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To the Jewish interpolation of the 1st century B.C. (about 60-40), i.e.

81Matritensis 10041 (begun in the year A.D 94 8), the words are omitted under the heading council of Constantinople but inserted under the heading council of Toledo, in the former MS., above the line and in a later hand, which shows conclusively how the interpolation crept in.

60Its use in the Eucharist of the undivided Church has been continued since the great schism, although the Eastern Church protests against the interpolation 1 Jevons, Introd.

61The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences.

40especially Ha,upt's note) and does not involve the interpolation of matter by the later redactor of Colossians and Ephesians (Holtzmann, Hausrath' and Bruckner, Reihenfolge d.

40If within the range5100-3700A, the constants are determined once for all, the formula seems capable of giving by interpolation results accurate to o 2 A, but as a rule the range to which the formula is applied will be much less with a corresponding gain in the accuracy of the results.

40When attention was called to the interpolation in the 9th century it became one cause of the schism between East and West.

41The interpolation really witnessed to a deep-lying difference between Eastern and Western theology.

41When attention was called to the interpolation in the 9th century it became one cause of the schism between East and West.

41" Now for the interpolation of the rest, I considered that the denominators I, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated.

42The next great advance on the Trigonometria artificialis took place more than a century and a half afterwards, when Michael Taylor published in 1792 his seven-decimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the Trigonometria to 10 places and then contracted to 7.

30The next great advance on the Trigonometria artificialis took place more than a century and a half afterwards, when Michael Taylor published in 1792 his seven-decimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the Trigonometria to 10 places and then contracted to 7.

30Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on Differences, Calculus Of; Infinitesimal Calculus; Interpolation; Vector Analysis.

20There is not only no room in history for this Median king of the Book of Daniel, but it is also highly likely that the interpolation of "Darius the Mede" was caused by a confusion of history, due both to the destruction of the Assyrian capital Nineveh by the Medes, sixty-eight years before the capture of Babylon by Cyrus, and also to the fame of the later king, Darius Hystaspis, a view which was advanced as early in the history of biblical criticism as the days of the Benedictine monk, Marianus Scotus.

10The most convenient and accurate formula of interpolation seems to be that discovered by J.

11To 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.

00It was Christianity that preserved Jewish apocalyptic, when it was abandoned by Judaism as it sank into Rabbinism, and gave it a Christian character either by a forcible exegesis or by a systematic process of interpolation.

00The attempts made during the 19th century to disprove the Pauline authorship now possess merely an historic interest, nor have the various hypotheses of more or less extensive interpolation won any serious support.

00"In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr Wallis, and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola (for instance, in this series of curves whose common base 0 or axis is x, and the ordinates respectively (I -xx)l, (i (I &c), I perceived that if the areas of the alternate curves, which are x, x 3x 3, x &c., could be interpolated, we should obtain the areas of the intermediate ones, the first of which (I -xx) 1 is the area of the circle.

002 3 4 The binomial theorem was thus discovered as a development of John Wallis's investigations in the method of interpolation.

00The conditions are thus similar to those which arise in interpolation (q.v.).

00In the case of a trapezette, for instance, the data are the magnitudes of certain ordinates; the problem of interpolation is to determine the values of intermediate ordinates, while that of mensuration is to determine the area of the figure of which these are the ordinates.

00One method is to construct a table for interpolation of x in terms of u, and from this table to calculate values of x corresponding to values of u, proceeding by equal intervals; a quadrature-formula can then be applied.

00For the methods involving finite differences, see references under DIFFERENCES, CALCULUS OF; and INTERPOLATION.

0053 would be explained if the narrative (which is not in Luke) may be held to be an interpolation by one not familiar with the localities.

00It only remains to give due honour to one of the most beautiful of legends, that of the deliverance of Adam's spirit from the nether world by the Christ, the earliest form of which is a Christian interpolation inA poc. Moses, ï¿½ 42 (cp. Malan, Adam and Eve, iv.

