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finite

finite

finite Sentence Examples

  • " The vulgar almost imagine him as a finite thing."

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  • God is not fully comprehensible by us, says Albert, because the finite is not able to grasp the infinite, yet he is not altogether beyond our knowledge; our intellects are touched by a ray of his light, and through this contact we are brought into communion with him.

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  • The invariant theory then existing was classified by them as appertaining to " finite continuous groups."

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  • There is a finite number of baseballs, beanbags, and balloons.

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  • Diogenes distinctly taught that the world is of finite duration, and will be renewed out of the primitive substance.

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  • Apart from God, the finite being has no reality, and we only have the idea of it from God.

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  • The world consists of a finite number of atoms, which have in their own nature a self-moving force or principle.

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  • The name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis.

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  • And so on for all the finite cardinals, which are thus defined successively.

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  • And so on for all the finite cardinals, which are thus defined successively.

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  • Here also it can be seen that the science of the finite ordinals is a particular subdivision of the general theory of classes and relations.

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  • Pain and sin must have been reduced to a minimum by God; though they are so ingrained in the finite that we have to make up our minds even to the endless sin and endless punishments of hell.

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  • They are infinite and perfect when the En Soph imparts his fullness to them, and finite and imperfect when that fullness is withdrawn from them.

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  • This finite number of forms is said to constitute the complete system.

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  • We have already seen that the Sephiric decade or the archetypal man, like Christ, is considered to be of a double nature, both infinite and finite, perfect and imperfect.

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  • The definitions of the finite ordinals can be expressed without use of the corresponding cardinals, so there is no essential priority of cardinals to ordinals.

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  • The definitions of the finite ordinals can be expressed without use of the corresponding cardinals, so there is no essential priority of cardinals to ordinals.

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  • He holds that freedom is the inalienable prerogative of the finite spirit; and this is the second point that distinguishes his theology from the heretical Gnosticism.

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  • The more celebrated and central thesis of the book - this finite universe, the best of all such that are possible - also restates positions of Augustine and Aquinas.

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  • He was also the first to consider the difficult problems involved in equations of mixed differences, and to prove that an equation in finite differences of the first degree and the second order might always be converted into a continued fraction.

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  • He admits two sources of knowledge - sensation and refiexion; and God is to him the Great First Cause, especially of our own existence (or of the existence of finite minds).

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  • This Essence is God, and includes within itself the finite unities of man, reason and nature.

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  • Proceeding in this way, we may be able to express P= Q as the sum of a finite number of terms k+m/r+n/r 2 +..

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  • In his conception of finite personality he recurs to something like the monadism of Leibnitz.

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  • Perpetuants.-Many difficulties, connected with binary forms of finite order, disappear altogether when we come to consider the (p1p2p3...) to where form of infinite order.

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  • Thus it is used to translate the Platonic 'SEa, Et50s, the permanent reality which makes a thing what it is, in contrast with the particulars which are finite and subject to change.

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  • The approach to the limit will therefore be by a series of jumps, each of which, however small, will be finite; i.e.

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  • Single binary forms of higher and finite order have not been studied with complete success, but the system of the binary form of infinite order has been completely determined by Sylvester, Cayley, MacMahon and Stroh, each of whom contributed to the theory.

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  • To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.

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  • But the definition of the cardinal number of a class applies when the class is not finite, and it can be proved that there are different infinite cardinal numbers, and that there is a least infinite cardinal, now usually denoted by o where to is the Hebrew letter aleph.

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  • If these equations could be assumed to hold when H is indefinitely small, it would follow that has a finite initial value, from which there would be no appreciable deviation in fields so weak that bH was negligibly small in comparison with a.

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  • According to Aristotle, "the first of Eleatic unitarians was not careful to say whether the unity which he postulated was finite or infinite, but, contemplating the whole firmament, declared that the One is God."

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  • God, he says, is to be regarded not as an absolute but as an Infinite Person, whose nature it is that he should realize himself in finite persons.

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  • He admits two sources of knowledge - sensation and refiexion; and God is to him the Great First Cause, especially of our own existence (or of the existence of finite minds).

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  • All forms of monism from Plotinus downwards tend to ignore personal individuality and volition, and merge all finite existence in the featureless unity of the Absolute; this, indeed, is what inspires the passion of the protest against monism.

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  • Now if n be any finite cardinal number, it can be proved that the class of those serial relations, which have a field whose cardinal number is n, is a relation-number.

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  • Now if n be any finite cardinal number, it can be proved that the class of those serial relations, which have a field whose cardinal number is n, is a relation-number.

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  • Is it finite, or is it for all practical purposes infinite?

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  • This idea that there are a finite number of jobs misses the point entirely of what makes a job.

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  • One or more of the electrons may be detached from the system by a finite force, the number so detachable depending on the valency of the atom; if the atom loses an electron, it becomes positively electrified; if it receives additional electrons, it is negatively electrified.

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  • A perfectly formless matter (materia prima) was regarded by him as the universal substratum and common element of all finite existences.

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  • The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences.

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  • Any space at every point of which there is a finite magnetic force is called a field of magnetic force, or a magnetic field.

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  • r= io, we get the ordinary expression of P/Q as an integer and a decimal; but, if P/Q were equal to 1/3, we could not express it as a decimal with a finite number of figures.

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  • We cannot, for instance, say that the fraction C _2 I is arithmetically equal to x+I when x= I, as well as for other values of x; but we can say that the limit of the ratio of x 2 - I to x - I when x becomes indefinitely nearly equal to I is the same as the limit of x+ On the other hand, if f(y) has a definite and finite value for y = x, it must not be supposed that this is necessarily the same as the limit which f (y) approaches when y approaches the value x, though this is the case with the functions with which we are usually concerned.

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  • If, on the other hand, the point be well immersed in the geometrical shadow, the earlier zones are altogether missing, and, instead of a series of terms beginning with finite numerical magnitude and gradually diminishing to zero, we have now to deal with one of which the terms diminish to zero at both ends.

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  • The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work.

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  • Similarly, a class of serial relations, called well-ordered serial relations, can be defined, such that their corresponding relation-numbers include the ordinary finite ordinals, but also include relation-numbers which have many properties like those of the finite ordinals, though the fields of the relations belonging to them are not finite.

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  • P. Gordan first proved that for any system of forms there exists a finite number of covariants, in terms of which all others are expressible as rational and integral functions.

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  • Of two or more binary forms there are also complete systems containing a finite number of forms. There are also algebraic systems, as above mentioned, involving fewer covariants which are such that all other covariants are rationally expressible in terms of them; but these smaller systems do not possess the same mathematical interest as those first mentioned.

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  • For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold's symbolical method in the calculus of invariants; and the like.

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  • " It became clear that in the system of perpetual Becoming and of the dialectical passing over of all forms into one another, the finite personality could scarcely raise a plausible claim to the character of a substance and to immortality in the religious sense."

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  • If m and n are finite cardinal numbers, the rational number m/n is the relation which any finite cardinal number x bears to any finite cardinal number y when n X x = m X y.

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  • If, on the other hand, the point be well immersed in the geometrical shadow, the earlier zones are altogether missing, and, instead of a series of terms beginning with finite numerical magnitude and gradually diminishing to zero, we have now to deal with one of which the terms diminish to zero at both ends.

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  • Integrating by parts in (II), we get J e = ikr d7 pc-11 / d (e r - ay= rJ Z d y - r / 1 dY, in which the integrated terms at the limits vanish, Z being finite only within the region T.

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  • r= io, we get the ordinary expression of P/Q as an integer and a decimal; but, if P/Q were equal to 1/3, we could not express it as a decimal with a finite number of figures.

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  • He may be said to furnish a further contribution to a metaphysical conception of evolution in his view of all finite individual things as the infinite variety to which the unlimited productive power of the universal substance gives birth.

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  • Given perfect information, frictionless markets, and other theoretical impossibilities, a finite amount of utility can be achieved in that way.

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  • His argument for the Being of God is based on the hypothesis that thought - not individual but universal - is the reality of all things, the existence of this Infinite Thought being demonstrated by the limitations of finite thought.

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  • In this all-important doctrine of the Sephiroth, the Kabbalah insists upon the fact that these potencies are not creations of the En Soph, which would be a diminution of strength; that they form among themselves and with the En Soph a strict unity, and simply represent different aspects of the same being, just as the different rays which proceed from the light, and which appear different things to the eye, are only different manifestations of one and the same light; that for this reason they all alike partake of the perfections of the En Soph; and that as emanations from the Infinite, the Sephiroth are infinite and perfect like the En Soph, and yet constitute the first finite things.

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  • Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

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  • Lastly Baxter attempted to prove that matter is finite.

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  • ', In Bobyleff's problem of the wedge of finite breadth, ch nS2 = ?a b' s h n S 2 = V b a a 1 u u b, (6)?

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  • - cos 2 na sin2ng s (9) sin 2 na - sin2n0 Putting n =I gives the case of a stream of finite breadth disturbed y a transverse plane, a particular case of Fig.

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  • The method of electrical images will enable the stream function, )' to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle 7r/m, where m is an integer (R.

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  • Taylor's Methodus Incrementorum Directa et Inversa (London, 1715) added a new branch to the higher mathematics, now designated the " calculus of finite differences."

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  • In 1734 he also published Prodromus philosophiae ratiocinantis de infinito et causa finali creationis, which treats of the relation of the finite to the infinite, and of the soul to the body, seeking to establish a nexus in each case as a means of overcoming the difficulty of their relation.

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  • In God there are three infinite and uncreated "degrees" of being, and in man and all things corresponding three degrees, finite and created.

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  • It defines the condition which must be fulfilled by the potential at any and every point in an electric field, through which p is finite and the electric force continuous.

