theorem 
of which a vertex is pole of the opposite base relatively 
tii a uuudric Hiirfucc, t lint bane being a conjugate ti i:mnk' 
irh.tivu to its section "f the i[ti;uliir, (M a rmijiiKiit' 1 ti'ii;i 
hetlron. Pappus's theorem. ("t I in- pmiHihition iii.it 
If a qiiudntiiKlu is Inscribi'tl in a < unjr, tin- |n<><ln< t <>f the 
distances of any point on the curve from one pair of op- 
posite sides in to the product of its distances from nn- 
utlicr such p;iit' in a constant mtjo : BO ciillt.-d owing to Ita 
connection with 1'nppua'B problem. (/>) One of the two 
propositions that the Hurface of a solid of revolution is 
ccmal to the product <-f tin* perimeter "f the Kenerntintf 
plane figure by the U'litfth of tho path described by the 
center of gravity, and that the volume of such a solid it* 
rr,u:il to the ari'ft of the plane HKiir multiplied by the 
mime length of path. Various other theorems contained 
In Mir collection of the Greek mathematician Pappus, of 
the third century, are Home-times called by his name. 
Particular theorem, a theorem which extends only to a 
particular quant ity. -Pascal's theorem., the proposition 
that the three intersections of pairs of opposite sides of a 
hexagon inscribed in a conic lie on a straight line: given 
by Uluise Pascal (1623-62) in 1640. The hexagon itself is 
called a rattcat'n hexagon or hexagram, and the straight 
line is called a Pascal'* line.- Ptcard's theorem. () 
The proposition that every function which In the whole 
plane of imaginary quantity except in /' straight lines is 
uniform and continuous, la equal to the sum of /> uniform 
functions, each of which has but one such line. (6) A cer- 
tain proposition concerning uniform functions connected 
by an algebraic relation. Pohlke'fl theorem, the prop- 
osition that any three limited straight lines drawn in a 
plane from one point form an oblique parallel projection 
of a system of three orthogonal and equal axes : given by 
II. K. 1'ohlke in 1853. Also known aa the fundamental thto- 
rein of axtmometry. Polsson's theorem, a rule for form- 
Ing Integrals of a partial differential equation from two 
given integrals. Polynomial theorem. Svtpolynwnial. 
Poncelet'S theorem, (a) The proposition that if there 
be a closed polygon Inscribed In a given conic and circum- 
scribed about another given conic, there Is an infinity of 
such polygons, (b) The proposition that a quantity of the 
forniR = Vu*-' 4- 1?' 1 * can not differ from aw + fin by more than 
Rtan* |, where a = cos(0 + c)/cot* 4v = sin (0 + cVcot' K 
- j{w 0), tan ** > a/a > tan 9. Both were given by Gen- 
eral J. V. Poncelet(l788-1877). Ptolemy's theorem, the 
proposition that if four points A, B, C, D lie on a circle 
In this cyclical order, then AB. CD + AD. BC = AC. DB. : 
given by the Egyptian (ireek mathematician of the second 
century, Claudius Ptolemy. Pulseux's theorem, the 
proposition that a function of a complex variable which 
is thoroughly uniform and satisfies an algebraic equation 
whose coefficients are rational integral functions of the 
same variable, is a rational function of that variable : 
named after V. A. Pulseux (1S2O-83X by whom it was 
given In 1861. Pythagorean theorem, the Pythagorean 
proposition (which see, under Pythagorean), Recipro- 
cal theorem, a theorem of geometry analogous to an- 
other theorem, but relating to planes instead of points, 
and vice versa, or In a plane to straight lines instead of 
points, and vice versa. Thus, Pascal's and Brlanchon's 
theorems are reciprocal to one another. Ribaucour's 
theorem, given a pseudospherical surface of unit curva- 
ture, if in every tangent plane a circle of unit radius be 
described about the point of contact as center, these cir- 
cles will be orthogonal to a family of pseuaospherlcal 
surfaces of unit radius belonging to a triple orthogonal 
system of which the other two families are envelops of 
spheres: given by A. Ribaucour in 1870. Riemann's 
theorem, a certain theorem relative to series of corre- 
sponding points for example, that two protective series 
of points He upon curves of the same deficiency. In it - 
generality the proposition is called the theorem yf Rie- 
m /.in and Koch, or of Riemann, Roch, and Nother. It was 
first given by G. F. B. Rlemann (1823-67) In 1857, generally 
demonstrated by Koch in 1865, and extended to surfaces 
by Nother in 1880. Robert's theorem, (a) The propo- 
sition that the geodesies joining any point on a quadric 
surface to two umbilics make equal angles with the lines 
of curvature at that point: given, with various other 
propositions relating to the asymptotic lines ami lines of 
curvature of uuadrics, by Michael Roberts in 1846. (6) The 
proposition that if a point be taken on each of the edges 
of any tetrahedron and a sphere be described through each 
vertex and the points assumed on the three adjacent edges, 
the four spheres will meet in a point: given by Samuel 
Roberts in 1881. Rodrigues's theorem, the proposition 
that 
t "~ M *+"* 
to 2 iv* 
~ l 
6277 
numbers at least aa small as p and prime to It : given in 
1876 by the eminent Irish m;ttht m;itician II. J. s. Smith 
*l). The theorem as generalized by I'aul Mansion 
in 1877 Is called Smith and Mansum't theorem. Stall (it s 
theorem, tin [H"p<>-itin that any Bernoulli number, I!.,, 
is equal to an integer minus 
2-' I a-'+0-'+ . A-', 
where a, 0, etc., are all the prime numbers one greater 
than the double of divisor* of n: given In 1M" by K. . 
