Mas singer, Emperor of the East, i

2. 2. In geom ., a demonstrable theoretical proposition. 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 observed 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 proposition, and as such substantially a statement that something is impo ible, while the kind of proposition called in geometry a problem is a statement that something is po ible; 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. R. Hamilton . Abel's theorem, the proposition that if we have several functions whose derivatives can be roots of the same algebraic equation having all its coefficients rational functions of one variable, we can always expre the sum of any number of such functions as the sum of an algebraic and a logarithmic function, provided we establish between the variables of the functions in question a certain number of algebraic relations: named after Niels Henrik Abel (1802-29), who first published it in 1826. - Addition theorem, a formula for a function of a sum of variables, such as sin (a + b) = sin a cos b + cos a sin b . Arbogast's theorem, a rule for the expansion of functions of functions, given in 1800 by L. F. A. Arbogast (1759-1803).- Aronhold's theorem, one of a number of propositions constituting the foundations of the theory of ternary cubics, given in 1849 by S. H. Aronhold (born 1819), the founder of modern algebra. - Bayes's theorem, the 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 po ible causes. uses. This fallacious rule was given by Rev. Thomas Bayes in 1763. - Becker's theorem, the proposition 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 G. F. Becker in 1886. -Beltrami'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 escribed circles. -Berger's theorem, one of a number of theorems relating to the limiting values of means of whole numbers, given by A. Berger in 1880. One of these theorems is that for n = the average sum of the divisors of n is Bernoulli's theorem. ( a ) The doctrine that the relative frequency of an event in a number of random trials tends 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 (1654-1705). (b) 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 pis the pre ure, p the density, V the potential of the forces, q the resultant velocity, A a certain quantity constant along a streamline, then dp S+V+q2 = A: IMG:content-0222.png:[ocr errors] given by Daniel Bernoulli (1700-82) in 1738.- Bertrand's theorem, the proposition that when a dynamical system receives a sudden impulse the energy actually acquired exceeds the energy by any other motion consistent with the conditions of the system and obeying the law of energy, by an amount equal to the energy of the motion which must be compounded with the supposed motion to produce the actual motion: an extension of a known proposition, given by J. L. F. Bertrand (born 1822).Betti's theorem, the proposition that the loci of the points of a surface for which the sum on the one hand and the difference on the other of the geodetic distances of two fixed curves on the surface are constant form an orthogonal system: given by E. Betti in 1858, and by J. We in gar ten in more general form in 1863. - Bézout's theorem, the 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. Bézout (1730-83) in 1779.Binet's theorem. (a) The proposition that the principal 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. See bino mial .-Bitonti's theorem, one of certain metrical theorems regarding the intersections of conics demonstrated by V. N. Bitonti in 1870.- Boltzmann's theorem, the proposition, proved by L. Boltzmann in 1868, that the mean living force of all the particles of a mixed gas will come to be the same. - Boole's theorem, the expansion (x+h)-(x) = B2(22-1)2! { $ ' (x+h)+' (x)} -B(24-1)4! {$ " ' ( x + h ) + $ " '( x )} +B.(2-1)6! { $v ( x +h) + (x)}-..., given by the eminent English mathematician George Boole (1815-64). - Bour's theorem, the proposition that helicoids are deformable into surfaces of revolution: given in 1862 by the French mathematician J. E. E. Bour (18321866). - Brian chon'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. Brian chon (born 1785, died after 1823) in 1806. It was the earliest application of polar reciprocals. - Budan's theorem, the proposition that if the roots of an algebraic equation are diminished 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 coefficients in pa ing from one to the other: given and demonstrated in 1811 by the French mathematician Budan.