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  • Combining this with the first law, for a Carnot cycle of finite range, if H is the heat taken in at 0', and H" is the heat rejected at 0", the work W done in the cycle is equal to the difference H' - H", and we have the simple relations, W/(0' - o") =H'/o' =H" o".

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  • of temperature (o' - o") is small, the figure ABCD may be regarded as a parallelogram, and its area W as equal to the rectangle BE XEC. This is accurately true in the limit when (0' - 0") is infinitesimal, but in practice it is necessary to measure specific heats, &c., over finite ranges of temperature, and the error involved is generally negligible if the range does not exceed a few degrees.

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  • It is not necessary in this example that AB, CD should be adiabatics, because the change of volume BC is finite.

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  • For a finite change it is necessary to represent the path by a series of small steps, which is the graphic equivalent of integration along the path represented by the given relation between v and 0, or p and 0.

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  • Since the change of energy is independent of the path, the finite change between any two given states may be found by integration along any convenient path.

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  • From the hypothesis of an external world a series of contradictions are deduced, such as that the world is both finite and infinite, is movable and immovable, &c.; and finally, Aristotle and various other philosophers are quoted, to show that the external matter they dealt with, as mere potentiality, is just nothing at all.

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  • In it the union between the finite self-consciousness and the infinite ego or God is handled in an almost mystical manner.

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  • Meanwhile, the same considerations had not been applied to time, so that in the days of Zeno of Elea time was still regarded as made up of a finite number of ` moments,' while space was confessed to be divisible without limit.

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  • Van der Waals, in a famous monograph, On the Continuity of the Liquid and Gaseous States (Leiden, 1873), has shown that the imperfections of equation (14) maybe traced to two_causes: (i.) The calculation has not allowed for the finite size of the molecules, and their consequent interference with one another's motion, and (ii.) The calculation has not allowed for the field of inter-molecular force between the molecules, which, although small, is known to have a real existence.

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  • The universal or infinite is one that realizes itself in finite particular minds and wills, not as accidents or imperfections of it, but as its essential form.

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  • It involves, it will be said, the reality of time, the dependence of the Infinite in the finite, and therewith a departure from the whole line of Hegelian thought.

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  • It is to be remedied not by giving up the idea of the Infinite but by ceasing to think of the Infinite as of a being endowed with a static perfection which the finite will merely reproduces, and definitely recognizing the forward effort of the finite as an essential element in Its self-expression.

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  • The general method of constructing formulae of this kind involves the use of the integral calculus and of the calculus of finite differences.

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  • In the case of statistical data there is the further difficulty that there is no real continuity, since we are concerned with a finite number of individuals.

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  • For the methods involving finite differences, see references under DIFFERENCES, CALCULUS OF; and INTERPOLATION.

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  • The receiving apparatus had what we may term a personal equation, for the break of contact could only take place when the membrane travelled some finite distance, exceedingly small no doubt, from the contact-piece.

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  • Since the curve represents a longitudinal disturbance in air it is always continuous, at a finite distance from the axis, and with only one ordinate for each abscissa.

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  • If we rest on the synthesis here described, the energy of the matter, even the thermal part, appears largely as potential energy of strain in the aether which interacts with the kinetic energy associated with disturbances involving finite velocity of matter.

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  • With reference to all such further refinements of theory, it is to be borne in mind that the perfect fluid of hydrodynamic analysis is not a merely passive inert plenum; it is also a continuum with the property that no finite internal slip or discontinuity of motion can ever arise in it through any kind of disturbance; and this property must be postulated, as it cannot be explained.

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  • Now the kinetics of a medium in which the parts can have finite relative motions will lead to equations which are not linear - as, for example, those of hydrodynamics - and the phenomena will be far more complexly involved.

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  • Now taking equation (72), and replacing tan B, as a variable final tangent of an angle, by tan i or dyldx, (75) tan 4) - dam= C sec n [I(U) - I(u)], and integrating with respect to x over the arc considered, (76) x tan 4, - y = C sec n (U) - f :I(u)dx] 0 But f (u)dx= f 1(u) du = C cos n f x I (u) u du g f() =C cos n [A(U) - A(u)] in Siacci's notation; so that the altitude-function A must be calculated by summation from the finite difference AA, where (78) AA = I (u) 9 = I (u) or else by an integration when it is legitimate to assume that f(v) =v m lk in an interval of velocity in which m may be supposed constant.

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  • - C = C cos n [5 (u 5) - S(ue)], ' y / e 0 A =tan - C sec n [I (u 0) - S] A now denoting any finite tabular difference of the function between the initial and final (pseudo-) velocity.

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  • The primeval Being is, as opposed to the many, the One; as opposed to the finite, the Infinite, the unlimited.

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  • It is, moreover, the Good, in so far as all finite things have their purpose in it, and ought to flow back to it.

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  • Thus log x is the integral function of 1/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as algebraical or circular functions.

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  • But the Fichtean teaching appeared on the one hand to identify too closely the ultimate ground of the universe of rational conception with the finite, individual spirit, and on the other hand to endanger the reality of the world of nature by regarding it too much after the fashion of subjective idealism, as mere moment, though necessitated, in the existence of the finite thinking mind.

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  • External things are produced by the will of the divine intelligence; they are caused, and caused in a regular order; there exists in the divine mind archetypes, of which sense experience may be said to be the realization in our finite minds.

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  • The relation between the divine mind and finite intelligence, at first thought as that of agent and recipient, is complicated and obscure when the necessity for explaining the permanence of real things comes forward.

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  • This apparent motion is due to the finite velocity of light, and the progressive motion of the observer with the earth, as it performs its yearly course about the sun.

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  • The limitation of power is introduced as in all optical instruments, by the finiteness of the length of a wave of light which causes the image of an indefinitely narrow slit to spread out over a finite width in the focal plane of the observing telescope.

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  • Thus if a molecule were set into vibration at a specified time and oscillated according to the above equation during a finite period, it would not send out homogeneous vibrations.

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  • itself, not in Aristotle's sense of any individual existing differently from anything else, but in the novel meaning of something existing alone, he concluded, logically enough from this mere misunderstanding, that there can be only one substance, and that, as no finite body or soul can exist alone, everything finite is merely a mode of one of the attributes of the one infinite substance which alone can exist by itself.

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  • Having, however, in consequence, lost his professorship at Jena, he gradually altered his views, until at length he decided that God is not mere moral order, but also reason and will, yet without consciousness and personality; that not mankind but God is the absolute; that we are only its direct manifestations, free but finite spirits destined by God to posit in ourselves Nature as the material of duty, but blessed when we relapse into the absolute; that Nature, therefore, is the direct manifestation of man, and only the indirect manifestation of God; and, finally, that being is the divine idea or life, which is the reality behind appearances.

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  • Schelling perceived that Hegel, in reducing everything to infinite mind, absorbed man's free but finite personality in God, and, in declaring that everything real is rational, failed to explain evil and sin: indeed, the English reader of T.

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  • But in a philosophy which reduces everything to phenomenal appearance except the self-interacting substance of God, there is no room for either the bodies or the souls of finite substances or human persons.

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  • Bradley, however, having satisfied himself, like Spinoza, by an abuse of the word " independent," that " the finite is self-discrepant," goes on to ask what the one Real, the absolute, is; and, as he passed from Herbart to Spinoza, so now he passes from Spinoza to Kant.

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  • 1855, professor of philosophy, Harvard) believes in the absolute like Green and Bradley, in " the unity of a single self-consciousness, which includes both our own and all finite conscious meanings in one final eternally present insight," as he says in The World and the Individual (1900; see also later works).

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  • Calderwood, in his Philosophy of the Infinite (1854), made the pertinent objection that, though thought, conception and knowledge are finite, the object of thought may be infinite.

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  • Hamilton, in fact, made the double mistake of limiting knowledge to what we can conceive, and confusing the determinate with the finite or limited.

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  • Further, to allow of a correction being applied for the finite length of the magnets the whole series of settings is repeated with the centre of the deflecting magnet at 40 cm.

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  • Unlike the Hindu, Xenophanes inclined to pantheism as a protest against the anthropomorphic polytheism of the time, which seemed to him improperly to exalt one of the many modes of finite existence into the place of the Infinite.

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  • characteristic is primarily the negation of the Finite.

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  • We find Philo Judaeus endeavouring to free the concept of the Old Testament Yahweh from anthropomorphic characteristics and finite determinations.

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  • On the principle that everything which is determined (finite) is "negated" ("determinatio est negatio"), God, the ultimate reality must be entirely undetermined.

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  • Like Hamilton, Mansel maintained the purely formal character of logic, the duality of consciousness as testifying to both self and the external world, and the limitation of knowledge to the finite and "conditioned."

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  • The first volume treats of the analysis of finite quantities, and the second of the analysis of infinitesimals.

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  • A people's religion completes and consecrates their whole activity: in it the people rises above its finite life in limited spheres to an infinite life where it feels itself all at one.

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  • Even philosophy with Hegel at this epoch was subordinate to religion; for philosophy must never abandon the finite in the search for the infinite.

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  • This is, however, an application of categories from a field where the conditions are finite to a sphere in which the circumstances are infinite."

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  • The state is the consummation of man as finite; it is the necessary starting-point whence the spirit rises to an absolute existence in the spheres of art, religion and philosophy.

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  • In the finite world or temporal state, religion, as the finite organization of a church, is, like other societies, subordinate to the state.

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  • For the Son of God, in the immediate aspect, is the finite world of nature and man, which far from being at one with its Father is originally in an attitude of estrangement.

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  • The earlier Hegelians had interpreted it in the sense that the world in its ultimate essence was not only spiritual but self-conscious intelligence whose nature was reflected inadequately but truly in the finite mind.

