theorbo 
board, ami tho III^MT In-siring th> ftOOOmiMUli- 
nifiit strings or "dijipiiMons," which wriv iln p- 
er in pitch, and wen- phm-d without Itfing 
stopped. The nmnnerand tuning nf tin- string v;ui ! 
considerably, as did tin: size and stiupe of tlir iiiNtruiurnt 
;is :i vvlinlr. llir (lirorlto WHS Illllctl lifted ill the SCVCtl- 
trriith century for BOOOflipntnMDt* of all kinds, and was 
HII important constituent of the orchestra of the j>< i \<l. 
M.iny lutes were made over Into thum-lms l>y I la- addition 
of 11 rU'CMlld Mrrk. Thr r*M-nti:iI <liltr! rtirrs lirt \vccil the 
TlimrliD, the archill te, and the rhitarrnnu npjuai to be 
rtiiuil). tliuiiKh their gem-nil slmpr \:iiir,| < m -i.i. i;i!.]\ ; 
and tin tcuii'-h \\nv u.-cd more or less interchangeably. 
Also called cithartt bijwja, or dnublf-twckfd lute. 
s.Mnr, Unit drlinht to touch the sterner wiry chord. 
The t'ythrmi, the I'andore, and the theorbo strike. 
Drayton, Polyolbion, iv. 361. 
theorem (the'o-rem), n. [== F. thcurbmc = Hp. 
= It. 
teorcma = <. theo- 
rem, < I Jp flu ore tna = Or. Qehpjjita, a sight, specta- 
cle, a principle contemplated, a rule, theorem, { 
fh-ufHtv, look at, view, contemplate, < 0rtyJf, a 
Hpectator, < OeaoQai, see, view. Of. theory.] 1. 
A universal demonstrable proposition, in the 
strict sense, a theorem must be true; It cannot be self- 
evident ; it must be capable of being rendered evident by 
necessary reasoning and not by Induction merely ; and it 
must be a universal, not a particular proposition. But a 
proposition the proof of which Is excessively easy or In- 
volves no genuine diagrammatic reasoning Is not usually 
called a theorem. 
The schoolmen had framed a number of subtile anil in- 
tricate axioms and theorem*, to save the practice of the 
Church. Bacon, Superstition (ed. 1887). 
By my thetrremg, 
Which your polite and terser gallants practise, 
I re-refine the court, and civilize 
Their barbarous natures. 
Ma*ariiHjer, Emperor of the East, i. 2. 
2. In gcom.) a demonstrable theoretical propo- 
sition. There is a traditional distinction between a 
problem, and a theorem, to the effect that a problem is 
practical, while a theorem is theoretical. Pappus, who 
makes this distinction, admits that it is not generally ob- 
served by the Greek geometers, and It has not been in 
general use except by editors and students of Euclid. It 
is recommended, however, by the circumstance that a 
theorem in the general and best sense is a universal propo- 
sition, and as such substantially a statement that some- 
thing is impossible, while the kind of proposition called in 
geometry a problem is a statement that something is pos- 
sible ; the former demands demonstration only, while the 
latter requires solution, or the discovery of both method 
and demonstration. 
I hope that it may not be considered as unpardonable 
vanity or presumption on my part if, as my own taste has 
always led me to feel a greater interest in methods than 
in results, so it is by methods, rather than by any theorems 
which can be separately quoted, that I desire and hope to 
be remembered. Sir W. Hamilton. 
Abel's theorem, the proposition that if we have several 
functions whose derivatives can be roots of the same al- 
gebraic equation having all its coefficients rational func- 
tions of one variable, we can always express the sum of 
any number of such functions as the sum of an algebraic 
and a logarithmic function, provided we establish be- 
tween the variables of the functions in question a certain 
number of algebraic relations: named after Niels Henrik 
Abel (1802-29), who flrst published it in 1826. Addition 
theorem, a formula for a function of a sum of variables, 
such as 
sin (a + b) = sin a cos 6 i cos a sin b. 
Arbogast's theorem, a rule for the expansion of func- 
tions of functions, given in 180U by JL F. A. Arbogast 
(1759-1803X Aronhold's theorem, one of a number of 
propositions constituting the foundations of the theory 
of ternary cubics, given in 1849 by 3. ll Aronhold (born 
1819), the founder of modern algebra. Bayes'S theo- 
rem, tho proposition that the probability of a cause is 
equal to the probability that an observed event would 
follow from it divided by the sum of the corresponding 
probabilities for all possible causes. This fallacious rule 
was given by Rev. Thomas Bayes In 1703. Becker's 
theorem, the proposit ion that in all moving systems there 
is a tendency to motions of shorter period, and that if 
there is a sufficient difference in the periods compared this 
tendency is a maximum : given by O. F. Becker In 1886. 
