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The form is infinitely finer in Matt.

3318We don't need our computers to be infinitely fast, just a whole lot faster than they are today.

287Truth varies infinitely under circumstances whose relative weight cannot be accurately gauged.

2515Estimates of Pilate's attitude at this point have varied infinitely, from Tertullian's, that he was " already in conviction a Christian " - jam pro sua conscientia Christianus- to Bacon's " jesting Pilate," who would not stay for a reply.

156It is often necessary to remind her that there are infinitely many things that the wisest people in the world cannot explain.

135We can still point to much in exact science that is absolutely his; and we can indicate infinitely more which is due to his inspiration.

136Probably I should not consciously and deliberately forsake my particular calling to do the good which society demands of me, to save the universe from annihilation; and I believe that a like but infinitely greater steadfastness elsewhere is all that now preserves it.

126Certainly it was true of him, in a far higher degree than of John Henry Newman, that the being of God and himself were to his mind two absolutely self-luminous truths - though both his God and his self were almost infinitely remote from Newman's.

106She almost equalled her husband in knowledge, and infinitely excelled him in talent and in tact.

1011She almost equalled her husband in knowledge, and infinitely excelled him in talent and in tact.

10112s = 277-h/A f (14) e = Af /h (15) The bands are thus of the same width as those due to two infinitely narrow apertures coincident with the central lines of the retarded and unretarded streams, the subject of examination being itself a fine luminous line.

832s = 277-h/A f (14) e = Af /h (15) The bands are thus of the same width as those due to two infinitely narrow apertures coincident with the central lines of the retarded and unretarded streams, the subject of examination being itself a fine luminous line.

83We don't need miniaturization to go to infinitely small, just really, really small.

84Dubuat considered that if water were a perfect fluid, and the channels in which it flowed infinitely smooth, its motion would be continually accelerated, like that of bodies descending in an inclined plane.

85Only when we have admitted the conception of the infinitely small, and the resulting geometrical progression with a common ratio of one tenth, and have found the sum of this progression to infinity, do we reach a solution of the problem.

72Firmness such as this secured for him the support of all constitutional elements, and after three years premiership his position was infinitely stronger than at the outset.

76According to common optics, where images are absolute, the diffraction pattern is supposed to be infinitely small, and two radiant points, however near together, form separated images.

76a disturbance limited to an infinitely thin slice of the medium, is supposed to fall upon a parallel grating, which again may With the best thickness so that in this case w=0 w = (2 w w= 0 be regarded as formed of infinitely thin wires, or infinitely narrow lines traced upon glass.

76But how can the infinitely active ego posit a limit to its own activity?

76a disturbance limited to an infinitely thin slice of the medium, is supposed to fall upon a parallel grating, which again may With the best thickness so that in this case w=0 w = (2 w w= 0 be regarded as formed of infinitely thin wires, or infinitely narrow lines traced upon glass.

76But this puts an infinitely worse face on the matter, and suggests, beside, that probably not even the other three succeed in saving their souls, but are perchance bankrupt in a worse sense than they who fail honestly.

54By night, of course, the perplexity is infinitely greater.

55But let it be observed, first, that to reduce the huge and confused mass of pre-existing law into the compass of these two collections was an immense practical benefit to the empire; secondly, that, whereas the work which he undertook was accomplished in seven years, the infinitely more difficult task of codification might probably have been left unfinished at Tribonian's death, or even at Justinian's own, and been abandoned by his successor; thirdly, that in the extracts preserved in the Digest we have the opinions of the greatest legal luminaries given in their own admirably lucid, philosophical and concise language, while in the extracts of which the Codex is composed we find valuable historical evidence bearing on the administration and social condition of the later Pagan and earlier Christian empire; fourthly, that Justinian's age, that is to say, the intellect of the men whose services he commanded, was quite unequal to so vast an undertaking as the fusing upon scientific principles into one new organic whole of the entire law of the empire.

43With an approximate date—or at least a year—and a first name, the chore would be infinitely easier than scouring decades for a nameless individual.

45It can be shown that, if the density of distribution of the stars through infinite space is nowhere less than a certain limit (which may be as small as we please), the total amount of light received from them (assuming that there is no absorption of light in space) would be infinitely great, so that the background of the sky would shine with a dazzling brilliancy.

45It can be shown that, if the density of distribution of the stars through infinite space is nowhere less than a certain limit (which may be as small as we please), the total amount of light received from them (assuming that there is no absorption of light in space) would be infinitely great, so that the background of the sky would shine with a dazzling brilliancy.

45To the infinitely actual there is necessary the possible; that which determines involves somewhat in which its determinations can have existence.

31The universe, then, is a living cosmos, an infinitely animated system, whose end is the perfect realization of the variously graduated forms. The unity which sunders itself into the multiplicity of things may be called the monas monadum, each thing being a monas or self-existent, living being, a universe in itself.

31It may be instructive to contrast this with the case of an infinitely narrow annular aperture, where the brightness is proportional to Jo 2 (z).

32A combination of the Babylonian with the Persian religion could only be effected by the degradation of the Babylonian deities into half-divine, half-daemonic beings, infinitely remote from the supreme God of light and of heaven, or even into powers of darkness.

32Moreover, did it do so, the action upon the human system would be infinitely less harmful than similar doses of copper or of lead.

33Dean cursed himself for not wanting to talk to the redheaded deputy, who would have been infinitely preferable to this obnoxious jerk.

34They skied together and to Dean it was a long and infinitely pleasant run.

34This conception of matter, as infinitely divisible and continuous, was taught by Anaxagoras more than four centuries before the Christian era, and in the philosophy of Aristotle the same ideas are found.

34The goddess of art of Akishino-dera, Nara, attributed to the 8th century, is the most graceful and least conventional of female sculptures in Japan, but infinitely remote from the feminine conception of the Greeks.

35Though he lacked the brilliant qualities of his rival Wallqvist, Nordin had the same alertness and penetration, and was infinitely more stable and disinterested.

21Although the above useful formulae, (io) to (15), are true only for an infinitely small magnet, they may be practically applied whenever the distance r is considerable compared with the length of the magnet.

21If we make the extreme suppositions of an infinitely small source and absolutely homogeneous light, there is no escape from the conclusion that the light in a definite direction is arbitrary, that is, dependent upon the chance distribution of apertures.

21Complete adaptation to an infinitely receding ideal is impossible, and relative adaptation depends on the distance between the actual and the ideal.

21This fact, together with the extraordinarily rare occurrence of such remains and meteoric particles in globigerina ooze, although there is no reason to suppose that at any one time they are unequally distributed over the ocean floor, can only be explained on the assumption that the rate of formation of the epilophic deposits through the accumulation of pelagic shells falling from the surface is rapid enough to bury the slowgathering material which remains uncovered on the spaces where the red clay is forming at an almost infinitely slower rate.

21Garrick was surrounded by many players of eminence, and he had the art, as he was told by Mrs Clive, " of contradicting the proverb that one cannot make bricks without straw, by doing what is infinitely more difficult, making actors and actresses without genius."

21But the years of his studious retirement were spent on a work of infinitely greater importance.

21As a politician he ended his career with his submission to Henry, who found in Roger of Salisbury a financier not less able and infinitely more acceptable to the nation.

21Although the above useful formulae, (io) to (15), are true only for an infinitely small magnet, they may be practically applied whenever the distance r is considerable compared with the length of the magnet.

21As a politician he ended his career with his submission to Henry, who found in Roger of Salisbury a financier not less able and infinitely more acceptable to the nation.

21A modern branch of mathematics having achieved the art of dealing with the infinitely small can now yield solutions in other more complex problems of motion which used to appear insoluble.

21Even the characteristic dualism of Gnosticism has already proved to be in part of Iranian origin; and now it becomes clear how from that mingling of late Greek and Persian dualism the idea could arise that these seven halfdaemonic powers are the creators or rulers of this material world, which is separated infinitely from the light-world of the good God.