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  • The completeness and self-consistency which our ideal requires can be realized only in a form of being in which subject and object, will and desire, no longer stand as exclusive opposites, from which it seemed at once to follow that the finite self could not be a reality nor the infinite reality a self.

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  • Bezout's method of elimination, the other on the number of integrals of an equation of finite differences.

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  • Thus we distinguish in the divine system beginning, middle and end; but these three are in essence one - the difference is only the consequence of our finite comprehension.

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  • No finite predicates can be applied to him; his mode of being cannot be determined by any category.

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  • The Mosaic account, then, is to be looked upon merely as a mode in which is faintly shadowed forth what is above finite comprehension.

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  • Metaphysically, all realities are parts of one ultimate reality; but logically, even philosophers think more often only of finite realities, existing men, dogs, horses, &c.; and children know that their parents exist long before they apprehend ultimate reality.

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  • But when, instead of the highly artificial expression ix-}-jy+kz, to denote a finite directed line, we employ a single letter, a (Hamilton uses the Greek alphabet for this purpose), and find that we are permitted to deal with it exactly as we should have dealt with the more complex expression, the immense gain is at least in part obvious.

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  • In the third-order complex the centre locus becomes a finite closed quartic surface, with three (one always real) intersecting nodal axes, every plane section of which is a trinodal quartic. The chief defect of the geometrical properties of these bi-quaternions is that the ordinary algebraic scalar finds no place among them, and in consequence Q:1 is meaningless.

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  • Their speculation as to the nature of God had led them gradually to separate him by an infinite distance from all creation, and to feel keenly the opposition of the finite and the infinite, the perfect and the imperfect, the eternal and the temporal.

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  • By it we perceive how God, the infinite, the absolute, the eternal, is yet not separated from the finite, the temporal, the relative, but, through the incarnation, enters into humanity.

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  • Unlike Thales, he was struck by the infinite variety in things; he felt that all differences are finite, that they have emerged from primal unity (first called epxn by him) into which they must ultimately return, that the Infinite One has been, is, and always will be, the same, indeterminate but immutable.

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  • Chasles any displacement whatever of the lamina in its own plane is equivalent to a rotation about some finite or infinitely distant point J.

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  • This point is called the centre of the given system of parallel forces; it is finite and determinate unless ~(P) = 0.

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  • We proceed to sketch the theory of the finite displacements of a rigid body.

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  • The composition of finite rotations about parallel axes is, a particular case of the preceding; the radius of the sphere is now infinite, and the triangles are plane.

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  • For this purpose the infinitesimal displacements of W the various joints are replaced by finite lengths proportional to them, and there- FIG.

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  • Again, when extraneous forces P act on the joints, the equation is Z(P.&P)+S.Os=-o, where op is the displacement of any joint in the direction of the corresponding force P. If ~(P. Op) =o, the stresses are merely indeterminate as before; but if ~ (P. op) does not vanish, the equation cannot be satisfied by any finite value of S, since Os =0.

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  • This means that, if the material of the frame were absolutely unyielding, no finite stresses in the bars would enable it to withstand the extraneous forces.

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  • The stresses in the bars would then be comparatively very great, although finite.

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  • It follows that the forces on any finite portion will satisfy the conditions of equilibrium which apply to the case of a rigid body (~ 4).

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  • In the case of a chain hanging freely under gravity it is usually convenient to formulate the conditions of equilibrium of a finite portion PQ.

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  • If the particles start from rest at a finite distance c, we have in (I6), C = 2p1C, and therefore ~=u=_,/~2M ~ (fr)

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  • For purposes of mathematical treatment a force which produces a finite change of velocity in a time too short to be appreciated is regarded as infinitely great, and the time of action as infinitely short.

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  • The effect of ordinary finite forces during the infinitely short duraticm of this impulse is of course ignored.

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  • The total impulse in any finite interval of time is the integral of the impulses corresponding to the infinitesimal elements 3t into which the interval may be subdivided; the summation of which the integral is the limit is of course to be understood in the vectorial sense.

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  • The question presents itself whether ther then is any other law of force, giving a finite velocity from infinity, under which all finite orbits are necessarily closed curves.

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  • mathematical points endowed with inertia coefficients, separated by finite intervals, and acting on one another with forces in the lines joining them subject to the law of equality of action and reaction.

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  • If, on the other hand, the body is finite, certain terminal conditions have to be satisfied.

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  • The solution of (40) which is finite for x = o is readily obtained in the form of a series, thus y=C(I_~-I-~1~_...) =CJo(z),, (4)

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  • In order to allow for the finite stiffness and strength of materials, the least distance of the centre of resistance inward from the nearest edge of the joint is made to bear a definite proportion to the depth of the joint measured in the same direction, which proportion is fixed, sometimes empirically, sometimes by theoretical deduction from the laws of the strength of materials.

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  • The essence of God, for instance, he held to be love; God, he said, can love nothing inferior to himself; but he cannot be an object of love to himself without going out, so to speak, of himself, without manifesting his infinity in a finite form; in other words, by becoming man.

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  • These effects, as Hertz showed, indicated the establishment of stationary electric waves in space and the propagation of electric and magnetic force through space with a finite velocity.

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  • Cudworth's ideas, like Plato's, have "a constant and never-failing entity of their own," such as we see in geometrical figures; but, unlike Plato's, they exist in the mind of God, whence they are communicated to finite understandings.

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  • He holds that we are rationally justified in affirming human immortality and the existence of a finite God who is to be a constitutional ruler, but not a despot, over the souls of men.

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  • The histories of philosophy may quite correctly describe his theory as the logical development of Descartes's doctrines of the one Infinite and the two finite substances, but Spinoza himself was never a Cartesian.

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  • The foundation of the system is the doctrine of one infinite substance, of which all finite existences are modes or limitations (modes of thought or modes of extension).

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  • Bostrom's philosophy is logically expressed and based on the one great conception of a spiritual, eternal, immutable Being, whose existence is absolute, above and external to the finite world of time and space.

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  • 6) at a finite distance from the axis (or with an infinitely O distant object, a point which subtends a finite angle at the system) is, in general, even then not sharply reproduced, if the pencil of rays issuing FIG.

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  • In the mathematical sense, however, this selection is arbitrary; the reproduction of a finite object with a finite aperture entails, in all probability, an infinite number of aberrations.

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  • This number is only finite if the object and aperture are assumed to be " infinitely small of a certain order"; and with each order of infinite smallness, i.e.

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  • The existence of an optical system, which reproduces absolutely a finite plane on another with pencils of finite aperture, is doubtful; but practical systems solve this problem with an accuracy which mostly suffices for the special purpose of each species of instrument.

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  • In order to render spherical aberration and the deviation from the sine condition small throughout the whole aperture, there is given to a ray with a finite angle of aperture u* (with infinitely distant objects: with a finite height of incidence the same distance of intersection, and the same sine ratio as to one neighbouring the axis (u* or h* may not be much smaller than the largest aperture U or H to be used in the system).

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  • ” If in the infinite continued fraction of the second class an?bn+i for all values of n, it converges to a finite limit not greater than unity.” 3.

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  • converges to an incommensurable limit if after some finite value of n the condition an

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  • - an verges to an incommensurable limit if after some finite value of n the condition a n ?b n +I is always satisfied, where the sign > need not always occur but must occur infinitely often.

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  • The cycle of its transformations and successive condensations constitutes the life of the universe, the mode of existence proper to finite and particular being.

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  • We may investigate the forces which act between finite portions of a liquid in the same way as we investigate the forces which act between finite portions of a solid.

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  • The first part establishes the laws of the elasticity of a finite portion of the solid subjected to a homogeneous strain, and deduces from these laws the equations of the equilibrium and motion of a body subjected to any forces and displacements.

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  • The second part endeavours to deduce the facts of the elasticity of a finite portion of the substance from hypotheses as to the motion of its constituent molecules and the forces acting between them.

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  • Laplace assumed that the liquid has uniform density, and that the attraction of its molecules extends to a finite though insensible distance.

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  • For insensible values it may become sensible, but it must remain finite even when z = o, in which case 0(o) = K.

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  • Integrating the first term within brackets by parts, it becomes - fo de Remembering that 0(o) is a finite quantity, and that Viz = - (z), we find T = 4 7rp f a, /.(z)dz (27) When c is greater than e this is equivalent to 2H in the equation of Laplace.

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  • Since 0(o) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply f 0(z)dz f: (z)dz, (34) and T= ref: z1,1,(z)dz (35) In Laplace's notation the second member of (34), multiplied by 27r, is represented by H.

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  • - g 2) t; and this ratio decreases without limit with the time, whatever be the initial (finite) ratio a 2: a1.

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  • Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself.

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  • Four of his books were of particular importance: Christian Nurture (1847), in which he virtually opposed revivalism and "effectively turned the current of Christian thought toward the young"; Nature and the Supernatural (1858), in which he discussed miracles and endeavoured to "lift the natural into the supernatural" by emphasizing the supernaturalness of man; The Vicarious Sacrifice (1866), in which he contended for what has come to be known as the "moral view" of the atonement in distinction from the "governmental" and the "penal" or "satisfaction" theories; and God in Christ (1849) (with an introductory "Dissertation on Language as related to Thought"), in which he expressed, it was charged, heretical views as to the Trinity, holding, among other things, that the Godhead is "instrumentally three - three simply as related to our finite apprehension, and the communication of God's incommunicable nature."

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  • He seems to waver between the opinion that finite individuals have no independent being and the opinion that they have it in an infinitesimal degree; and the conception of " degrees of existence " in the essay on Virtue is not developed to elucidate the point.