c. von staudt (1788-187).- Steiner's theorem, one of a 
large number of propositions In geometry Riven by Jakob 
Mrim-r (17IK1- 18&H), who was probably the greatest geo- 
metrical genius that ever lived ; but the necessities of 
life prevented the publication of by far the greater part 
of his discoveries, until his health was shattered, and most 
of those that were printed (in 1820 and the following years) 
were given without proofs, and remained an enigma to 
mathematicians until 1862, when I.ulul Cremona demon- 
strated most of them. Stirling's theorem, the prop- 
osition that 
given by James Stirling (1690- 1770). Sturm's theorem, 
a proposition in the theory of equations for determining 
the number of real rooU of an equation between given 
limits : given by the French mathematician J. C. K. Sturm 
(1803 - 66) in 1835. Sylow's theorem. Sec Conchy''. theo- 
rem((>), above. Sylvester's theorem, (a) An extension 
of Newton's rule on the limits of the roots of an algebraic 
equation, tin The proposition that every quaternary cubic 
Is the sum of the cubes of five linear forms, (e) The prop- 
osition that if A,, A 9 , etc., are the latent roots of a matrix 
in, then 
given by the great algebraist J. J. Sylvester (born 1814). 
Tanner's theorem, a property of pfafflans, 
given by H. U. L. Tanner In 1879. Taylor's theorem, 
a formula of most extensive application in analysis, dis* 
covered by Dr. Brook Taylor, and published by him in 1715. 
It is to the following effect : let u represent any function 
whatever of the variable quantity x; then If x receive any 
increment, as A, let u become '; then we shall have u' = 
du_ A d-u A^ d'u A' du A' 
Hx ' I + ~axf ' i! + ~dx* TsT + "d* 1 " ' fH* + 
where d represents the differential of the function u. 
Theorem of aggregation. See aygregatian. Uni- 
versal theorem, a theorem which extends to any quan- 
tity without restriction. Wallis's theorem, the prop- 
osition that 
ir/2 = (2',f3'X(4 1 /5').(6 1 /7').(8'/(>'X etc., 
named after the discoverer, John Wallls (1616-1708). 
Weierstrasa's fundamental theorem, the proposition 
that every analytical function subject to an addition 
theorem is either an algebraic function, or an algebraic 
function of an exponential, or an algebraic function of the 
Weierstrasslan function <> : given by Karl Weierstrass 
(born 1816).- Weingarten'g theorem. See Bettft theo- 
rem, above. Wilson's theorem, the proposition that If 
p is a prime number, the continued product 1.2.3. . . 
( /' 1) increased by 1 is divisible by p, and if not, not : 
discovered by Judge John Wilson (1741-93), and published 
by Waring. Wronskl's theorem, an expansion for a 
function of a root of an equation. Yvon-Villarceau's 
theorem, a general proposition of dynamics, expressed 
by the formula 
Rolle's theorem, the proposition that between any two 
real roots of an equation, algebraic or transcendental, if 
the first derived equation is finite and continuous in the 
interval, It must vanish an odd number of times : given 
iniosflby Michel Uolle (1662-1719). Scherk's theorem, 
the proposition that the Eulerian numbers iti Arabic no- 
tation end alternately with 1 and fi. Scho' nemann s 
theorem, the proposition that if four points of a rigid 
body slide over four fixed surfaces, all the normals to sur- 
faces that are loci of other points of the body pass through 
two fixed straight lines: published under Steiner's aus- 
pices in 1855, but not noticed, and rediscovered by A. 
