the series (7) either consists of a single term or it is made up of a cycle of primitive n^^ roots of unity, \ A« A?-i wi , wi , wi ,...., wi ; (18> that is to say, no term in (8) after the first is equal to the first, but u)\ = wj. Also, if (let it Vje kejjt in view that n is prime) the cycle that contains all the primitive n^'^ roots of unity be Wl , Wi , Wi ,...., wi , (9) and if Ci be the sum of tlie terms in the cycle (8), the form of F (x) is- F (X) = xd- (piCr + P2C2 + .... + PmCm) X^ -^ -\- ^^ (qiCi + qiC-z + etc.) x'^-^ + etc. ^ where each of the expressions in the series C\, C-z, C3, etc., is what the immediately preceding term becomes by changing wi into "'I 1 Cm through this change becoming Ci ; and ;?!, 2^2, Qi, etc., are clear of oji. For, assuming that there is a term wo ii^ (7) additional to wi, we may take 0*2 to be the first term in (9) after cui that occurs in (7) ; and it may be considered to be wi , which may be otherwise written A . . (ui . Then, if F (x) be written ^ (aii), we have by hypothesis OF THE HIGHER DEGREES, WITH APPLICATIONS. 89 A . ■ . A A tp {u)\) ^ (p (lui ). Therefore, by §12, changing wi into coi ,

nato forms 1 " _i_ of R ; and let ^^ , ^ , K + i , etc., be respectively what J^ , h^ , etc., become in passing from ri to r„ ^ i . Also let the 7n jmrticular cognate forms of i?, obtained by changing J in 7-« + i successively into the different to*'* roots of J^ + i , be ''a + 1 ,'■« + •: , . . . . , Ta + ,a ■ (15) I These tei-ms are all unequal. For, because J is a principal surd in '/'i , and r-) is what ri becomes when J is changed into a surd whose value is ^i J , wi being a primitive vi^^'^ root of unity, the \iew may be taken that r-i involves no suids additional to those found in rj , except the primitive ?n^^ root of unity wi . Therefoie ?'i — r-^ involves no surds distinct from primitive ■m'''^ roots of unity that are not found in the simplified ex])ression 9-] . Therefore r^ — r-2 is in a simple state. 1 1 Let ?•« + 2 he what Ta + i becomes by changing J into a»iJ Then Va + i — Va + -j is a particular cognate form of the generic expression under which the simplified expression ri — rz falls. Therefore ?•« + i — ■J'a + 2 cannot be zero , for, if it were, ri — r^ would, by Cor. 2, Prop. lit'., be zero ; which, by the j)roposition, is impossible. Hence, the first two terms in (15) are unequal. In like manner all the terms in (15) are unequal. §24. Cor. 2. Let Xj = be the equation whose roots are the terms in (14). When Xi is modified according to §21, it is, by §16, m clear of the surd J . Should it involve any nurds tliat are not 1 roots of unity, take z a surd of the highest rank not a root of 1 unity in Xi : and, when z is changed successively into the different c^^ roots of the determinate base 2i , let Xx, Ti, A-,, .... ,xt~'\ (16) be respectively what X\ becomes. Any term in (16), as A'l , being selected, the m roots of the equation A'l = are unequal particular 92 PRINCIPLES OF THE SOLUTION OF EQUATIONS I cognate forms of R. For, z^ being a c^^ j-oot of si distinct from z_^ , let Ta + 1 be what n becomes when z becomes z ; the ex- 1 i_ pressions J , Ai , etc., at the same time becoming J , ha + i, etc. Then we may put c — 1 c — 2 Xi = a;'^ + (Jsi <^ + dz^ + etc.) a;"*-! + etc. ; (17) 1 where b, d, etc., are clear of z . Therefore, because ri is a root of the equation Xi = 0, 1 m — 1 j - (AiJ"^ + etc.) \ "^ c — \ c — 1 1 m — \ + {6s c -I- dz c -f etc.) j — (hiJ"^^ + etc.) I "» -1 -f etc. = 0. 1 1 ' ( m ' 1 ' ) All the surds in this equation occur in the simplitied expression ri . Therefore, by Prop. II., OT — 1 > m 1 m — i - (A« + iJ~^^ + etc.) !"" m a + \ \ c—1 C— 2 -J CT — 1 + (bz <= + dz + etc)S - (ha + iZl '« -f etc.) V" " ^ +etc. = 0. 2 2 / m ^ o + 1 I Therefore"^ {ha + iJ^ -|- etc.) or r,, + i is a root ot the equation c — 1 Xi = a;™ + {bz^ " + etc.) .t'"-i + etc. = 0. (18) Therefore also, by Cor. Prop. II., all the terms in (15) are roots of that equation. And, by Cor. 1, the terms in (15) are all unequal. Therefore the equation Xi = has vi unequal particular cognate forms of B for its roots. §25. Cor. 3. No two of the exj)ressions in (16), as Xi and Xi, are identical with one another. For, in order that Xi and Xi might be identical, the coefficients of the several powers of x in Xi would need to be equal to those of the corresponding powers of x in Xi ; but, if OF THE HIGHER DEGREES, WITH APPLICATIONS. 93 1 one of the coefficients of X\ be selected in which ;: is present, this coeflicient can be shown to be unequal to the corresponding coefficieut in Xi in the same way in whicli the terms in (15) were proved to be all unequal. §26. Cor. 4. Any two of the terms in (16), as Xi and Xi . being selected, the equations Xi = and Xi = have no root in common. For, suppose, if possible, that these equations have a root in common. Taking the forms of Xi and Xi in (17) and (18), since ri is a root of the equation A'l = 0, c — 1 ^ + (^^2 ' + ^*^«-) ''7"' + et«- = 0- (19) m — 1 1 All the surds in this equation except z occur in ri . It is impossible 1 1 1 1 that z can occur in ri ; for, z occurs in ri ; and z, = O-^z > 1 Oi being a primitive c^^ root of unity ; but this equation, if both z " 1 c and «., occurred in ri , would be of the inadmissible type (3). 1 Since z does not occur in ri , it is a principal (see §2) surd in (19). We may, therefore, keeping in view that ri is the expression (1) in w . which J is a principal surd, arrange (19) thus, 1 m — 1 c — 1 c — 2 m . m . c c f (-^1 ) = ^1 (P^-2 + P-^-2 + ^*<^-) m — 2 e — 1 c — 2 m c c + ^1 (?i-^ + '?--2 + etc.) + etc. = 0; (20) 1 where pi , gi , etc., are clear of z^ . Then, wi being a primitive 1 m^^ root of unity such that, by changing J into the 7n^^ root of J 1 whose value is wiJ , ri becomes r-2 , 94 PRINCIPLES OF THE SOLUTION OF EQUATIONS 1 w. — 1 c — 1 m — 1 HI — 1 C — 1 + '"l -*! ('?'1»2 + ^^^-^ + *^^^' (21) The coefficients of the several poMers of J in 97 (J ) cannot Le 1 all zero ; for, if they were, we should have, from (21), w (t'^i J ) = ^■ This means that r-i is a root of the equation X\ = 0. But in like manner all the terms in (14) would be roots of that equation, and J^i would be identical with X; which, by Cor. 3, is impossible. 1 J. Since the coefficients of the different powers of J in ^ (J are 1 not all zero, the equation (20) gives us, by §5, a)J =^1,0) beini; 1 an m}^ root of unity, and ^1 involving only surds in

^^^2 > etc., we reach a term Xg into which no surds
enter that are not roots of unity, the vie . . . . I roots of the equation
Xg = being unequal particular cognate forms of R. Should Xg
modified according to §21, not be rational, its form, by Prop. IV.,
putting d for 7nc .... I, is
'^e=Xd—{p^Oi+ -\-pmOm)xf^-'^ + {qiCi+ .... +5'™C^)x'^-2-f etc. ;
where, one of the roots occurring in Xg being the primitive n^^ root
of unity coi , the coefficients pi , qi , etc., are clear of Wi ; and Ci is
the sum of the cycle of primitive n^^ roots of unity (8) containing
fi I
s or terms ; and, the cycle (9) containing all the primitive
wth roots of unity, the change of coi into Wi causes (7i to become C-> ,
and C2 to become C3 , and so on. Cm becoming Ci . As was explained
at the close of §20, the cycle (8) may be reduced to a single term,
which is then identical with Ci . It will also not be forgotten that
the roots of \inity such as the 7i* here sj)oken of are, according to §1,
subject to the condition that the numbers such as n are prime. When
Ci in Xg is changed successively into Ci , C2 , etc., let Xg become
^t , -^e ) -^e , • • • • ) ^e (22)
96 PRINCIPLES OF THE SOLUTION OF EQUATIONS
If Xf_ 4. 1 1)6 the continued jiroduct of the terras in (22), the dm roots
of the equation Xe -|- 1 = can be shown to be iinequal particular
<;• )gnate forms of R. For, no two terms in (22) as X^, and X^ are
identical ; because, if they were, X^ would i-emain unaltered bj the
a a
change of wi into wi ; which, by Prop. IV., because (t»i is not a t^erm
in the cycle (8), is impossible. It follows that no two of the equations
Xg = 0, Xe = 0, etc., have a root in common. For, if the equations
A'e = 0, and X^ = had a root in common, since Xg and Xg are not
identical, Xg would have a lower measure involving only surds found
in Xp , because the surds in Xg are the same with those in Xg . Let
(f (x) be this lower measure of Xg , and let ri be a root of the equa-
tion f (^) = ^- Then, by Cor. Prop. II., all the d roots of the
equation Xg = ax-e roots of the equation ^ (x) = ; which is
impossible. In the same way it can be proved that no equation in
the series Xg = 0, Xg = 0, etc., has equal roots. Since no one of
these equations has equal roots, and no two of them have a root in
<.ommon, tlie dm roots of the equation Xg + 1 ^ are unequal | ar-
ticular cognate forms of R. Also X^ + i. modified according to
§21, is clear of the primitive ?^*^ roots of unity. Should Xg + 1 not
be rational, we can deal with it as we did with Xg . Going on in
this way, we ultimately reach a rationed exjjression X^ such that the
dm . . . . g roots of the equation X^ = are unequal particular
c'gnate forms of JR. This equation must be identical with the equa-
tion F (x) ^ of which ri is a root. For, by Prop. III., the equation
F (x) =0 has for its roots the unequal particular cognate forms of R.
Therefore, because the roots of the equation Xj := are all unequal
and are at the same time particular cognate forms of R, X^ must be
either a lower measure of F (x) or identical with F (x). But F (x),
V>eing ii-reducible, has no lower measure. Therefore Xz is identical
with F (x). Therefore, the equation F (x) ^ being the X^^ degree,
X = mc .... Im . . . . g. Hence X is a multiple of m. This is the
1
result arrived at when ri involves a surd of the highest rank J not
a root of unity. Should Vi involve no surds except i-oots (see §1) of
unity, we should then have set out from Xg regarded as identical with
X — r\ . The result woidd have been X = m . . . . g. Therefore X
is a multiple of m ; and, because m is here the number of cycles of s
terms each, that make up the series of the 2Jrimiti\e n^^ roots of unity,
tiis = n — 1. Therefore iV is a multiple of a measure of n — 1.
§29. Cor. Let X be a prime number. Then, if ri involve a surd
1
■of the highest rank J not a root (see §1) of unity, X = m ; for,
OF THE HIGHER DEGREES, WITH APPLICATIONS. 97
the series of integers vi, c, etc., of which iV" is the continued product,
is reduced to its fii'st term. If ri involve only surds that are roots of
unity, n — 1 is a multiple of iV ; ior 2^ = m . . . . g ; therefore,
because iV is prime, it is equal to m ; but ms = n — 1 ; therefore
The Solvable Irreducible Equation of the m^'^ Degree, m Prime.
§30. The priLciples that have been established may be illustrated
by an examination of the solvable irreducible rational equation of the
^th (Jegree F (x) = 0, m being prime. Two cases may be distinguished,
though it will be found that the roots can in the two cases be brought
under a common form ; the one case being that in which the simplified
root ri is, and the other that in which it is not, a rational function of
roots of unity, that is, according to §1, of rooth of unity having the
denominators of their indices prime numbers. The equation F(^x) =
may be said to be in the former case of the first class, and in the latter
■of the second class.
The Equation F (x) = of the First Class.
§31. In this case, by Cor. Prop. YL, r^ being modified according to
§21, if one of the roots involved in ri be the primitive n^^ root of
unity wi , « — 1 is a multiple of m. Also the expression written
Xf in Prop. YI. is reduced to x — ^i , so that
n = piCi + P2C2 + + PmCm .