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  • ' In abandoning the theory of ideas - that is to say, the theory of figures and numbers, the possessions of universal mind, eternally existent out of space and time, which figures and numbers when they pass into space and time as the heritage of finite minds are regarded as things - Speusippus had the approval, as of the Platonists generally, so also of Aristotle.

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  • In such a case the two systems must be regarded as a single more complex one, the absorbed vibration becomes large, though remaining always finite, and the transmitted vibration undergoes a remarkable change in its period.

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  • From the basis of immediate experience or perception thought proceeds by comparison and abstraction, establishing connexions among facts, but remaining in its nature mediate and finite.

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  • It is impossible that there should be a God, for if so he would of necessity be finite.

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  • But a finite God, a God that is known, is no God.

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  • But I distinguish the two finite causes self and not-self Panther ism.

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  • They are not mere modifications of this cause or properties, as with Spinoza, - they are free forces having their power or spring of action in themselves, and this is sufficient for our idea of independent finite reality.

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  • This is the moment of the finite.

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  • The finite and the infinite become two real correlatives in the relation of cause and product.

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  • This is the third and highest stage of development, the relation of the finite and the infinite.

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  • The East typifies the infinite, Greece the finite or reflective epoch, the modern era the stage of relation or correlation of infinite and finite.

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  • Y g Y g infinite or causes, correlative and reciprocally finite, to which absolute.

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  • The correlation of the ideas of infinite and finite does not necessarily imply their correality, as Cousin supposes; on the contrary, it is a presumption that finite is simply positive and infinite negative of the same - that the finite and infinite are simply contradictory relatives.

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  • The answer to this is that in the case of contradictory statements - A and not A - the latter is a mere negation of the former, and posits nothing; and the negation of a notion with positive attributes, as the finite, does not extend beyond abolishing the given attributes as an object of thought.

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  • This is a mere paralogism; we can never infer either absolute or infinite from relative or finite.

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  • It is, moreover, to be noticed that the points at infinity may be all or any of them imaginary, and that the points of intersection, whether finite or at infinity, real or imaginary, may coincide two or more of them together, and have to be counted accordingly; to support the theorem in its universality, it is necessary to take account of these various circumstances.

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  • Both Plato and Aristotle devoted much thought to the discussion as to which is most truly real, the finite objects of sense, or the universal idea of each thing laid up in the mind of God; what is the nature of that unity which lies behind the multiplicity and difference of perceived objects ?

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  • A similar relation exists in thought between the various grades of species and genera; the highest genus is the " infinitely infinite," each subordinated genus being infinite in relation to the particulars which it denotes, and finite when regarded as a unit in a higher genus.

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  • If human understanding cannot fully solve the infinite problem of the universe, man may at least see that at no stage of his finite experience is he necessarily the sport of chance, and that he can practically secure his own wellbeing.

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  • We find ourselves, when we try, compelled to lose our positive ideas of finite spaces in the negative idea of Immensity or Boundlessness, and our positive ideas of finite times in the negative thought of Endlessness.

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  • for the cause of an event is what we find we cannot help having; yet it is a demand for what in the end the mind cannot fully grasp. Locke is content to trace the idea of " cause and effect," as far as mere natural science goes, to our " constant observation " that " qualities and finite substances begin to exist, and receive their existence from other beings which produce them."

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  • Anything, as far as " constant observation " tells us, might a priori have been the natural cause of anything; and no finite number of " observed " sequences, per se, can guarantee universality and necessity.

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  • For " our perceptions of the particular existence of finite beings without us " go beyond mere probability, yet they are not purely rational.

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  • As to extent, it may be said, in a general way, that while no definite limits can be set to the possible extent of the universe, or the distance of its farthest bodies, it seems probable, for reasons which will be given under Star, that the system to which the stars that we see belong, is of finite extent.

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  • Romer, on the other hand, deserves full credit for originating the transit-circle and the prime vertical instrument; and he earned undying fame by his discovery of the finite velocity of light, made at Paris in 1675 by comparing his observations of the eclipses of Jupiter's satellites at the conjunctions and oppositions of the planet.

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  • 35) attributes to him two treatises, of which one contrasts the eternity of ideas with the finite nature of things, and the other is an attempt to reconcile Plato and Aristotle.

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  • The divine omnipotence is quantitatively represented by the sum of the forces of nature, and qualitatively distinguished from them only as the unity of infinite causality from the multiplicity of its finite phenomena.

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  • In his earlier days he called it a feeling or intuition of the universe, consciousness of the unity of reason and nature, of the infinite and the eternal within the finite and the temporal.

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  • The specific feature of Christianity is its mediatorial element, its profound feeling of the striving of the finite individual to reach the unity of the infinite whole, and its conception of the way in which Deity deals with this effort by mediatorial agencies, which are both divine and human.

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  • (2) In like manner the Many must be numerically both finite and infinite - numerically finite, because there are as many things as there are, neither more nor less; numerically infinite, because, that any two things may be separate, the intervention of a third thing is necessary, and so on ad infinitum.

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  • But an infinite distance (which Zeno fails to distinguish from a finite distance infinitely divided) cannot be traversed in a finite time.

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  • The "infinity" of the premise is an infinity of subdivisions of a distance which is finite; the "infinity" of the conclusion is an infinity of distance.

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  • Thus Zeno again confounds a finite distance infinitely divided with an infinite distance.

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  • Lord Rayleigh has pointed out that all theories are defective in that they disregard the fact that one at least of the media is dispersive, and that it is probable that finite reflection would result at the interface of media of different dispersive powers, even in the case of waves for which the refractive indices are absolutely the same.

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  • Religious thinkers are faced by the problem of the Creator and the created, and the necessity for formulating a close relationship between God and man, the Infinite and Perfect with the finite and imperfect.

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  • Campbell) so far as to postulate a divine element in human beings, so definitely bridging over the gap between finite and infinite which was to some extent admitted by the bulk of early Christian teachers.

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  • In support of such a view are adduced not only the metaphysical difficulty of postulating any relationship between the infinite and the purely finite, but also the ethical 1 P. G., tom.

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  • how a merely human being could appreciate the nature of or display divine goodness - and the epistemological problem of explaining how finite mind can cognize the infinite.

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  • Daub in his Judas Ishcarioth argued that a finite evil presupposes an absolute evil, and the absolute evil as real must be in a person.

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  • The central projection, of which the centre is the middle point of the entrance pupil on the plane focused for, will show in weaker systems, or those very much stopped down, a certain finite depth of definition; that is to say, the totality of points, which lie out of the plane focused for, and which are projected with circles of confusion so small that they appear to the eye as sharp points, will include the sharp object relief, and determine the depth of definition of the lens.

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  • For finite intelligences there was an inevitable incompleteness so far as knowledge of matters of fact was concerned.

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  • It was conjectured that this number is normal, meaning that it contains all finite bit strings.

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  • Apply the algebra of finite automata to design systems and to solve simple problems on creating acceptors for particular languages.

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  • It is called algebraic Curves over a Finite Field and is currently 644 pages.

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  • These are automatically computed by finite difference approximation (either forward or central differences ).

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  • arithmetic progression of consecutive primes for any given (finite) length?

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  • article entitled " A finite element study of the mechanics of sports mouthguards " .

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  • Necessary algorithmic problems are then solved by converting types to finite tree automata.

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  • Content Introduction to theory of machines: [4 lectures] deterministic finite automata, regular languages, nondeterministic finite automata.

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  • Application domains include digital circuit design, image filter design, and finite state automata induction.

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  • The maximum number of states that the finite automaton requires is set by default to 500.

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  • The finite scheme of things, to use a homely phrase, is thus actually lifting itself by its own bootstraps.

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  • Furthermore, they can replace heavy 3D numerical calculations (for example finite element calculations) with high accuracy.

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  • Kolmogorov's proposal outlined strives for the firmer and less contentious ground expressed in finite combinatorics and effective computation.

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  • To a considerable extent, finite and infinite combinatorics are parts of the same subject.

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  • commutator factor group is finite can only have finite nilpotent quotient groups.

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  • A group whose commutator factor group is finite can only have finite nilpotent quotient groups.

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  • competepproach is tempered by the many competing demands placed upon finite police resources.

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  • conjugate gradient algorithm in a finite element me.. .

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  • insect sense organs Funded by EU FP5, we have used finite element modeling to understand the design of sensors in insect cuticle.

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  • They therefore represent a finite scientific and economic resource and are a notable determinant of landscape character.

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  • disjoint sets Let X be a finite set of n elements.

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  • distraction from the task of identifying the particular output of the spammers within the finite.

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  • eigenvalues of finite dimensional unitary operators of natural quantum mechanical systems.

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  • You will be helping to alleviate human famine by taking less out of the World's finite resources.

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  • Microscopically, however, quantal effects of the large, yet finite, ensemble of strongly interacting fermions need to be considered.

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  • finite in size.

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  • finite in duration, and souls in Purgatory will be assured of final salvation.

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  • Furthermore, it is not possible for the infinite and finite and finite to mix, making an infinite finite.

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  • Content Introduction to theory of machines: [4 lectures] deterministic finite automata, regular languages, nondeterministic finite automata, regular languages, nondeterministic finite automata.

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  • finite automaton requires is set by default to 500.

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  • finite verb.

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  • finite approximations approaching strict equivalence in the limit.

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  • finite lifespan.

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  • finite elements for short wave modeling.

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  • The interpretation we produced was itself either finite or denumerably infinite.

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  • Since only finite regions are considered, the algebras are called local algebras.

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  • It should be remembered that titles and rhythm descriptions for such pieces are not always finite.

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  • Although electricity travels fast, its speed is still finite and over a wire it is slower than in a vacuum.