22For the period preceding British rule the records have not been so well preserved, but there is ample evidence to show that famine was just as frequent in its incidence and infinitely more deadly in its effects under the native rulers of India.

22And 1 In a letter dated the 4th of April 1882, referring to the publication of his drama Mary Stuart, Swinburne wrote, to Edmund Clarence Stedman: "Mary Stuart has procured me two satisfactions which I prefer infinitely to six columns of adulation in The Times and any profit thence resulting.

22If the distance between the disks could be made infinitely small each time, then the multiplier r would be 2, and the charge would be doubled each time.

22either amongst those Jews who looked to the coming of a Messiah or, Such a view on the part of men who knew Arabic infinitely better than the most accomplished European Arabist will ever do, may well startle us.

22In Nubia, owing to the poverty of the country and its scanty population, the proportion of monuments surviving is infinitely greater than in Egypt.

22Literary Records.In estimating the sources of information regarding pre-Christlan Egypt, the native sources, first opened to us by Champollion, are infinitely the most important.

22This seemed to satisfy the official, who was obviously embarrassed over the entire debacle and infinitely relieved that at least half their problem was solved without any accusatory newspaper headlines.

23The Hungarian diet frantically opposed every Austrian alliance as endangering the national independence, but to any unprejudiced observer a union with the house of Habsburg, even with the contingent probability of a Habsburg king, was infinitely preferable to the condition into which Hungary, under native aristocratic misrule, was swiftly drifting.

23He would have plunged England into war with Spain in 1572, when the risks would have been infinitely greater than in 1588, and when the Huguenot influence over the French government, on which he relied for support, would probably have broken in his hands.

23By monarchians of the former class Christ was held to be a mere man, miraculously conceived indeed, but constituted the Son of God simply by the infinitely high degree in which he had been filled with Divine wisdom and power.

23It Is Therefore Infinitely More Commodious To Determine The Commencement Of The Year By A Fixed Rule Of Intercalation; And Of The Various Methods Which Might Be Employed, No One, Perhaps Is On The Whole More Easy Of Application, Or Better Adapted For The Purpose Of Computation, Than The Gregorian Now In Use.

23But even if--imagining a man quite exempt from all influences, examining only his momentary action in the present, unevoked by any cause--we were to admit so infinitely small a remainder of inevitability as equaled zero, we should even then not have arrived at the conception of complete freedom in man, for a being uninfluenced by the external world, standing outside of time and independent of cause, is no longer a man.

11Abandoning the conception of cause, mathematics seeks law, that is, the property common to all unknown, infinitely small, elements.

11Clerk Maxwell supposed two compartments, A and B, to be filled with gas at the same temperature, and to be separated by an ideal, infinitely thin partition containing a number of exceedingly small trap-doors, each of which could be opened or closed without any expenditure of energy.

12But in all moral qualities the brilliant adventurer of the r 5th was infinitely superior to the brilliant adventurer of the r9th century.

12Thought and extension are peaceable attributes in this one substance; there are infinitely many other attributes, but these only are known to us.

12In fact a little consideration of the theory of probabilities will show it to be infinitely probable that such an object should really have some movement of rotation, no matter by what causes the nebula may have originated.

12To the student of the original texts Lancelot is an infinitely less interesting hero than Gawain, Perceval or Tristan, each of whom possesses a well-marked personality, and is the centre of what we may call individual adventures.

12Interment in the wall-recess or loculus, though infinitely the most common, was not the only mode employed in the catacombs.

12The poles at the ends of an infinitely thin uniform magnet, or magnetic filament, would act as definite centres of force.

12The integrations may also be effected by means of polar coordinates, taking first the integration with respect to 4) so as to obtain the result for an infinitely thin annular aperture.

12If the origin of light be treated as infinitely small, and be seen in focus, whether with the naked eye or with the aid of a telescope, the whole of the light in the absence of obstacles would be concentrated in the immediate neighbourhood of the focus.

12If we suppose the diameter of the lens to be given (2R), and its focal length f gradually to increase, the original differences of phase at the image of an infinitely distant luminous point diminish without limit.

12On the other hand, infinitely thin films of silver which can be produced chemically on glass surfaces are absolutely opaque.

12He is living, active intelligence, the principle of motion and creation, realizing himself in the infinitely various forms of activity that constitute individual things.

12The angels are infinitely numerous20.

12If the origin of light be treated as infinitely small, and be seen in focus, whether with the naked eye or with the aid of a telescope, the whole of the light in the absence of obstacles would be concentrated in the immediate neighbourhood of the focus.

12At the same time that we are earnest to explore and learn all things, we require that all things be mysterious and unexplorable, that land and sea be infinitely wild, unsurveyed and unfathomed by us because unfathomable.

12These had an infinitely wider renown in their day, but modern criticism has restored the balance in his favour, and is even in danger of erring in the opposite direction.

13This is tantamount to an assumption that A is infinitely small.

13This is tantamount to an assumption that A is infinitely small.

13The propositions maintained in the argument are - "(1) That something has existed from eternity; (2) that there has existed from eternity some one immutable and independent being; (3) that that immutable and independent being, which has existed from eternity, without any external cause of its existence, must be self-existent, that is, necessarily existing; (4) what the substance or essence of that being is, which is self-existent or necessarily existing, we have no idea, neither is it at all possible for us to comprehend it; (5) that though the substance or essence of the self-existent being is itself absolutely incomprehensible to us, yet many of the essential attributes of his nature are strictly demonstrable as well as his existence, and, in the first place, that he must be of necessity eternal; (6) that the self-existent being must of necessity be infinite and omnipresent; (7) must be but one; (8) must be an intelligent being; (9) must be not a necessary agent, but a being endued with liberty and choice; (to) must of necessity have infinite power; (I I) must be infinitely wise, and (12) must of necessity be a being of infinite goodness, justice, and truth, and all other moral perfections, such as become the supreme governor and judge of the world."

00Thus, according to the canons of the ancient philosophy, justice is done to all the factors of our problem - God remains as Father, the infinitely remote and absolute source of all; as Son, the Word who is.

00It originated a rotatory movement, which arising in one point gradually extended till the whole was in motion, which motion continues and will continue infinitely.

00It need not be infinitely small, or even small compared with ordinary standards; thus in astronomy such vast bodies as the sun, the earth, and the other planets can for many purposes be treated merely as points endowed with mass.

00Chasles any displacement whatever of the lamina in its own plane is equivalent to a rotation about some finite or infinitely distant point J.

00The limiting position J of this centre, when the two positions are taken infinitely close to one another, is called the instantaneous centre.

00By imagining the successive positions to be taken infinitely close to one another we derive the theorem stated.

00The investigation fails when P+Q~O, since it leads to an infinitely small resultant acting in an infinitely distant line.

00An infinitely small rotation about any axis is conveniently represented geometrically by a length AB measures along the axis and proportional to the angle of rotation, with the convention that the direction from A to B shall be related to the rotation as is the direction of translation to that of rotation in a righthanded screw.

00The analogy between the mathematical relations of infinitely small displacements on the one hand an-d those of force-systems on the other enables us immediately to convert any theorem in the one subject into a theorem in the other.

00For example, we can assert without further proof that any infinitely small displacement may be resolved into two rotations, and that the axis of one of these can be chosen arbitrarily.

00In the analogous theory of infinitely small displacements of a solid, a null-line is a line such that the lengthwise displacement of any point on it is zero.

00There is a corresponding kinematic peculiarity, in that the connection is now not strictly rigid, an infinitely small relative displacement being possible.

00Work.The work done by a force acting on a particle, in any infinitely small displacement, is defined as the product of the force into the orthogonal projection of the displacement on the direction of the force; i.e.

00In the same way, the work dne by a force acting on a rigid body in any infinitely small displacement of the body is the scalar product of the force into the displacement of any point on the line of action.

00To see this, let A, B be the displaced positions of A, B, and let i~ be the infinitely small angle between AB and AB.

00The total work done by two concurrent forces acting on a particle, or on a rigid body, in any infinitely small displacement, is equal to the work of their resultant.