The m roots of the equation F (x) = being ri , r^ , etc , we must
shave
n = Pi C\ 4- P2O2 -^ + PmOm ,
r2 = PmCi + 2hG2 + + 2'>m-lCm , i ^^^^
rm= V% Oi + psC2 + 4- IhCm .
For, by Prop. II., because ri is a root of the equation F (x) = 0, all
the expressions on the right of the equations (23) are roots of that
■equation. And no two of these expressions are equal to one another.
For, take the first two. If these were equal, we should have
{Pm — Pi) Ci-\- {201 — P2 ) Ci + etc. = . Therefore, by §13,
each of the terms pm — P\ , f\ — f'l , etc., is zero. This makes
!P\ 1 P2 i etc., all equal to one another. Therefore ri = — pi ; so
that the primitive w*^ root of unity is eliminated from ri ; which, by
.§21, is impossible. Hence the values of the m roots of the equation
F (x) = are those given in (23).
98 PRINCIPLES OF THE SOLUTION OF EQUATIONS
/
§32. Let ri be one of the pai'ticular cognate forms of the generic
exin-ession B under which the simplified exi)ression ri falls. Then,
because, by Prop. II., all the particular cognate forms of H are roots
of the equation F (x) = 0, rj is equal to one of the m terms ri , r2 ,
etc., say to r, . I will now show that the changes of the surds
involved that cause ri to become ri , whose value is r^ , cause r2 to
receive the value Vn + i , and r^ to receive the value r^ ^ -j , and so on.
This may appear obvious on the face of the equations (23) ; but, to
jjrevent misunderstanding, the steps of the deduction are given. Any
changes made in ri must transform Ci into Cg , one of the m terms
Ci , C2 , etc. In passing from ri to ri , while Ci becomes Cg , let r,
/ / /
become ro , and 2h become pi , and p2 become po , and so on. The
change that causes Ci to become Cg transforms C'2 into C'^ + 1 , and
C3 into Cs + 2 , and so on. Therefore, it being understood that
pm + 1 ) Cm + 1 ) etc., are the same as pi , C\ , etc., respectively,
n = piGs + PiCs + 1 + etc.,
and r-i = pmCs-\- piCg + 1 + etc. j
which may be otherwise written
' ' ' \
n = Pm + 2-,0i + p,n + 3 - J C'2 + etc., I
r2=Pm + i — sCi -\- pm + i — sC-i, + etc, '
Therefore, form (24) and (23),
' /
Gi{l)m + 2 — z—Pm+2 — z) + Ci {pm + Z-t— Pm + Z-s) + etc. = 0.
Therefore, by §13, ^,„ + 2 _ s = jo^ + 2 - z , ;Jm + 3 _ « = it?m + 3 - *, etc.
Hence the second of the equations (24) becomes
r-i = Pm + i-xCi -\- p,n + o-zC2-\- etc. = r2 + 1.
Thus r-2, is transformed into r^ + \ . In like manner rs receives the
value r^ + 2 , and so on.
§33. By Cor. Prop. VL, the primitive n^^ root of unity being one of
those involved in. ri , n — 1 is a multiple of m. In like manner, if
the primitive a"^ root of unity be involved in ri , a — 1 is a multiple
of 7)1, and so on. Therefore, if t\, be the primitive m*^ root of unity,
ti is distinct from all the roots involved in ri .
OF THE HIGHER DEGREES, WITH APPLICATIONS.
99
§34. From this it follows that, if tlie circle of roots ri , r^ , . . . . ,
Tm y be arranged, beginning with Tc , in the order Tc , Vg + i, ?'c + 2 >
etc., and again, beginning with Vg , in the order ?'«,»'« + 1 , r, + 2 , etc.,
and if, tf being one of the primitive m*^ roots of unity,
re + T-c + 1 «i + re + 2 «i + etc. = r, + r^ + i i!i + r, + 2 «;i;*+etc.(25)
re = Ts . It is understood that in the series r^ , r,. ^ 1 , etc., when r„,
is reached, the next in order is ri , so that Vm. + 1 is the same as ri ,
and so on. In like manner r^ ^ 1 is the same as ri , and so on. Since
ri , r2 , etc., do not involve the primitive m^^ root of unity h , we can,
by §12, substitute for ti in (25) successively the different primitive
^th roots of unity. Let this be done. Then, by addition,
mrc — (ri + 7*2 + etc.)^ rnvg — (ri + »*2 + etc.). Therefore r^ =r4 .
§35. Proposition VII. Putting
^1 = n + i5ir2 -h t^rs +
-^ 2 4
-^2 = n -i- t^-i + t^rz-\-
+ ^,
+ K
2 (m — 1)
-1 -2
n + ^1 ^2 + t^ rs +
the terms,
• • • • + h^m ,
'^m — 1 ,
■ (26)
Jl, A.2, A3, .... , Jm-l, (27)
are the roots of a rational irreducible equation of the (m — 1)*^ degree
^ (x) =: 0, which may be said to be auxilianj to the equation
F {x) = 0.
For, let A be the generic expression of which Ai is a particular
cognate form ; and let A' denote any one indifferently of the m — 1
particular cognate forms of A in (27). Because, by §33, the primitive
m*^ root of unity does not enter into ri , r2 , etc., no changes made
in ri , r2 , etc., affect ti . Also, by §32, if ri becomes r^ , r2 becomes
**« + 1 7 ''3 becomes r^ + 2 , and so on. Therefore the expression
(r, +

^'2 ,
etc., are not rational. We may take the primitive nP^ root of unity
wi to be present in these coefficients. But wi occurs in ri , r-j , etc.,
and therefore also in A^ , only in the expressions Ci , C2 , etc.
Therefore Ai ^ di Ci -\- .... -j- dm Cm ', where d\ , etc., are clear of
(x)i . The coefficients di , d2 , etc., cannot all be equal ; for this would
make A^ = — di ; which, by §21, is impossible. Hence m unequal
OF THE HIGHER DEGREES, WITH APPLICATIONS. 101
values of the generic expression J are obtained by changing Ci snc-
oessively into Oy , C2 , etc., namely,
ill C'l + ckC-z + .... + d,n C,„ ,
d-mPi -\- d\ C2 -\- ■ ■ ■ ■ + dm —I C),i ,
d2 Ci + c/3 C2 + . . . . + (h On, .
To show that these expressions are all unequal, take the first two.
If these were equal, we should have
(dnv - ih ) Ci + {di - d.2 ) C2 + etc. = .
Therefore, by §13, d„i — di = , di — di = , and so on ; which,
because di , d^ , etc., are not all equal to one another, is impossible.
Since then J has at least 7^^ unequal particular cognate furms, Ji is,
by Prop. III., the root of a rational irreducible equation of a degree
not lower than the m*^ ; which, by Prop. VII., is impossible.
Therefore k\ , k-i ., etc., are rational. Hence each of the expi'essions
in (27) is a rational function of ^i .
§37. Cor. Any expression of the type ^i + ki ty + ^'3 'i + etc.,
which is such that all the unequal particular cognate forms of the
generic expression under which it falls are obtained by substituting
for t\ successively the different primitive vi^^ I'oots of unity, while
ki , kii , etc., remain unaltered, is a rational function of ti . For, in
the Proposition, Ai or k\ -\- k-i ti + etc. was shown to be a i-ational
function of t\ , the conclusion being based on the circumstance that
Ji satisfies the condition specified.
§38. Proposition IX. If g be the sum of the roots of the equation
■F{x) = 0,
\ 2 _3_
r2 = ~{g + J, + ai J^ -t- oi Jj +
+ eiJ^'" +^iJ/" ); (29)
For, z being one of the whole numbers, 1, 2, .... , m — 1, put
p, = (n + t\ 1-2 + tl'rz + etc.) (ri + h^-^ i\ n + etc.)— . (30)
Multiply the first of its factors by t\' and the second by t\ . Then
Pz = (»'2 + tin -j- tin + etc.) {vi + «i ra + t[ r^ + etc.)-^. (31)
Hence pz does not alter its value when we change vi into r-z , ?'2 into
Tz , and so on. In like manner it does not alter its value when we
102 PRINCIPLES OF THE SOLUTION OP EQUATIONS
change n into ?■„ , To into Ta + i, and so on. Therefore, by §33, pz is
not changed by any alterations that may be made in ri , ^2 , etc.,
while ti remains unaltered. Consequently, if p^ be a particular
cognate form of P, all the unequal particular cognate forms of P are
obtained by substituting for t^ successively in p^ the different primi-
tive 711^^ roots of unity, while ?'i , ?-2 , etc., remain unaltered. There-
fore, by Cor., Prop. VIII., p^ is a rational function of h . When
z= 2, let pz = «! ; when z = S, let 2h = &i , and so on. Then, from
1 2 1 _3_
(26) and (30), J^"" = «i ^^ ' ^3'" = ^1 "^/" ^"'^ ^'^ ^^- ^^*' ^''^™
(27), since g is the sum of the roots of the equation F (x) = ,
J- -L JL
*•! =— (^ + ^1 + ^2 +••••+ -:J,, _ 1 ).
2 13 1
By putting ai A for A , &i J for J and so on, this becomes
(29). Because a^ , bi , etc., are rational functions of ^i , while Ji , the
root of a rational irreducible equation of thf (m — 1)'^'^ degree, is also
a rational function of ti , the coefficients «i , 61 , etc., involve no surd
the prime number m be odd, the
A
1 m + :
that is
not subordinate to J
§39.
Proposition X. If
expressions
1 1 1
A A , A .
1 "m — 1 ' 2
(32)
are the roots of a rational equation of the I j degree.
By §32, when ri , is charged into Vg , r^ becomes r^ + 1 , r^ becomes
r^ ^ 2 , ai^d so on. Hence the terms rir^ , r^Vz , .... r^ri , form a
cycle, the sum of the terms in which may be denoted by the symbol
S2 . In like manner the sum of the terms in the cycle r^ r^ , ^2 r4 ,
.... , r^ r2 , may be written 2.3 . And so on. In harmony with
this notation, the sura of the m terms ri , ^2 , etc., may be written 2i .
Now ri can only be changed into one of the terms rj , r2 , etc. ; and
we have seen that, when it becomes 7\ , r^ becomes ? / + 1 , and so on.
Such changes leave the cycle ri r2 , r^, rz , etc., as a whole unaltered.
OF THE HIGHER DEGREES, WITH ArPLICATIONS. 103
Therefore, by Prop. III., S2 is the root of a simple equation, or has a
rational value. In like manner each of the expressions
2i , S2 5 S3 , . . . . , 'Sw , (33)
has a rational value. From (26), by actual multiplication,
1 1
K'' ^J-i = 2i + (2^) ^1 + (-^^ ^' + ^*"-
But S2 , S3 , etc., are respectively identical with 2,^5 S,«_i , etc.
Therefore
1 1
^r KH-x = 2' + (^•^) {h + h') + {l}^){tl + ir')+etc.(34)
Hence, since the terms in (33) are all rational, and since the terms in
1 1
(32) are respectively what A A becomes by changing t\ succes-
sively into the — — terms t\ , t\ , etc., the terms in (32) are the
(m — 1\*^
— - — I degree.
§40. For the solution of the equation aj" — 1 ^ , n being a prime
number such that m is a prime measure oi n — 1 , it is necessary to
obtain the solution of the equation of the m'^ degree which has for
one of its roots the sum of the .terms in a cycle of primitive
^th roots of unity. This latter equation will be referred to as the
reducing Gaussian equation of the vi^^ degree to the equation
.x'^ — 1 = .
§41. Proposition XI. When the equation F {x) = is the re-
ducing Gaussian (see §40) of the m^^ degree to the equation
x" — 1 = 0, each of the — - — expressions in (32) is equal to n.
Let the sum of the primitive w*^ roots of unity forming the cycle
(8), which sum has in preceding sections been indicated by the
symbol G\ , be the root r\ of the equation F (x) = . This implies,
since s is the number of the terms in (8), that vis ^ n — 1 . Let
us reason first on the assumption that the cycle (8) is made up of
pairs of reciprocal roots wi and wi~ , and so on. Then, because the
cycle consists of — pairs of reciprocal roots, C\ or r\ is the sum of
104 PRINCIPLES OF THE SOLUTION OF EQUATIONS
s^ terms, each an n*^ root of unity. Among these unity occurs s
times. Let u)\ occur hi times ; and let wi the second term in (8),
occur h' times. iSince a»i may be made the first term in the cycle
(8), it must, under the new arraugement, present itself in the value
of Ti , precisely where wi previously appeared. That is to say,
h' ^ h\ . In like manner each of the terms in (8) occurs exactly
hi times in the expression for n . The cycle (9) being that which
contains all the primitive n^^ roots of unity, let us, adhering to the
notation of previous sections, suppose that, when wi is changed inta
ii)\ , G\ or ri becomes C-z or r2 , C2 or r-i becomes C3 or Vz , and so on.