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  • For them, the realities of a very finite planet simply do not equate with their conventional wisdoms of infinite growth.

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  • Indeed, we only have a sense of the infinite, or what is not finite, or what is not finite, from various created finite viewpoints.

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  • As speculative computations may be infinite, while mandatory ones remain finite, while mandatory ones remain finite, divergence should be defined with the greatest care.

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  • floating point types only. static T min () throw(; Returns the minimum finite value.

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  • A universe with nonzero size and finite energy density appears from a quantum fuzz.

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  • hadron nucleus interactions are incorporated as well as vaccum polarization and finite size effects.

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  • He warns hoteliers that there is probably a finite window for charging.

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  • impulse response will consist of a series of pulses of finite width.

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  • Hence, the input impulse response will consist of a series of pulses of finite width.

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  • Also, since all operations are performed with finite precision, the results are of limited precision and necessarily inexact.

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  • For Aristotle all the elements in an actual finite exist simultaneously, whereas a potential infinite is realized over time by addition or division.

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  • The question is if G-d's knowledge is infinite, how can something infinite be placed in something finite, the written torah.

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  • The finite nature of the human mind cannot grasp and comprehend the infinite.

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  • It is beyond finite minds to understand the infinite.

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  • The effect of finite efficiency for the quantum interrogation is delineated for the various schemes.

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  • Invariant Theory of Finite Groups This introductory lecture will be concerned with polynomial invariant Theory of Finite Groups This introductory lecture will be concerned with polynomial invariants of finite groups which come from a linear group action.

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  • These are usually called ' closed universes ' and they have a finite total lifetime.

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  • To produce a finite straight line continuously in a straight line.

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  • localized in any finite region of space-time no matter how large it is.

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  • In fact GAP supports finite fields with elements represented via discrete logarithms only up to a given size.

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  • A finite group G is called monomial (or M -group) if each ordinary irreducible character of G is monomial.

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  • musculature of any living organs can be accurately represented through finite element discretization.

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  • But it can lead to the more efficient use of our finite natural resources.

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  • You are a finite concept that represents the negation of your own infinite essence.

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  • Current astronomical observations can't really tell us whether the world really did have an infinite past or just a finite one.

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  • Since there is an infinite ordinal, every finite ordinal is a set and the first infinite ordinal ω is a set.

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  • Since there is an infinite ordinal, every finite ordinal is a set and the first infinite ordinal ω is a set.

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  • The size evolution of polytetrahedral packings enables us to study the development of bulk liquid structure in finite systems.

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  • The ChemShell QM/MM procedures are most commonly with finite clusters, especially for highly polar materials where cutoffs cannot be used.

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  • polynomial invariants of finite groups which come from a linear group action.

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  • Or are they actually not infinitely precise, given the practical limitations of measuring them with a finite universe?

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  • We present a statistical finite state model that combines prosodic, linguistic and punctuation class features.

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  • At a stroke MVNO could remove the absolute barrier to entry arising from the finite available radio spectrum.

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  • reasoning abilities over finite state systems.

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  • Finite generalized tetrahedron groups with a cubic relator 2003/21 Vincent Schmitt.

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  • self-consistent solution based on finite element analysis.

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  • God is not finite, not sinful, etc.

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  • Another area of interest is the animals: these are finite subsets of a regular tiling.

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  • tableau system for model-checking the linear time mu-calculus on finite systems.

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  • topology optimization problems using different finite element meshes.

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  • In order to make such models tractable, we require that they have finite domains.

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  • transfinite mathematics of the actual infinite is translated into real finite terms, the results are nonsensical.

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  • undamped vibration of any linear structure, for example by Finite Element methods.

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  • But it can lead to the more efficient use of our finite natural resources.

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  • variance of the resource distribution is finite.

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  • Such tentative evidence is not yet sufficiently weighty to change the author's convictions that the universe must be finite in all respects.

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  • do y'all know what a Finite State Machine is?

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  • We have 62=W12+W22+032=P2 (N) 2 Y 2+P2 (A I - 3) 2 If AN, AD be both finite, we learn from (7) that there is no direction perpendicular to the primary (polarized) ray in which the secondary light vanishes.

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  • So long as the particles are small no such vanishing of light in oblique directions is observed, and we are thus led to the conclusion that the hypothesis of a finite AN and of vibrations in the plane of polarization cannot be reconciled with the facts.

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  • " The vulgar almost imagine him as a finite thing."

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  • " It became clear that in the system of perpetual Becoming and of the dialectical passing over of all forms into one another, the finite personality could scarcely raise a plausible claim to the character of a substance and to immortality in the religious sense."

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  • The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work.

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  • Apart from God, the finite being has no reality, and we only have the idea of it from God.

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  • Special experiments were made to determine the work done against resistances outside the vessel of water, which amounted to about 006 of the whole, and corrections were made for the loss of heat by radiation, the buoyancy of the air affecting the descending weights, and the energy dissipated when the weights struck the floor with a finite velocity.

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  • The reason why the whole of the energy of the current is not available is that heat must always be generated in a wire in which a finite current is flowing, so that, in the case of a battery in which the whole of the energy of chemical affinity is employed in producing a current, the availability of the energy is limited only on account of the resistance of the conductors, and may be increased by diminishing this resistance.

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  • God, he says, is to be regarded not as an absolute but as an Infinite Person, whose nature it is that he should realize himself in finite persons.

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  • In his conception of finite personality he recurs to something like the monadism of Leibnitz.

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  • When such a system is worked out in full detail, it essays or ought to essay a proof of the following points: (I) God or the Absolute necessarily exists; (2) He necessarily is what He is; (3) He or it necessarily manifests itself in the finite, (4) and necessarily manifests itself in just this finite which we know from experience.

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  • Pain and sin must have been reduced to a minimum by God; though they are so ingrained in the finite that we have to make up our minds even to the endless sin and endless punishments of hell.

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  • If the principle of the universe is impersonal or unconscious, personal consciousness in finite spirits comes to wear the appearance of a blunder.

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  • Diogenes distinctly taught that the world is of finite duration, and will be renewed out of the primitive substance.

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  • At the same time Aristotle precludes the idea of a natural development of the mental series by the supposition that man contains, over and above a natural finite soul inseparable from the body, a substantial and eternal principle (voi) which enters into the individual from without.

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  • The world consists of a finite number of atoms, which have in their own nature a self-moving force or principle.

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  • He may be said to furnish a further contribution to a metaphysical conception of evolution in his view of all finite individual things as the infinite variety to which the unlimited productive power of the universal substance gives birth.

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  • Later on he develops the materialistic view of Epicurus, only modifying it so far as to conceive of matter as finite.

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  • Since a finite number of particles is only susceptible of finite transpositions, it must happen (he says), in an eternal duration that every possible order or position will be tried an infinite number of times, and hence this world is to be regarded (as the Stoics maintained) as an exact reproduction of previous worlds.

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  • The more celebrated and central thesis of the book - this finite universe, the best of all such that are possible - also restates positions of Augustine and Aquinas.

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  • (See Infinitesimal Calculus.) He also discovered a method of deriving one curve from another, by means of which finite areas can be obtained equal to the areas between certain curves and their asymptotes.

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  • His argument for the Being of God is based on the hypothesis that thought - not individual but universal - is the reality of all things, the existence of this Infinite Thought being demonstrated by the limitations of finite thought.

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  • The thinker who sees man confronted by the infinite non-moral forces presumed by natural pantheism inevitably predominating over the finite powers of men may appear to the modern Christian theologian or to the evolutionist as a hopeless pessimist, and yet may himself have concluded that, though the future holds out no prospect save that of annihilation, man may yet by prudence and care enjoy a considerable measure of happiness.

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  • He holds that freedom is the inalienable prerogative of the finite spirit; and this is the second point that distinguishes his theology from the heretical Gnosticism.

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  • The name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis.

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  • This Essence is God, and includes within itself the finite unities of man, reason and nature.

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  • As creation involves intention, desire, thought and work, and as these are properties which imply limit and belong to a finite being, and moreover as the imperfect and circumscribed nature of this creation precludes the idea of its being the direct work of the infinite and perfect, the En Soph had to become creative, through the medium of ten Sephiroth or intelligences, which emanated from him like rays proceeding from a luminary.

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  • In this all-important doctrine of the Sephiroth, the Kabbalah insists upon the fact that these potencies are not creations of the En Soph, which would be a diminution of strength; that they form among themselves and with the En Soph a strict unity, and simply represent different aspects of the same being, just as the different rays which proceed from the light, and which appear different things to the eye, are only different manifestations of one and the same light; that for this reason they all alike partake of the perfections of the En Soph; and that as emanations from the Infinite, the Sephiroth are infinite and perfect like the En Soph, and yet constitute the first finite things.

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  • They are infinite and perfect when the En Soph imparts his fullness to them, and finite and imperfect when that fullness is withdrawn from them.

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  • We have already seen that the Sephiric decade or the archetypal man, like Christ, is considered to be of a double nature, both infinite and finite, perfect and imperfect.

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  • With these definitions it is now possible to prove the following six premisses applying to finite cardinal numbers, from which Peano 2 has shown that all arithmetic can be deduced i.

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  • Here also it can be seen that the science of the finite ordinals is a particular subdivision of the general theory of classes and relations.

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  • Owing to the correspondence between the finite cardinals and the finite ordinals, the propositions of cardinal arithmetic and ordinal arithmetic correspond point by point.

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  • But the definition of the cardinal number of a class applies when the class is not finite, and it can be proved that there are different infinite cardinal numbers, and that there is a least infinite cardinal, now usually denoted by o where to is the Hebrew letter aleph.