-Bürmann's theorem, a formula for developing one function in terms of another, by an application of Lagrange's theorem. - Cagnoli's theorem, in spherical trigon., the formula for the sine of half the spherical exce in terms of the sides: given by the Italian astronomer Andrea Cagnoli (1743-1816). - Cantor's theorem, the proposition that if for every value of a greater than a and le than b the formula holds that limit (An sin na + Bn cos nx) = 0, then also limit An = 0 and limit Bn = 0: given by G. Cantor in 1870.-Carnot's theorem. (a) The proposition that if the sides of a triangle ABC (produced if nece ary) cut a conic, AB in C' and C", AC in B' and B", BC in A' and A", then ΑΒ' × ΑΒ ́ × BC x BC" X CA' X CA" = CB' × CB" × BA' X BA" X AC × AC". (b) The proposition that in the impact of inelastic bodies vis viva is always lost. (c) The proposition that in explosions vis viva is always gained. These theorems are all due to the eminent mathematician General L. N. M. Carnot (1753-1823), who published ( a ) in 1803 and (b) and (c) in 1786. (d) The proposition that the ratio of the maximum 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 (1796-1832): often called Carnot's principle . - Casey's theorem, the proposition that if S1 = 0, S = 0, S3 = 0 are the equations of three circles, and if 11, 12, 1, are respectively the lengths of the common tangents from contact to contact of the last two, the first and last, and the first two, then the equation of a circle which touches all three circles is VIS + V1282 + VIS =0: 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 proce go through its whole cycle of values as many times as it has zeros or roots within that contour. (b) 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 divisible by a power of a prime, it contains a group whose order is that power - is called Cauchy and Sylow's theorem, or simply Sylow's theorem , because proved by the Norwegian L. Sylow in 1872. (c) The rule for the development of determinants according to binary products of a row and a column. (d) The false proposition that the sum of a convergent series whose terms are all continuous functions 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 Cavendish (1731-1810). - Cayley's theorem, the proposition that every matrix satisfies an algebraic equation of its own order: also called the principal proposition of ma trices: given by the eminent English mathematician Arthur Cayley. Cesaro's theorem, the proposition that if the vertices A, B, C of one triangle lie respectively on the sides (produced if nece ary) B'C', C'A', A'B' of a second triangle, which sides cut the sides of the first triangle in the points A", B", C" respectively, and if S be the area of the first triangle, S' that of the second, then IMG:content-0223.png:[blocks in formation] theorem the product AB' × BC × CA': given by Giovanni Ceva in 1678.- Chasles's theorem, the proposition that of a unidimensional family of conics in a plane the number which satisfy a simple condition is expre ible in the form αμ + βν , where a and ẞ depend solely on the nature of the condition, while is the number of conics of the family pa ing through an arbitrary point, and is the number touched by an arbitrary line: given in 1864 by M. Chasles (1793-1880) 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 compre ion or ellipticity is equal to the ratio of the equatorial centrifugal force le the exce of polar over equatorial gravity to the mean gravity: given in 1743 by Alexis Claude Clairaut (1713-65). - Clapeyron's theorem, the proposition that if a portion of a horizontal beam supported at three points A, B, C has uniform loads w1 and w2 on the parts AB and BC respectively, the lengths of which are respectively l, and 12, and if a, β, y are the bending moments at the three points of support, then al1 + 26(11 + 1) + 2 = (w1l + w2): given by B. P. E. Clapeyron (1799-1868): otherwise called the theorem of three moments. - Clausen's theorem. Same as Staudt's theorem. - Clausius's theorem, the proposition that the mean kinetic energy of a system stationary motion is equal to its given J. E. Clausius (born 1822) in 1870: otherwise called the theorem of the virial. - Clebsch's theorem, the proposition that a curve of the nth order with (n-1) (n-2) double points is capable of rational parametric expre ion: 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). - Coriolis's theorem, the kinematical proposition that the acceleration of a point relative to a rigid system is the resultant of the absolute acceleration, the acceleration of attraction, and the acceleration of compound centrifugal force: named from its author, G. G. Coriolis (1792-1843). - Cotesian theorem. Same as Cotes's properties of the circle (which see, under circle). Coulomb's theorem, the proposition that when a conductor is in electrical equilibrium the whole of its electricity is on the surface: given by C. A. Coulomb (17361806). - Crocchi's theorem, the proposition that if denotes what (x1+X2 + + am ) becomes when the coefficients of the development are replaced by unity, and if sp = x + x + x ++x, then IMG:content-0224.png:[ocr errors][ocr errors][ocr errors][ocr errors][merged small][ocr errors] given by Morgan W. Crofton in 1868. Certain symbolic expansions and a proposition in least squares are also so termed.