0023, an interpolation) for the Passover of the Crucifixion; and the ministry would thus be reduced to a single year.

0018 or r9) =01.202.4 [this last is a later interpolation from Eusebius] in the Acts of Pilate.

00" With desire have I desired to eat this Passover, before I suffer and places the bread after the wine, unless indeed the Pauline interpolation comprises the whole of verse 19.

00Interpolation then always has a motive.

00Fraudulent interpolation, whether the fraud be pious or otherwise, does occur, but is comparatively rare.

00Interpolation is then a voluntary alteration, but in practice it is often hard to distinguish from other changes in which its motive is absent.

00Interpolation is sometimes due to an inopportune use of knowledge, as when a quotation or a narrative is made to agree with what the interpolator has read elsewhere.

00Spectroscopic Measurements and Standards of Wave-Length.- All spectroscopic measurement should be reduced to wavelengths or wave-frequencies, by a process of interpolation between lines the wave-lengths of which are known with sufficient accuracy.

00By means of these, spectroscopists are enabled to measure by interpolation the wave-length of any line they may wish to determine.

00Interpolation is easy in the case of all observations taken with a grating.

00The chief difficulty is the reference to a certain sophist, Dionysius, but this is probably an interpolation.

007-9, which is plainly an interpolation, perhaps from the margin, upon the qualifications of episcopoi.

00The famous correspondence produced next year in evidence against her at the conference of York may have been, as her partisans affirm, so craftily garbled and falsified by interpolation, suppression, perversion, or absolute forgery as to be all but historically worthless.

00Windthorst thereupon raised the question in the Reichstag, but the Prussian government refused to take any notice of the interpolation on the ground that there was no right in the constitution for the imperial authority to take cognizance of acts of the Prussian government.

00So, on the other hand, there is no single verse or clause which can be plausibly made out to be an interpolation by Zaid at the instance of Abu Bekr, Omar, or Othman.

00I, 2, 9-11; the narrative of E, on the other hand, has in part disappeared owing to the interpolation of later material, in part has been retained in xxiv.

00Against the Portuguese claim it is argued that the Villancico corresponding to Joao de Lobeiro's poem is an interpolation in the Spanish text, that Portuguese prose was in a rudimentary stage of development at the period when--ex hypothesi - the romance was composed, and that the book was very popular in Spain almost a century before it is even mentioned in Portugal.

00They rejected as an illegitimate interpolation the eternal subject of development, and, instead of one continuing God as the subject of all the predicates by which in the logic the absolute is defined, assumed only a series of ideas, products of philosophic activity.

00As the word " interpolation " implies, Hermann did not maintain the hypothesis of a congeries of independent " lays."

00The interpolation (if it is one) is probably due to local interests.

00But in neither form is it free from later interpolation; and its untrustworthiness is shown by its conflicting with data 1 I is now generally recognized, thanks to Volquardsen and others, that Ephorus is the principal authority followed by Diodorus, except in the chapters relating to Sicilian history.

0017-21a suggest that it is a later interpolation, such as writings on church discipline were 1 The same motive occurs in the preface to Irenaeus's treatise, Adv.

0034 f., 39-45), which are especially cited in support of the exilic standpoint of the writer, are just those which, on other grounds, show signs of later interpolation.

0040-45 display obvious signs of interpolation.

00we have memoirs relating to the proof of the theorem that every numerical equation has a real or imaginary root, the memoir on the Hypergeometric Series, that on Interpolation, and the memoir Determinatio attractionis - in which a planetary mass is considered as distributed over its orbit according to the time in which each portion of the orbit is described, and the question (having an implied reference to the theory of secular perturbations) is to find the attraction of such a ring.

00She is not mentioned in the Iliad or the Odyssey, but in Hesiod (Theogony, 409) she is the daughter of the Titan Perses and Asterie, in a passage which may be a later interpolation by the Orphists (for other genealogies see Steuding in Roscher's Lexikon).

