theorem 
named from G. Dostor, by whom it was given in 1870. 
Certain corollaries from this in regard to the ellipse 
and hyperbola are also known as Dostor's theorems. 
Du Bois Reymond's theorem, the proposition that if 
/a is a function of limited variation between a. = A and 
a = B, and if <K, n) is such a function that/ A <f>(, )da 
(where 6 is any number between A and B) has its modulus 
less than a fixed quantity independent of b and of n, and 
that when n increases indefinitely the integral tends to- 
ward a fixed limit G for all values of b between A and B, 
then /",/. #, Xl wil1 tend uniformly to G/(A + 0)if 
B > A; and to G/(A - 0) if B < A : named from the German 
mathematician Paul du Bois Reymond. Dupin S theo- 
rem the proposition that three families of surfaces cut- 
ting 'one another orthogonally cut along lines of curva- 
ture: given by Charles Dupin (1784-1873). Earnshaw's 
theorem, the proposition that an electrified body placed 
in an electric field cannot be in stable equilibrium. 
Eisenstein's theorem, the proposition that when y in 
the algebraic equation fyx, y) = is developed in powers 
of x, the coefficients, reduced to their lowest terms, have 
a finite number of factors in the denominator: given in 
1862 by F. G. M. Eisenstein (1823-52).-Euler'S theo- 
rem, (a) The proposition that at every point of a surface 
the radius of curvature p of a normal section inclined at 
an angle t to one of the principal sections is determined 
by the equation 
so that in a synclastic surface p, and p, are the maximum 
and minimum radii of curvature, but in an anticlastic 
surface, where they have opposite signs, they are the two 
minima radii. (b) The proposition that in every polyhe- 
dron (but it is not true for one which enwraps the center 
more than once) the number of edges increased by two 
equals the sum of the numbers of faces and of summits, 
(c) One of a variety of theorems sometimes referred to 
as Euler's, with or without further specification : as, the 
theorem that (xd/Ax + yA/Ay)rf(x, )' = Vfa y) ; the 
theorem, relating to the circle, called by Kuler and others 
Fermat i geometrical theorem; the theorem on the law of 
formation of the approximations to a continued fraction ; 
the theorem of the 2, 4, 8, and 16 squares ; the theorem 
relating to the decomposition of a number into four posi- 
tive cubes. All the above (except that of Fermat) are due 
to Leonhard Euler (1707 -83X Exponential theorem. 
SeeraooTUiiittai. Fagnanos theorem, a theorem given 
by Count G. C. di Fagnano (1682-1766) in 1716, now gen- 
erally quoted under the following much-restricted form: 
the difference of two elliptic arcs AA', aa', whose extremi- 
ties A and a, A' and a' form two couples of conjugate 
points, is equal to the difference of the distances from the 
center of the curve to the normals passing through the 
extremities of one of the two arcs. Fassbender's theo- 
rem, the proposition that if a, p, y are the angles the bi- 
sectors of the sides of a triangle make with those sides, 
then cot a 4 ; cot p + cot y = 0. Format's theorem, (o) 
The proposition that if p is a prime and a is prime to 
p, then a f ~ * 1 is divisible by p. Thus, taking p = 7 
and a = 10, we have 999999 divisible by 7. The following 
is commonly referred to as Format's theorem generalized : 
if a is prime to n and <n is the totient of n, or number of 
numbers as small and prime to it, then o^" 1 is di- 
visible by . This and the following are due to the won- 
derful genius of Pierre Fermat (1608-65). (6) One of a 
number of arithmetical propositions which Fermat, owing 
to pressure of circumstances, could only jot down upon 
the margin of books or elsewhere, and the proofs of which 
remained unknown for the most part during two centuries, 
and which are still only partially understood especial- 
ly the following, called the last theorem of Fermat: the 
equation x -I- y = z, where n is an odd prime, has 
no solution in integers, (c) The proposition that, if from 
the extremities A and B of the diameter of a circle lines 
AD and BE be 
drawn at right an- 
gles to the diame- 
ter, on the same 
side of it, each 
equal to the 
straight line AI or 
BI from A or B to 
the middle point 
of the arc of the 
semicircle, and if 
through any point 
C in the circumference, on either side of the diameter 
AB, lines DCF, ECG be drawn from D and E to cut AB 
(produced if necessary) in F and G, then AG a + BF 2 = AB 2 : 
distinguished as Fermats geometrical theorem. This is 
shown in the figure by arcs from A as a center through G 
and from B as a center through F meeting at H on the 
circle, (d) The proposition that light travels along the 
quickest path. Feuerbach's theorem, the proposition 
that the inscribed and three escribed circles of any tri- 
angle all touch the circle through the mid-sides : given 
in 1822 by K. W. Feuerbach (1800-34). The circle, often 
called the Feuerbach or nine-point circle, also passes 
through the feet of perpendiculars from the vertices 
upon the opposite sides and through the points midway 
between the orthocenter and the vertices. Its center bi- 
sects the distance between the orthocenter and the cen- 
ter of the circumscribed circle. Fourier's theorem, 
the theorem that every rectilinear periodic motion is re- 
solvable into a series of simple harmonic motions hav- 
ing periods the aliquot parts of that of their resultant : 
named after the French mathematician Baron J. B. J. 