00It follows from the preceding statements that any two systems of forces which are statically equivalent, according to the principles of ~ 4, 8, will (to the first order of small quantities) do the same amount of work in any infinitely small displacement of a rigid body to which they may be applied.

00It is also evident that the total work done in two or more successive infinitely small displacements is equal to the work done in the resultant displacement.

00The work of a couple in any infinitely small rotation of a rigid body about an axis perpendicular to the plane of the couple is equal to the product of the moment of the couple into the angle of rotation, proper conventions as to sign being observed.

00This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axe~.

00If a line is a null-line with respect to the wrench (X, Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero, and its coordinates are accordingly subject to the further relation Lf+M~+Nl+XX+Yp+Zvo, (5~

00It is evident, in the first place, that in any displacement common to the two surfaces, the work of the two equal and opposite normal pressures will cancel; moreover if, one of the surfaces being fixed, an infinitely small displacement shifts the point of contact from A to B, and if A be the new position of that point of the sliding body which was at A, the pro jectior of AA on the normal at A is of the second order.

00In any infinitely small deformation of the frame as thus modified, the virtual work of the forces 5, together with that of the original extraneous forces, must ~ranish this determines S.

00The form (I) is deduced from it by putting itt, and taking t to be infinitely small.

00For 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.

00The effect of ordinary finite forces during the infinitely short duraticm of this impulse is of course ignored.

00The impidse of a force in any infinitely small interval of time & is the product of the force into &; it is to be regarded as a vector.

00If X, Y, Z are the components of force, then considering the changes in an infinitely short time 3t we have, by projection on the co-ordinate axes, i3(mu) =Xi5t, and so on, or du dv dw m-~jj=X, m~=Y, m~=Z.

00The following table gives the period, for various amplitudes a, in terms of that of oscillation in an infinitely small arc about a vertical axis half-way between the points of attachment of the upper string.

00The kinematical relations above explained now lead to the conclusion that in calculating the effect of extraneous forces in an infinitely short time t we may take moments about an axis passing through the instantaneous position of G exactly as if G were fixed; moreover, the result will be the same whether in this process we employ the true velocities of the particles or merely their velocities relative to G.

00q,,) tobe generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (II) with respect tot over the infinitely short duration of the impulse ~-=Q.

00The quantities R, e, * are the coefficients in the expression ROr+M+4.~ for the work done in an infinitely small displacement; viz.

00Condition of Stability of Position.If the materials of a structure were infinitely stiff and strong, stability of position at any joint would be insured simply by making the centre of resistance fall within the joint under all possible variations of load.

00This expression shows that as a increases y increases until when Wa = gc, y becomes infinitely large.

00is infinitely sadder.

00These investigations led him to the announcement of the fundamental law of action between elements of current, or currents in infinitely short lengths of linear conductors, upon one another at a distance; summed up in compact expression this law states that the action is proportional to the product of the current strengths of the two elements, and the lengths of the two elements, and inversely proportional to the square of the distance between the two elements, and also directly proportional to a function of the angles which the line joining the elements makes with the directions of the two elements respectively.

00If Coke's reports show completer mastery of technical details, greater knowledge of precedent, and more of the dogged grasp of the letter than do Bacon's legal writings, there can be no dispute that the latter exhibit an infinitely more comprehensive intelligence of the abstract principles of jurisprudence, with a richness and ethical fulness that more than compensate for their lack of dry legal detail.

00The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis (the symmetrical axis of the system) are infinitely small, i.e.

00If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their " perpendicular height of incidence," i.e.

00If the object point 0 be infinitely distant, u, and u 2 are to be replaced by h l and h 2, the perpendicular heights of incidence; the " sine condition " then becomes sin u',/h 1 = sin u' 2 /h 2 .

006) 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.

00from it and traversing the system is made infinitely narrow by reducing me aperture stop; such a pencil consists of the rays which can pass from the object point through the now infinitely small entrance pupil.

00This 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.

00Each of the four co-ordinates, n', x', y are functions of %, rj, x, y; and if it be assumed that the field of view and the aperture be infinitely small, then, n, x, y are of the same order of infinitesimals; consequently by expanding ', ii', x', y in ascending powers of E, rt, x, y, series are obtained in which it is only necessary to consider the lowest powers.

00Freedom from aberration for two axis points, one of which is infinitely distant, is known as " Herschel's condition."

00In 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).

00For infinitely distant objects the radius of the chromatic disk of confusion is proportional to the linear aperture, and independent of the focal length (vide supra," Monochromatic Aberration of the Axis Point "); and since this disk becomes the less harmful with an increasing image of a given object, or with increasing focal length, it follows that the deterioration of the image is proportional to the ratio of the aperture to the focal length, i.e.

00- 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.

00But he was infinitely generous and affectionate, and spent his enormous fortune liberally.

00If it is a risky matter to plant out the robust young fry of trout under an age of three months, it would seem to be an infinitely more speculative proceeding to plant out the delicate week-old larvae of sea-fishes in an environment which teems with predaceous enemies.

00One sage, most learned of all, assents, but intimates that the scene of this glory will be, not the paternal kingdom, but another infinitely more exalted, and that the child will adopt the faith which his father persecutes.

00Laplace investigated the force acting on the fluid contained in an infinitely slender canal normal to the surface of the fluid arising from the attraction of the parts of the fluid outside the canal.

00If we next write f II (f) d f =, P(z), (24) then 27rmagz) will represent - (I) The work done by the attractive force while a particle m is brought from an infinite distance to a distance z from an infinitely thin stratum of the substance whose mass per unit of area is Z o; (2) The attraction of a particle m placed Q 2 at a distance z from the plane surface of an infinite solid whose density is a.

00If n +4 be positive, the attraction of infinitely distant parts contributes to the result; while if n+4 be negative, the parts in immediate contiguity act with infinite power.

00The Magnitude Of A Drop Delivered From A Tube, Even When The Formation Up To The Phase Of Instability Is Infinitely Slow, Cannot Be Calculated A Priori.

00For Example, In The Case Of Water Delivered From A Glass Tube, Which Is Cut Off Square And Held Vertically, A Will Be The External Radius; And It Will Be Necessary To Suppose That The Ratio Of The Internal Radius To A Is Constant, The Cases Of A Ratio Infinitely Small, Or Infinitely Near Unity, Being Included.

00When the ellipse differs infinitely little from a circle, the equation of the meridian line becomes approximately y = a+c sin (x/a) where c is small.

00Let us conceive an infinitely long circular cylinder of liquid, at rest (a motion common to every part of the fluid is necessarily without influence upon the stability, and may therefore be left out of account for convenience of conception and expression), and inquire under what circumstances it is stable or unstable, for small displacements, symmetrical about the axis of figure.

00The time of vibration is of course itself a function of the nature of the fluid and of the size of the drop. By the method of dimensions alone it may be seen that the time of infinitely small vibrations varies directly as the square root of the mass of the sphere and inversely as the square root of the capillary tension; and it may be proved that its expression is - V C?

00The Langobards, German in their faults and in their strength, but coarser, at least at first, than the Germans whom the Italians had known, the Goths of Theodoric and Totila, found themselves continually in the presence of a subject population very different from anything which the other Teutonic conquerors met with among the provincials - like them, exhausted, dispirited, unwarlike, but with the remains and memory of a great civilization round them, intelligent, subtle, sensitive, feeling themselves infinitely superior in experience and knowledge to the rough barbarians whom they could not fight, and capable of hatred such as only cultivated races can nourish.

00Panurge has almost all intellectual accomplishments, but is totally devoid of morality: he is a coward, a drunkard, a lecher, a spiteful trickster, a spendthrift, but all the while infinitely amusing.

00Most fortunately for Europe, the Teutonic races already settled in Gaul rallied to the defence of the empire against invaders infinitely more barbarous than themselves.

00Whatever in literature revealed the hearts of men was infinitely precious to him; and for this reason he professed almost a cult for St Augustine.