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  • Similarly, a class of serial relations, called well-ordered serial relations, can be defined, such that their corresponding relation-numbers include the ordinary finite ordinals, but also include relation-numbers which have many properties like those of the finite ordinals, though the fields of the relations belonging to them are not finite.

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  • If m and n are finite cardinal numbers, the rational number m/n is the relation which any finite cardinal number x bears to any finite cardinal number y when n X x = m X y.

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  • Under the general heading "Fundamental Notions" occur the subheadings "Foundations of Arithmetic," with the topics rational, irrational and transcendental numbers, and aggregates; "Universal Algebra," with the topics complex numbers, quaternions, ausdehnungslehre, vector analysis, matrices, and algebra of logic; and "Theory of Groups," with the topics finite and continuous groups.

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  • Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

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  • Lastly Baxter attempted to prove that matter is finite.

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  • The invariant theory then existing was classified by them as appertaining to " finite continuous groups."

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  • P. Gordan first proved that for any system of forms there exists a finite number of covariants, in terms of which all others are expressible as rational and integral functions.

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  • Of any form az there exists a finite number of invariants and covariants, in terms of which all other covariants are rational and integral functions (cf.

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  • This finite number of forms is said to constitute the complete system.

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  • Of two or more binary forms there are also complete systems containing a finite number of forms. There are also algebraic systems, as above mentioned, involving fewer covariants which are such that all other covariants are rationally expressible in terms of them; but these smaller systems do not possess the same mathematical interest as those first mentioned.

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  • Single binary forms of higher and finite order have not been studied with complete success, but the system of the binary form of infinite order has been completely determined by Sylvester, Cayley, MacMahon and Stroh, each of whom contributed to the theory.

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  • Perpetuants.-Many difficulties, connected with binary forms of finite order, disappear altogether when we come to consider the (p1p2p3...) to where form of infinite order.

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  • Thus it is used to translate the Platonic 'SEa, Et50s, the permanent reality which makes a thing what it is, in contrast with the particulars which are finite and subject to change.

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  • Any space at every point of which there is a finite magnetic force is called a field of magnetic force, or a magnetic field.

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  • If these equations could be assumed to hold when H is indefinitely small, it would follow that has a finite initial value, from which there would be no appreciable deviation in fields so weak that bH was negligibly small in comparison with a.

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  • One or more of the electrons may be detached from the system by a finite force, the number so detachable depending on the valency of the atom; if the atom loses an electron, it becomes positively electrified; if it receives additional electrons, it is negatively electrified.

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  • To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.

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  • All forms of monism from Plotinus downwards tend to ignore personal individuality and volition, and merge all finite existence in the featureless unity of the Absolute; this, indeed, is what inspires the passion of the protest against monism.

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  • God is not fully comprehensible by us, says Albert, because the finite is not able to grasp the infinite, yet he is not altogether beyond our knowledge; our intellects are touched by a ray of his light, and through this contact we are brought into communion with him.

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  • A perfectly formless matter (materia prima) was regarded by him as the universal substratum and common element of all finite existences.

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  • The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences.

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  • He was also the first to consider the difficult problems involved in equations of mixed differences, and to prove that an equation in finite differences of the first degree and the second order might always be converted into a continued fraction.

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  • Proceeding in this way, we may be able to express P= Q as the sum of a finite number of terms k+m/r+n/r 2 +..

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  • The approach to the limit will therefore be by a series of jumps, each of which, however small, will be finite; i.e.

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  • We cannot, for instance, say that the fraction C _2 I is arithmetically equal to x+I when x= I, as well as for other values of x; but we can say that the limit of the ratio of x 2 - I to x - I when x becomes indefinitely nearly equal to I is the same as the limit of x+ On the other hand, if f(y) has a definite and finite value for y = x, it must not be supposed that this is necessarily the same as the limit which f (y) approaches when y approaches the value x, though this is the case with the functions with which we are usually concerned.

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  • For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold's symbolical method in the calculus of invariants; and the like.

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  • If, however, as in practice, the light be heterogeneous, the source of finite area, the obstacles in motion, and the discrimination of different directions imperfect, we are concerned merely with the mean brightness found by varying the arbitrary phase-relations, and this is obtained by simply multiplying the brightness due to a single aperture by the number of apertures (n) (see Interference Of Light, § 4).

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  • Hence, in accordance with the rule for compounding vector quantities, the resultant vibration at B, due to any finite part of the primary wave, is represented in amplitude and phase by the chord joining the extremities of the corresponding arc (U2-0.1).

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  • Integrating by parts in (II), we get J e = ikr d7 pc-11 / d (e r - ay= rJ Z d y - r / 1 dY, in which the integrated terms at the limits vanish, Z being finite only within the region T.

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  • According to Aristotle, "the first of Eleatic unitarians was not careful to say whether the unity which he postulated was finite or infinite, but, contemplating the whole firmament, declared that the One is God."

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  • ', In Bobyleff's problem of the wedge of finite breadth, ch nS2 = ?a b' s h n S 2 = V b a a 1 u u b, (6)?

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  • - cos 2 na sin2ng s (9) sin 2 na - sin2n0 Putting n =I gives the case of a stream of finite breadth disturbed y a transverse plane, a particular case of Fig.

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  • The method of electrical images will enable the stream function, )' to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle 7r/m, where m is an integer (R.

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  • Taylor's Methodus Incrementorum Directa et Inversa (London, 1715) added a new branch to the higher mathematics, now designated the " calculus of finite differences."

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  • Emphasis was made to fall on the reason, the conscience and the will of the finite personality; and just as these were found to be native in him they were held to be immanent in the cause of his universe.

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  • In 1734 he also published Prodromus philosophiae ratiocinantis de infinito et causa finali creationis, which treats of the relation of the finite to the infinite, and of the soul to the body, seeking to establish a nexus in each case as a means of overcoming the difficulty of their relation.

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  • In God there are three infinite and uncreated "degrees" of being, and in man and all things corresponding three degrees, finite and created.

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  • It defines the condition which must be fulfilled by the potential at any and every point in an electric field, through which p is finite and the electric force continuous.

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  • The Poisson equation cannot, however, be applied in the above form to a region which is partly within and partly without an electrified conductor, because then the electric force undergoes a sudden change in value from zero to a finite value, in passing outwards through the bounding surface of the conductor.

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  • Combining this with the first law, for a Carnot cycle of finite range, if H is the heat taken in at 0', and H" is the heat rejected at 0", the work W done in the cycle is equal to the difference H' - H", and we have the simple relations, W/(0' - o") =H'/o' =H" o".

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  • of temperature (o' - o") is small, the figure ABCD may be regarded as a parallelogram, and its area W as equal to the rectangle BE XEC. This is accurately true in the limit when (0' - 0") is infinitesimal, but in practice it is necessary to measure specific heats, &c., over finite ranges of temperature, and the error involved is generally negligible if the range does not exceed a few degrees.

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  • It is not necessary in this example that AB, CD should be adiabatics, because the change of volume BC is finite.

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  • For a finite change it is necessary to represent the path by a series of small steps, which is the graphic equivalent of integration along the path represented by the given relation between v and 0, or p and 0.

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  • Since the change of energy is independent of the path, the finite change between any two given states may be found by integration along any convenient path.

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  • From the hypothesis of an external world a series of contradictions are deduced, such as that the world is both finite and infinite, is movable and immovable, &c.; and finally, Aristotle and various other philosophers are quoted, to show that the external matter they dealt with, as mere potentiality, is just nothing at all.

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  • In it the union between the finite self-consciousness and the infinite ego or God is handled in an almost mystical manner.

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  • Meanwhile, the same considerations had not been applied to time, so that in the days of Zeno of Elea time was still regarded as made up of a finite number of ` moments,' while space was confessed to be divisible without limit.

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  • Van der Waals, in a famous monograph, On the Continuity of the Liquid and Gaseous States (Leiden, 1873), has shown that the imperfections of equation (14) maybe traced to two_causes: (i.) The calculation has not allowed for the finite size of the molecules, and their consequent interference with one another's motion, and (ii.) The calculation has not allowed for the field of inter-molecular force between the molecules, which, although small, is known to have a real existence.

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  • The universal or infinite is one that realizes itself in finite particular minds and wills, not as accidents or imperfections of it, but as its essential form.

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  • It involves, it will be said, the reality of time, the dependence of the Infinite in the finite, and therewith a departure from the whole line of Hegelian thought.

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  • It is to be remedied not by giving up the idea of the Infinite but by ceasing to think of the Infinite as of a being endowed with a static perfection which the finite will merely reproduces, and definitely recognizing the forward effort of the finite as an essential element in Its self-expression.

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  • The general method of constructing formulae of this kind involves the use of the integral calculus and of the calculus of finite differences.

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  • In the case of statistical data there is the further difficulty that there is no real continuity, since we are concerned with a finite number of individuals.

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  • For the methods involving finite differences, see references under DIFFERENCES, CALCULUS OF; and INTERPOLATION.

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  • The receiving apparatus had what we may term a personal equation, for the break of contact could only take place when the membrane travelled some finite distance, exceedingly small no doubt, from the contact-piece.

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  • Since the curve represents a longitudinal disturbance in air it is always continuous, at a finite distance from the axis, and with only one ordinate for each abscissa.

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  • If we rest on the synthesis here described, the energy of the matter, even the thermal part, appears largely as potential energy of strain in the aether which interacts with the kinetic energy associated with disturbances involving finite velocity of matter.

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  • With reference to all such further refinements of theory, it is to be borne in mind that the perfect fluid of hydrodynamic analysis is not a merely passive inert plenum; it is also a continuum with the property that no finite internal slip or discontinuity of motion can ever arise in it through any kind of disturbance; and this property must be postulated, as it cannot be explained.