Fourier (1768-1830). Fundamental theorem of alge- 
bra, the proposition that every algebraic equationhas 
a root, real or imaginary. Fundamental theorem of 
arithmetic, the proposition that any lot of things the 
count of which in any order can be terminated is such 
that the count in every order can be terminated, and 
ends with the same number. Galileo's theorem, the 
proposition that the area of a circle is a mean propor- 
tional between the areas of two similar polygons one cir- 
cumscribed about the circle and the other isoperimetrical 
with it : given by Galileo Galilei (1564-1642). Gaussian 
or Gauss's theorem, a name for different theorems re- 
lating to the curvature of surfaces, especially for the 
theorem that the measure of curvature of a surface de- 
6276 
pends only on the expression of the square of a linear 
element in terms of two parameters and their differential 
coefficients. Geber'S theorem, the proposition that in 
a spherical triangle ABO, right angled at C. if b is the leg 
opposite B, then cos B = cos b sin A : believed to have 
been substantially given by an Arabian astronomer, Jabir 
ibn Aflah of Seville, probably of the twelfth century. 
Geiser's theorem, the proposition that two forms whose 
elements correspond one to one are projective : given by 
C F. Geiser in 1870. Goldbaeh's theorem, the propo- 
sition that every even number is the sum of two primes : 
named after C. Goldbach (1690-1764), by whom it is said 
to have been given. Graves's theorem, the proposi- 
tion that a pen stretching a thread loosely tied round an 
ellipse will describe a confocal ellipse : not properly a 
theorem but an immediate corollary from a theorem by 
Leibnitz, drawn by Dr. Graves in 1841, and named after 
him as his most important achievement. Green's theo- 
rems certain theorems of fundamental importance in 
the theory of attractions, discovered by George Green 
(1793-1841). They are analytical expressions of the fact 
that the accumulation of any substance within a given 
region is the excess of what passes inward through its 
boundary over that which passes outward. Guldin'B 
theorems, two theorems expressing the superficies and 
solid contents of a solid of revolution: named after a 
Swiss mathematician, Guldin (1577-1643); but the theo- 
rems are ancient. Hachette's theorem, the proposition 
that any ruled surface has normal to it along any genera- 
tor a hyperbolic paraboloid having for directrices of its 
generators three normals to the regulus through three 
points of its given generator : given in 1832 by J. N. P. 
Hachette (1769-1834). Hauber's theorem, the logical 
proposition that if a genus be divided into species in two 
ways, and each species in one mode of division is entirely 
contained under some species in the second mode, then 
the converse also holds : given in 1829 by K. F. Hauber 
(1775-1851). Henneberg's theorem, the proposition 
that the necessary and sufficient condition that a minimal 
surface admitting a plane curve as its geodesic should be 
algebraic, is that this line should be the development 
of an algebraic curve : given in 1876 by L. Henneberg. 
Herschel's theorem, (a) The development 
Format's Geometrical Theorem. 
given in 1820 by Sir J. F. W. Herschel (1792-1872). (b) 
The proposition that forced vibrations follow the period 
of the exciting cause. Hess'S theorem, the proposition 
that the herpolhode has neither cusp nor inflection : given 
by W. Hess in 1880, and constituting an important correc- 
tion of notions previously current among mathematicians. 
See herpolhode. Hippocrates's theorem, the proposi- 
tion that the area of a lune bounded by a semicircle and 
a quadrantal circular arc curved the same way is equal 
to that of the isosceles right triangle whose hypotenuse 
joins the cusps of the lune : named from its discoverer, 
the great Greek mathematician Hippocrates of Chios. 