-Culmann's theorem, the proposition that the corresponding sides of two funicular polygons which are in equilibrium under the same system of forces cut one another on a straight line. - D'Alembert's theorem, the proposition that every algebraic equation has a root: named from Jean le Rond d'Alembert (1717-83). See also D'Alem bert's principle , under principle . - Dandelin's 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 G. P. Dandelin (1794-1847), who gave it in 1827; but 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 contact is a stereographic projection of a spherical conic is by Dandelin. - Darboux's theorem, the proposition that if y is a function of z having superior and inferior limits within a certain interval of values of x, and if this interval is cut up into partial intervals Io, I... Ir , in which the largest values of y are respectively Mo, M.,.. Mk, then ΣΜΙ will tend toward a fixed limit as the number of intervals is increased, without reference to the mode of di ection: named from its author, J. G. Darboux. - De Moivre's theorem. (a) The proposition that (cos + i sin 0) = cos ηθ + i sin ηθ : better called De Moivre's formula . (b) Same as De Moivre's property of the circle (which see, under circle ). (c) A certain proposition in probabilities. All these are by Abraham De Moivre (1667-1754). - Des argues's theorem. (a) The proposition 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'B'C' are so placed that the three straight lines through corresponding vertices meet in a point, then also the three points of intersection of corresponding sides (produced if nece ary) lie in one straight line, and conversely. Both were discovered by Gérard Des argues (1593-1662). - Des cartes's theorem. Same as Des cartes's rule of signs (which see, under rule1 ). - Diophantus's theorem, the proposition that no sum of three squares of integers is a sum of two such squares: given by a celebrated Greek arithmetician, probably of the third century. Dostor's theorem, the proposition that in a plane triangle, where b , e are two of the sides, A the angle included between them, and & the inclination of the bisector of this angle to the side opposite, b+c tan 8 = tanA; b-c 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 fa is a function of limited variation between a = A and a = B, and if (a, n) is such a function that få (a, n)da (where b is any number between A and B) has its modulus le than a fixed quantity independent of band of n, and that when n increases indefinitely the integral tends toward a fixed limit G for all values of b between A and B, then få fa. (a, n)da will tend uniformly to Gf (A + 0) if B> A, and to Gf (A-0) if B the algebraic equation f( x , y) = 0 is developed in powers of x, the coefficients, reduced to their lowest terms, have a finite number of factors in the denominator: given in 1852 by F. G. M. Eisenstein (1823-52). - Euler's theorem. (a) The proposition that at every point of a surface the radius of curvature p of a normal section inclined at an angle o to one of the principal sections is determined by the equation 1/p=cos2 0 (1/p1) + sin20(1/22); so that in a synclastic surface p1 and p2 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 poly he 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/dx + yd/dy) f(x, y)" = nr f(x, y)"; the theorem, relating to the circle, called by Euler and others Fermat's 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 positive cubes. All the above (except that of Fermat) are due to Leon hard Euler (1707-83). - Exponential theorem. See exponential . - Fagnano's theorem, a theorem given by Count G. C. di Fagnano (1682-1766) in 1716, now generally quoted under the following much-restricted form: the difference of two elliptic arcs AA', aa ' , whose extremities 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 pa ing through the extremities of one of the two arcs.- Fa bender's theorem, the proposition that if a, ß , y are the angles the bisectors of the sides of a triangle make with those sides, then cot a + cot 3 + cot y = 0.