Mannheim in I860 (whence long called Mannheim'* the- 
orem); but Schoncmann's paper was reprinted in Bor- 
< hanlt's Journal in 1880. Slonlmsky's theorem, the 
proposition that if the successive multiples of a number 
expressed in the Arabic notation are written regularly 
under one another, there are only 28 different columns of 
figures whicli have to be added to the last figures of the 
successive multiples of a digit to get the numbers written 
in any n ninil n liiuni. - Sluze's theorem, the proposi- 
tion that the volume of the solid generated by the revo- 
lution of a common cisaoid about its asymptote is equal 
to the volume of the juiehor-riiiK' Kent-rated by the revolu- 
tion of the primitive circle about the same axis. This 
theorem, which is true for any kind of cissoid. and is sus- 
ceptible of further ranenllation, was given in Ides by the 
Baron de sluze (162 - sr). Smith's theorem the propo- 
sition that S (1, 1) (2, 2) ... (n, n) = 41. *i . . . n, 
where tin- left-hand side is a symmetrical determinant, 
(p, q) denoting the greatest common divisor of the Inte- 
gers p and q, and ./.p being the totient of p, or number of 
where c is the velocity, r the radius vector of the point 
whose mass Is m and Its coordinates x, y. z, while X, Y Z 
are the components of the force, /the force, and A the 
distance of two particles : given in 1872 by A. J. F. YTon- 
Villarceau (1813-83). It much resembles the theorem 
of the virial. = Syn. See inference. 
theorem (the'o-rem), v. t. [< theorem, M.] To 
reduce to or formulate as s theorem. [Bare.] 
To attempt theorising on such matters would profit lit- 
tle ; they are matters which refuse to be theoremed and 
diagramed, which Logic ought to know that she cannot 
speak of. Carlyte. 
theorematic (the'o-re-mat'ik), a. [< Gr. 6eu- 
ptlftartK^, of or pertaining to a theorem, < 6cu- 
prifia, a theorem: see theorem.] Pertaining to 
a theorem ; comprised in a theorem ; consisting 
of theorems: as, theorema tic truth. 
theorematical (the'o-re-mat'i-kal), a. [< theo- 
rematic + -/.] Same as theorematic. 
theorematist (the-o-rem'a-tist), n. [< Gr. Oeu- 
ptlfia(T-), a theorem, + -is<.] One who forms 
theorems. 
theoremic (the-o-rem'ik), a. [< theorem + -c.] 
Theorematic. 
theoretic (the-o-ret'ik), a. and n. [= F. theo- 
rttique, < NL. "theoretical, < Gr. deuprrrutdf, of or 
pertaining to theory, < ffcupia, theory: see the- 
<iry.~} I, a. Same as theoretical. 
For, spite of his fine theoretic positions. 
Mankind is a science defies definitions. 
Burnt, Fragment Inscribed to C. J. Fox. 
II. M. Same as theoretics. .*>'. //. //</</#, 
Time and Space, $ 68. [Rare.] 
theoretical (the-o-ret'i-kal), a. [< theoretic + 
-nl.] 1. Having the object of knowledge (0cu- 
prrr6) as its ciui: concerned with knowledge 
only, not with accomplishing anything or pro- 
ducing anything; purely scientific; speculative. 
theoricon 
This Is the original, proper, and best meaning of the word. 
Aristotle divides nil knowledge into productive tart) and 
unproductive (xcience), and the latter Into that which alms 
at accomplishing something (practical science) and that 
which alms only at understanding Its object, which is the. 
ontical tcicnrc. Thin distinction, which has descended to 
-in times (but with practical science and art joined toge- 
ther), diminishes in Importance as science advances, all 
the sciences finding practical applications. 
Weary with the pursuit of academical studies, he I('.,l 
lins) no longer confined himself to the search of theoreti- 
cal knowledge, but commenct-d, the scholar of humanity, 
to study nature In her works, and man in society. 
Langhorne, On Collins's Ode, The Manners. 
2. Dealing with or making deductions from im- 
perfect theory, and not correctly indicating the 
real facts as presenting themselves in experi- 
ence. All the practical sciences that have been pursued 
with distinguished success proceed by deductions from 
hypotheses known not to be strictly true. This Is the ana- 
lytical method, of which modern civilization is the fruit. 