On the same grounds on which every term in (8) occui-s the same
2
number of times in the value of ri , each term in the cycle of terms
whose sum is C^ occurs the same number of times ; and so on.
Therefore
ri = s + hi Ci + h2 C2 + .... + h,„Cm •
n = s -\- A,„(7i -\- hi d -\- .... + hm-l Cm ,
rm = s -\- hi Ci -\- hs Co + . . . ■ + hi Cm .
Therefore, keeping in view (11), -1 = ms ^ {hi -\- h2 -\- . . . . -\- hm)-
But s^ — s is the number of the terms in the value of r^ which are
primitive n* roots of unity. And this must be equal to
s {hi -\- .... + hyi).
Therefore
Ai + A2 + • . • . + hm = s — I . • . I\ = ms -{- 1 — s = n — s.
Again, because ri is made up of pairs of reciprocal roots, and because
therefore unity does not occur among the s- terms of which ri r2is
the sum,
ri 7-2 = X-'i Ci + A'2 C2 + • • • • + ^m C'm ,
T-2 ^3 = km C\ -\- kxCi -\- .... + ^m — 1 Gin >
"M }
I'm ''1 = ^'2 C\ 4- ^'.3 ^2 +••••+ ^'1 Cr,
where ki , k^ , etc., ai-e whole numbers whose sum is s. Therefore
2*2 = — s. In like manner each of the terms in (33) except the first
is equal to — s . Therefore (34) becomes
] 1
J J = {n — s) — s (ti + * etc.,
we have
h = qi -\- q2 Ci -\- qs C2 -i- .... + qm Gm - 1 ;
where g-i , q2 , etc., are known rational quantities. But, by §13, the
rational coefficients qi — k^ , q^ , etc., are all equal to one another.
Therefore k\ = qi — 5-2 . In like manner k-i , k-i , etc., are known.
Therefore, from (36), di , A2 , etc., are known. Therefore, from
(35), ri is known.
§44. Proposition XIII. The law established in Prop. X falls
under the following more general law. The m — 1 expressions in
each of the groups
r (37)
and so on, are the roots of a rational equation of the (m — 1)**^ degree.
1
1
1
1
1
1 -^
(<
m
^7
A '^
^m-2' ••
A '^
^r>)
2
1
2
1
2
1
(<
a"^
K"
A~^
^«t-4' ••
A "*
<•)
8
1
3
1
3
1
(<
^r
m
^:-)
OF THE HIGHER DEGREES, WITH APPLICATIONS. 107
The m — 1 terms in the first of the groups (37) are the
2
terms in (32) each taken twice. Therefoi'e, by Prop. X., the law
enunciated in the present Proposition is established so far as this
groupe is concerned. The general pi-oof is as follows. By (30) in
ni — z 1
§38, taken in connection with (26), p„i — z ^, ^= ^ _ • There-
z 1
fore A J ^ = p„j _ 2 Ji . But, by §38, pm _ « is a rational
function of ^i ; and, by Prop. YIII., Ji is a rational function of ti .
z 1
Therefore A, A is a rational function of U . Also from the
manner in which p-m — z is formed, when t\ in jd,„ _ 2 ^^i is changed
z 1
sucessively into h h , . . . . , h , the expression A A _ is
changed successively into the m — 1 terms of that one of the groups
(37) whose first term is zl, A . Therefore the terms in that
^ ' i m — z
group are the roots of a rational equation.
§45. Cor. The law established in the Proposition may be brought
under a yet wider generalization. The expression
A, J., A A
m
\ ^2 ^3 •••• ^m-1 (38)
is the root of a rational equation of the (m — 1)*^ degree, if
a + 26 + 3c + .... + (?/i — \) s = Wm ,
W being a whole number. Por, by (30) in connection with (26),
1 2 1 1
^., = p2 ^j J ^3 = Pz A , and so on. Therefore (38) has
the value
a + 2b + 3e + .... + {m — 1) *
b c bo W
(pipi ....} Aj^ "* , or (;?2it?3 ....) ^1 .
This is a rational function of ti , and therefore the root of a rational
equation of the (m — 1)'^ degree.
108 principles of the solution of equations
The Equation F{x) = of the Second Class.
§46. We now suppose that the simplified root r^ of the rational
irreducible equation F (x) ^ of the m}^ degree, m prime, involves,
when modified according to §21, a principal surd not a root of unity.
It must not be forgotten that, when we thus speak of roots of unity,
we mean, according to §1, roots which have prime numbers for the
denominators of their indices. In this case conclusions can be estab-
lished similar to those reached in the case that has been considered.
The root r\ is still of the form (29). The equation F (x) = has
still an auxiliary of the (m — 1)*^ degree, whose roots are the m***
powers of the expressions
1 2 3 m — 2 ?>i — 1
^1 , «i ^1 , Oi ^1 , , ei ^1 , hi A^ , (39)
though the auxiliary here is not necessarily irreducible. Also, sub.
1 1
stituting the expressions in (39) for A A , etc., in (37), the law
of Proposition XIII. still holds, together with corollary in §45.
§47. By Cor. Prop. VI., the denominator of the index of a surd of
1
the highest rank in r\ is m. Let A be such a surd. By §21, the
1
coefficients of the difierent powers of A in r\ cannot be all zero.
We may take the coefficient of the first power to be distinct from zero
1 1
1 ki — —
and to be — for, if it were — , we might substitute s for kiA ,
m m ' ^ 1
1
and so eliminate A from rj , introducing in its room the new surd
1
s "* with — for the coefficient of its first power. We may then put
m
1 3 m— 2 m—1
n =~ {g + A^ + ai A^ + + ei j/'' + h A^ ); (40)
1 1
where g , a\ , etc., are clear of A . When A is changed succes-
1
sively into A , ti A , ti A , etc., let
ri,r2, Tm, (41)
OP THE HIGHER DEGREES, WITH APPLICATIONS. 109
be respectively what r^ becomes, ti being a primitive m^^ root of unity.
By Pro)). VI., the terms in (41) are the roots of the equation
F (x) = 0. Taking r„ , any one of the particular cognate forms of
1 1
B, let J , «n , etc., be respectively what J , o-i , etc., become in pass-
1
ing fi'om ri to r„ ; and when A is changed successively into the
different ?«* roots of the determinate base J„, let r^ become
/ //
Tn, rn, r„, .... , rT~^^ • (42)
By Prop. II., the terms in (42) are roots of the equation F (x) = ;
and, by §23, they are all imequal. Therefore they ai'e identical, in
some order, with the terms in (41). Also, the sum of the terms in
(41) is g. Therefore g is rational.
5:^48. Proposition XIV. In ri , as expressed in (40), J is the
only principal (see §2 ) surd.
1
Suppose, if possible, that there is in r\ a principal surd z distinct
1 1
from J . And first, let z be not a root of unity. (It will be kept
in view that when, in such a case, we speak of roots of unity, the
denominators of their indices are understood, according §1, to be prime
1 1
numbers.) When « is changed into s^ , one of the other c*'^ roots
-of z\ , let rj , ai , etc., become respectively n , ai , etc. Then
1 2
m in
mn , = g + J^ + «i J^ + etc (43)
By Prop. II., Ti is equal to a term in (41), say to r^ . And, by §48,
putting tn ~ 1 for ^J - "^
1 2
m-Tn = 9 + tn-i ^^'^ + tl_i ai J^" + etc. (44)
Therefore,
1 2
J^*" (1 - tn-i) + ^1™ Ki - «i «„ _i) + etc. = 0. (45)
110 PRINCIPLES OF THE SOLUTION OF EQUATIONS
ThLs equation involves no surds except those found in the simplified
expression ri , together with the primitive m^^ root of unity. There-
fore the expression on the left of (45) is in a simple state. Therefore,
1
by §8, the coefficients of the different powers of J are separately
r r
zero. Therefore ^„_i = 1, ai ^ ai , 6i = bi , and so on. But, as
1
was shown in Prop. V. , z being a principal surd not a root of unity
1
in the simplified expression ai , ai cannot be equal to ax unless z
can be eliminated from ai without the introduction of any new surd.
1
In like manner bi cannot be equal to 6i unless z can be eliminated
from bi . And so on. Therefore, because ai = ai , and 6i = bi ,
1
and so on, z admits of being eliminated from ?-i without the intro-
duction of any new surd, which, by §21, is impossilile. Next, let
z^ be a root (see §1) of unity, which may be otherwise written 0i
Let the difierent pi'imitive c*^ roots of unity be Oi , 6^ , etc. ; and,
when 0\ is changed successively into 6i , 0^, , etc., let ?'i become suc-
cessively ri , Ti, etc. Suppose it possible that the c — 1 terms
X
''i > ^'i > etc., are all equal. Since z is a principal surd in rj , we
(J \ f> 2
may put n = hO^ + k0^ -\- ....-{- I ; where h, k, etc., are
clear of 0i . Therefore (c — 1) n = c^ — (h -{- k -{- etc.) Thus
z may be eliminated from ri without the introduction of any new
surd ; which by §21 is impossible. Since then the tern.s ri , ri , etc.,
are not all equal, let ri and ri be unequal. Then ri is equal to a term
in (41) distinct from ri , say to ?•„. Expressing mri and m?-„ as in
(43) and (44), we deduce (45) ; which, as above, is imjwssible.
1
§49. Proposition XV. Taking ri,^^, J , etc., as in §47, an
1 c
equation t A = p J (46)
OF THE HIGHER DEGHEliS, WITH APPLICATIONS. Ill
can be formed ; where t is an m^^ root of unity, and c is a whole
number less than m but not zero, and p involves only surds subordi-
1 1
- m in
nate (see §3) to J or J
By §47, one of the terms in (42) is equal to ri . For our argument
it is immaterial which be selected. Let r„ = ri . Therefoi'e
m — 1 m — 2 1
{K J, + e.„ -i„ +.... + J, )
m — 1 in — 2 _1_
- (A, j7^+ ., j"'"^^ . . . . + j;^ ) = . (47)
1
The coefficients of the difterent powers of J here are not all zero,
for the coefficient of the first power is unity. Therefore by §5, an
1
equation tA = h subsists, t being an 7n^^ root of unity, and ^i in-
1
volving only surds exclusive of A that occur in (47). By Prop.
1
XIV., J is a surd of a higher rank (see §3) than any surd in (47)
except J . Therefore we may put
1 2 m — 1
l^= d + di J^ + c/.j J^ +....+ d„, _. 1 J^ ;
1
where d, di , etc., involve only surds lower in rank than J^ . Then
1
J„= ^7 = (c^ + c/i J^"* + etc.)'-
1 2
^ d -\- d\ J + d-i J + etc.;
1
I I —
where d , di, etc., involve only surds lower in rank than J . By
1
§8, since J is a surd in the simplified expressions ri , the coefficients
d — J„ , c?i , etc., in the equatiuu
8
112 PKINCIPI.KS oF THE SOLUTION OF EQUATIONS
1
{d - Jn) + di A^ + d^ J^ + etc. = (48)
are separately zero. Therefore {d -\- di A + etc.)'" = d . And,
tx being a primitive m^^^ root of unity,
{d + di h ^"' + etc.)'" = d -^ d ti A^'" + etc. = d.