Holdltch's theorem, the proposition that if a rod moves 
in a plane so as to return to its first position, and if A, B, 
C are any points fixed upon it, the distances AB, BC, CA 
being denoted by c, a, b, and if (AX (BX (C) are the areas 
described by A, B, C respectively, then 
o(A) + XB) + c(C) = Trabc : 
given by the Rev. Hamnet Holditch (born 1800). Ivory's 
theorem, the proposition that the attraction of any homo- 
geneous ellipsoid upon an external point is to the attrac- 
tion of the confocal ellipsoid passing through that point 
on the corresponding point of the first ellipsoid, both at- 
tractions being resolved in the direction of any principal 
plane, as the sections of the two ellipsoids made by this 
S lane and this according to whatever function of the 
istance the attractions may vary. Jacobi's theorem. 
(a) The proposition that a function (having a finite num- 
ber of values) of a single variable cannot have more than 
two periods. (6) The proposition that an equilibrium el- 
lipsoid may have three unequal axes, (c) One of a variety 
of other propositions relating to the transformation of 
Laplace's equation, to the partial determinants of an ad- 
junct system, to infinite series whose exponents are con- 
tained in two quadratic forms, to Hamilton's equations, to 
distance-correspondences {or quadric surfaces, etc. All 
are named from their author, K. G. J. Jacob! (1804-51). 
Joachlmsthal's theorem, the proposition that if a 
line of curvature be a plane curve, its plane makes a con- 
stant angle with the tangent plane to the surface at any 
of the points where it meets it : given in 1846 by F. Jo- 
achimsthal (1818-61). Jordan's theorem, the proposi- 
tion that functions of n elements which are alternating 
or symmetrical relatively to some of them have fewer 
values than those which are not so; but this has excep- 
tions when ?i is small. Lagrange's theorem, (a) A rule 
for developing in series the values of an implicit function 
known to differ but little from a given explicit function : 
if z = x + afz, then 
theorem 
theorem. Laurent's theorem, a rule for the develop- 
ment of a function in series, expressed by the formula 
where the modulus of x is comprised between R and R': 
given by P. A. Laurent (1813-54). Legendre's theo- 
rem, the proposition that if the sides of a spherical tri- 
angle are very small compared with the radius of the 
sphere and a plane triangle be formed whose sides are 
proportional to those of the spherical triangle, then each 
angle of the plane triangle is very nearly equal to the 
corresponding angle of the spherical triangle less one 
third of the spherical excess. This is near enough the 
truth for the purposes of geodesy : given by A. M. Legendrc 
(1752-1833). Leibnitz's theorem, a proposition con- 
cerning the successive differentials of a product : namely, 
that 
d 
3 uv = (D + D*)" mi 
dXH 
is equal to the same after development of (D + Vv)" by 
the binomial theorem, where D denotes differentiation as 
if u were constant, and Dv differentiation as if v were con- 
stant. Lejeune-Dlrichlet's theorem, a proposition dis- 
covered by the German arithmetician P. G. Lejeune-Di- 
richlet (1805-59), to the effect that any irrational may be 
represented by a fraction whose denominator in is a whole 
number less than any given number n with an error less 
than mn. Lexell's theorem, one of two propositions 
expressing relations between the sides and angles of poly- 
gons: given in 1775 by A. J. Lcxell (1740-84). Lhuilier's 
theorem, the proposition that if a, 6, c are the sides of a 
spherical triangle and E the spherical excess, then 
tan 3 JE = tan i(o + 6 + c) x tan J(a + 6 - c) 
xtanj(a-o-t c) x tanJ(-<* + M c): 
given by S. A. J. Lhuilier (1750-1840). Listing's theo- 
rem, an equation between the numbers of points, lines, 
surfaces, and spaces, the cyclosis, and the periphraxis of a 
figure in space: given in 1847 by J. B. Listing. Also called 
the census theorem. Lueroth's theorem, the proposi- 
tion that a Riemann's surface may in every case be so con- 
structed that there shall be no cross-lines except be- 
tween consecutive sheets. McClintock's theorem, a 
very general expansion formula by E. McClintock. 
MacCullagh's theorem, the proposition that a trian- 
gle being inscribed in an ellipse, the diameter of its cir- 
cumscribed circle is equal to the product of the elliptic 
diameters parallel to the sides divided by the product 
of the axes : discovered by the Irish mathematician 
James MacCullagh (1809-47), and published in 1866. 