- Fermat's theorem. (a) The proposition that if p is a prime and a is prime to p , then ap -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 Fermat's theorem generalized: if a is prime to n and on is the totient of n, or number of numbers as small and prime to it, then ab " -1 is divisible by n. This and the following are due to the wonderful genius of Pierre Fermat (1608-65). (b) One of a number of arithmetical propositions which Fermat, owing to pre ure 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 - especially the following, called the last theorem of Fermat : the equation x + yn = zn, 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 angles to the diameter, 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 Fermat's Geometrical Theorem. C in the circumference, on either side of the diameter AB, lines DCF, ECG be drawn from Dand E to cut AB (produced if nece ary) in F and G, then AG2 + BF2 = AB2: distinguished as Fermat's 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 triangle 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 pa es 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 bisects the distance between the orthocenter and the cen ter of the circumscribed circle. Fourier's theorem, the theorem that every rectilinear periodic motion is resolvable into a series of simple harmonic motions having periods the aliquot parts of that of their resultant: named after the French mathematician Baron J. B. J. Fourier (1768-1830). - Fundamental theorem of algebra, the proposition that every algebraic equation has 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 proportional between the areas of two similar polygons one circumscribed about the circle and the other isoperimetrical with it: given by Galileo Galilei (1564-1642). -Gau ian or Gau 's theorem, a name for different theorems relating to the curvature of surfaces, especially for the theorem that the measure of curvature of a surface de pends only on the expre ion 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 ABC, right-angled at C. if b is the leg opposite B, then cos B = cos bsin 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.- Goldbach's theorem, the proposition 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 proposition 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 theorems, certain theorems of fundamental importance in the theory of attractions, discovered by George Green (1793-1841). They are analytical expre ions of the fact that the accumulation of any substance within a given region is the exce of what pa es inward through its boundary over that which pa es outward. - Guldin's theorems, two theorems expre ing the superficies and solid contents of a solid of revolution: named after a Swi mathematician, Guldin (1577-1643); but the theorems are ancient. - Hachette's theorem, the proposition that any ruled surface has normal to it along any generator 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 IMG:content-0225.png:[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors] where the modulus of x is comprised between R and R': given by P. A. Laurent (1813-54). - Legendre's theorem, the proposition that if the sides of a spherical triangle 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 le one third of the spherical exce . This is near enough the truth for the purposes of geodesy: given by A. M. Legendre (1752-1833).- Leibnitz's theorem, a proposition concerning the succe ive differentials of a product: namely, that IMG:content-0226.png:[blocks in formation] is equal to the same after development of (Du + Do)" by the binomial theorem, where Du denotes differentiation as if u were constant, and De differentiation as if u were constant. Lejeune-Dirichlet's theorem, a proposition discovered by the German arithmetician P. G. Lejeune-Dirichlet (1805-59), to the effect that any irrational may be represented by a fraction whose denominator m is a whole number le than any given number n with an error le than mn.-Lexell's theorem, one of two propositions expre ing relations between the sides and angles of polygons: given in 1775 by A. J. Lexell (1740-84). - Lhuilier's theorem, the proposition that if a , b , c are the sides of a (1775-1851). Henneberg's theorem, the proposition spherical triangle and E the spherical exce , then that the nece ary 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 2.2 fex = f1+(1+1)+(1+1) 0이 x given in 1820 by Sir J. F. W. Herschel (1792-1872). (b) The proposition that forced vibrations follow the period of the exciting cause. - He 's theorem, the proposition that the herpolhode has neither cusp nor inflection: given by W. He in 1880, and constituting an important correction of notions previously current among mathematicians. See herpolhode. - Hippo crates's theorem, the proposition that the area of a lune bounded by a semicircle and