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  • Now the kinetics of a medium in which the parts can have finite relative motions will lead to equations which are not linear - as, for example, those of hydrodynamics - and the phenomena will be far more complexly involved.

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  • Formulae of the calculus of finite differences enable us from the chronograph records to infer the velocity and retardation of the shot, and thence the resistance of the air.

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  • w /nd2 = C, (16) S(V) - S(v) = for cos i to be undistinguishable from unity, equation (16) becomes In a problem of direct fire, where the trajectory is flat enough (19) v(di/dt=g, or di/dt=g/v; so that we can put (20) Ai/ At.t = g/v, if v denotes the mean velocity during the small finite interval of time At, during which the direction of motion of the shot changes through Ai radians.

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  • Now taking equation (72), and replacing tan B, as a variable final tangent of an angle, by tan i or dyldx, (75) tan 4) - dam= C sec n [I(U) - I(u)], and integrating with respect to x over the arc considered, (76) x tan 4, - y = C sec n (U) - f :I(u)dx] 0 But f (u)dx= f 1(u) du = C cos n f x I (u) u du g f() =C cos n [A(U) - A(u)] in Siacci's notation; so that the altitude-function A must be calculated by summation from the finite difference AA, where (78) AA = I (u) 9 = I (u) or else by an integration when it is legitimate to assume that f(v) =v m lk in an interval of velocity in which m may be supposed constant.

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  • - C = C cos n [5 (u 5) - S(ue)], ' y / e 0 A =tan - C sec n [I (u 0) - S] A now denoting any finite tabular difference of the function between the initial and final (pseudo-) velocity.

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  • The primeval Being is, as opposed to the many, the One; as opposed to the finite, the Infinite, the unlimited.

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  • It is, moreover, the Good, in so far as all finite things have their purpose in it, and ought to flow back to it.

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  • As a single soul (world-soul) it belongs in essence and destination to the intelligible world; but it also embraces innumerable individual souls; and these can either submit to be ruled by the nous, or turn aside to the sensual and lose themselves in the finite.

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  • Thus log x is the integral function of 1/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as algebraical or circular functions.

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  • But the Fichtean teaching appeared on the one hand to identify too closely the ultimate ground of the universe of rational conception with the finite, individual spirit, and on the other hand to endanger the reality of the world of nature by regarding it too much after the fashion of subjective idealism, as mere moment, though necessitated, in the existence of the finite thinking mind.

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  • External things are produced by the will of the divine intelligence; they are caused, and caused in a regular order; there exists in the divine mind archetypes, of which sense experience may be said to be the realization in our finite minds.

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  • The relation between the divine mind and finite intelligence, at first thought as that of agent and recipient, is complicated and obscure when the necessity for explaining the permanence of real things comes forward.

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  • This apparent motion is due to the finite velocity of light, and the progressive motion of the observer with the earth, as it performs its yearly course about the sun.

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  • The limitation of power is introduced as in all optical instruments, by the finiteness of the length of a wave of light which causes the image of an indefinitely narrow slit to spread out over a finite width in the focal plane of the observing telescope.

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  • Thus if a molecule were set into vibration at a specified time and oscillated according to the above equation during a finite period, it would not send out homogeneous vibrations.

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  • itself, not in Aristotle's sense of any individual existing differently from anything else, but in the novel meaning of something existing alone, he concluded, logically enough from this mere misunderstanding, that there can be only one substance, and that, as no finite body or soul can exist alone, everything finite is merely a mode of one of the attributes of the one infinite substance which alone can exist by itself.

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  • Having, however, in consequence, lost his professorship at Jena, he gradually altered his views, until at length he decided that God is not mere moral order, but also reason and will, yet without consciousness and personality; that not mankind but God is the absolute; that we are only its direct manifestations, free but finite spirits destined by God to posit in ourselves Nature as the material of duty, but blessed when we relapse into the absolute; that Nature, therefore, is the direct manifestation of man, and only the indirect manifestation of God; and, finally, that being is the divine idea or life, which is the reality behind appearances.

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  • Schelling perceived that Hegel, in reducing everything to infinite mind, absorbed man's free but finite personality in God, and, in declaring that everything real is rational, failed to explain evil and sin: indeed, the English reader of T.

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  • But in a philosophy which reduces everything to phenomenal appearance except the self-interacting substance of God, there is no room for either the bodies or the souls of finite substances or human persons.

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  • Bradley, however, having satisfied himself, like Spinoza, by an abuse of the word " independent," that " the finite is self-discrepant," goes on to ask what the one Real, the absolute, is; and, as he passed from Herbart to Spinoza, so now he passes from Spinoza to Kant.

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  • 1855, professor of philosophy, Harvard) believes in the absolute like Green and Bradley, in " the unity of a single self-consciousness, which includes both our own and all finite conscious meanings in one final eternally present insight," as he says in The World and the Individual (1900; see also later works).

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  • Not only so, but in his review of Cousin (" Philosophy of the Unconditioned," in Discussions, pp. 12-15), he made conception the test of knowledge, argued that " the mind can conceive, and consequently can know, only the limited, and the conditionally limited," that " to think is to condition," that all we know either of mind or matter is " the phenomenal," that " we can never in our highest generalizations rise above the finite," and concluded that we cannot conceive or know the unconditioned, yet must believe in its existence.

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  • Calderwood, in his Philosophy of the Infinite (1854), made the pertinent objection that, though thought, conception and knowledge are finite, the object of thought may be infinite.

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  • Hamilton, in fact, made the double mistake of limiting knowledge to what we can conceive, and confusing the determinate with the finite or limited.

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  • Further, to allow of a correction being applied for the finite length of the magnets the whole series of settings is repeated with the centre of the deflecting magnet at 40 cm.

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  • Unlike the Hindu, Xenophanes inclined to pantheism as a protest against the anthropomorphic polytheism of the time, which seemed to him improperly to exalt one of the many modes of finite existence into the place of the Infinite.

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  • characteristic is primarily the negation of the Finite.

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  • We find Philo Judaeus endeavouring to free the concept of the Old Testament Yahweh from anthropomorphic characteristics and finite determinations.

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  • On the principle that everything which is determined (finite) is "negated" ("determinatio est negatio"), God, the ultimate reality must be entirely undetermined.

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  • Like Hamilton, Mansel maintained the purely formal character of logic, the duality of consciousness as testifying to both self and the external world, and the limitation of knowledge to the finite and "conditioned."

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  • The first volume treats of the analysis of finite quantities, and the second of the analysis of infinitesimals.

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  • A people's religion completes and consecrates their whole activity: in it the people rises above its finite life in limited spheres to an infinite life where it feels itself all at one.

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  • Even philosophy with Hegel at this epoch was subordinate to religion; for philosophy must never abandon the finite in the search for the infinite.

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  • This is, however, an application of categories from a field where the conditions are finite to a sphere in which the circumstances are infinite."

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  • The state is the consummation of man as finite; it is the necessary starting-point whence the spirit rises to an absolute existence in the spheres of art, religion and philosophy.

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  • In the finite world or temporal state, religion, as the finite organization of a church, is, like other societies, subordinate to the state.

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  • For the Son of God, in the immediate aspect, is the finite world of nature and man, which far from being at one with its Father is originally in an attitude of estrangement.

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  • The earlier Hegelians had interpreted it in the sense that the world in its ultimate essence was not only spiritual but self-conscious intelligence whose nature was reflected inadequately but truly in the finite mind.

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  • The completeness and self-consistency which our ideal requires can be realized only in a form of being in which subject and object, will and desire, no longer stand as exclusive opposites, from which it seemed at once to follow that the finite self could not be a reality nor the infinite reality a self.

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  • The scope of his researches was described by Arthur Cayley, his friend and fellow worker, in the following words: "They relate chiefly to finite analysis, and cover by their subjects a large part of it - algebra, determinants, elimination, the theory of equations, partitions, tactic, the theory of forms, matrices, reciprocants, the Hamiltonian numbers, &c.; analytical and pure geometry occupy a less prominent position; and mechanics, optics and astronomy are not absent."

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  • Bezout's method of elimination, the other on the number of integrals of an equation of finite differences.

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  • Thus we distinguish in the divine system beginning, middle and end; but these three are in essence one - the difference is only the consequence of our finite comprehension.

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  • No finite predicates can be applied to him; his mode of being cannot be determined by any category.

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  • The Mosaic account, then, is to be looked upon merely as a mode in which is faintly shadowed forth what is above finite comprehension.

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  • Metaphysically, all realities are parts of one ultimate reality; but logically, even philosophers think more often only of finite realities, existing men, dogs, horses, &c.; and children know that their parents exist long before they apprehend ultimate reality.

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  • But when, instead of the highly artificial expression ix-}-jy+kz, to denote a finite directed line, we employ a single letter, a (Hamilton uses the Greek alphabet for this purpose), and find that we are permitted to deal with it exactly as we should have dealt with the more complex expression, the immense gain is at least in part obvious.

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  • In the third-order complex the centre locus becomes a finite closed quartic surface, with three (one always real) intersecting nodal axes, every plane section of which is a trinodal quartic. The chief defect of the geometrical properties of these bi-quaternions is that the ordinary algebraic scalar finds no place among them, and in consequence Q:1 is meaningless.

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  • Their speculation as to the nature of God had led them gradually to separate him by an infinite distance from all creation, and to feel keenly the opposition of the finite and the infinite, the perfect and the imperfect, the eternal and the temporal.

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  • By it we perceive how God, the infinite, the absolute, the eternal, is yet not separated from the finite, the temporal, the relative, but, through the incarnation, enters into humanity.