00In fine, the interpolation of a Median Darius must be regarded as the most glaring historical inaccuracy of the author of Daniel.

00was bracketed as an interpolation.

00But the unity of thought and atmosphere is such as to show that the work is one whole (subject no doubt to a certain amount of redaction and interpolation), and that the apocalyptic part was composed as an introduction to the rest.

00The tale, as told in the 1476 chronicle, is clearly an interpolation, for it comes immediately after a distinct statement that "God had helped the Confederates, and that with great labour they had defeated the knights and Duke Leopold," while the passage immediately following joins on to the former quite naturally if we strike out the episode of the "true man," who is not even called Winkelried.

00Assuming this, and rejecting the evidence of the 1476 chronicle as an interpolation and full of mistakes, and that of the song as not proved to have been in existence before 1531, Herr Burkli comes to the startling conclusion that the phalanx formation of the Austrians, as well as the name and act of Winkelried, have been transferred to Sempach from the fight of Bicocca, near Milan (April 27, 1522), where a real leader of the Swiss mercenaries in the pay of France, Arnold Winkelried, reall y met his death in very much the way that his namesake perished according to the story.

00Interpolation.

00It may be regarded as derived from a wholly dependent sporogonium not unlike that of some of the simpler Bryophyta; the latter are assumed to have arisen from primitive Algal forms, in which, as the first step in the interpolation of the second generation in the life cycle, the fertilized ovum gave rise to a group of swarm spores, each of which developed into a new sexual plant.

00This chapter, on " things and their qualities," looks like an interpolation in an analysis of mere " ideas."

00It is an amplification and interpolation, by means of spurious decretals, of the canonical collection in use in the Church of Spain in the 8th century, all the documents in which are perfectly authentic.

00The word parabola was used by Archimedes, who was prior to Apollonius; but this may be an interpolation.

0026 a is with Kneucker to be rejected as an interpolation.

00bilinear interpolation can result in fields being smeared over the output grid.

00bilinear interpolation resampling takes a weighted average of four pixels in the original image nearest to the new pixel location.

00There are three common methods for resampling: nearest neighbor, bilinear interpolation, and cubic convolution.

00No interpolation is done, scaling is done by simple-minded pixel duplication or removal.

00linear interpolation.

00These codes also allow some data manipulation, eg adding spectra, multiplying by a constant, interpolation.

00polynomial interpolation is shown to give equivalent results.

00quadratic interpolation to be used.

00spline interpolation.

00The fusion of the data from LR images into a higher resolution image is then carried out through thin plate spline interpolation.

00trilinear interpolation there are at least 8 floating point operations required for each interpolation evaluation.

00This interpolation can, of course, be accomplished with the aid of a micrometer-microscope whose optical axis is normal to the plate, provided that the plate is mounted on slides which enable the observer to bring the reseau-squares successively under the microscope.

00What is said of this affair interrupts the original context of chap. xviii., to which the insertion has been clumsily fitted by an interpolation in the second half of ver.

00For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS; EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY.

00To 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.

00The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences.

00Reference may also be made to the special articles mentioned at the commencement of the present article, as well as to the articles on Differences, Calculus Of; Infinitesimal Calculus; Interpolation; Vector Analysis.

00especially Ha,upt's note) and does not involve the interpolation of matter by the later redactor of Colossians and Ephesians (Holtzmann, Hausrath' and Bruckner, Reihenfolge d.

00To the Jewish interpolation of the 1st century B.C. (about 60-40), i.e.

00It was Christianity that preserved Jewish apocalyptic, when it was abandoned by Judaism as it sank into Rabbinism, and gave it a Christian character either by a forcible exegesis or by a systematic process of interpolation.

00The attempts made during the 19th century to disprove the Pauline authorship now possess merely an historic interest, nor have the various hypotheses of more or less extensive interpolation won any serious support.