Beltraml's theorem, the proposition that the center 
of a circle circumscribed about a triangle is the center of 
gravity of the centers of the inscribed and fsmlu-u i in-lrs. 
Berger's theorem, one of a number of theorems re- 
lating to the limiting values of means of whole numbers, 
given by A. Berger in 1880. One of these theorems is that 
for ii ' the average sum of the divisors of ;t Is ,> -/j. 
Bernoulli's theorem, (a) The doctrine that the relative 
frequency of an event in a number of random trials U-nds 
as that number is increased toward the probability of it, or 
its relative frequency in all experience. This fundamental 
principle, which is not properly a theorem, was given by 
Jacob Bernoulli (1654- 1705X (6) The proposition that the 
velocity of a liquid flowing from a reservoir is equal to 
what it would have if it were to fall freely from the level 
in the reservoir ; or, more generally, if p is the pressure, 
p the density, V the potential of the forces, q the resultant 
velocity, A a certain quantity constant along a stream- 
line, tlit-n -. 
(x) \ 
'"(* + A) + *'"(*) \ 
6275 
proposition, given by J. L. K. Bertrand (born 1822X 
Bettl'a theorem, the proposition that the loci of the 
I ii >ii its of a surfaec for wbieh the hum on the one hand and 
tin- iliit.-n iireon the otber of the geodetic distances of two 
fixed curves on the surface are constant form an orthogonal 
system : given by K. lletti in 1*5K, and by .1. \\ i in 
In more general form in IWM. Bezout's theorem, tin 
proposition that the degree of the equation resulting from 
the elimination of a variable between two equations Is 
equal to the product of the degrees of these equations, 
which was shown by E. Bezout (1730-83) In 1779. 
Binet's theorem, (a) The proposition that the princi- 
pal axes for any point of a rigid body are normals to 
three quadric surfaces through that point confocal with 
the central ellipsoid: given by J. P. M. Binet (1786-1856) 
in 1811. (b) The generalized multiplication theorem of 
determinants (1812). Binomial theorem. Hee bino- 
mial. Bltontl'B theorem, one of certain metrical theo- 
rems regarding the intersections of conies demonstrated 
by v. N. Bltonti in Is7n. Boltzmann'a theorem, the 
proposition, proved by I. Boltzmann in 1HU8, that the 
mean living force of all the particles of a mixed gas will 
come to be the same. Boole's theorem, the expansion 
* (* + A) $(*) =B, (2' 1)2 ! 
B.(2' 1)41 \ 
4B.(z" 1)61 
given by the eminent English mathematician George 
Boole (1815-64). Bour's theorem, the proposition that 
helicoids are deformablc into surfaces of revolution : given 
in 1862 by the French mathematician J. E. E. Bour (1832- 
1866). Brlanchon's theorem, the proposition that the 
lines joining opposite vertices of a hexagon circumscribed 
about a conic meet in one point : given by C. J. Brianchou 
(born 1785, died after 1823) in 1806. It was the earliest ap- 
plication of polar reciprocals. Sudan's theorem, the 
Sroposltion that if the roots of an algebraic equation are 
iminished first by one number and then by another, there 
cannot be more real roots whose values lie between those 
numbers than the number of changes of sign of the co- 
efficients in passing from one to the other : given and 
demonstrated In 1811 by the French mathematician Bu- 
diui. -Burmann'B theorem, a formula for developing 
one function in terms of another, by an application of 
Lagrange's theorem. Cagnoll'B theorem, in spherical 
trigon,, the formula for the sine of half the spherical ex- 
cess in terms of the sides : given by the Italian astrono- 
mer Andrea Cagnoli (1743-1816). Cantor's theorem, 
the proposition that if for every value of x greater than a 
and less than b the formula holds that limit (A.- sin nx 
-f B cos nx) = 0, then also limit A = and limit 1!,, 
= 0: given by O. Cantor in 1870. Camot'B theorem. 
(a) The proposition that if the sides of a triangle ABC 
(produced if necessary) cut a conic, AB In C* and C", AC 
In B' and B", BC In A' and A", then AB' x AB" x BC 1 x 
BC" x CA' x CA" = CB' x CB" x BA' x BA" x ACT x AC'. 
(b) The proposition that in the Impact of Inelastic bodies 
vis viva is always lost, (c) The proposition that In ex- 
plosions vis viva is always gained. These theorems are 
all due to the eminent mathematician General L. K. M. 