00While Gulliver is infinitely the most famous and popular of Swift's works, it exhibits no greater powers of mind than many others.

00The theory refers to radiation homogeneous at all points within a single closed boundary maintained at uniform temperature; in the actual case we have a double boundary, one the sun's surface, and the other infinitely remote, or say, non-existent, and at zero temperature; and it is assumed that the density of radiation in the free space varies inversely as the squares of the distance from the sun.

00Mosquitoes, butterflies, spiders, beetles and ants are infinitely numerous, and some of the species are indescribably troublesome.

00And it must be said that his time was better employed in original investigations than it would have been had he spent it in observations made even with the best of instruments, infinitely better than if he had spent it on those of the observatory, which, however good originally, were then totally unfit for the delicate requirements of modern astronomy.

00Sensation might arise, for aught we know, so far as causality leads us, not from a world of forces at all, but from a will like our own, though infinitely more powerful, acting upon us, partly furthering and partly thwarting us.

00Similarly by continual subdivision we reach the idea of the infinitely small.

00Thus a point, which is by definition infinitely small, is as compared with a line a unit: the line is infinite, made up of an infinite number of points, any pair of which have an infinite number of points between them.

00This is Spinoza's theory of the infinitely infinite," the limiting notion of infinity being of a numerical, quantitative series, each term of which is a qualitative determination itself quantitatively little, e.g.

00in length) is qualitatively limited when regarded as an infinitely small unit of a plane.

00A 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.

00His " latitudinarianism " was the result of extraordinary reverence for truth, and a perception that knowledge may be sufficient for the purposes of human life while it falls infinitely short of speculative completeness.

00By the truce of Andrussowo (February 1, 1667) Vitebsk, Polotsk and Polish Livonia were restored to Poland, but the infinitely more important Smolensk and Kiev remained in the hands of the Muscovite together with the whole eastern bank of the Dnieper.

00But in his general view of ethical principles as being, like mathematical principles,' essentially truths of relation, Clarke is quite in accordance with Locke; while of the four fundamental rules that he expounds, Piety towards God, Equity, Benevolence and Sobriety (which includes self-preservation), the first is obtained, just as Locke suggests, by " comparing the idea " of man with the idea of an infinitely good and wise being on whom he depends; and the second and third are axioms self-evident on the consideration of the equality or similarity of human individuals as such.

00It will be seen that the reasoning applies to the case of any one body or system of bodies, the dimensions of which are not regarded as infinitely small compared with the distance of the attracting body.

00Let the parallel dotted lines represent rays of light we regard as infinitely distant, a star for example.

00The theory rests on three positions: that satisfaction is necessary on account of God's honour and justice; that such satisfaction can be given only by the peculiar personality of the God-man; that such satisfaction is really given by the voluntary death of this infinitely valuable person.

00(I) If the Existent is Many, it must be at once infinitely small and infinitely great - infinitely small, because the parts of which it consists must be indivisible and therefore without magnitude; infinitely great,, because, that any part having magnitude may be separate from any other part, the intervention of a third part having magnitude is necessary, and that this third part may be separate from the other two the intervention of other parts having magnitude is necessary, and so on ad infinitum.

00But an infinite distance (which Zeno fails to distinguish from a finite distance infinitely divided) cannot be traversed in a finite time.

00Thus Zeno again confounds a finite distance infinitely divided with an infinite distance.

00It is indeed difficult to understand how so acute a thinker should confound that which is infinitely divisible with that which is infinitely great, as in (I), (2), (5), and (6); that he should identify space and 'magnitude, as in (3); that he should neglect the imperfection of the organs of sense, as in (4); that he should deny the reality of motion, as in (7); and that he should ignore the relativity of speed, as in (8): and of late years it has been thought that the conventional statements of the paradoxes, and in particular of those which are more definitely mathematical, namely (5), (6), (7), (8), do less than justice to Zeno's acumen.

00The former should be regarded as asserting that the whole is, not temporally, but logically, subsequent to the part, and that therefore there is an infinite regress in the notion of a whole which is infinitely divisible - a view which at any rate demands a serious refutation.

00Finally, the last paradox may be interpreted as a valid refutation of the doctrine that space and time are not infinitely divisible.

00Selligue had no particular comprehension of the problem, for his achromatic single systems were simply telescope objectives corrected for an infinitely distant point, and were placed so that the same surface was turned towards the object in the microscope objective as in the telescope objective; although contrary to the telescope, the distance of the object in the microscope objective is small in proportion to the distance of the image.

00Dean cursed himself for not wanting to talk to the redheaded deputy, who would have been infinitely preferable to this obnoxious jerk.

00With an approximate dateâ€”or at least a yearâ€”and a first name, the chore would be infinitely easier than scouring decades for a nameless individual.

00This seemed to satisfy the official, who was obviously embarrassed over the entire debacle and infinitely relieved that at least half their problem was solved without any accusatory newspaper headlines.

00They skied together and to Dean it was a long and infinitely pleasant run.

00The Celtic artists drew inspiration from the infinitely subtle mutability of nature.

00These include That the spammers are infinitely adaptable in being able to get their messages across.

00Singapore, for example a supposedly highly authoritarian state has an infinitely better education system for developing creative brains.

00UPDATE 2004/08/31: ow tolouene bogy Next up, Stuart, with his infinitely more poetic Two Bone Eulogy.

00copycat behavior by pupils made teaching " infinitely more difficult " .

00Then what about infinitely differentiable functions that were non-analytic?

00Magnitude, on the other hand, was usually described as being continuous, or being divisible into parts that are infinitely divisible into parts that are infinitely divisible.

00The geometric continuum is infinitely divisible; segments, no matter how small or how large, can arise.

00Confound it, I could have earned ten thousand dollars with infinitely less trouble.

00eloquence of the great American orator, but his effective range was infinitely wider.

00Von Koch's first results were on infinitely many linear equations in infinitely many unknowns.

00They are distinguished from other goods by five characteristics: digital goods are nonrival, infinitely expansible, discrete, aspatial, and recombinant.

00You can make a simple, complete game without much effort and it's almost infinitely extensible.

00Newton ' s fluxions produced wonderful mathematical results but many were wary of his use of infinitely small increments.

00His fulness is an infinite fulness; and he is infinitely sweet, as well as infinitely full.

00The same offense may be a serious thing to you, it is a far more serious thing to an infinitely holy God.

00impermeable, infinitely thin membrane that lets you swim in the digital world.

00Whilst being a small town, it is infinitely improbable you will end up anywhere near where you are trying to get to.

00The area occupied by the spokes is very irregular, being marked out by the supports of the branch, which are infinitely variable.

00Such restrictions may seem cruel, but are infinitely preferable to letting the pony get laminitis.

00linear equations in infinitely many unknowns.

00linear equations in infinitely many unknowns.

00Eve n m ore satisfying was that my landing was infinitely better tha n m y take-off!

00First of all, because they were a great toy, an infinitely malleable artistic medium of creation.

00The Infobahn carries digital information: and digital information is infinitely mutable.

00Eve n m ore satisfying was that my landing was infinitely better tha n m ore satisfying was that my landing was infinitely better tha n m y take-off!

00over-the-top special effects: it's infinitely more fun than renting the video.

00More important, the vital shipping lane into London became infinitely less perilous.

00Newcastle is a spacious, extended, infinitely populous place.

00Time is infinitely precious, with the golden opportunities it presents - just once, in so many cases.

00Or are they actually not infinitely precise, given the practical limitations of measuring them with a finite universe?

00A visit to your local Habitat is infinitely preferable to shopping online.

00Are there infinitely many sets of 3 consecutive primes in arithmetic progression.

00There are infinitely more reals than floats, so some corners must be cut.

00This may occur using very deeply recursive ML functions, but it may be because the functions are accidentally infinitely recursive.

00The material was then edited into elaborate circles of contradiction derived by riding the crest of an infinitely regressive, irrational thought.

00Wind power seems like the perfect answer to our energy problems - it is clean, non-polluting and infinitely renewable.