Therefore,
1 1 _2_
(d + di ti Jj"' + etc.) = ti {d + di J^"' + d, J^ + etc.),
ti being one of the m^^ roots of unity. In the same way in which
1
the coefficients of the different powers of J in (48) ai-e sejmrately
zero, eacli of the expressions d (1 — ti ), d\ {ti — h), etc., must be
a
zero. But not more than one of the m — 1 factoz'S, t\ — ^i ,
2 a,
h — ^1 J etc., call be zero. Therefore not rnoie than one of the
7)1 — 1 terms di , d-i , etc., is distinct from zero. Suppose if possible
1
that all these terms are zero. Then t A = d. Therefore the
n
_1_
different powers of A can be expressed in terms of the surds in-
1 2
volved in d and of the m^^^ root of unity. Substitute for A , A
etc., in (47), their values thus obtained. Then (47) becomes
w- l _i_
Q-{rn\"' +..••+ V) = 0^ (49)
where Q involves no surds, distinct from the prindtive ?«*^ root
1
of unity, that are not lower in rank than A^ ; which, because
_i_
the coefficient of the first power of A^ in (49) is not zero, is, by §8,
impossible. Hence there must be one, while at the same there can be
only one of the m — I terms, c/j , d-2 , etc., distinct from zero. Let
OF Tin: H Hi HER DEGREES, WITH APPLICATIONS. 1 1 ."{
Jc he the term that is not zero. Then tl — t1 = 0. Therefoio
1 t^ is not zero. Therefore d = 0. Tlierefore, putting p for d,; ,
\ c
, m m
n •' 1
§50. Cor. By the proj^osition, values of the different powers of
1
J can be obtained as follows ;
n
1 ^_ 2 « 3 z
tJ = 1) ^, ,t^-i = q ^, ,t^ ^ = k J^ , etc.; (50
where p, q, etc., involve only surds that occur in Ji or J„ ; and c, *<, z,
etc., are whole number.s in the series 1, 2, m — 1. No two f)f
the numbers c, s, etc., can be the same ; for they are the [)roducts,
with multiples of the prime number m left out, of the terms in the
series 1,2, . . . . , m — 1, by the whole number c which is less than
m. Therefore the series c, s, a, etc., is the series I, 2, . . . . , w — 1,
in a certain oi'der.
§51. Proposition XVI. If r,j be one of the particular cognate
forms of R, the expressions
1 2 m — 2 m— 1
«j",<2a„j"^, .... ,/,— V„ J '^ ,P—^h„ J^, (51)
are severally equal, in some order, to those in (39), t being one of the
m^^ roots of unity.
By §47, one of the terms in (42) is equal to ?■] . For our argument
it is immaterial which be chosen. Let Vn = ri . By Oor. Pro]).
XV., the equations (50) subsist. Substitute in (47) the values of the
1
different powers of A so obtained. Then
c s
{t-^pA^ -f «- - qc'n ^/" + etc.)
1 2
- (j;" + «i j/" + etc.) = 0. (52)
c s
m m ......
By Cor. Proji. XV., the series J , J , etc., is ulentieal, m some
1 2 1 -
.■II • - '"■ , ■'"■ , 4 1 1 r. I • <■'"'•
order, with tlie series J. , J , etc. Also, by v:i>, since J^ is a
114 PRINCIPLES OF THE SOLVTION OF EQUATIONS
1
surd occurring in the simplified expression ry , and since besides J
there are in (52) no surds, distinct from the primitive tn}^ root of
unity, that are not lower in rank than A , if the equation (52)
1
were arranged according to the powers of J^ lower than the m**^,
1
the coefficients of the different powers of J would be separately
1
zero. Hence J is equal to that one of the expressions,
c s
t- 1 p A^ , t- 2 qan A^ , etc. (53)
in which A is a factor. In like manner a\ A is equal to that one
2
of the expressions (53) in which A is a factor. And so on. There
1 2
fore the terms J , «i J , etc., forming the series (39), are sever-
ally equal, in some order, to the terms in (53), which are those
forming the series (51.)
§52. Proposition XVII. The equation F {x) =■ ^ has a rational
auxiliary (Compare Prop. VII.) equation w (a;) ^ 0, whose roots are
the m^"^ powers of the terms in (39).
Let the unequal particular cognate forms of the generic expression
A under which the simplified expression Ji falls be
Ji , ^2 ,...., -Ic . (54)
By Prop. XVI., there is a value t of the m^^^ root of unity for
which the expressions
1 2 TO — 2 m — 1
t a;, fi a, aJ, .... , <— 2 ., j/", ^—1 h, A^ (55)
are severally equal, in some order, to those in (39). Therefore A^ is
equal to one of the terms
J, , ^M Ji ,. . . .,ei Ji , //I Ji . (56)
OF THE HIGHER DEGREES, WITH APPLICATIONS. 115
In like manner each of the terms in (54) is eqnal to a term in (56).
And, because the terms in (54) are unequal, they are severally equal
to different terms in (56). By Prop. III., the tei'ms in (54) are the
roots of a rational irreducible equation, say ^'i [x) = 0, Rejecting
from the series (56) the roots of the equation i (x) = 0, therefore
«« = «.v.
§55. Proposition XIX. Let the terms in (39) be written
respectively
1 j_ _i_ _i_
The symbols J, , <^^ , <\ , etc., are employed instead of ^j , -^^t ^j , etc.,
because this latter notation might suggest, what is not necessarily
true, that the terms in (56) are all of them particular cognate forms
of the generic expression under whicli Ji falls. Then (compare Prop.
XIII.) the m — I expressions in each of the groups
1 ji
1 1
m m
1 1
m . m
Tn. in in in ^ iii, "^
(J, ,5
2 1
m m
— 2 ' 2
til m
TO _ 4 ' 3 7ft — e
1 1
'^ 1 ^1 ')
TO — 1 I '
2 _]_
TO— 1 2 " I
(59)
(^;
» m TO
HI _ S ' 2 m — (
3 OT —
,)
and so on, are the roots of a rational equation of the (m — 1)*^ degree.
m — 1
Also (compare Prop. X.) the first — - —
terms in the first of the
jvth
groups (59) are the roots of a rational equation ot the I — 1
degree.
In the enunciation of the proposition the remark is made that the
series (54) is not necessarily identical with the series
^1 , ^2 , '^3 , • ■ • • > ^m— 1-
The former consists of the unequal particular cognate forms of J ; the
latter consists of the roots of the auxiliary equation *

*/ «, and si = j) — q y/ z ; vi, n, z, }■> and q
being rational ; and the surd y/ s being irreducible.
By Propositions XIII. and XIX., the terms in (66) are the roots
of a quadratic. Therefore Ji A^ and Jg A^ are the roots of a quad-
ratic. Suppose if possible that Ji J3 is the root of a quadratic. By
i 3
Propositions IX. and XIX., J3 = ei A{ . Therefore ei Ji is the
root of a quadratic. From this it follows (Prop. III.) that there are
not more than two unequal terms in the series,
el At , el A2, 63 ^3 , ei Ai. (69)
But suppose if possible that el Ai = 62 A2 . Then, < being one of the
fifth roots of unity, tei Ai =60 ^2 But, by Propositions IX. and
14 4 1
XIX., A2 = hi aJ . Therefore, tei jf = 62 hi aI Ai . There-
OF THE HIGHER DEGREES, WITH APPLICATIONS.
123
fore, by §8, ei = 0. Therefore one of the roots of the auxiliary-
biquadratic is zero ; which because the auxiliary biquadratic js
assumed to be irreducible, is impossible. Therefore ei Ji and e-z J2
are unequal. In the same way all the terms in (69) can be shown to
be unequal ; which, because it has been jjroved that there are not more
than two luiequal terms in (69), is impossible. Therefore Ai J3 is not
the root of a quadratic equation. Therefore the product of two of the
roots, Ai and J4 , of the auxiliary biquadratic is the root of a quad-
ratic equation, while the product of a different pair, Ji and J3 , is not
the root of a quadratic. But the only terms which the roots of an
irreducible biquadratic can assume consistently with these conditions
are those given in (68).
§68. Proposition XXV. The surd y/ si can have its value ex-
pressed in terras of ^/s and ^/z.
By Pi'opositions XIII. and XIX, the terras of the first of the groups
(67) are the roots of a biquadratic equation. Therefore their fifth
powers
aIa,, Al A,, (70)
aIas
A'I Ai
are the roots of a biquadratic. From the values of Ji , A2 , As and
Ai in (68), the values of the terms in (70) may be expressed as
follows :
y ('1)
+ (^4 + i^5 ^/ ^) v/ si + (^6 + ^7 v/ «) v/ s v/ si ,
aIAi= F- Fiy z-\- (Fo- Fs^ z) ^ si
- (Fi - F,V z)^ s -(Fc-F,^z)^s^s,,
aIA2= F- Fi^ z- {F2 - /s k/ -) s/ si
+ {F4, - F,^ z) ^ s - (F, - Fi s/^)ys^si,
AlAi= F+ F,y z- (F^-h F^^ z) ^ s
-{Fi + F,^z) ^ si + {F, + F-, y/z)s/s^s,,^
where F, F\ , etc., are rational. Let - (Ji J3 ) be the sum of the
four expressions in (70). Then, because these expressions are the
roots of a biquadratic, - ( Ji zJg ) or iF + iF-j ^/ s s/ Si , must be
rational. Suppose if possible that v'si cannot have its value expressed
in terms of v/ « and ^/ z. Then, because ^ s ,/ si is not rational,
F^ = 0. By (68), this implies that n = 0. Let
{A\As
-f {Li + Lr, y/ z) y/ 61 + (/^6 + L; ^ z) s/ s ^ «i ,
124 PRINCIPLES OF THE SOLUTION OF EQUATIONS
wliere L, L\ , etc., ai'e rational. Then, as above, Z7 := 0. Keeping
in view that n = 0, this means that m^ q ^ 0. But q is not zero,
for this would make \/ s := \/ si ; whicli, because we are reasoning
on the hypothesis that y/ si cannot have its value expressed in terms
of >/ s and y/ z, is impossible. Therefore m is zero. And it wa.s
shown that n is zero. Therefore J^ = ^ s, and J3 = — ^8.
Therefore Ji J3 = — -v/ {2>^ — q^ ^) ) which, because it has been
proved that Ji J3 is not the root of a quadratic equation, is impossible.
Hence \/ si cannot but be a rational function of ^Z s and ^Z z.
§69. Proposition XXVI. The form of s is
A (1 + e2 ) + h s/(l + e2 ), (72)
h and e being rational, and 1 + e^ being the value of z.
By Prop. XXV., y/ si= v-f c ^ s, v and c being rational
functions of ^/ z. Therefore si = v^ + c' s -\- 'Ivc y/ s. By Prop.
XXIV., x/ s is irreducible. Therefore vc = 0. But c is not zero,
for this would make y/ si = v, and thus ^ Si would be the root of a
quadratic equation. Therefore u = 0, and -y/ si = c v^ s =
(ci -f C2 -s/ ») v/ s, ci and C2 being rational. Therefore
^ (ssi) = V ip''- r ^) = (ci + C2 ^/ s) (i^ + ? x/ z)
= (ci f -\- c-i q z) -\- ^ z (ci q -\. c-iii) = F -^ Q ^ z.
Here, since p^ — (f' z is rational, either /' = or ^ = 0. As the
latter of these alternatives would make ^/ {^p^ — ■^s and Ax .
That is to say, A\ is a rational function oi y/ s, y/ S\ and -^ z. But
it was shown that \/ Si -^ s = he ^ z. Therefore A\ is a rational
function of ^ s and ^Z s. We may therefore put
A^ = K + A^' J, + K" J4 + K'" A^ J4 ,
126 PRINCIPLES OF THE SOLUTION OF EQUATIONS
K, K', K" and K'" being rational. But the terms A\ , Ao , A^ ,
A-i circulate with J^ , J2 , A^ , J3 . Therefore
^2 = A' + A^' Jo 4. K" J3 + K'" Jo J3 ,
Ai = K + K' J4 + K" Ji + K'" Ji J4 ,
^3 = A^ + K' J3 + K" J2 + K"' Jo Js ,
These are Abel's values.
§72. Keeping in view the values of A\ , A-i , etc., in (G7), and also
that % ■= \ -^ 6' ., and s = As + /i ^/ 2:, any rational values that may
be assigned to m, n, e, h, K, K', K" and K'" make r\ , as presented
in (74), the root of an equation of the fifth degree. For, any rational
values of m, n, etc., make the values of *S'i , »S'o , etc., in ^62, rational.
§73. It may be noted that, not only is the expression for Vi in (74)
the root of a quintic equation whose aiixiliaiy biquadratic is irre-
ducible, but on the understanding that the surds y/ s and ^ z in
Ji may be reducible, the expression for r\ in (74) contains the roots
both of all equations of the fifth degree whose auxiliary l)iquadratics
have their roots rational, and of all that have quadratic sub-
auxiliaries. It is unecessary to offer proof of this.
§74. The equation x^ — 10a;3 -f Sa;^ + lOa^ + 1 = is an
example of a solvable quintic vvith its auxiliary biquadratic irre-
ducible. One of its roots is
1 2 3 4
(o being a primitive fifth root of unity. It is obvious that this root
satisfies all the conditions that have been jjoir.ted out in the preceding
analysis as necessary. A root of an equation of the seventh degree
of the same character is
1 2 3 4 5 fi
u) being a primitive seventh root of unity. The general form under
which these instances fall can readily be found. Take the cycle that
contains all the j^rimitive {ni'Y^ roots of unity,
0,0^, 0^\ etc. (75)
m being prime. The number of terms in the cycle is {m — 1)^.