Carnot (1753-1823), who published (a) In 1803 and (b) and 
(c) In 1786. (<i) The proposition that the ratio of the max- 
imum mechanical effect to the whole heat expended in an 
expansive engine is a function solely of the two temper- 
atures at which the heat is received and emitted: given 
In 1824 by Sadi Carnot (1790-1832) : often called Carnot 'i 
principle. Case fa theorem, the proposition that If 
S , = 0, S, = 0, 8., = are the equations of three circles, 
and if /,, I , /, are respectively the lengths of the com- 
mon tangents from contact to contact of the last two, the 
flrst and last, and the first two, then the equation of a 
circle which touches all three circles is 
given by Daniel Bernoulli (1700 -82) in 1733. Bertrand's 
theorem, the proposition that when a dynamical system 
receives a sudden impulse the energy actually aei|iiired 
exceeds the enemy by any other motion consistent with 
tin- conditions of the system and obeying the law of en- 
ergy, by an amount equal to (he energy of the motion 
which must be compounded with the supposed motion to 
produce the actual motion: an extension of a known 
given by John Casey in 1866. Catalan's theorem, the 
proposition that the only real minimal ruled surface is the 
square-threaded screw-surface x = a arc tan (y z) : named 
after E. c. Catalan (born 1814). Cauchy's theorem, 
(a) The proposition that if a variable describes a closed 
contour In the plane of imaginary quantity, the argument 
of any synectic function will In the process go through 
its whole cycle of values as many times as it has zeros or 
roots within that contour. (6) The proposition that If 
the order of a group Is divisible by a prime number, then 
It contains a group of the order --of that prime. The 
extension of this that if the order of a group Is di- 
visible by a power of a prime, it contains a group whose 
order is that power is called Cauchu and Sy/<w' theorem, 
or simply Sylme'* theorem, because proved by the Norwe- 
gian L. Sylow in 1872. If) The rule for the development 
of determinants according to binary products of a row 
and a column, (rf) The false proposition that the sum of 
a convergent series whose terms are all continuous film - 
tions of a variable is itself continuous, (e) Certain other 
theorems are often referred to as Cauchy's, with or without 
further specification. All these propositions are due to 
the extraordinary French analyst, Baron A. L. Cauchy 
(1789-1857). Cavendish's theorem, the proposition 
that if a uniform spherical shell exerts no attraction on 
an interior particle, the law of attraction is that of the 
inverse square of the distance : given by Henry Caven- 
dish (mi-1810). Cayley's theorem, the proposition 
that every matrix satisfies an algebraic equation of Its 
own order : also called the prinfipal proposition of ma- 
trices: given by the eminent English mathematician Ar- 
thur Cayley. Cesaro'a theorem, the proposition that if 
the vertices A, B, C of one triangle lie respectively on the 
sides (produced if necessary) B'C, CA', A B' of a second 
triangle, which sides cut the sides of the flrst triangle in 
the points A", B", C" respectively, and if S be the area of 
the flrst triangle, S' that of the second, then 
CB". BA". AC" - AB". BC". CA" 
_ A&BOjU s- B 
\ l:.i;< .c-A-'SJ' " 
given by E. Cesaro in 1885. It is an extension of Ceva's 
theorem. Ceva'S theorem, the projmsitton that if the 
straight lines connecting a point with the vertices of 
a triangle AKi' meet the opposite sides in A', B', C". the 
product of the segments CB' x BA' x AC' is equal to 
theorem 
the product All / lir - ('A : given by (iiovannl C'eva In 
UI78. Chasles's theorem, thu proposition that of a 
unidlmenilonal family of conies In a plane the number 
which satisfy a simple riindiiii.n is expressible in the form 
aM . 0i', where a and B depend solely on the nature of the 
K. n, ulnIi-M it tin- number of conies of the family 
passing through an arbitrary point, and v is the number 
I..IL li.-d I.) 1111 arbitrary line : given In 1-1:1 by VI. Chasles 
(l73-lKu) without proof. Clairaut's theorem, the 
proposition that if the level surface of the earth Is an 
elliptic spheroid symmetrical about the axis of rotation. 
then the compression or clllptlclty Is equal to the ratio 
of ) the equatorial centrifugal force less the excess of 
polar over equatorial gravity to the mean gravity : given 
in 1743 by Alexis Claude Clairaut (1718-65). Clapey- 
ron's theorem, the proposition that If a portion of a 
horizontal beam supported at three points A, B, C has uni- 
form loads ur, and if ... on the parts AB and BC respectively, 
the lengths of which are respectively I, and / , and if 
a, 0, ? are the bending moments at the three points of 
support, then 
given by B. P. E. Clapeyron (1799-1868): otherwise called 
the theorem of three moment. Clausen's theorem. 