00For the JinkOshotO-ki, by its strong advocacy of the mikados administrative rights as against the ustirpations of military feudalism, may be said to have sowed the seeds of Japan~s modern polity; and the Taihei-ki, by its erudite diction, skilful rhetoric, simplification of old grammatical constructions and copious interpolation of Chinese words, furnished a model for many imitators and laid the foundations of Japans 19th-century style.

00"In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr Wallis, and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola (for instance, in this series of curves whose common base 0 or axis is x, and the ordinates respectively (I -xx)l, (i (I &c), I perceived that if the areas of the alternate curves, which are x, x 3x 3, x &c., could be interpolated, we should obtain the areas of the intermediate ones, the first of which (I -xx) 1 is the area of the circle.

00" Now for the interpolation of the rest, I considered that the denominators I, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated.

002 3 4 The binomial theorem was thus discovered as a development of John Wallis's investigations in the method of interpolation.

00Its use in the Eucharist of the undivided Church has been continued since the great schism, although the Eastern Church protests against the interpolation 1 Jevons, Introd.

00The interpolation really witnessed to a deep-lying difference between Eastern and Western theology.

00Matritensis 10041 (begun in the year A.D 94 8), the words are omitted under the heading council of Constantinople but inserted under the heading council of Toledo, in the former MS., above the line and in a later hand, which shows conclusively how the interpolation crept in.

00a forged interpolation, based on another document, not by Mary The whole affair has been obscured and almost inextricably entangled, as we shall see, by the behaviour of Mary's accusers.

00That part of the letter then, it is argued by many, is a forged interpolation based on Crawford's notes and memories.

00In The Mystery of Mary Stuart the evidence for an early forged letter was presented with confidence; the interpolation of forgeries based on Crawford's declaration was more dubiously suggested.

0013 (A.V.), which critics are generally agreed in regarding as an interpolation, and which, while used in the Greek and the Protestant churches, is omitted in the Roman rite.

00The treatment of an angle as generated by rotation, the investigation of the relations between trigonometrical ratios and circular measure, the application of interpolation to trigonometrical tables, and the general use of graphical methods to represent continuous variation, all imply an analytical onlook, and must therefore be deferred to this stage.

00The conditions are thus similar to those which arise in interpolation (q.v.).

00In the case of a trapezette, for instance, the data are the magnitudes of certain ordinates; the problem of interpolation is to determine the values of intermediate ordinates, while that of mensuration is to determine the area of the figure of which these are the ordinates.

00An alternative method, which is in some ways preferable, is to complete the table of differences by repeating the differences of the highest order that will be taken into account (see Interpolation), and then to use central-difference formulae.

00One method is to construct a table for interpolation of x in terms of u, and from this table to calculate values of x corresponding to values of u, proceeding by equal intervals; a quadrature-formula can then be applied.

00For the methods involving finite differences, see references under DIFFERENCES, CALCULUS OF; and INTERPOLATION.

0053 would be explained if the narrative (which is not in Luke) may be held to be an interpolation by one not familiar with the localities.

00It only remains to give due honour to one of the most beautiful of legends, that of the deliverance of Adam's spirit from the nether world by the Christ, the earliest form of which is a Christian interpolation inA poc. Moses, Ã¯¿½ 42 (cp. Malan, Adam and Eve, iv.

00He attempted the quadrature of the circle by interpolation, and arrived at the remarkable expression known as Wallis's Theorem (see Circle, Squaring Of).

00In the Septuagint column he drew attention to those passages for which there was no Hebrew equivalent by prefixing an obelus; but where the Septuagint had nothing corresponding to the Hebrew text he supplied the omissions, chiefly but not entirely from the translation of Theodotion, placing an asterisk at the beginning of the interpolation; the close of the passage to which the obelus or the asterisk was prefixed was denoted by the metobelus.

