (SeeAnomalyandOrbit.)

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 meridianaftermean noon; the − signbeforemean noon.

Table of the Equation of Time.

EQUATOR(Late Lat.aequator, fromaequare, 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.écurie, a stable, through its older formescurie, 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 (seeHousehold,Royal).

EQUIDAE,the family of perissodactyle ungulate mammals typified by the horse (Equus caballus); seeHorse. 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 genusEquus. 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 familyLophiodontidae(seePerissodactyla), that it has now become a very general practice to include the whole “phylum” in the familyEquidae. TheEquidae, in this extended sense, together with the extinctPalaeotheriidae, 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 earlierEquidaethe 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 inlength; 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,bandc) 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 ofHipparion. Not unfrequently, however, the lower ends of the splint-bones carry a small expansion, representing the phalanges.

Remains of horses indistinguishable fromE. caballusoccur 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 probablyE. 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 ofEquusoccur in the Pleistocene and from that continent others reached South America during the same epoch. In the latter country occursHippidium, 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 ArgentineOnohippidium, 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.Onohippidiumis 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 forHippidium.

Species from the Pliocene of Texas and the Upper Miocene (Loup Fork) of Oregon were at one time assigned toHippidium, but this is incorrect, that genus being exclusively South American. The namePliohippushas been applied to species from the same two formations on the supposition that the foot-structure was similar to that ofHippidium, 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 generaProtohippusandMerychippusof 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. InMerychippus, 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 genusAnchitherium. From the well-knownHipparion, orHippotherium, 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 asNeohipparion, we reach small horses which are now generally regarded as a lateral offshoot from theMerychippustype. 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 ofcin 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 inH. antilopinumof India the lateral toes had disappeared. If this be so, we have the development of a monodactyle foot in this genus independently ofEquus.

The foregoing genera constitute the subfamilyEquinae, or theEquidaeas 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 generaDesmatippusandAnchippusorParahippus. 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 andthe 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 isAnchitherium, typically from the Middle Miocene of Europe, but also represented by one species from the Upper Miocene of North America. The EuropeanA. aurelianensewas of the size of an ordinary donkey. The cheek-teeth are of the type shown inaof 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 AmericanMesohippus, 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 ofMesohippusare included in the genusMiohippus, but others consider that the two genera are distinct.

MesohippusandMiohippusare connected with the earliest and most primitive mammal which it is possible to include in the familyEquidaeby means ofEpihippusof the Uinta or Upper Eocene of North America, andPachynolophus, orOrohippus, 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 byHyracotherium(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 ofAnchitherium; 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 inEquus. FromHyracotherium, which is closely related to the Eocene representatives of the ancestral stocks of the other three branches of the Perissodactyla, the transition is easy toPhenacodus, the representative of the common ancestor of all the Ungulata.

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