a quadrantal circular are 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 Hippo crates of Chios. Holditch's theorem, the proposition that if a rod moves in a plane so as to return to its first position, and if A, B, Care any points fixed upon it, the distances AB, BC, CA being denoted by c, a , b , and if (A), (B), (C) are the areas described by A, B, C respectively, then a (A)+ b (B) + c(C) = παbe : given by the Rev. Hamnet Holditch (born 1800). -Ivory's theorem, the proposition that the attraction of any homogeneous ellipsoid upon an external point is to the attraction of the confocal ellipsoid pa ing through that point on the corresponding point of the first ellipsoid, both attractions being resolved in the direction of any principal plane, as the sections of the two ellipsoids made by this plane-and this according to whatever function of the distance the attractions may vary. - Jacobi's theorem. (a) The proposition that a function (having a finite number of values) of a single variable cannot have more than two periods. (b) The proposition that an equilibrium ellipsoid 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 adjunct system, to infinite series whose exponents are contained in two quadratic forms, to Hamilton's equations, to distance-correspondences for quadric surfaces, etc. All are named from their author, K. G. J. Jacobi (1804-51). - Joachimsthal's theorem, the proposition that if a line of curvature be a plane curve, its plane makes a constant angle with the tangent plane to the surface at any of the points where it meets it given in 1846 by F. Joachimsthal (1818-61). - Jordan's theorem, the proposition 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 exсерtions when n 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 IMG:content-0227.png:[blocks in formation] tan2 E = tan( a + b + c) x tan ( a + b -c) x tan ( a - b +c) x tan(a+b+c): given by S. A. J. Lhuilier (1750-1840). - Listing's theorem, 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 proposition that a Riemann's surface may in every case be so constructed that there shall be no cro -lines except between consecutive sheets. - McClintock's theorem, a very general expansion formula by E. McClintock.MacCullagh's theorem, the proposition that a triangle being inscribed in an ellipse, the diameter of its circumscribed 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 1855.Maclaurin and Bra i ken ridge's theorem, the proposition 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 pa through the fixed points is a conic: given by Colin Maclaurin and G. Bra i ken ridge in 1735.- Maclaurin's general 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 formula of the differential calculus, for the development of a function according to ascending powers of the variable: named after the Scotch mathematician Colin Maclaurin (1698-1746). It is an immediate corollary from Taylor's theorem, and is written 1 2! Fx = FO + Fox + 2 F0.x2 + 3 F0.x3 +.... Malus's theorem, the law of double refraction: given in 1810 by E. L. Malus (1775-1812). - Mannheim's theorem. Same as Schönemann'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), especially that if three straight lines drawn from a point O are cut by a fourth line in the points A, B, C in order, then (OA) BC - (OB) AC +(OC)2AB = 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 Menelaus, of the first century. - Meusnier's theorem, the proposition that the radius of curvature of an oblique section 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 proposition in statics. - Miquel'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 vers a, 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,... an , etc., be given, and a corresponding series of functions, 4., 41,... ψn, etc., of the form IMG:content-0228.png:[blocks in formation] sinx=( r +r1 + c)/a, and sin x = √ ( rr-c)/a, rand 1 being the focal radii of the extremities, c the chord, and a the semiaxis major. (b) A proposition relating 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 expre ed 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 mono dromic function fz can always be found having for critical points a, a,... an, etc., and such that fz = + == n + ψη =..., being a function for which a is not a critical point: given by G. Mittag-Leffler.- Multinomial theorem. See multinomial . - Newton's theorem. ( a ) The proposition that if in the 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. (b) 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 theorem of which a vertex is pole of the opposite base relatively to a quadric surface, that base being a conjugate triangle relative to its section of the quadric, is a conjugate tetrahedron.