0023, an interpolation) for the Passover of the Crucifixion; and the ministry would thus be reduced to a single year.

0018 or r9) =01.202.4 [this last is a later interpolation from Eusebius] in the Acts of Pilate.

00Thus the number lox in the arithmetical column corresponds to 10 8 (1.0001) x in the geometrical column; the intermediate numbers being obtained by interpolation.

00" With desire have I desired to eat this Passover, before I suffer and places the bread after the wine, unless indeed the Pauline interpolation comprises the whole of verse 19.

00Interpolation then always has a motive.

00Fraudulent interpolation, whether the fraud be pious or otherwise, does occur, but is comparatively rare.

00Interpolation is then a voluntary alteration, but in practice it is often hard to distinguish from other changes in which its motive is absent.

00Interpolation is sometimes due to an inopportune use of knowledge, as when a quotation or a narrative is made to agree with what the interpolator has read elsewhere.

00Spectroscopic Measurements and Standards of Wave-Length.- All spectroscopic measurement should be reduced to wavelengths or wave-frequencies, by a process of interpolation between lines the wave-lengths of which are known with sufficient accuracy.

00By means of these, spectroscopists are enabled to measure by interpolation the wave-length of any line they may wish to determine.

00Interpolation is easy in the case of all observations taken with a grating.

00The most convenient and accurate formula of interpolation seems to be that discovered by J.

00If within the range5100-3700A, the constants are determined once for all, the formula seems capable of giving by interpolation results accurate to o 2 A, but as a rule the range to which the formula is applied will be much less with a corresponding gain in the accuracy of the results.

00The chief difficulty is the reference to a certain sophist, Dionysius, but this is probably an interpolation.

007-9, which is plainly an interpolation, perhaps from the margin, upon the qualifications of episcopoi.

00The famous correspondence produced next year in evidence against her at the conference of York may have been, as her partisans affirm, so craftily garbled and falsified by interpolation, suppression, perversion, or absolute forgery as to be all but historically worthless.

00Windthorst thereupon raised the question in the Reichstag, but the Prussian government refused to take any notice of the interpolation on the ground that there was no right in the constitution for the imperial authority to take cognizance of acts of the Prussian government.

00So, on the other hand, there is no single verse or clause which can be plausibly made out to be an interpolation by Zaid at the instance of Abu Bekr, Omar, or Othman.

00I, 2, 9-11; the narrative of E, on the other hand, has in part disappeared owing to the interpolation of later material, in part has been retained in xxiv.

00Against the Portuguese claim it is argued that the Villancico corresponding to Joao de Lobeiro's poem is an interpolation in the Spanish text, that Portuguese prose was in a rudimentary stage of development at the period when--ex hypothesi - the romance was composed, and that the book was very popular in Spain almost a century before it is even mentioned in Portugal.

00They rejected as an illegitimate interpolation the eternal subject of development, and, instead of one continuing God as the subject of all the predicates by which in the logic the absolute is defined, assumed only a series of ideas, products of philosophic activity.

00As the word " interpolation " implies, Hermann did not maintain the hypothesis of a congeries of independent " lays."

00The interpolation (if it is one) is probably due to local interests.

00But in neither form is it free from later interpolation; and its untrustworthiness is shown by its conflicting with data 1 I is now generally recognized, thanks to Volquardsen and others, that Ephorus is the principal authority followed by Diodorus, except in the chapters relating to Sicilian history.

0017-21a suggest that it is a later interpolation, such as writings on church discipline were 1 The same motive occurs in the preface to Irenaeus's treatise, Adv.

0034 f., 39-45), which are especially cited in support of the exilic standpoint of the writer, are just those which, on other grounds, show signs of later interpolation.

0040-45 display obvious signs of interpolation.