Maclaurin and Braikenridge's theorem, the propo- 
sition that n fixed points and n-1 fixed lines in one plane 
being given, the locus of the vertex of an n-gon whose 
other vertices lie on the fixed lines while its sides pass 
through the fixed points is a conic : given by Colin Mac- 
laurin and G. Braikenridge in 1735. Maclaurin's gen- 
eral theorem concerning curves, the proposition that 
if through any point O a line be drawn meeting a curve in 
n points, and at these points tangents be drawn, and if any 
other line through O cut the curve in R, R', R", etc., and 
the system of n tangents in r, r', r", etc.. then the sum of 
the reciprocals of the lines OR is equal to the sum of the 
reciprocals of the lines Or. Maclaurin's theorem, a 
formnla of the differential calculus, for the development 
of a function according to ascending powers of the vari- 
able : named after the Scotch mathematician Colin Mac- 
laurin (1698-1746). It is an immediate corollary from Tay- 
lor's theorem, and is written 
(6) The proposition that the order of a group is divisible 
by that of every group it contains : also called the fun- 
damental theorem of substitutions. Both by J. L. Lagrange 
(1736-1813). Lambert's theorem, (a) The proposition 
that the focal sector of an ellipse is equal to 
Area ellipse 
- 
, where 
- 
- F"0.*3 
Malus's theorem, the law of double refraction : given 
in 1810 byE. L Malus (1775-1812). Mannheim's theo- 
rem. Same as Schonemann's theorem (which see, below). 
Mansion's theorem. Same as Smith's theorem (which 
see, below). Matthew Stewart's theorem, one of 
sixty-four geometrical propositions given in 1746 by 
the philosopher Dugald Stewart's father (1717-85), es- 
pecially that if three straight lines drawn from a point 
O are cut by a fourth line in the points A, B, C in or- 
der, then (OA)'BC - (OB)-AC + (OC)"AB = AB. BC. CA. 
Menelaus's theorem, the proposition that if a triangle 
QRS is cut by a transversal in C, A, and B, the product of 
the segments QA, RB, SC is equal to the product of the 
segments SA, QB, RC : given by the Greek geometer Mene- 
laus, of the first century. Meusnier's theorem, the 
proposition that the radius of curvature of an oblique sec- 
tion of a surface is equal to the radius of curvature of the 
normal section multiplied by the cosine of the inclination 
to the normal : given in 1775 by J. B. M. C. Meusnier de 
la Place (1754-93). Minding's theorem, a certain prop- 
osition in statics. Miguel's theorem, the proposition 
that if five straight lines and five parabolas are so drawn 
in a plane that each of the latter is touched by four of the 
former, and vice versa, then the foci of the parabolas lie on a 
circle : given by A. Miquel. Mittag-Leffler's theorem, 
the proposition that if any series of isolated imaginary 
quantities, a,,, a , , . . . a,,, etc. , be given, and a correspond- 
ing series of functions, ij/ , ^i, >("', etc '> ' tne form 
"in }*=! v/(r+ r'+c)/a, and sin ix'=Jl / ( 1 " + '"' -")/> 
r and rl being the focal radii of the extremities, c the 
chord, and a the semiaxis major, (b) A proposition re- 
lating to the apparent curvature of the geocentric path of 
a comet. Both are named from their author, J. H. Lambert 
(1728-77). Lancret's theorem, in solid geometry, the 
proposition that along a line of curvature the variation 
in the angle between the tangent plane to the surface and 
the osculating plane to the curve is equal to the angle 
between the two osculating planes. Landen's theorem, 
the proposition that every elliptic arc can be expressed 
by two hyperbolic arcs, and every hyperbolic arc by two 
elliptic arcs: given in 1755 by John Landen (1719-90). 
Laplace's theorem, a slight modification of Lagrange's 
a monodromic function /z can always be found having for 
critical points t , o,, . . . , etc., and such that 
<t>n being a function for which a is not a critical point : 
given byG. Mittag-Leffler. Multinomial theorem. See 
multinow ial. Newton's theorem, (a) The proposition 
that if in tile plane of a conic two lines be drawn through 
any point parallel to any two fixed axes, the ratio of the 
products of the segments is constant: given by Sir Isaac 
Newton (1642 - 1726) in 1711. (6) The proposition that the 
three diagonals of a quadrilateral circumscribed about a 
circle are all bisected by one diameter of the circle. 
Painvin's theorem, the proposition that a tetrahedron 