00CL: It's not quite the same as transputers, it's not infinitely scalable.

00But the stoutest hearts can not survive forever in the face of superior numbers and infinitely superior weapons.

00At this point, they seem to inhabit parallel universes which extend infinitely but don't touch.

00This conception of matter, as infinitely divisible and continuous, was taught by Anaxagoras more than four centuries before the Christian era, and in the philosophy of Aristotle the same ideas are found.

00The infinite universe is infinitely full of matter.

00Clerk Maxwell supposed two compartments, A and B, to be filled with gas at the same temperature, and to be separated by an ideal, infinitely thin partition containing a number of exceedingly small trap-doors, each of which could be opened or closed without any expenditure of energy.

00But in all moral qualities the brilliant adventurer of the r 5th was infinitely superior to the brilliant adventurer of the r9th century.

00Firmness such as this secured for him the support of all constitutional elements, and after three years premiership his position was infinitely stronger than at the outset.

00But, if we should take the view that nature is infinitely extended - part of the " Antithesis " in Kant's first " Antinomy " - relative necessity breaks down on the last analysis, since boundless nature may overwhelm that sequence which we thought most securely established.

00Thought and extension are peaceable attributes in this one substance; there are infinitely many other attributes, but these only are known to us.

00In fact a little consideration of the theory of probabilities will show it to be infinitely probable that such an object should really have some movement of rotation, no matter by what causes the nebula may have originated.

00To the student of the original texts Lancelot is an infinitely less interesting hero than Gawain, Perceval or Tristan, each of whom possesses a well-marked personality, and is the centre of what we may call individual adventures.

00Though he lacked the brilliant qualities of his rival Wallqvist, Nordin had the same alertness and penetration, and was infinitely more stable and disinterested.

00To express in any language or to illustrate by any images, from a purely objective standpoint, the infinitely complicated movements of the actual world, is a task far beyond human capacity.

00These had an infinitely wider renown in their day, but modern criticism has restored the balance in his favour, and is even in danger of erring in the opposite direction.

00The form is infinitely finer in Matt.

00Interment in the wall-recess or loculus, though infinitely the most common, was not the only mode employed in the catacombs.

00The poles at the ends of an infinitely thin uniform magnet, or magnetic filament, would act as definite centres of force.

00The Hungarian diet frantically opposed every Austrian alliance as endangering the national independence, but to any unprejudiced observer a union with the house of Habsburg, even with the contingent probability of a Habsburg king, was infinitely preferable to the condition into which Hungary, under native aristocratic misrule, was swiftly drifting.

00According to common optics, where images are absolute, the diffraction pattern is supposed to be infinitely small, and two radiant points, however near together, form separated images.

00The integrations may also be effected by means of polar coordinates, taking first the integration with respect to 4) so as to obtain the result for an infinitely thin annular aperture.

00It may be instructive to contrast this with the case of an infinitely narrow annular aperture, where the brightness is proportional to Jo 2 (z).

00If we make the extreme suppositions of an infinitely small source and absolutely homogeneous light, there is no escape from the conclusion that the light in a definite direction is arbitrary, that is, dependent upon the chance distribution of apertures.

00If we suppose the diameter of the lens to be given (2R), and its focal length f gradually to increase, the original differences of phase at the image of an infinitely distant luminous point diminish without limit.

00the determination of the motion in an infinitely extended elastic solid due to a locally applied periodic force.

00On the other hand, infinitely thin films of silver which can be produced chemically on glass surfaces are absolutely opaque.

00Dubuat considered that if water were a perfect fluid, and the channels in which it flowed infinitely smooth, its motion would be continually accelerated, like that of bodies descending in an inclined plane.

00After three abortive campaigns Mahmud was compelled, infinitely against his will, to summon to his assistance the already too powerful pasha of Egypt, Mehemet Ali, whom he had already employed to suppress the rebellious Wahhabis in Arabia.

00Truth varies infinitely under circumstances whose relative weight cannot be accurately gauged.

00Certainly it was true of him, in a far higher degree than of John Henry Newman, that the being of God and himself were to his mind two absolutely self-luminous truths - though both his God and his self were almost infinitely remote from Newman's.

00The goddess of art of Akishino-dera, Nara, attributed to the 8th century, is the most graceful and least conventional of female sculptures in Japan, but infinitely remote from the feminine conception of the Greeks.

00Complete adaptation to an infinitely receding ideal is impossible, and relative adaptation depends on the distance between the actual and the ideal.

00Estimates of Pilate's attitude at this point have varied infinitely, from Tertullian's, that he was " already in conviction a Christian " - jam pro sua conscientia Christianus- to Bacon's " jesting Pilate," who would not stay for a reply.

00This fact, together with the extraordinarily rare occurrence of such remains and meteoric particles in globigerina ooze, although there is no reason to suppose that at any one time they are unequally distributed over the ocean floor, can only be explained on the assumption that the rate of formation of the epilophic deposits through the accumulation of pelagic shells falling from the surface is rapid enough to bury the slowgathering material which remains uncovered on the spaces where the red clay is forming at an almost infinitely slower rate.

00He is living, active intelligence, the principle of motion and creation, realizing himself in the infinitely various forms of activity that constitute individual things.

00To the infinitely actual there is necessary the possible; that which determines involves somewhat in which its determinations can have existence.

00The universe, then, is a living cosmos, an infinitely animated system, whose end is the perfect realization of the variously graduated forms. The unity which sunders itself into the multiplicity of things may be called the monas monadum, each thing being a monas or self-existent, living being, a universe in itself.

00But how can the infinitely active ego posit a limit to its own activity?

00Garrick was surrounded by many players of eminence, and he had the art, as he was told by Mrs Clive, " of contradicting the proverb that one cannot make bricks without straw, by doing what is infinitely more difficult, making actors and actresses without genius."

00The angels are infinitely numerous20.

00But the years of his studious retirement were spent on a work of infinitely greater importance.

00A combination of the Babylonian with the Persian religion could only be effected by the degradation of the Babylonian deities into half-divine, half-daemonic beings, infinitely remote from the supreme God of light and of heaven, or even into powers of darkness.

00Even the characteristic dualism of Gnosticism has already proved to be in part of Iranian origin; and now it becomes clear how from that mingling of late Greek and Persian dualism the idea could arise that these seven halfdaemonic powers are the creators or rulers of this material world, which is separated infinitely from the light-world of the good God.

00He would have plunged England into war with Spain in 1572, when the risks would have been infinitely greater than in 1588, and when the Huguenot influence over the French government, on which he relied for support, would probably have broken in his hands.

00By monarchians of the former class Christ was held to be a mere man, miraculously conceived indeed, but constituted the Son of God simply by the infinitely high degree in which he had been filled with Divine wisdom and power.

00It Is Therefore Infinitely More Commodious To Determine The Commencement Of The Year By A Fixed Rule Of Intercalation; And Of The Various Methods Which Might Be Employed, No One, Perhaps Is On The Whole More Easy Of Application, Or Better Adapted For The Purpose Of Computation, Than The Gregorian Now In Use.

00For the period preceding British rule the records have not been so well preserved, but there is ample evidence to show that famine was just as frequent in its incidence and infinitely more deadly in its effects under the native rulers of India.

00Moreover, did it do so, the action upon the human system would be infinitely less harmful than similar doses of copper or of lead.

00And 1 In a letter dated the 4th of April 1882, referring to the publication of his drama Mary Stuart, Swinburne wrote, to Edmund Clarence Stedman: "Mary Stuart has procured me two satisfactions which I prefer infinitely to six columns of adulation in The Times and any profit thence resulting.

00If the distance between the disks could be made infinitely small each time, then the multiplier r would be 2, and the charge would be doubled each time.

00Cleanthes, pointing out that from a nature thoroughly evil we can never prove the existence of an infinitely powerful and benevolent Creator, hazards the conjecture that the deity, though all-benevolent, is not all-powerful.