Let 0^, be the {m -f 1)'^ term in the cycle (75), O-z the {2m + 1)**^
term, and so on. Then the root of an equation of the ?/i'^ degree,
including the instances above given, is
[Read be/ore the Canadian Institute, March 3rd, 1883].
RESOLUTION
SOLYABLE EQUATIONS OF THE FIFTH DEGREE,
BY GEORGE PAXTON YOUNG,
Toronto, Canada.
CONTENTS.
1. Sketch of the method employeJ. General statement of the
ciitei'ion of solvability of an equation of the fifth degree. §2-5.
2. Case in which ui u^ = U2 ws . The roots determinable in terins
of the coefficients pi , p2 , etc., even while particular numerical values
have not been assigned to the coefficients. Three verifying instances ;
one, in which the auxiliary biquadratic is irreducible ; a second, in
which there is a quadratic sub-auxiliary ; a third, in which the roots of
the auxiliary biquadratic are all rational. §6-10.
3. Deduction, in the case in which ui u^ = ic-y us , of the equation
p' = j where p' is a rational function of the coefficients jJi , p2 , «tc.
Verifying instances. §11-13.
4. The trinomial quintic x^ + />4 « + Po = 0- Form which the
criterion of solvability here takes. Example. §14-16.
5. When any relation is assumed between the six unknown quantitie.--,
the roots of the quintic can be found in terms of pi , p-z , etc. §17.
. 6. The general case. §18.
§1. By means of the laws established in the paper entitled " Prin-
ciples of the Solution of Equations of the Higher Degrees," which is
concluded in the present issue of the Journal of Mathematics, a
criterion of the solvability of equations of the fifth degree may be
found, and the roots of solvable quintics obtained in terms of given
numerical coefficients. In certain classes of cases, the roots can be
determined in terms of coefficients to which particular numerical
values have not been assigned, but which are only assumed to be so
related as to make the equations solvable.
11
128 resolution of solvable equations
Sketch of the Method Employed.
§2. Let ri , ri , r% , ?"4 , rs , be the roots of the solvable irreducible
equation of the fifth degree wanting the second term,
F (x) = x^ + ^^2 «3 + p3 a;2 -\- Pi X -\- po = 0. (1)
It was proved in the " Principles " that
i
ill
n = h (^r + ^2 + ^I + ^J )'
where Ji , J-z , J3 , /J4 are the roots of a biquadratic equation
auxiliary to the equation F (x) = 0. It was also shown that the
root can be expressed in the form
n = h (4 + «i 4 -h ei4 + ^^1 4 )' (2)
where ai , e\ , hi , involve only surds occurring in J"; and no surds
occur in Ji exce|)t ^{hz -\- h x/ ^) and its subordinate y/ z ; z being
equal to 1 + e- , and h and e being i-ational. As in the •' Pi-inciples,"
1111
we may put 5ui , = J^ , 5u2 = J.f , 5ii3 = J3 , 5ui = J^ . Then
n — Ml -f ii2 + W3 + W4 . (3)
Let -s'l be the sum of the roots of the equation F (x) = 0, S2 the sum
of their squares, and so on. Also let
- ('^1 U3 ) = Wi U3 + Zio Ml -f Uz Ui + '2*-l ^*2 >
- (Wi Mo ) = Ml ?(2 + it-I ICi + ■2<'3 «1 + Ui ?'3 ,
N-/ 2 2, 22, 22,22, 22
- (Ml Mj ?/4} = Ml ?t3 2«.4 -)- ?/2 ^i M3 + M3 M4 Uo + M4 M2 ^*1 J
„, - (w5) = zti + ul + M3 + 4 ; }^ (4)
S2 = 10 (mi M4 + M2 M3 ), ,S'3 =15 \-(ui W3 ) i ,
Si = 20 {l (ui U2 )} + y3. (^Sl) -f 60 ICi M2 ^t3 M4 ,
S, = 5 \l(ul}} + I (.So S3) + 50 {^(uiulul)}- ^
§3. It was proved in the " Princii^les " that ?ii ?t4 and M2 M3 are the
I'oots of a quadratic equation. But
25 Ml iii = hi Ji , and 25 M2 M3 = ai ei Ai .
Therefore, because «i , ci , Ai , involve no surds that are not sub-
1
ordinate to Jj^ , ^ s is the only surd that can appear in ui M4 and
2 U3 . Consequently we may put
OF THE FIFTH DEGREE.
129
m Ui = g -\- a *

*/ S,
UiU2 = k -\- C y/ z
-\- *

*3 ,
k = -^\{pz). (10)
It will be convenient to retain the symbols g and k, whose values are
given in (6) and (10). Again, because w^ u% = ^ ^ ^ ^ ^ we
M2 W3
have, from (5) and (8),
(9)
130 RESOLUTION OF SOLVABLE EQUATIONS
^z u^=^-^^^\k + c V z + {0 + 2 ; a"*

etc., without definite numerical values being assigned to p-i , pz ^ etc. This I proceed to show. u\ = B- - B' ^ z-{- ^ si, icz = B - - 5' v/ 2 — v/ «l . 132 RESOLUTION OF SOLVABLE EQUATIONS §7. By (5), because mi u^ = u^ U3 , a = 0. Thus, one of the six unknown quantities is determined, while we have still the six equations (15) to work with. It might be sufficient to say, that, from six equations five unknown rational quantities can be found. I will recur to this idea ; but in the meantime the following line of reasoning may be pursued. From (11), ^ = . Therefore equatioa (12) becomes gj}^ = - 20 (F _ c2 2) + 5^3 . (16) Also, because a = 0, equations (7) being kept in view, u^ = 1 1 g2 . 2 {k^ — c'z) {k + c^z)-g^ (k — cs/z) + 2{k-\-c^z) {B'—

+ B" + ^"Vs) Vs. . •. %2 = y^ j 2 (Z;2 _ c2 s) - ^3 } + 2chez (0^ _ ^2 ^) and ^y = c { 2 (A;2 _ c2 «) + ^3 I + 2Me (02 _ ^2 ^). .■.u\= \\k { 2 (A;2 _ c2 2) _ ^3 j ^ 2chez (6^ -

| which may be written M = 5kcz — F, y (20) and iV" = 2 (k'^ + c2 2) — ^3 _ 4 ^c, I •which may be written ]V ^ F — 4: kc. J The two equations (19) give us 0|if2 _ ~ JV^2 _4e2 (^2 _c2 s)2| = e^2 I J/ _ 2e (/5;2 _ c^z) j ^ j iI/2 — « iV^2 — 4e2 (F _c2 2)2 1 = ey2 ^7; Therefore r" (^l) _0__ if - 2e (F - c2 2) OF THE FfFTH DEGREE. 133 Equating the value of rr^ — ; — ^^^r — obtained from (21) with {)■'■+■ (f^ z -\- '2tffz that derived from the last two of equations (15), 2kcz j Jf — 2e (A;2 — c2 3) 1 2 _^ i\r2 « + 2iVJ 3f — 2e {k"- —c'-z)] i^M — 2e {k- — c- z) [2 ^ ^■^z-\- 'INz j M — 2e (k- — o'- z) j ' ^^^^ The coefficients p^ , 7^3 , etc., in the equation F (x) = 0, being given, ff and k are known by (6) and (10). Therefore, by (16), c'^ z is known. Then (22) will be found to be a quadratic equation determinative of c. For, keeping in view the value of F in (20), (22) may be written ^.2 _j_ g2 2; — ^3 2kc^ z |4 (F -f c2 zf -f 7^2 j _ 8y^/>c _ 16^ (^2 _ c2 z) (ce) ] 4 (A;2 _ c2 zf — 16/t2 c2 2 _ P2 j c + 8y^c2 zF—^ [W — c^ z) F (ce) Because g, k, c^ z and F are known, this equation is of the form H {ce) = Kc + L, where H, K, L, are known. Therefore, since c^ e^ = c^ z — c^ , c2 (7/2 + A'2 ) ^ 2A'Zc + (7:2 _ 7f2 c2 s) = ; from which c is known. Therefore, since c^ z is know^n, ::: is known. Therefore e is known. Therefore, by (21), Q and

*
etc. This elimination has been performed, under the dii-ection of the
author of the paper, by Mr. Warren Reid of Toronto, with the
following result. Putting P, as in §7, for 2 {Jc^ -{- c^ z) — g^ , let
A = — 2ytc2 zg^ j 8 (F + c^ s) — 3^3 | ^
B= f \ 16F c2 « + 4 (^2 + c2 zY — 5/ (/!;2 + c2 2) + / J,
2) = — 4 (F — c2 «) j — ^6 4. 3^3 (^2 4. c2 ^) - 2 (;fc2 _ ^^ ^f [ ,
A^= — ^kc^z[2,2kc^z{k'^—ch)—P |jOj/+8/l^(A;2-c2^)-4^^3 j]
j^5^2+8>t(y!;2_c2«)— 4^•/ J [— 32^2c23+(7^J 4(^2+c23)_^3 J]
+ 64>;;c2 zP {k"^ — c2 2),
7) = - 16y5:c2 zg^ |4 (F + c^ z) -g^ ]
4. 4P (/J;2 _ c2 2) jj95 ^2 + 8A; (F - c2 2) — 4A;^3 j.
Then, since 10*7 = — p2 , and 20^- = — ;j3 , and
20c2 z = Pig — ^g^ + 20F ,
OF THE FIFTH DEGREE. 137
the quantities A, B, D, Ai , Bx , Di , are known rational functions
of p2 , pz , etc. And
(^2 + 2)2 ) (^2 _ 2)2 c2 z) — {Bl + Z)2) (^2 _ 2)2 gZ «)
+ 4 f ^^ (52 + 2)2) _ ^^ ^^ (^2 + 2)2 I
{yl5 (^2 _ 2)2 c2 2) _ Ax Bx (^2 _ 2)2 c2 «) [ = 0. (25)
§12. To verify this I'esult, the Gaussian equation in §8 may be used.
Here
_ IP /IP + 112 X i9>^ 118 X 3
/ 2^
2° X 512 y 56 / 2* X 517
7? _ 11! / 11^ , 3^ X 72 X IP _ 9 X 35 X 11^ , 11^ \
56 V 2* X 59 "^ 2* X 512 8 X 512 •" "512 /
9 X IF
D =
A,=
4 X 516
112 X 31/ ,,, . 7 X 27 X IP 3P X 1P\
-^I^(- 11' + 8 8--)
3 X 31 X IP
4 X 516 •
11« .,. . „,. IP
26 X
-51^(19 + 31)=^,-^
IP IP X 109
A= 5n^(- 53+ 44 X 41 - 19 X 31) = -^^l
IP /63 X IP 1P\ IP X 19 X 31
i).= -
/ 63 X IP _ 21l\ _
\ 2 X 56 "56^^
4 X 512 V 2 X 56 56 / 8 X 51*
IP X 26
"~ 517
Therefore
m _L 7)2 9 ^ 11^' X 41 2 , ^2 IP X 11029
^ + ^ = 8 K 530 ' ^^ + ^^ = 2*^ X 53.^
9 X IP* X 89
^2 _ 2)2 c2 s = —
26 X 535
^2 ^2,._ 11^x40139
^1 — i^i c s — 210 X 53'
138 RESOLUTION OF SOLVABLE EQUATIONS
By the substitution of these values, equation (25) becomes
1156 y 34
926 5136 1 62653332 - 2886277 x 13600357 \ =
1156 V 3*
^,, ^,,, \ 39254397600889 - 39254397600889 I = 0.
226 X 5136 { J
§13. As an additional verification, the equation
a;5 -)_ lOx^ — 80a;2 + 145x — 480 =
may be taken. Here, by §9,
g = - I, k = 4, k'^ - c'^ z = 7, k-^ -^ c^ z = 25.
Therefore
^ = 23 X 32 X 7 X 29, 5 = — 2 X 5 X 17 X 29,
i) = 23 X 3 X 7 X 29,
A^= — 29 X 3* X,141, ^1 = 2* X 3 X 17 X 2393,
D^= — 27 X 32 X 13 X 19.
B'-i- Z)2 = 22 X 292 X 14281,
^2_^ i)2 ^ 28 X 32 X 5 X 338016989,
^2_ i)2 c2 ^ ^ 0,
^2_ Z)2 c2 3 = 214 X 36 X 5 X 7 X 172 X 277.