Same as Stavdt't theorem. Clausius's theorem, the 
proposition that the mean kinetic energy of a system 
in stationary motion is equal to Its virial : given by R. 
J. E. Clanslus (born 1822) In 1870: otherwise called the 
theorem of the virial. Clebsch'B theorem, the propo- 
sition that a curve of the nth order with Un 1) (n 2) 
double points is capable of rational parametric expression : 
given in 1866 by R. F. A. Clebsch (1833-72). Clifford's 
theorem, the proposition that any two lines in a plane 
meet in a point, that the three points so determined by 
three lines taken two by two lie on a circle, that the four 
circles so determined by four lines taken three by three 
meet in a point, that the five points so determined by 
five lines taken four by four lie on a circle, that the six 
circles so determined by six lines taken five by five meet 
In a point, and so on indefinitely : given in 1871 by W. K. 
Clifford (1845-79). CorlOlis'B theorem, the kinematl- 
cal proposition that the acceleration of a point relative to 
a rigid system is the resultant of the absolute accelera- 
tion, the acceleration of attraction, and the acceleration 
of compound centrifugal force : named from Its author, Q. 
ci. Corioiis (1792-1843X Cotesian theorem. Same as 
Cftten'g properties of the circle (which see, under circle). 
Coulomb's theorem, the proposition that when a con- 
ductor is in electrical equilibrium the whole of Its elec- 
tricity Is on the surface: given by C. A. Coulomb (1736- 
1806). Crocchl's theorem, the proposition that if K/> 
denotes what (x, + ,+ + xm)f becomes when the 
coefficients of the development are replaced by unity, and 
given by L. Crocchl in 1880. Crofton's theorem, the 
proposition that if L be the length of a plane convex con- 
tour, O its inclosed area, du> an element of plane external 
to this, and t the angle between two tangents from the 
point to which dw refers, then 
/( - sin *) d. = }L - n : 
given by Morgan W. Crofton In 1868. Certain symbolic 
expansions and a proposition in least squares are also so 
termed. Culmann'a theorem, the proposition that the 
corresponding sides of two funicular polygons which are In 
equilibrium under the saraesystem of forces cut one another 
on a straight line. D'Alembert'a theorem, the proposi- 
tion that every algebraic equation has a root : named from 
Jean le Rond d'Alembert (1717-83). See also D'Alem- 
bert'i principle, under principle. Dandelln'a theorem, 
the proposition that if a sphere be Inscribed in a right 
cone so as to touch any plane, Its point of contact with 
that plane is a focus and the intersection with that plane 
of the plane of the circle of contact of sphere and cone is 
a directrix of the section of the cone by the first plane : 
named from (. P. Dandelin (1794-1847), who gave it in 
1827 : hut he Is said to have been anticipated by Quetelet. 
The theorem that the locus of a point on the tangent of a 
fixed conic at a constant distance from the point of con- 
tact is a stereograph ic projection of a spherical conic is 
by Dandelin. Darboux's theorem, the proposition that 
if V is a function of x having superior and Inferior limits 
within a certain interval of values of x, and if this inter- 
val iscut up into partial intervals !, I,, . . . I*, in which 
the largest values of y are respectively M , M,, . . . M*, 
then MI will tend toward a fixed limit as the num- 
ber of Intervals is increased, without reference to the 
mode of dissection : named from its author, J. G. Dar- 
boux De Molvre'a theorem, (a) The proposition that 
(cos 9 -t- i sit ) = cos n + i sin n : better called De 
Moivre't fortpula. (b) Same as De Mnirre'i property of the 
circle (whicR see, under circle), (c) A certain proposition 
in probabilities. All these are by Abraham De Moivre 
(1667-1754). Desargues's theorem, (a) The propo- 
sition that when a quadrilateral is inscribed in a conic 
every transversal meets the two pairs of opposite sides 
and the conic in three pairs of points in involution. 
(b) The proposition that if two triangles ABC and A I; ( 
are so placed that the three straight lines through cor- 
responding vertices meet In a point, then also the three 
points of intersection of corresponding sides (produced if 
necessary) lie in one straight line, and conversely. Both 
were discovered by Girard Desargues (1593- 1662). Des- 
cartes's theorem. Same as Detcartet'i rule of ami 
(which see. under rwfri). Diophantus's theorem, the 
uroposition that no sum of three squares of integers Is a 
sum of two such squares: given by a celebrated tf reek arith- 
metician, probably of the third century- - Dostor'B theo- 
rem, the proposition that in a plane triangle, where 6, c 
are two of the sides. A the angle included between them, 
and the inclination of the bisector of this angle to the 
side opposite, 
. 
O C 