00We can still point to much in exact science that is absolutely his; and we can indicate infinitely more which is due to his inspiration.

00either amongst those Jews who looked to the coming of a Messiah or, Such a view on the part of men who knew Arabic infinitely better than the most accomplished European Arabist will ever do, may well startle us.

00In Nubia, owing to the poverty of the country and its scanty population, the proportion of monuments surviving is infinitely greater than in Egypt.

00Literary Records.In estimating the sources of information regarding pre-Christlan Egypt, the native sources, first opened to us by Champollion, are infinitely the most important.

00But let it be observed, first, that to reduce the huge and confused mass of pre-existing law into the compass of these two collections was an immense practical benefit to the empire; secondly, that, whereas the work which he undertook was accomplished in seven years, the infinitely more difficult task of codification might probably have been left unfinished at Tribonian's death, or even at Justinian's own, and been abandoned by his successor; thirdly, that in the extracts preserved in the Digest we have the opinions of the greatest legal luminaries given in their own admirably lucid, philosophical and concise language, while in the extracts of which the Codex is composed we find valuable historical evidence bearing on the administration and social condition of the later Pagan and earlier Christian empire; fourthly, that Justinian's age, that is to say, the intellect of the men whose services he commanded, was quite unequal to so vast an undertaking as the fusing upon scientific principles into one new organic whole of the entire law of the empire.

00Infinitely greater as a designer was Alfred Stevens, whose influence on English craftsmen might be regarded as almost comparable to that of Michelangelo on that of his Italian contemporaries.

00The propositions maintained in the argument are - "(1) That something has existed from eternity; (2) that there has existed from eternity some one immutable and independent being; (3) that that immutable and independent being, which has existed from eternity, without any external cause of its existence, must be self-existent, that is, necessarily existing; (4) what the substance or essence of that being is, which is self-existent or necessarily existing, we have no idea, neither is it at all possible for us to comprehend it; (5) that though the substance or essence of the self-existent being is itself absolutely incomprehensible to us, yet many of the essential attributes of his nature are strictly demonstrable as well as his existence, and, in the first place, that he must be of necessity eternal; (6) that the self-existent being must of necessity be infinite and omnipresent; (7) must be but one; (8) must be an intelligent being; (9) must be not a necessary agent, but a being endued with liberty and choice; (to) must of necessity have infinite power; (I I) must be infinitely wise, and (12) must of necessity be a being of infinite goodness, justice, and truth, and all other moral perfections, such as become the supreme governor and judge of the world."

00Thus, according to the canons of the ancient philosophy, justice is done to all the factors of our problem - God remains as Father, the infinitely remote and absolute source of all; as Son, the Word who is.

00It originated a rotatory movement, which arising in one point gradually extended till the whole was in motion, which motion continues and will continue infinitely.

00It need not be infinitely small, or even small compared with ordinary standards; thus in astronomy such vast bodies as the sun, the earth, and the other planets can for many purposes be treated merely as points endowed with mass.

00Chasles any displacement whatever of the lamina in its own plane is equivalent to a rotation about some finite or infinitely distant point J.

00The limiting position J of this centre, when the two positions are taken infinitely close to one another, is called the instantaneous centre.

00By imagining the successive positions to be taken infinitely close to one another we derive the theorem stated.

00The investigation fails when P+Q~O, since it leads to an infinitely small resultant acting in an infinitely distant line.

00An infinitely small rotation about any axis is conveniently represented geometrically by a length AB measures along the axis and proportional to the angle of rotation, with the convention that the direction from A to B shall be related to the rotation as is the direction of translation to that of rotation in a righthanded screw.

00The analogy between the mathematical relations of infinitely small displacements on the one hand an-d those of force-systems on the other enables us immediately to convert any theorem in the one subject into a theorem in the other.

00For example, we can assert without further proof that any infinitely small displacement may be resolved into two rotations, and that the axis of one of these can be chosen arbitrarily.

00In the analogous theory of infinitely small displacements of a solid, a null-line is a line such that the lengthwise displacement of any point on it is zero.

00There is a corresponding kinematic peculiarity, in that the connection is now not strictly rigid, an infinitely small relative displacement being possible.

00Work.The work done by a force acting on a particle, in any infinitely small displacement, is defined as the product of the force into the orthogonal projection of the displacement on the direction of the force; i.e.

00In the same way, the work dne by a force acting on a rigid body in any infinitely small displacement of the body is the scalar product of the force into the displacement of any point on the line of action.

00To see this, let A, B be the displaced positions of A, B, and let i~ be the infinitely small angle between AB and AB.

00The total work done by two concurrent forces acting on a particle, or on a rigid body, in any infinitely small displacement, is equal to the work of their resultant.

00It follows from the preceding statements that any two systems of forces which are statically equivalent, according to the principles of ~ 4, 8, will (to the first order of small quantities) do the same amount of work in any infinitely small displacement of a rigid body to which they may be applied.

00It is also evident that the total work done in two or more successive infinitely small displacements is equal to the work done in the resultant displacement.

00The work of a couple in any infinitely small rotation of a rigid body about an axis perpendicular to the plane of the couple is equal to the product of the moment of the couple into the angle of rotation, proper conventions as to sign being observed.

00This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axe~.

00If a line is a null-line with respect to the wrench (X, Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero, and its coordinates are accordingly subject to the further relation Lf+M~+Nl+XX+Yp+Zvo, (5~

00It is evident, in the first place, that in any displacement common to the two surfaces, the work of the two equal and opposite normal pressures will cancel; moreover if, one of the surfaces being fixed, an infinitely small displacement shifts the point of contact from A to B, and if A be the new position of that point of the sliding body which was at A, the pro jectior of AA on the normal at A is of the second order.

00In any infinitely small deformation of the frame as thus modified, the virtual work of the forces 5, together with that of the original extraneous forces, must ~ranish this determines S.

00The form (I) is deduced from it by putting itt, and taking t to be infinitely small.

00For 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.

00The effect of ordinary finite forces during the infinitely short duraticm of this impulse is of course ignored.

00The impidse of a force in any infinitely small interval of time & is the product of the force into &; it is to be regarded as a vector.

00If X, Y, Z are the components of force, then considering the changes in an infinitely short time 3t we have, by projection on the co-ordinate axes, i3(mu) =Xi5t, and so on, or du dv dw m-~jj=X, m~=Y, m~=Z.

00The following table gives the period, for various amplitudes a, in terms of that of oscillation in an infinitely small arc about a vertical axis half-way between the points of attachment of the upper string.

00The kinematical relations above explained now lead to the conclusion that in calculating the effect of extraneous forces in an infinitely short time t we may take moments about an axis passing through the instantaneous position of G exactly as if G were fixed; moreover, the result will be the same whether in this process we employ the true velocities of the particles or merely their velocities relative to G.

00q,,) tobe generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (II) with respect tot over the infinitely short duration of the impulse ~-=Q.

00The quantities R, e, * are the coefficients in the expression ROr+M+4.~ for the work done in an infinitely small displacement; viz.

00Condition of Stability of Position.If the materials of a structure were infinitely stiff and strong, stability of position at any joint would be insured simply by making the centre of resistance fall within the joint under all possible variations of load.

00This expression shows that as a increases y increases until when Wa = gc, y becomes infinitely large.

00It was not worth while to master and economize the resources of this earth, to utilize the good and ameliorate the evils of this life, while every one agreed, in theory at any rate, that the present was but a bad prelude to an infinitely worse or infinitely better future.

00is infinitely sadder.

00These investigations led him to the announcement of the fundamental law of action between elements of current, or currents in infinitely short lengths of linear conductors, upon one another at a distance; summed up in compact expression this law states that the action is proportional to the product of the current strengths of the two elements, and the lengths of the two elements, and inversely proportional to the square of the distance between the two elements, and also directly proportional to a function of the angles which the line joining the elements makes with the directions of the two elements respectively.