By the substitution of these values, equation (25) becomes
2" X 3« X 6 X 7 X 17' X 29* j 277 x 1428P
+ 5' X 7 X 338016989 — 2' x 3 x 141 x 2393 x 14281 J = 0.
7'he Trinomial Quintic x^ -\- p^x -\- p;, =: 0.
§14. In this case, by (6) and (10), 9 = 0, and ^ = 0. Therefore,
by (11), A ^ - ^^ ^^' ~ ^' ^\ Therefore, by (12),
20he (^2 _ ^2 ^)
p^ = 5^ — ' -f 15a2 z. (26)
Also, by §3, ^ - -4: '>—o?z'^c+2hecz{e-(p'^)Y Therefore, by (13), '
p,= - ^J^[0^ - ^'^ z) + Uacz. (27)
OF THE FIFTH DEGREE. 139
Hence the quintic becomes
F{x) = x' -\-\ 5^ ^-— ^ + 15a2 z\x
+ j _ ^ {pi-^-^z) ^ Uacz 1=0. (28).
The criterion of solvability of a trinomial quintic of the kind under
consideration is therefore that the coefficients p^ and p^ be related in
the manner indicated in the form (28) ; while at the same time the
last four of the equations (15), modified by putting g = k = 0,
subsist between the rational quantities a, c, e, h, 0, f. From these
data, the three following equations may be deduced, v being put for
Sev^ — izv" + z {?> ^ ie) V — z^ = Q, ^
% + ^J^= 350.,
a ac
iv
(29)
{ze + 4sv — 8v-')=(— 3z + |^,) {s + 4v (e — 1) + Sv^]. i
The first of these equations is obtained from a comparison of the two
equations (9), the second is obtained by putting ^^^ and p^ respectively
equal to the values they have in (28) ; and the third is obtained by
putting j9^ equal to the coefficient of the first power of x in (28).
§15. If any rational values of e and v can be found satisfying the
first of equations (29), let such values be taken. Then, from the
second and third of (29), a- and ac can be found. Therefore a and c
are known. Therefore, by (21), ^ and ^ are known. Therefore, by
(9), h is known. In this way all the elements for the solution of the
quintic are obtained,
§16. For example, the three equations (29) are satisfied by the values..
1 5 „ 25
4 , , 45 X 253
a = 5, . •. = 0, (p = — , A
75' 16
When these values are substituted in (28), the quintic becomes
a,5 + ^ 4. 3750 _ 0.
140 RESOLUTION OF SOLVABLE EQUATIONS
Then the values of u^, u^, u^, w^, obtained from the expression for
5 ■
Ui, in §3, are
Ui
1*2
=^{-'V(4)-;^^/(4-7^)}.
W3 =
4^{-'+7(4)+;^v/(4-74)}
Hence, rj = w^ + m^ + Wj + ^t, = — 1.52887 — 2.25035 -f
2.48413 — 3.65639 = — 4.95148.
When any Relation is Assumed between the Six
Unknown Quantities.
§17. In the case in which ui u^ was taken equal to U2 u^ a relation
was in fact assumed betwixt the six unknown quantites a, c, e, h, $, tp ;
for, as we saw, to put ii\ u^ — ui uz is tantamount to putting a = 0.
Hence, as was noticed in §7, we had only five unknown quantities
to be found from six equations. Now, when any relation whatever
is assumed betwixt the six unknown quantities, the root of the quintic
can be found in terms of the given coefficients P2 , Pz , etc., without
any definite numerical values being assigned to the coefficients,
because six rational quantities can always be found from seven
equations.
The General Case.
§18. We have hitherto been dealing with solvable quintics, assumed
to be subject to some condition additional to what is involved in their
solvability. We have now to consider how the general case is to be
dealt with. That is to say, we here make no supposition regarding
the equation of the fifth degree F {x) = except that it wants the
second term and is solvable algebraically. In this case it is impossible
to find the roots in terms of the coefficients j^^, 2hf ^^c, while these
coefficients retain their general symbolic forms. But the equations in
§3 enable us to find the roots when the coefficients receive any definite
numerical values that render the equation solvable. For, we have
the six equations (15) to determine the six unknown quantities
a, c, e, h, 0, *

*lex, Hudson, var.
angustifolium, Cray.
NAIADACE^.
Potamogeton natans, Ij.
" amplifolius, Tuckerm.
" lucens. L., var. minor.
" perfoliatus, L.
" compressus, L.
" pauciflorus, Pursh.
" pectinatus, L.
ALfSMACEvE.
Alisma plantngo, L., var. Ameri-
can um, Cray.
Sagittaria variabilis, Engelm.
HYDKOCIIAKIDACEiE.
Anacharis Canadensis, Planchon.
Vallisneria spiralis, L.
ORCHIDACE.E.
Orchis spectabilis, L.
ORCUWACEM— Continued.
* Habenaria tridcntata, Lindl. Mill-
grove.
* " virescens, Spi-eng. Prin-
ce's Island.
* " viridis, K. Br., var. brac-
teata, Reichenbach.
Mountain at head of
Queen Street.
* " hyperborea, R. Br. Sul-
phur Spring.
" Hookeri, Torr.
* " orbiculata, Torr.
* " leucojjhaia, Cray. Mill-
grove.
* " psychodes, Cray. Mill-
grove.
* " fimbriata, R. Br. Land's
Farm.
Coodyera pubeFceiis, R. Br.
* Spiranthes cernua, Richardson. The
Dell, Ancaster.
* Pogonia ophioglossoides, Nutt.
Millgrove.
* Calypso borealis, Salisb. Lake
Medad.
* Corallorhiza iunata, R. Br. Prin-
ce's Island.
" odontorhiza, Nutt.
" multitlora, Nutt.
Cypripedium parviilorum, Salisb.
" pubescens, Willd.
* " spectabile, Swartz.
Lake Medad.
* " acaule, Ait. Millgrove.
AMARYLLID.VCEiE.
* Hypoxys erecta, L. Prince's Is-
land.
I RID ACE J^..
Iris versicolor, L.
Sisyrinchium Bermudiaua, L., var.
anceps, Gray.
DIORCOREACE.^.
Dioscorea villosa. L. Near Dundas
Marsh.
SMILACE.E.
Smilax hispida, Muhl.
" herbacea, L.
LILIACE.E.
Trillium granditiorum, Salisb.
" erectum, L.
" erectum, L., var. album,
Pursh.
154
PROCEEDINGS OF THE CANADIAN INSTITUTE.
lAUXCKM— Continued.
Medeola Virginica, L.
Uvularia gran !orfolk.
Ramex sanguineus, L., occurs at London and Barrie.
Uimus racemosa, Thomas, occurs at St. Thomas.
Juniperus Sabina, L., var. procumbens, Pursh., which I formei'ly reported as
occurring, proves to be J. Virginiana, L.
In the discussion which followed, Mr. Geo. E. Shaw, Mr.
T. Mackenzie, Mr. Henry Montgomery, Mr. James Bain, jun.^
and the reader of the paper took part.
Mr. Fred. Phillips read a paper on " The Antiquity of the
Negro Race," the object of which was to show that the negro
race made its appearance before the white races.
A discussion ensued, in which the President, Mr. John Not-
man, and Mr. Montgomery took part.
TWELFTH ORDINARY MEETING.
The Twelfth Ordinary Meeting of the Session i883-'84 was
held on Saturday, February 2nd, 1884, Dr. Geo. Kennedy,
Third Vice-President, in the chair.
The minutes of last meeting were read and confirmed.
The following list of donations and exchanges received
since last meeting was read : —
1. The Financial Reform Almanack for 1884 ; presented by the Cobden Club.
TWELFTH ORDINARY MEETING. 157
2. Museum of Comparative Zo5logy at Harvard College, Vol. XI., Nos.
5, 6, 7.
3. Proceedings of the Ainerioau Academy of Arts and Sciences, Vol. XI., pp.
45—210.
4. Journal of the Franklin Institute for February, 1884.
5. Science Record, January 15, 1884.
6. Science, for January 25, 1884.
7. Proceedings of the Academy of Natural Sciences of Philadelphia, Part 2,
June to October, 188o.
8. Nye Alcyonider Corgonider, og Pennatulider, tilhorende Norges Fauna ;
from the Royal Museum of Bergen. (Norwegian Fauna.)
Prof. G. P. Young then read a paper entitled, " The Real
Correspondents of Imaginary Points."
After the reading of the paper, remarks were made upon
the subject by Prof. Galbraith and Mr. Alfred Baker.
THIRTEENTH ORDINARY MEETING.
The Thirteenth Ordinary Meeting of the Session i883-'84
was held on Saturday, February 9th, 1884, the President in
the Chair.
The minutes of last meeting were read and confirmed.
The following list of donations and exchanges received
since last meeting was read : —
1. Transactions of the New York Academy of Sciences, Vol- II., Nos. 3 to 8,
Contents and Title Page, Vol. I.
2. Annals of the New York Academy of Sciences, Nos. 12 and 13, Vol. II.
3. The Canadian Practitioner, for February, 1884.
4. Science, Vol. III., No. 52, for February, 1884.
5. Memoires et Compte Rendu des Travaux de la Societe des Ingenieurs
Civils, November, 1883.
6. Bulletin of the Museum of Comparative Zoology at Harvard College, Vol.
XL, iNo. 8.
It was moved and seconded " That the Council be a Com-
mittee, with power to add to their number, to arrange for the
reception and entertainment of such members of the British
Association as may visit Toronto during the month of Sep-
tember." — Carried.
"158 PROCEKDINGS OF THE CANADIAN INSTITUTE.
Mr. W. H. VanderSmissen then read a paper by the Rev
Prof. Campbell of Montreal, on
THE KHITAN LANGUAGES; THE AZTEC AND ITS
RELATIONS.
My translation of the Hittite Inscriptions found at Hamath and
Jerabis, in Syria, is the only one yet publislied with an explanation
of the process by which it was accomplished. The Rev. Dunbar I
Heath has sent me copies of his papers in which the Hamath
inscriptions are translated as Chaldee orders for musical services, but
no process is hinted at by the learned author. In the discussion
which followed the reading of one of these papers, a well-known
Semitic scholar remarked, " that so long as no pi'inciple was laid
down anil explained as to the system by which the characters had
been transliterated, it would be impossible to express an opinion on
the value of the proposed reading." Whatever may be the merits of
my translation, it does not make default in this respect. The pro-
cess is simple and evident. The phonetic values of the Aztec hiero-
glyphic system are transferred to corresponding hieroglyphic charac-
ters in the Hittite inscriptions. Common Hittite symbols are the
arm, the leg, the shoe, the house, the eagle, the tish. These are also
found as Me.xican hieroglyphics. There is nothing to tell us what
their phonetic values are in Hittite, because hardly any other remains
of the Hittite language have survived. But in Aztec we know that
these values are the lirst syllables of the words they represent. Thus
an arm being called neitl, gives the phonetic value ite for the hiero-
glyphic representing an arm. A leg being called meztli, furnishes
me. A shoe gives ca from cactli ; a house, also, ca from calli ; an
eagle, qua from qunuhli ; and a fish, ini from michin. But the
question has been raised, " What ])0ssible connection can there be
between the Hittites or Khita of ancient Syria and the Aztecs of
Mexico f As well might we ask what connection can there be
between Indian Brahmins and Englishmen ; between European
Osmanli and Siberian Yakuts. Geogi'apliical separation in such case,
is simply the result of a movement tliat has been going on from early
^ges. Men are not plants nor mere animals to be restricted to floral
and faunal centres. The student of history, who has followed the
Hunnic and Monsolian hordes in their devastatinj' course across two
THIRTEENTH ORDINARY MEETING. 159
continents, will not be surprised to find that well-known Iroquois
scholar, the Abbd Ouoq, suggesting the relationship of the Iroquois
with the wandering and barbarous Alans and Huns. Still less sur-
prise should be experienced when the more cultured Aztecs of Mexico
are connected with an ancient Old World civilization. Aztec history
does not begin till the 11th century of our era, and even that of the
Toltecs, who pz'eceded the Aztecs, and were of the same or of an
allied race, goes no farther back than the 8th. The period of their
connection with Old World history as a displaced Asiatic people is
thus too early to be accounted for by the invasions of the Mongols,
but coincides with the eastern movements of the Khitan, who, after
centuries of warfare on the borders of Siberia, disappeai'ed from the
historian's view in 1123. It is certainly a coincidence that the
Aztecs should claim to be of the noble race of the Citin, and that citli,
the hare, or, in the plural, citin, should be the totem or heraldic
device of their nation.