00we have memoirs relating to the proof of the theorem that every numerical equation has a real or imaginary root, the memoir on the Hypergeometric Series, that on Interpolation, and the memoir Determinatio attractionis - in which a planetary mass is considered as distributed over its orbit according to the time in which each portion of the orbit is described, and the question (having an implied reference to the theory of secular perturbations) is to find the attraction of such a ring.

00She is not mentioned in the Iliad or the Odyssey, but in Hesiod (Theogony, 409) she is the daughter of the Titan Perses and Asterie, in a passage which may be a later interpolation by the Orphists (for other genealogies see Steuding in Roscher's Lexikon).

00There is not only no room in history for this Median king of the Book of Daniel, but it is also highly likely that the interpolation of "Darius the Mede" was caused by a confusion of history, due both to the destruction of the Assyrian capital Nineveh by the Medes, sixty-eight years before the capture of Babylon by Cyrus, and also to the fame of the later king, Darius Hystaspis, a view which was advanced as early in the history of biblical criticism as the days of the Benedictine monk, Marianus Scotus.

00In fine, the interpolation of a Median Darius must be regarded as the most glaring historical inaccuracy of the author of Daniel.

00was bracketed as an interpolation.

00But the unity of thought and atmosphere is such as to show that the work is one whole (subject no doubt to a certain amount of redaction and interpolation), and that the apocalyptic part was composed as an introduction to the rest.

00The tale, as told in the 1476 chronicle, is clearly an interpolation, for it comes immediately after a distinct statement that "God had helped the Confederates, and that with great labour they had defeated the knights and Duke Leopold," while the passage immediately following joins on to the former quite naturally if we strike out the episode of the "true man," who is not even called Winkelried.

00Assuming this, and rejecting the evidence of the 1476 chronicle as an interpolation and full of mistakes, and that of the song as not proved to have been in existence before 1531, Herr Burkli comes to the startling conclusion that the phalanx formation of the Austrians, as well as the name and act of Winkelried, have been transferred to Sempach from the fight of Bicocca, near Milan (April 27, 1522), where a real leader of the Swiss mercenaries in the pay of France, Arnold Winkelried, reall y met his death in very much the way that his namesake perished according to the story.

00Here the principle of interpolation has reached its maximum.

00It may be regarded as derived from a wholly dependent sporogonium not unlike that of some of the simpler Bryophyta; the latter are assumed to have arisen from primitive Algal forms, in which, as the first step in the interpolation of the second generation in the life cycle, the fertilized ovum gave rise to a group of swarm spores, each of which developed into a new sexual plant.

00This chapter, on " things and their qualities," looks like an interpolation in an analysis of mere " ideas."

00It is an amplification and interpolation, by means of spurious decretals, of the canonical collection in use in the Church of Spain in the 8th century, all the documents in which are perfectly authentic.

00The word parabola was used by Archimedes, who was prior to Apollonius; but this may be an interpolation.

0026 a is with Kneucker to be rejected as an interpolation.

00Values of 1 or 2 will cause linear or quadratic interpolation to be used.

00This allowed the interpolation of the irregularly spaced height data onto a regular raster grid.

00The fusion of the data from LR images into a higher resolution image is then carried out through thin plate spline interpolation.

00For trilinear interpolation there are at least 8 floating point operations required for each interpolation evaluation.

00a forged interpolation, based on another document, not by Mary The whole affair has been obscured and almost inextricably entangled, as we shall see, by the behaviour of Mary's accusers.

01That part of the letter then, it is argued by many, is a forged interpolation based on Crawford's notes and memories.

01In The Mystery of Mary Stuart the evidence for an early forged letter was presented with confidence; the interpolation of forgeries based on Crawford's declaration was more dubiously suggested.

01The treatment of an angle as generated by rotation, the investigation of the relations between trigonometrical ratios and circular measure, the application of interpolation to trigonometrical tables, and the general use of graphical methods to represent continuous variation, all imply an analytical onlook, and must therefore be deferred to this stage.

01Here the principle of interpolation has reached its maximum.

01

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