In some cases the hypotheses are so tar from the truth that 
the results have to receive corrections. In such cases the 
uncorrected result is called theoretical, the corrected re- 
mit practical. 
What logic was to the philosopher legislation was to 
the statesman and moralist, a practical, as the other was 
a theoretical, casuistry. 
StuNu, Medieval and Modem Hist, p. 211. 
3. In Kantian terminology, having reference 
to what is or is not true, as opposed to practi- 
cal, or having reference to what ought or may 
innocently be done or left undone. -Theoretical 
agriculture, arithmetic, chemistry. See the nouns. 
Theoretical cognition, cognition either not In the Im- 
perative mood or not leading to such an imperative; 
knowledge of what the laws of nature prescribe or admit, 
not of what the law of conscience prescribes or permits. 
Theoretical geometry. See geometry. Theoretical 
Intellect. See intellect, 1. Theoretical logic. Same 
as abstract logic (which see, under l<*jic). Theoretical 
meteorology, philosophy, proposition, reality, rea- 
son, etc. See the nouns. 
theoretically (the-o-ret'i-kal-i), adr. In a the- 
oretic manner ; in or by theory ; from a theoret- 
ical point of view ; speculatively : opposed to 
practically. 
theoretician (the'p-re-tish'an), n. [< theoretic 
+ -ian.J A theorist; a theorizer; one who is 
expert in the theory of a science or art. 
theoretics (the-o-ret'iks), n. [PI. of theoretic 
(see -tcs). ] The speculative parts of a science. 
With our Lord himself and his apostles, as represented 
to us In the New Testament, morals come before contem- 
plation, ethics before theoretic*. H. B. Wilton. 
theoric 1 ! (the'o-rik), a. and n. [I. a. = F. the- 
orique = Sp. teorico = Pg. theorico = It. tcorico, 
< ML. theoriciis, < Gr. BeapiKof, of or pertaining 
to theory, < Bcupla, theory: see theory. II. w. 
Also theorick, thenrique, < ME. theorik, theorike, < 
OF. theorique, F. theorique = Sp. teorica = Pg. 
theorica = It. teorica, < ML. thcorica (sc. ars), < 
Gr. Beu/Mnof, of or pertaining to theory: see I.] 
1. a. Making deductions from theory, especially 
from imperfect theory; theorizing. Also (Aeon- 
cat. 
Your courtier theoric is he that hath arrived to his 
farthest, and doth now know the court rather by specula- 
tion than practice. I!. Jonson, Cynthia's Revels, If. 1. 
A man but young, 
Yet old In judgment ; theoric and practlc 
In all humanity. 
ilatrinycr and Field, Fatal Dowry, 11. 1. 
II. n. 1. Theory; speculation; that which 
is theoretical. 
The bookish theoric, 
Wherein the toged consuls can propose 
As masterly as he ; mere prattle, without practice, 
Is all his soldiership. Shot., Othello, I. 1. 24. 
An abstract of the theorick and practlck In the /Escula- 
plan art. B. Jonton, Volpone, II. 1. 
2. A treatise or part of a treatise containing 
scientific explanation of phenomena. 
The 4 pin tie shal ben a theorik to declare the moevynge 
of the celestial bodies with the causes. 
Chaucer, Astrolabe, Prol. 
theoric 2 (the-or'ik), a. [< Gr. Oeuptxuf. of or per- 
taining to public spectacles, ra fcupua, or rb 8ru- 
ptxov, the theoric fund (< ffeupia, a viewing: see 
theory. Cf. theoric 1 ).] Of or pertaining to 
public spectacles, etc Theoric fund. In Athenian 
antiq. , same as theoricon. 
theoricalt (the-or'i-kal), a. [< Uteorici + -a/.] 
Same as theoric 1 . 
I am sure wisdom hath perfected natural disposition In 
you, and given you not only an excellent theoricaf discourse, 
but an actual reducing of those things Into practice which 
are better than you shall find here. 
Rev. T. Admnt, Works, III., p. xlL 
theoricallyt (the-or'i-kal-i), adv. Theoretically ; 
speculatively. 
He is very musicsll, both theoricatty and practically, 
and he had a sweet voyce. 
Aubrey, Lives (William Holder). 
theoricon (the-or'i-kon), H. [< Gr. Ocuput6i>, 
neut. of Bcupixof, of or pertaining to public 