Since I wrote the article on the Khitan Languages, in which I
traced the Chinese Khitan backwards to central Siberia about the
sources of the Yenisei, where, according to Malte Brun, the Tartars
called their mounds Li Katei, or the tombs of the Cathayans, I have
received from Mr. VI. YouferoflF, of the Imperial Society of Geo-
gi-aphy at St. Petersburg, copies of the chief inscriptions from that
region. These triumphantly confirmed my supposition that the
Katei and the Khita or Hittites were the same people, by presenting
characters occupying a somewhat intermediate position in form
between the Hittite hieroglyphics and the more cursive script of our
Mound Builders. The rude representations of animals and other
natural objects accompanying some of the inscriptions are precisely
of the type furnished by the Davenport Stone. One inscription,
which I deciphered and the translation of which is now before the
Imperial Society of Geography, relates the victory of Sekata, a
Khitan monarch, the Sheketang of the Cliinese hostorians, over two
revolted princes or chiefs dwelling at Uta or Utasa in Siberia. As
in the case of the Syrian Hittite inscriptions, I have translated the
Siberian one by means of the Japanese, using the Basque, the Aztec,
and other languages of the Khitan family, for confirmation. What-
ever foreign infiuences may have done to modify the physical features,
the character, language, religion, and arts of the Japanese, and, in
lesser measure, of the Coreans, there can be no doubt that these are
13
160 PUOCEEDINGS OF THE CANADIAN INSTITUTE.
at basio Hittite or Khitan. Already at the commencement of my
Hittite studies I had noted the agreement of many characters in the
Corean alphabet with those of Hamath and Jerabis on the one hand,
and, on the other, with those on our mound tablets. The Rev. John
Edwards of Atoka with great kindness procured for me, from a mem-
ber of the Japanese Imperial Household at Tokio, a work on the
ancient writing of the Japanese. One of tlie forms of writing exhi-
bited in this work and occupying much space is very similar to the
Corean, and is undeniably of the same oi-igin. I have not yet had
time to investigate the volumes tlioroughly, Ijut as they appear to
contain samples of ancient alphabets with guesses at their significa-
tion rather than complete inscriptions, little progress may be antici-
pated V)y means of them. Nevertheless the existence in Japan of a
syllabary of so Hittite a type as the Corean in ancient times is con-
firmatoiy of the Khitan origin of the Japanese. As for the relations
of American civilizations, such as those of the Mexicans, Muyscas,
and Peruvians, with that of Japan, I need only refer to the writings
of so accurate and judicious an observer as Humboldt.
Returning to the Hlttites of Syria, who figure so largely in the
victorious annals of the Egyptian Pharaohs and Assyrian kings, and
whose empire came to an end towards the close of the 8th century
B.C., we find that, although apart from my own conclusions no defi-
nite opinion has been reached regarding their language beyond the
mere fact that it was Turanian, guesses have been made by scholars
whose hypotheses even are worthy of consideration. Professor Sayce
believes the Hittite language to have been akin to that furnished by
the ancient Tannic inscriptions of Armenia. The Vannic language,
according to Lenormant, belongs to the Alarodian family, of which
the best known living example is the Georgian of the Caucasus.
Now it is the Caucasus that I have made the starting point of Hit-
tite migration, which terminated at Biscay in the west, and in the
east, reaching the utmost bounds of Northern Asia, overflowed into
America. Not only the Georgians, I unhesitatingly assert, but most
of the other Caucasian families, the Circassians, Lesghians, and
Mizjeji at least, should be classed as Alarodians, or better still as
Khitan. So far I have found no evidence from ancient Caucasian
inscriptions, though such I believe have been discovered ; but an
evidence as conclusive is furnished by the languages of the Caucasian
families I liave named as compared with those which are presum-
THIRTEENTH ORDINARY MEETING. IGl
ably of Hittite origin in the Old World and in the New. lu die
remainder of this paper, I propose chiefly to set forth the relations of
the Aztec language, by means of vvliieh I transliterated the Hittite
inscriptions, with the Caucasian tongues, which of all Khitan forms
of speech are in closest geographical propinquity to the ancient habi-
tat of the Hittite nation. Before doing so I may set forth the prin-
cipal members of the Khitan family at the present day.
THE KHITAN FAMILY.
1. Old World Division.
Basque.
Caucasian = Georgian, Lesghian, Circassian, Mizjeji.
Siberian = Yeniseian, Yukahirian, Koriak, Tchuktchi, Karntchadale.
Japanese =.Japanese, LooChoo, Aino, Corean.
2. Amehican Division.
Dacotah.
Huron-Iroquois inchiding Cherokee.
Choctaw-Muskogee including Natchez.
Pawnee including Ricaree and Caddo.
Paduca =Shoshonese, Comanche, Ute, &c.
Yuma =Yuma, Cuchan, Maricoi^a.
Pueblos =Zuni, Tequa, &c.
Sonera = Opata, Cora, Tarahumara, &c.
Aztec including Niquirian.
Lenca =Guajiquiro, Opatoro, Intibuca.
Chibcha or Muysca.
Peruvian = Quicliua, Aymara, Cayubaba, Sapibocono, Atacameno, &c.
Chileno = Araucanian, Patagonian, P\iegian, &c.
The Nahuatl, or language of the Aztecs, as distinguished from
other tribes of diverse speech inhabiting Mexico, has long been a
subject of no little difficulty to philologists. It is not that its gram-
matical construction is peculiar, but because its vocabulary exhibits
combinations of letters or sounds that have come to be regarded as its
almost peculiar property. The most important of these is the sound
represented by tl, whether it be initial, medial or final. The Aztecs
of Nicaragua drop the tl altogether or reduce it to ^ ; hence some
writers have su])posed theirs to be the true form of the language, and
the literary tongue of Mexico a corruption. Upon this an argument
has been founded for the southern origin of the Nahua race. But,
as Dr. Buschmann and others have shewn, a mere casual survey of
the languages of more northern peoples, the Sonora and Pueblo
tribes, and the great Paduca family, reveals the fact that they con-
162 PROCEEDINGS OF THE CANADIAN INSTITUTE.
tain a considerable proportion of Aztec words, and that in them, as in
the Nahuatl of Nicaragua, the Aztec tl disappears or is converted into
t, d, k, s, r or I. Here therefore it is claimed by others is an argu-
ment for the northern derivation of the Mexicans.
If we carry forward the work of comparison, having regard to cer-
tain laws of phonetic change, we shall find, as I profess to have done,
that the vocabulary, and to a large extent the grammar, of the Aztecs
are those of all the greater families in point of culture and warlike
character of the Northern and Southern Continents. Nor do the
Aztec and its related American languages form a family by them-
selves. They have their countei'parts, as I have indicated, in many
regions of the Old World. If my classification of these languages
be just, there should, among a thousand other subjects of interest, be
found some explanation of the great peculiarity of Aztec speech to
which I have referred.
The Aztec combination tl appears, although to no very great ex-
tent, in the Koriak, Tchuktchi, and Kamtchatdale dialects. It has
no place in Corean, Japanese, or Aino, and only isolated instances of
its use are found in the- Yukahirian and Yeniseian languages. Of
the fovir Caucasian tongues which pertain to the Khitan family, two,
the Georgian, and Mizjeji, are almost as destitute of such a sound as
the Corean and Japanese ; while the Circassian and Lesghian vocabu-
laries, by their frequent employment of tl, reproduce in great measure
the characteristic feature of the Nahuatl. It is altogether wanting
in the Basque, and is a combination foreign to the genius of that
language. Yet there is no simpler task in comparative philology
than to show the radical unity of the Basque and Lesghian forms of
speech. Such a comparison, as well as one of the Lesghian dialects
among themselves and with the other Caucasian languages, will en-
able us to decide whether the tl of the Lesghian and Circassian forms
part of an original phonetic system, or is an expedient, naturally
adopted by speakers whose relaxed vocal organs made fsome other
sound diflicult or impossible, to stave ofi" the process of phonetic decay
by substituting for such sound the nearest equivalent of which they
■were capable.
In order first of all to exhibit the common origin of the Basque
and the Lesghian, I submit the following comparison of forms, the
relations of which are apparent to the most casual observer. The
Lesghian vocabulary is that of Klaproth, contained in his Asia Poly-
THIRTEENTH ORDINARY MEETING.
163
glotta ; the Basque is derived from the dictionaries of Yan Eys and
Lecluse. It will be observed that the Lesghian almost invariably
differs from the Basque : —
1. In substituting m for initial h.
2. In dispensing with initial vowels ; or, when they cannot be dis-
pensed with, in prefixing to them b or p, t or d.
3. In generally rendering the Basque aspirate, together with ch and
g, by the correspondingly harder forms g, k and q.
4. In occasionally adding final I or r.
(The last named letters I and r are interchangeable in the Khitan
as they are in all other families of speech.)
COMPARISON OF BASQUE AND LESGHIAN.
RULB 1.
English.
Basque.
Lesghiak.
beard
bizar
mussiir, muzul
head
buru
mier, maar
nail
behatz
maats
back
bizkhar
machol, michal
to-morrow
bihar
michar (Georgian)
Rule 2, a.
skia
achala
quli
hand
ahurra
kuer
river
uharre
chyare, uor
thunder
ehurzuria, curciria
gurgur
hair
ileak
ras
cold
otzo
zoto
no
ez
zn
left hand
ezquerra, ezker
kuzal, kisil
milk
eznea
sink
star
izarra
suri
day
eguna
kini
Rule 2, J.
deer
oreina
burni
clothes
aldar
paltar
child
aurra
durrha
stone
arri, harri
tsheru, gnl
Rule 3.
great
handi
kundi
house
eche
akko
hail
harri
goro
smoke
gue
kui
tooth
hortz
kertschi
leaf
orri
kere
finger
erhi
kilish
Rule 4.
rain
uria
kural
son
seme
chimir
great
zabala
chvallal
The following, though generally agreeing, present same exceptions
to the above rules.
164
PROCEEDINGS OF THE CANADIAN INSTITUTE.
English.
Basque.
Lbrghian.
heaven
cpru
ser
birfj
chori
zur
red
ffori, gorri
hiri
blue, green
urdin
crdj'n
death
heriotze
haratz
old
agure, zar,
zahar
herau, etshru
throat
cinzur
seker
wliite
churia, zuria
tchalasa
wood
zura
zul
leg
aziai
uttur
tree
zuliatsa
giiet, hueta
fire
su
zo
higli
gan
okanne
tongue
mia
mas
A comparison of the Basque with tlie other Caucasian languages,
Georgian, Circassian, and Mizjeji, would display similar relations
with some modification of the laws of phonetic change.
If now we ask what the Basque does with the Lesghian tl, we shall
find that it represents that sound chiefly by the letters r and I.
This equivalency of tl, and sometimes of ntl, to r and I also appears
in comparing the Lesghian dialects among themselves or with other
Caucasian languages.
COMPARISON OF LESGHIAN FORMS IN tl WITH OTHER CAU-
CASIAN AND BASQUE FORMS.
English.
hair
bone
wood
tomorrow
night
sheep
maize
goat
six
nail
low
eight
sun
flesh
forehead
easy
loins
water
butter
hair
earth
Lesghian.
Other Forms.
tlozi
tlusa
ras, Lesghian.
rekka "
thludi
redu-kazu "
shishatla
shile "
ret'lo
rahle "
betl
bura "
zoroto-roodl
tzozal-lora "
antle
arle
antlko
ureekul
niatl
tluksr
mare, Mizjeji
loehun
bitlno
bar, barl "
mitli
maleh
" beri, Lesqhian.
marra, Circassian
yti
tlokva
gin
illech
intlaugu
illesu "
errecha, Basque.
tlono
errainac
htli
ur "
yetl
tlozi
guri
ileac "
rati
lurra, laur "
THIRTEENTH ORDINARY MEEEING. 165
The following represent the exceptions to the rule both iu form
and in numerical proportion : —
English. Lksohian. Other Forms.
yellow tlelii dula, Lfsfjhian.
day tl.Y"! thyal, tolizul "
horn tlar adar, Baaque.
knee tlon belaun "
From the preceding examples it appears that the Lesghian sounds
represented by tl, thl, nil, are the equivalents of r and I generally,
and sometimes of ff or t. The latter excej^tion probably finds its
explanation in Basque, for in the dialects of that language an occa-
sional permutation of r and I into t and d takes place. Thus iJeki
to take away, becomes ireki, and iduzki the sun, becomes iruzki,
while p.lur snow, sometimes assumes the form edar, and belar grass,
that of hedar. The last exception cited, that in which the Lesghian
tlo)i is compared with the Basque heJaun. is really no e.xception, for
elaun is the true representation of tlon, the initial h being prosthetic
to the root, as is frequently the case in Basque. A.mong many
examples that might be given, I may simply cite hdar the ear, as
compared with the Mizjeji lerh.