-Pappus's theorem. (a) The proposition that if a quadrangle is inscribed in a conic, the product of the distances of any point on the curve from one pair of opposite sides is to the product of its distances from another such pair in a constant ratio: so called owing to its connection with Pappus's problem. (b) One of the two propositions that the surface of a solid of revolution is equal to the product of the perimeter of generating numbers at least as small as p and prime to it: given in 1876 by the eminent Irish mathematician H. J. S. Smith (1826-83). The theorem as generalized by Paul Mansion in 1877 is called Smith and Mansion's theorem. - Staudt's theorem, the proposition that any Bernoulli number, Bu, is equal to an integer minus 2-1+-+-+-1, where a, ß, etc., are all the prime numbers one greater than the double of divisors of n : given in 1840 by K. G. C. von Staudt (1798-1867). - Steiner's theorem, one of a plane figure by the length of the path described by the large number of propositions in geometry given by Jakob center of gravity, and that the volume of such a solid is equal to the area of the plane figure multiplied by the same length of path. Various other theorems contained in the collection of the Greek mathematician Pappus, of the third century, are sometimes called by his name.Particular theorem, a theorem which extends only to a particular quantity. - 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 Blaise Pascal (1623-62) in 1640. The hexagon itself is called a Pascal's hexagon or hexagram , and the straight line is called a Pascal's line. - Picard's theorem. (a) The proposition that every function which in the whole plane of imaginary quantity except in pstraight lines is uniform and continuous, is equal to the sum of puniform functions, each of which has but one such line. (b) A certain proposition concerning uniform functions connected by an algebraic relation. - Pohlke's theorem, the proposition 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 H. K. Pohlke in 1853. Also known as the fundamental theo rem of axonometry. - Poi on's theorem, a rule for forming integrals of a partial differential equation from two given integrals.- Polynomial theorem. See polynomial . -Poncelet's theorem. ( a ) The proposition that if there be a closed polygon inscribed in a given conic and circumscribed about another given conic, there is an infinity of such polygons. (b) The proposition that a quantity of the form R = Vu2 + v cannot differ from au + Bv by more than Rtanje, where a = cos (0+)/coste, p = sin (0 + €)/coste, =(0-0), tanu / v>tan 0. Both were given by General J. V. Poncelet (1788-1877). - 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 Greek mathematician of the second century, Claudius Ptolemy. - Puiseux'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. Puiseux (1820-83), by whom it was given in 1851.-Pythagorean theorem, the Pythagorean proposition (which see, under Pythagorean ). - Reciprocal theorem, a theorem of geometry analogous to another theorem, but relating to planes instead of points, and vice vers a, or in a plane to straight lines instead of points, and vice vers a. Thus, Pascal's and Brian chon's theorems are reciprocal to one another.-Ribaucour's theorem, given pseudo spherical surface unit curvature, if in every tangent plane a circle of unit radius be described about the point of contact as center, these circles will be orthogonal to a family of pseudo spherical 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 corresponding points-for example, that two projective series of points lie upon curves of the same deficiency. In its generality the proposition is called the theorem of Riemann and Roch, or of Riemann , Roch , and Nother . It was first given by G. F. B. Riemann (1826-67) in 1857, generally demonstrated by Roch in 1865, and extended to surfaces by Nother in 1886. - Robert's theorem. (a) The proposition that the geodesics 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 and lines of curvature of quadrics, by Michael Roberts in 1846. (b) The proposition that if a point be taken on each of the edges of any tetrahedron and a sphere be described through each Steiner (1796-1863), who was probably the greatest geometrical genius that ever lived; but the nece ities 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 1826 and the following years) were given without proofs, and remained an enigma to mathematicians until 1862, when Luigi Cremona demonstrated most of them. Stirling's theorem, the proposition that IMG:content-0229.png:[blocks in formation] given by James Stirling (1696-1770). - Sturm's theorem, a proposition in the theory