00If Coke's reports show completer mastery of technical details, greater knowledge of precedent, and more of the dogged grasp of the letter than do Bacon's legal writings, there can be no dispute that the latter exhibit an infinitely more comprehensive intelligence of the abstract principles of jurisprudence, with a richness and ethical fulness that more than compensate for their lack of dry legal detail.

00The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis (the symmetrical axis of the system) are infinitely small, i.e.

00If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their " perpendicular height of incidence," i.e.

00If the object point 0 be infinitely distant, u, and u 2 are to be replaced by h l and h 2, the perpendicular heights of incidence; the " sine condition " then becomes sin u',/h 1 = sin u' 2 /h 2 .

006) 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.

00from it and traversing the system is made infinitely narrow by reducing me aperture stop; such a pencil consists of the rays which can pass from the object point through the now infinitely small entrance pupil.

00This 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.

00Each of the four co-ordinates, n', x', y are functions of %, rj, x, y; and if it be assumed that the field of view and the aperture be infinitely small, then, n, x, y are of the same order of infinitesimals; consequently by expanding ', ii', x', y in ascending powers of E, rt, x, y, series are obtained in which it is only necessary to consider the lowest powers.

00Freedom from aberration for two axis points, one of which is infinitely distant, is known as " Herschel's condition."

00In 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).

00For infinitely distant objects the radius of the chromatic disk of confusion is proportional to the linear aperture, and independent of the focal length (vide supra," Monochromatic Aberration of the Axis Point "); and since this disk becomes the less harmful with an increasing image of a given object, or with increasing focal length, it follows that the deterioration of the image is proportional to the ratio of the aperture to the focal length, i.e.

00- 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.

00But he was infinitely generous and affectionate, and spent his enormous fortune liberally.

00If it is a risky matter to plant out the robust young fry of trout under an age of three months, it would seem to be an infinitely more speculative proceeding to plant out the delicate week-old larvae of sea-fishes in an environment which teems with predaceous enemies.

00One sage, most learned of all, assents, but intimates that the scene of this glory will be, not the paternal kingdom, but another infinitely more exalted, and that the child will adopt the faith which his father persecutes.

00Laplace investigated the force acting on the fluid contained in an infinitely slender canal normal to the surface of the fluid arising from the attraction of the parts of the fluid outside the canal.

00If we next write f II (f) d f =, P(z), (24) then 27rmagz) will represent - (I) The work done by the attractive force while a particle m is brought from an infinite distance to a distance z from an infinitely thin stratum of the substance whose mass per unit of area is Z o; (2) The attraction of a particle m placed Q 2 at a distance z from the plane surface of an infinite solid whose density is a.

00Hence if we write K =27rf o,/i(z)dz, H =27rf o zi/i(z)dz, the pressure of a column of the fluid itself terminating at the surface will be p2{K+1H(I/Rid-I/R2)}, and the work done by the attractive forces when a particle m is brought to the surface of the fluid from an infinite distance will be mp{K+zH(I/Ri+I/Ro)} If we write (.0 J then 27rmpo(z) will express the work done by the attractive forces, while a particle m is brought from an infinite distance to a distance z from the plane surface of a mass of the substance of density p and infinitely thick.

00If n +4 be positive, the attraction of infinitely distant parts contributes to the result; while if n+4 be negative, the parts in immediate contiguity act with infinite power.

00The Magnitude Of A Drop Delivered From A Tube, Even When The Formation Up To The Phase Of Instability Is Infinitely Slow, Cannot Be Calculated A Priori.

00For Example, In The Case Of Water Delivered From A Glass Tube, Which Is Cut Off Square And Held Vertically, A Will Be The External Radius; And It Will Be Necessary To Suppose That The Ratio Of The Internal Radius To A Is Constant, The Cases Of A Ratio Infinitely Small, Or Infinitely Near Unity, Being Included.

00When the ellipse differs infinitely little from a circle, the equation of the meridian line becomes approximately y = a+c sin (x/a) where c is small.

00Let us conceive an infinitely long circular cylinder of liquid, at rest (a motion common to every part of the fluid is necessarily without influence upon the stability, and may therefore be left out of account for convenience of conception and expression), and inquire under what circumstances it is stable or unstable, for small displacements, symmetrical about the axis of figure.

00The time of vibration is of course itself a function of the nature of the fluid and of the size of the drop. By the method of dimensions alone it may be seen that the time of infinitely small vibrations varies directly as the square root of the mass of the sphere and inversely as the square root of the capillary tension; and it may be proved that its expression is - V C?

00The Langobards, German in their faults and in their strength, but coarser, at least at first, than the Germans whom the Italians had known, the Goths of Theodoric and Totila, found themselves continually in the presence of a subject population very different from anything which the other Teutonic conquerors met with among the provincials - like them, exhausted, dispirited, unwarlike, but with the remains and memory of a great civilization round them, intelligent, subtle, sensitive, feeling themselves infinitely superior in experience and knowledge to the rough barbarians whom they could not fight, and capable of hatred such as only cultivated races can nourish.

00Panurge has almost all intellectual accomplishments, but is totally devoid of morality: he is a coward, a drunkard, a lecher, a spiteful trickster, a spendthrift, but all the while infinitely amusing.

00Most fortunately for Europe, the Teutonic races already settled in Gaul rallied to the defence of the empire against invaders infinitely more barbarous than themselves.

00Whatever in literature revealed the hearts of men was infinitely precious to him; and for this reason he professed almost a cult for St Augustine.

00While Gulliver is infinitely the most famous and popular of Swift's works, it exhibits no greater powers of mind than many others.

00The theory refers to radiation homogeneous at all points within a single closed boundary maintained at uniform temperature; in the actual case we have a double boundary, one the sun's surface, and the other infinitely remote, or say, non-existent, and at zero temperature; and it is assumed that the density of radiation in the free space varies inversely as the squares of the distance from the sun.

00Mosquitoes, butterflies, spiders, beetles and ants are infinitely numerous, and some of the species are indescribably troublesome.

00And it must be said that his time was better employed in original investigations than it would have been had he spent it in observations made even with the best of instruments, infinitely better than if he had spent it on those of the observatory, which, however good originally, were then totally unfit for the delicate requirements of modern astronomy.

00Sensation might arise, for aught we know, so far as causality leads us, not from a world of forces at all, but from a will like our own, though infinitely more powerful, acting upon us, partly furthering and partly thwarting us.

00Similarly by continual subdivision we reach the idea of the infinitely small.

00Thus a point, which is by definition infinitely small, is as compared with a line a unit: the line is infinite, made up of an infinite number of points, any pair of which have an infinite number of points between them.

00This is Spinoza's theory of the infinitely infinite," the limiting notion of infinity being of a numerical, quantitative series, each term of which is a qualitative determination itself quantitatively little, e.g.

00in length) is qualitatively limited when regarded as an infinitely small unit of a plane.

00A 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.

00His " latitudinarianism " was the result of extraordinary reverence for truth, and a perception that knowledge may be sufficient for the purposes of human life while it falls infinitely short of speculative completeness.

00By the truce of Andrussowo (February 1, 1667) Vitebsk, Polotsk and Polish Livonia were restored to Poland, but the infinitely more important Smolensk and Kiev remained in the hands of the Muscovite together with the whole eastern bank of the Dnieper.

00But in his general view of ethical principles as being, like mathematical principles,' essentially truths of relation, Clarke is quite in accordance with Locke; while of the four fundamental rules that he expounds, Piety towards God, Equity, Benevolence and Sobriety (which includes self-preservation), the first is obtained, just as Locke suggests, by " comparing the idea " of man with the idea of an infinitely good and wise being on whom he depends; and the second and third are axioms self-evident on the consideration of the equality or similarity of human individuals as such.

00It will be seen that the reasoning applies to the case of any one body or system of bodies, the dimensions of which are not regarded as infinitely small compared with the distance of the attracting body.

00Let the parallel dotted lines represent rays of light we regard as infinitely distant, a star for example.

00The theory rests on three positions: that satisfaction is necessary on account of God's honour and justice; that such satisfaction can be given only by the peculiar personality of the God-man; that such satisfaction is really given by the voluntary death of this infinitely valuable person.