Turning now to the Aztec, on the supposition that it is related to
the Basque and Caucasian languages, we naturally expect to find on
comparison a coincidence of roots and even of words following upon
the recognition of tl and ntl as the equivalents of r and I in these
forms of s])eech. The fact that the Aztec alphabet is deficient in
the letter r favours such an expectation. But our comparison must
be made with due caution. Any one who has examined a Mexican
dictionary, such as that of Molina, must have been struck with the
remarkable preponderance of words commencing with the letter t
over those beginning with any other letter of the alphabet. These
wor'ls comprise consideral)ly more than one third of the whole lexi-
con. A certain explanation of this is found in the fact that the two
particles te and tla possess, the former an indefinite personal, and the
latter a substantive, signification, and thus enter largely into the
structure of compound words. Whatever its grammatical value in
Aztec, however, it appears, on comparing the Aztec vocabulary with
its related forms of speech, that initial t or te, which leaving tl out of
account still occupies one fifth of the lexicon, is frequently prosthetic
to the root.
The following are some of the chief laws of phonetic change derived
166
PROCEEDINGS OF THE CANADIAN INSTITUTE.
from a comparison of the Aztec and Lesghian languages. These
may be found operating to almost as great an extent in the Lesghian
dialects among themselves : —
1. The Aztec combinations tl, ntl, are either rendered in Lesghian by
the same sounds, or by r or I. In some cases in which phonetic
decay has set in, the Aztec tl is either omitted or represented by
a dental. The Lesghian occasionally renders the Aztec I and
U by tl.
2. The interchange of p and m, which appeared in comparing the
Basque and the Lesghian, for the Aztec is deficient in the sound
of h, characterizes a comparison of the Aztec with the Caucasian
languages.
3. A similar interchange of n and I, or the ordinary equivalents of 1,
such as marked the Iroquois in comparison with the Basque
occasionally charactei'izes the relations of the Aztec and Caucas-
ian tongues.
4. The Lesghian, as already indicated, persists in the rejection of
initial vowels, and the same is generally true of reduplications
and medial aspirates.
5. As in many Aztec words initial t forms no part of the root, but is
a prosthetic particle, it finds no place in such cases in the corres-
ponding Lesghian term.
6. The Lesghian occasionally strengthens a word by the insertion of
medial r before a guttural, for which of course there can be no
provision in Aztec.
I have not thought it desirable to burden this paper with laws
relating to other changes, as the relation of the compared words will
be sufficiently apj^arent ; but, for the purpose of illustration, I have
added corresponding terms from other Khitan languages exempli-
fying the rules set forth.
COMPARISON OF AZTEC AND LESGHIAN FORMS.
English. Aztec. Puoxbtic Chakge. Lesghian. Illdstrations.
water atl ar al htli ur, Basque
low tlatzintli latzili, latziri tlukur liuchtliu, Koriak
day tlacatli lacali, lacari tlyal, djekul alluchal, teluchtat, Koriak
knee tlanquaitl lancail, laucair tlon zangar, Basque
cconcor, Quichua
deer niazatl mazal, mazar niitli mool, Yuma
earth tlalli ralli, larri rati lurra, Basqiie
night tlalli " " retlo, rahle neillhe, Choctaw
yesterday yalliua alhua hutl hooriz, Dacotah
ice cctl eel, cer zer, zar kori, Japanese
wind ehecatl eheeal, eliecar cliuri gygalkei, Koriak
sheep ichcatl ichcal, it-hear klr achuri, Basque
ccaora, Aymara
THIRTEENTH ORDINARY MEETING.
167
English.
Atzec
Phonetic Change.
Lesohian.
mud
zoquitl
zokil, zokir
zchur
stone
tetl
tel, ter
tsheru
dust
teulitli
teuhli, teuhn
clmr
grass
quilitl
kilil, kirir
cher, gulu
star
citlalli
cilalii, cirarri
suri
hair
tzontli
tzoli, tzori
tshara
skin
CU.ltl
cnal, cuar
quli
eye
ixtli
ishli, ishri
chuli
wood
quauitl
kauil, kauir
zul
"
"
kauit
guet, hueta
foot
icxitl
icshil, icshir
kash
year
xiuitl
shiuil, shiuir
thahel
god
teotl
teol, teor
saal, zalla
clothes
tlatqtl
ratkl, latkr
]Kiltar, retelkum
cold
cecuiztli
'•eeuizli, cecuizri
cliuatzala
mountain
tejietl
tepel, teper
dubura
moon
metztli
metzli, metzri
moots, bars
leg
mutztli
"
maho
hand
niaitl
mail, mair
ku mur
honey
neeutll
neculi, necuri
nutzi, nuzo
bread
tlaxcalli
lashcalli, rashcalli
zulha
copper
tepuztli
tepuzli, tepuzri
dupsi
mouth
camatl
carnal, camar
sumun, moli
belly
xillantli
shillal, shillar
siarad
feather
yhuitl
ywil, ywir
bel, pala
rain
quiahuitl
kiavil, kiavir
gvaral
woman
eihuatl
cival, civar
tshaba
bird
to-totl
tol, tor
adjari, zur
name
to-caitl
call, cair
zyer, zar
beard
te-nchalli
nchalli, ncharri
muzul, raussur
river
at-oyatl
oy.il, oyar
uor, chyare
throat
t-uzquitl
uzkii, uzkir
seker
back
to-puztli
puzli, puzri
luachol
sun
to-natiuh
natiuL
mitzi
evening
te-otlac
olak, orak
sarrach, Mizjeji
snow-
cepayauitl
payauil, payauir
marchala
man
maceualli
maceualli
murgul
small
tlocoton, tzoeoton locoton, fzocoton
chitina
sand
xalli
shalli, sharri
keru
shouUlers
aoolli
acolli, acorri
hiro
son
tepil-tzin
tepil, tepir
timir, chimir
woman, wife
tenamie
tenamie
ganabi
nsh
michiu
michin
migul, besuro
to-day
axcan
ashcan
djekul
give
maca
maca
beekish
stone
topecat
topecat
teb
black
caputztic
caputztic
kaba
bard
tepitztie
tepitztie
debehase
old
veue
veue
vochor
green
quiltic
kiltie
sholdisa
great
yzachi
izachi
zekko
"
yzaehipul
izaehipul
chvallal
dog
chichi
ehichi
choi
no
amo
amo
anu
I
ne
ne
na
than
te
te
duz
he
ye, yehua
he, heua
heich
Illustrations.
ehulu, Corean
tol
turo, Quichua
kyraii, Fe;ti.seiart/
zirari, Aino
thorok, Corean
ccaia, Quichua
okalua, Iroquois
kullu, Quichna
zuliaitz, Basque
ochsita, Iroquois
osera, "
chail, koil, Yukahiri
aldarri, aldagarri, Basqut
hutseelu, xeteliur, Yuraa
neit-ti|ipel, Koriak
muarr, Shoslwnese
ouitsa, Iroquois
masseer. Shoshonese
miski, Qjiickua
mitzi, Japanese
lagul, Yukahiri
rajali, Yeniseian
tup, thep, Yeniseian
tetinpulgun, KamtchatdaU
si'iii, Quichua
homal-galgen, Koriak
kolid, Kamlchatdnle
puru, Quichua
kutil-kislieu, Koriak
si pi, Corean
suiigwal, Shoshonese
tori, Japanese
chareigtsh, Kamtchatdalt
tegiiala, Sonera
h-inuDCkquell, Shoshonese
hahuiri, Aymara
eztarri, Basque
bizkhar, "
kapteher, Koriak
nitc;hi, Japanese
inti, Quichua.
simrek, Iroquois
pukoelli, Yvktihiri
pagolka. Koriak
birklijar.jat, Yeniseian
mailik, I'ujunl
cikadang, Dakotah
iskitini, Choctaw
challa, Aymara
callachi, "
comerse, Yuma
tiperie, Sonora
kanafe, Corean
mughat, pughutsi, Shosho7ieBe
hichuru, Aymara
taehan, Mizjeji
eman, eiiiak, Basque
ti\n, Shoshonese
shupitkat, Uacotah
kiliichii, Japanese
vucha, Arancanian
apachi, Aymara
sherecat, Ducotah
Lashka, "
zabal, Basque
cocoelii, Sonora
ama, Quichua
ni, Basque
na, Aymara
zu, Basque
ta, Aymara
hau, Basqzte
uca, Aymara
108
PKOCEKDINGS OF THE CANADIAN INSTITUTE.
The Geoi'gian does not exhibit the A.ztec tl, but, as it is regarded
by Professor Sayce as the living language most likely to represent
the speech of the ancient Hittites, a brief comparison of its forms
with those of the Aztec may not be out of place. Like the Lesghian
it is impatient of initial vowels, and it generally agrees with that
language in the laws of phonetic change, adding, however, this pecu-
liarity, the occasional insertion of v before /. The v seems generally
to represent ii,, or some similar vowel sound, and is probably such a
corruption of the original as appears in the Samivel of Pickwick
compared with the orthodox Samuel.
COMPARISON OF AZTEC AND GEORGIAN FORMS.
English.
AZTKC.
Phonetic Chance.
Georgian.
Illustrations.
fowl
tototl
totol, totor
dedali
totolin, Soiiora
red
chichi'itic
chichiltic
tziteli
tsatsal, Kamtchatdale
blood
eztii
ezli, ezri
sisehli
odol, Basiiue
ehri, Dncotah
house
calli
calli
sachli
cari, caliki, Sonora
moimtam
quautla
kaula, kaura
gora
kkollo, Aymara
horn
quac^uauitl
kakaul, kakaur
akra
quajra, "
sheep
ichcatl
ichcal, ichoar
tschchuri
ccaora, "
wind
ehecatl
ehecal, ehecar
kari
helcala, Sonora
heart
yullotl
yullol, yuUor
gulu
gullugu, Kamtchatdale
girl
ocuel
ocuel '
ukui-za, kali
okulosolia, Choctaw
dog
yzcuintli
izkili, izkiri
dzagli, dji
ogori
schari, Shoshonese
nose
yacatl
hacal, hacar
zchviri
surra, Basque
cher. Puehlon
hair
tzontli
tzoli, tzori
tzvere (beard)
tsheron, Kamtchatdale
moon
metztli
metzli, metzri
mtvare
inuarr, Shoshonese
silver
teo-quitlatl
kilal, kiiar
kvartshili
cilarra, Basque
shoulder
te-piiztli
puzli, puzri
mchari
buhun, Lesghian
tomorrow
muztii
mnzli, rauzri
michar
mayyokal, Vuma
leg
metztli
metzli, metzri
muciili
ametehe, "
to kill
niiclia
miclia
mokluli
wakerio, enkerio, Iroquois
mother
nantii
nali, nari
nana
nouriia, Iroquois
snow
cepayauitl
cepayauil, cepayauir
tovli
repaliki, Sonora
suake
coliuatl
coval, eovar
gveli
toeweroe, Shoshonese
boy
tepii-tziu
tepil
shvili
tiperic, Sonora
lightning
tlapetlani
lapelani
elvai
illappa, Quichua
wilhyap, i'uma
leaf
iatla-pallo
iala-pallo, iala-)iarro
pur-zeli
bil-tlel, Kamtchatdale
small
tzocoton
tzocoton
katou
cikadaug, Dacotah
man
oquichtli
okichli, okichri
ankodj
oonquieh, Iroquois
oiakotsh,
Koriak
aycootch, Yuma
guru.
Aino
ccari, Quichua
The Circassian language abounds in labials, and thus finds its best
American representatives among the Dacotah dialects. Neverthe-
less it presents many words which come under the same general laws
in relation to the Aztec that have characterized the Lesghian and
Georgian.
THIUTEENTH ORDINARY MEIOTING.
1G9
COMPARISON OF AZTEC AND ClilCASSIAN FOKMS.
Illustrations.
nape, Davotah
niaslipa, Shoshonese
sliupitcat, Dacntah
yu pikha, Shoslwnese
tekay, tekash, Darotah
itaku, Itakisa, "
tsliiikyhetch, Knriak
culmba, Muysca
tap.sut, Ainn
gejiuca, Muysca
ibusu, ,/ai>anese
kuchi-liiiu, Japanese
niku, Japanese
raku.
errecha, Basque
arraiigya, Yukahiri
ja<'ucl, Ynma
akual-iie>iuta, Natchez
kflgols, h'amchatdale
odul, liasii>ie
huila, ,*