of equations for determining the number of real roots of an equation between given limits: given by the French mathematician J. C. F. Sturm (1803-55) in 1835. - Sylow's theorem. See Cauchy's theo rem (b), above. - Sylvester's theorem. (a) An extension of Newton's rule on the limits of the roots of an algebraic equation. (b) The proposition that every quaternary cubic is the sum of the cubes of five linear forms. (c) The proposition that if A1, A2, etc., are the latent roots of a matrix m, then IMG:content-0230.png:[merged small][ocr errors] the or icon This is the original, proper, and best meaning of the word. Aristotle divides all knowledge into productive ( art ) and unproductive ( science ), and the latter into that which aims at accomplishing something ( practical science ) and that which aims only at understanding its object, which is the oretical science. This distinction, which has descended to our times (but with practical science and art joined together), diminishes in importance as science advances, all the sciences finding practical applications. Weary with the pursuit of academical studies, he [Col lins) no longer confined himself to the search of theoreti cal knowledge, but commenced, 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 imperfect theory, and not correctly indicating the real facts as presenting themselves in experience. All the practical sciences that have been pursued with distinguished succe proceed by deductions from hypotheses known not to be strictly true. This is the analytical method, of which modern civilization is the fruit. In some cases the hypotheses are so far from the truth that the results have to receive corrections. In such cases the uncorrected result is called theoretical , the corrected result practical. What logic was to the philosopher legislation was to the statesman and moralist, a practical, as the other was a theoretical , casuistry. Stubbs, Medieval and Modern 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 IMG:content-0231.png:[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small] π/2 = (22/32).(42/52).(62/72).(82/92), etc., named after the discoverer, John Wallis (1616-1703). We i ers tra 's fundamental theorem, the proposition that every analytical function to an theorem is either an algebraic function, or an algebraic function of an exponential, or an algebraic function of the We i ers tra i an function 8: given by Karl We i ers tra (born 1815). - We in gar ten's theorem. See Betti's theo rem , above. Wilson's theorem, the proposition that if pis a prime number, the continued product 1.2.3. (p-1) increased by 1 is divisible by p , and if not, not: discovered by Judge John Wilson (1741-93), and published by Waring.-Wronski's theorem, an expansion for a function of a root of an equation. - Yvon-Villarceau's theorem, a general proposition of dynamics, expre ed IMG:content-0232.png:[blocks in formation] where is the velocity, r the radius vector of the point whose ma is m and its coördinates x , y , z, while X, Y, Z are the components of the force, f the force, and a the distance of two particles: given in 1872 by A. J. F. YvonVillarceau (1813-83). It much resembles the theorem of the virial. = Syn. See inference. theorem (the'ō-rem), v. t. [ theorem , n.] n .] To 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 logic ). - Theoretical meteorology, philosophy, proposition, reality, reason, etc. See the nouns. theoretically (the-ō-ret'i-kal-i), adv . In a theoretic manner; in or by theory; from a theoretical point of view; speculatively: opposed to practically. theoretician (theō-re-tish'an), n. [ theoretic +-ian.] A theorist; a theorizer; one who is expert in the theory of a science or art. theoretics (the-ō-ret'iks), n. [Pl. of theoretic (see-ics).] The speculative parts of a science. With our Lord himself and his apostles, as represented to us in the New Testament, morals come before contemplation, ethics before theoretics . H. B. Wilson . theoriclt (theo-rik), a. and n. [I. a. = F. thé orique = Sp. teórico =Pg. theorico = It. teorico , theoric us , θεωρικός , of or pertaining to theory, θεωρία , theory: see theory . II. n. Also the or ick , theorique , theorik , theorike , theorique , F. théorique = Sp. teorica = Pg. theoric a = It. teorica , theoric a (sc. ars ), θεωρικός , of or pertaining to theory: see I.] I. a . Making deductions from theory, especially from imperfect theory; theorizing. Also the or i cal . Your courtier theoric is he that hath arrived to his farthest, and doth now know the court rather by speculation than practice. B. Jonson, Cynthia's Revels, ii. 1. A man but young, Yet old in judgment; theoric and practic In all humanity.