00(I) If the Existent is Many, it must be at once infinitely small and infinitely great - infinitely small, because the parts of which it consists must be indivisible and therefore without magnitude; infinitely great,, because, that any part having magnitude may be separate from any other part, the intervention of a third part having magnitude is necessary, and that this third part may be separate from the other two the intervention of other parts having magnitude is necessary, and so on ad infinitum.

00But an infinite distance (which Zeno fails to distinguish from a finite distance infinitely divided) cannot be traversed in a finite time.

00Thus Zeno again confounds a finite distance infinitely divided with an infinite distance.

00It is indeed difficult to understand how so acute a thinker should confound that which is infinitely divisible with that which is infinitely great, as in (I), (2), (5), and (6); that he should identify space and 'magnitude, as in (3); that he should neglect the imperfection of the organs of sense, as in (4); that he should deny the reality of motion, as in (7); and that he should ignore the relativity of speed, as in (8): and of late years it has been thought that the conventional statements of the paradoxes, and in particular of those which are more definitely mathematical, namely (5), (6), (7), (8), do less than justice to Zeno's acumen.

00The former should be regarded as asserting that the whole is, not temporally, but logically, subsequent to the part, and that therefore there is an infinite regress in the notion of a whole which is infinitely divisible - a view which at any rate demands a serious refutation.

00Finally, the last paradox may be interpreted as a valid refutation of the doctrine that space and time are not infinitely divisible.

00Selligue had no particular comprehension of the problem, for his achromatic single systems were simply telescope objectives corrected for an infinitely distant point, and were placed so that the same surface was turned towards the object in the microscope objective as in the telescope objective; although contrary to the telescope, the distance of the object in the microscope objective is small in proportion to the distance of the image.

00There are infinitely more reals than floats, so some corners must be cut.

00This may occur using very deeply recursive ML functions, but it may be because the functions are accidentally infinitely recursive.

00The material was then edited into elaborate circles of contradiction derived by riding the crest of an infinitely regressive, irrational thought.

00Wind power seems like the perfect answer to our energy problems - it is clean, non-polluting and infinitely renewable.

00CL: It 's not quite the same as transputers, it 's not infinitely scalable.

00The Celtic artists drew inspiration from the infinitely subtle mutability of nature.

00But the stoutest hearts can not survive forever in the face of superior numbers and infinitely superior weapons.

00At this point, they seem to inhabit parallel universes which extend infinitely but do n't touch.

00I have infinitely more patience than I did with my oldest child, whom I had at 24, almost twenty years ago.

00Socks are soft, easy to manipulate, and infinitely useful long past the baby shower.

00However, your chances of acquiring a cat that harbors respiratory disease is infinitely higher when you select your animal from a shelter or pet store where exposure to other cats and animals is high and difficult to regulate.

00The rustic textures and natural color palette is infinitely appealing and fits in with a relaxed lifestyle that is perfect for families.

00This process is infinitely more gentle and comfortable for the client than the traditional permanent makeup procedure.

00It's a stain, so it lasts infinitely longer than a powder, and even beats out a powder layered over a long-lasting cream blush, in my experience.

00They use fresh, natural ingredients that are locally available, but they are also infinitely flexible when it comes to incorporating new and different ingredients introduced from other cultures.

00Not only is organic cotton environmentally responsible, but your toddler will be infinitely comfortable in organic cotton.

00Instead, the convenience of less expensive "hand wash" items is infinitely more appealing.

00As you move up the levels from JC Penney to Macy's to Nordstrom and Bloomindales and on to places such as Saks and Neiman Marcus, you will note that the merchandise becomes more unique and infinitely more costly.

00The first thing to know is that although many people regard their dog as their child or their friend, dogs and animals in general are infinitely more sensative than most people when it comes to recognizing and accepting impending death.

00That was infinitely more appealing to male audiences and it was achieved not through nudity, but through an undressing that mimicked the disrobing which preceded a sexual encounter.

00These youthful examples are infinitely superior to worn-out lifted stock, and yield quantities of vigorous cuttings for February and March propagation.

00It can be colored, molded, imprinted and infinitely customized.

00Concrete is infinitely moldable, providing unique designs for every space.

00It's good to know the types of clothes that look great on a man, but a guide to what not to wear for men will likely be infinitely more helpful!

00Many women find that slipping a piece of sexy lingerie beneath their clothes instantly elevates their confidence and makes them feel infinitely more alluring and even a bit mysterious.

00Fast forward to a more fashionable age, when every bit of a plus size woman's wardrobe is infinitely chicer and more thoughtfully designed than ever before.

00Rain hats are infinitely kinder to hairstyles than hoods and a fashionable way to keep your head dry.

00The game is infinitely more interesting with Nintendo Wi-Fi Connection.

00Complexity: Compared to the plug and play mechanism incorporated by consoles, PC's take infinitely more work to get up and running perfectly.

00Tossing something infinitely more complex in front of a new gamer's face will leave confused looks, grumpy gamers, and bankrupt companies.

00Tanks: It has not yet been determined how to achieve Platinum status in this infinitely destructive (and enjoyable) mini-game that will charge you with shooting shells at all those other tanks on the screen.

00The psychological changes and challenges of puberty are made infinitely greater if its timing is off.

00Using pancake mix from your own homemade pancake recipe is not rocket science, just infinitely satisfying.

00The Infinitely Adjustable Umbrella Sports Chair has an umbrella that attaches to either side of the backrest and can both rotate and tilt in any direction.

00The belted bikini is super sassy and infinitely sexy.

00Dark washes are infinitely more versatile than lighter ones.

00Some are infinitely more casual than others, but it's just as easy to get all dressed up and still remain fresh - even when the mercury really rises.

00Wearing a cashmere cardigan is quite easy in cold weather, but it's infinitely more fun to style this sweater when the weather is warm and the sleeves are short.

00Dating websites can almost infinitely widen your social pool.

00Sure, a diamond may last forever, but it's infinitely more important for the marriage to last.

00Even if it's affordable to you, though, it's infinitely more pleasing to score a designer purse for less.

00Cork is infinitely more comfortable than wood and keeps the shoe comfortable and relaxed.

00So you won't be the first to wield this method of tattoo concealment, as it is infinitely more glamorous than wearing a turtleneck to your wedding!

00Like the knots themselves, Celtic knot tattoos are infinitely variable and allow for complete self-expression.

00The Breitling Chronomat men's watch, designed with premium grade stainless steel and 18K yellow gold, is infinitely beautiful.

00In other words, yoga is an infinitely adaptable instrument that can and should be tailored to each person's abilities, needs, and aspirations.

00Humans are highly visual and adding images to something as simple as a newsletter can make such a project infinitely more attractive to your readers.

00Likewise, choosing kettlebells that are too light in weight will not tax your system enough to build muscle, though it is infinitely better for you if the weights are too light than if they are too heavy.

00Easy French sentences can prove infinitely helpful when it comes to navigating the French language.

00Try buying small slices of bread so that you and your family can make mini-sandwiches, which are infinitely more interesting than your run of the mill large-slice sammies.

00Incorporating girlish designs in women's clothes was nothing new, but it was only in this form of sleepwear that girlishness could be combined with womanliness in a way that was infinitely sexy.

00All of Adea's underwear and layering tops are much less restrictive than the average daywear line, making them infinitely more comfortable.

00The result is something infinitely more intriguing and worthy than your typical bedroom attire.

00It's infinitely better to spend more money on something quality that won't compromise your skin's comfort for fashion.

00Anyone looking to do any video or sound editing whatsoever will find that having a wide range of MP3 sound effects readily available can be infinitely useful.

00The new sound was softer, more radio-friendly and infinitely more pop-oriented.

00At the same time, there's something magical about creating a tiny tile graphic that can be infinitely repeated with no detectable edges.

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The word usage examples above have been gathered from various sources to reflect current and historial usage. They do not represent the opinions of YourDictionary.com.