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  • Unlike Thales, he was struck by the infinite variety in things; he felt that all differences are finite, that they have emerged from primal unity (first called epxn by him) into which they must ultimately return, that the Infinite One has been, is, and always will be, the same, indeterminate but immutable.

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  • Chasles any displacement whatever of the lamina in its own plane is equivalent to a rotation about some finite or infinitely distant point J.

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  • This point is called the centre of the given system of parallel forces; it is finite and determinate unless ~(P) = 0.

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  • We proceed to sketch the theory of the finite displacements of a rigid body.

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  • rotation will bring a point from C to C and that the second will bring it back to C; the result is therefore equivalent to a rotation about OC. We note also that if the given rotations had been effected in the inverse order, the axis of the resultant rotation would have been OC, so that finite rotations do not obey the commutative law.

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  • The composition of finite rotations about parallel axes is, a particular case of the preceding; the radius of the sphere is now infinite, and the triangles are plane.

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  • For this purpose the infinitesimal displacements of W the various joints are replaced by finite lengths proportional to them, and there- FIG.

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  • Again, when extraneous forces P act on the joints, the equation is Z(P.&P)+S.Os=-o, where op is the displacement of any joint in the direction of the corresponding force P. If ~(P. Op) =o, the stresses are merely indeterminate as before; but if ~ (P. op) does not vanish, the equation cannot be satisfied by any finite value of S, since Os =0.

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  • This means that, if the material of the frame were absolutely unyielding, no finite stresses in the bars would enable it to withstand the extraneous forces.

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  • The stresses in the bars would then be comparatively very great, although finite.

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  • It follows that the forces on any finite portion will satisfy the conditions of equilibrium which apply to the case of a rigid body (~ 4).

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  • In the case of a chain hanging freely under gravity it is usually convenient to formulate the conditions of equilibrium of a finite portion PQ.

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  • If the particles start from rest at a finite distance c, we have in (I6), C = 2p1C, and therefore ~=u=_,/~2M ~ (fr)

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  • For purposes of mathematical treatment a force which produces a finite change of velocity in a time too short to be appreciated is regarded as infinitely great, and the time of action as infinitely short.

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  • The effect of ordinary finite forces during the infinitely short duraticm of this impulse is of course ignored.

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  • The total impulse in any finite interval of time is the integral of the impulses corresponding to the infinitesimal elements 3t into which the interval may be subdivided; the summation of which the integral is the limit is of course to be understood in the vectorial sense.

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  • The question presents itself whether ther then is any other law of force, giving a finite velocity from infinity, under which all finite orbits are necessarily closed curves.

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  • mathematical points endowed with inertia coefficients, separated by finite intervals, and acting on one another with forces in the lines joining them subject to the law of equality of action and reaction.

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  • Equations of Motion in Generalized Co-ordinates.Suppose we have a dynamical system composed of a finite number of material particles or rigid bodies, whether free or constrained in any way, which are subject to mutual forces and also to the action of any given extraneous forces~ The configuration of such a system can be completely specified by means of a certain number (n) of independent quantities, called the generalized coordinates of the system.

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  • If, on the other hand, the body is finite, certain terminal conditions have to be satisfied.

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  • The solution of (40) which is finite for x = o is readily obtained in the form of a series, thus y=C(I_~-I-~1~_...) =CJo(z),, (4)

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  • In order to allow for the finite stiffness and strength of materials, the least distance of the centre of resistance inward from the nearest edge of the joint is made to bear a definite proportion to the depth of the joint measured in the same direction, which proportion is fixed, sometimes empirically, sometimes by theoretical deduction from the laws of the strength of materials.

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  • The essence of God, for instance, he held to be love; God, he said, can love nothing inferior to himself; but he cannot be an object of love to himself without going out, so to speak, of himself, without manifesting his infinity in a finite form; in other words, by becoming man.

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  • These effects, as Hertz showed, indicated the establishment of stationary electric waves in space and the propagation of electric and magnetic force through space with a finite velocity.

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  • Cudworth's ideas, like Plato's, have "a constant and never-failing entity of their own," such as we see in geometrical figures; but, unlike Plato's, they exist in the mind of God, whence they are communicated to finite understandings.

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  • He holds that we are rationally justified in affirming human immortality and the existence of a finite God who is to be a constitutional ruler, but not a despot, over the souls of men.

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  • The histories of philosophy may quite correctly describe his theory as the logical development of Descartes's doctrines of the one Infinite and the two finite substances, but Spinoza himself was never a Cartesian.

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  • The foundation of the system is the doctrine of one infinite substance, of which all finite existences are modes or limitations (modes of thought or modes of extension).

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  • Bostrom's philosophy is logically expressed and based on the one great conception of a spiritual, eternal, immutable Being, whose existence is absolute, above and external to the finite world of time and space.

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  • 6) at a finite distance from the axis (or with an infinitely O distant object, a point which subtends a finite angle at the system) is, in general, even then not sharply reproduced, if the pencil of rays issuing FIG.

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  • In the mathematical sense, however, this selection is arbitrary; the reproduction of a finite object with a finite aperture entails, in all probability, an infinite number of aberrations.

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  • This number is only finite if the object and aperture are assumed to be " infinitely small of a certain order"; and with each order of infinite smallness, i.e.

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  • with each degree of approximation to reality (to finite objects and apertures), a certain number of aberrations is associated.

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  • The existence of an optical system, which reproduces absolutely a finite plane on another with pencils of finite aperture, is doubtful; but practical systems solve this problem with an accuracy which mostly suffices for the special purpose of each species of instrument.

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  • In order to render spherical aberration and the deviation from the sine condition small throughout the whole aperture, there is given to a ray with a finite angle of aperture u* (with infinitely distant objects: with a finite height of incidence the same distance of intersection, and the same sine ratio as to one neighbouring the axis (u* or h* may not be much smaller than the largest aperture U or H to be used in the system).

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  • ” If in the infinite continued fraction of the second class an?bn+i for all values of n, it converges to a finite limit not greater than unity.” 3.

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  • converges to an incommensurable limit if after some finite value of n the condition an

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  • - an verges to an incommensurable limit if after some finite value of n the condition a n ?b n +I is always satisfied, where the sign > need not always occur but must occur infinitely often.

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  • The cycle of its transformations and successive condensations constitutes the life of the universe, the mode of existence proper to finite and particular being.

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  • We may investigate the forces which act between finite portions of a liquid in the same way as we investigate the forces which act between finite portions of a solid.

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  • The first part establishes the laws of the elasticity of a finite portion of the solid subjected to a homogeneous strain, and deduces from these laws the equations of the equilibrium and motion of a body subjected to any forces and displacements.

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  • The second part endeavours to deduce the facts of the elasticity of a finite portion of the substance from hypotheses as to the motion of its constituent molecules and the forces acting between them.

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  • Laplace assumed that the liquid has uniform density, and that the attraction of its molecules extends to a finite though insensible distance.

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  • For insensible values it may become sensible, but it must remain finite even when z = o, in which case 0(o) = K.

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  • Integrating the first term within brackets by parts, it becomes - fo de Remembering that 0(o) is a finite quantity, and that Viz = - (z), we find T = 4 7rp f a, /.(z)dz (27) When c is greater than e this is equivalent to 2H in the equation of Laplace.

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  • Since 0(o) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply f 0(z)dz f: (z)dz, (34) and T= ref: z1,1,(z)dz (35) In Laplace's notation the second member of (34), multiplied by 27r, is represented by H.

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  • - g 2) t; and this ratio decreases without limit with the time, whatever be the initial (finite) ratio a 2: a1.

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  • Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself.

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  • Four of his books were of particular importance: Christian Nurture (1847), in which he virtually opposed revivalism and "effectively turned the current of Christian thought toward the young"; Nature and the Supernatural (1858), in which he discussed miracles and endeavoured to "lift the natural into the supernatural" by emphasizing the supernaturalness of man; The Vicarious Sacrifice (1866), in which he contended for what has come to be known as the "moral view" of the atonement in distinction from the "governmental" and the "penal" or "satisfaction" theories; and God in Christ (1849) (with an introductory "Dissertation on Language as related to Thought"), in which he expressed, it was charged, heretical views as to the Trinity, holding, among other things, that the Godhead is "instrumentally three - three simply as related to our finite apprehension, and the communication of God's incommunicable nature."

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  • He seems to waver between the opinion that finite individuals have no independent being and the opinion that they have it in an infinitesimal degree; and the conception of " degrees of existence " in the essay on Virtue is not developed to elucidate the point.

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  • More exactly, it may be said that the Platonism of Plato's maturity included the following principal doctrines: (i.) the supreme cause of all existence is the One, the Good, Mind, which evolves itself as the universe under certain eternal immutable forms called " ideas"; (ii.) the ideas are apprehended by finite minds as particulars in space and time, and are then called " things"; (iii.) consequently the particulars which have in a given idea at once their origin, their being, and their perfection may be regarded, for the purposes of scientific study, as members of a natural kind; (iv.) the finite mind, though it cannot directly apprehend the idea, may, by the study of the particulars in which the idea is revealed, attain to an approximate notion of it.

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  • ' In abandoning the theory of ideas - that is to say, the theory of figures and numbers, the possessions of universal mind, eternally existent out of space and time, which figures and numbers when they pass into space and time as the heritage of finite minds are regarded as things - Speusippus had the approval, as of the Platonists generally, so also of Aristotle.

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  • In such a case the two systems must be regarded as a single more complex one, the absorbed vibration becomes large, though remaining always finite, and the transmitted vibration undergoes a remarkable change in its period.

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  • From the basis of immediate experience or perception thought proceeds by comparison and abstraction, establishing connexions among facts, but remaining in its nature mediate and finite.

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