The rules then are sine of the middle part = product of tangents of adjacent parts = product of cosines of opposite parts.
He then took the degree of doctor in divinity, defending as his thesis the two propositions: Nullum fidei Christianae dogma, in Sacris Scripturis traditum, est rectae rationi dissentaneum, and Sine actionum humanarum libertate nulla potest esse religio.
a (iI) We may find here the value of this when we have a train of waves in which the displacement is represented by a sine curve of amplitude a, viz.
Finally, by abstraction from the individual things of sense, the mind is able to contemplate the universal apart from its accompaniments (animal sine homine, asino, et aliis speciebus); these subjective existences are the universalia post rem of the Nominalists and Conceptualists.
the direction of the wave in the first medium), and the normal to the surface separating the two media, at the point in which the incident ray meets it; (2) The sine of the angle between the normal and the incident ray bears to the sine of the angle between the normal and the refracted ray a ratio which is constant for the same pair of media.
The enforcement of these reforms, however, was postponed sine die owing to the revolution which transformed the Ottoman Empire into a constitutional state; and the powers, anticipating an improvement in the administration of Macedonia by the new government, withdrew their military officers in the summer of 1908.
But he considered that the change ought to be so made that o should be the logarithm of unity and io,000,000,000 that of the whole sine, which.
Mechain in 1790, but of the people and to elicit their binding response, his object in was not recognized as periodic until found by Tuttle in 1858, summoning a contio was merely to bring the people together either for their instruction or for a declaration of his will as expressed in an edict ("contionem habere est verba facere ad populum sine ulla rogatione," Gell.
in 1727, and by various other works, including Moses's Sine Principio, 1730; The Confusion of Tongues and Trinity of the Gentiles, 1731; Power Essential and Mechanical, or what power belongs to God and what to his creatures, in which the design of Sir I.
He found out the formula for deriving the sine of a multiple angle, knowing that of the simple angle with due regard to the periodicity of sines.
Napier's logarithms are not the logarithms now termed Napierian or hyperbolic, that is to say, logarithms to the base e where e= 2.7182818 ...; the relation between N (a sine) and L its logarithm, as defined in the Canonis Descriptio, being N=10 7 e L/Ip7, so that (ignoring the factors re, the effect of which is to render sines and logarithms integral to 7 figures), the base is C".
ZI /t = - (a - s) M'Q 2 sine cos ° - EQ sin() =[ - (a - (3)M'U+E]V (8) Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid - aM' (Ç_vR), - (3M' (-- E -FUR), - EC' dR, (9) so that its equations of motion are M (Ç - vR) _ - aM' (_vR) - (a - $) M'VR, (io) M (Ç+uR) = - OM' (dV+U R) - (a - ()M'UR - R, '(II) C dR = dR + (a - Q)M'UV+0V; (12) and putting as before M+aM'=ci, M+13M' = c2, C+EC'=C3, ci dU - c2VR=o, dV +(c1U+E)R=o, c 3 dR - (c 1 U+ - c 2 U)V =o; showing the modification of the equations of plane motion, due to the component E of the circulation.
On the 28th the Hungarian parliament adjourned sine die, pending the settlement of the crisis, without having voted the estimates for 1910, and without there being any prospect of a meeting of the delegations.
The sine of an angle can never be greater than unity; and consequently under the most favourable circumstances only 1/m 2 ir 2 of the original light can be obtained in the m u ' spectrum.
The sine table, however, was afterwards published on this scale under the name of Thesaurus Mathematicus (Frankfort, 1613) by B.
Along the path of a particle, defined by the of (3), _ c) sine 2e, - x 2 + y2 = y a 2 ' (Io) sin B' de' _ 2y-c dy 2 ds ds' on the radius of curvature is 4a 2 /(ylc), which shows that the curve is an Elastica or Lintearia.
This deflecting force is directly proportional to the velocity and the mass of the particle and also to the sine of the latitude; hence it is zero at the equator and comes to a maximum at the poles.
It further follows, as in the analogous case of light, that there is a certain angle termed the critical angle, whose sine is found by dividing the less by the greater velocity, such that all rays of sound meeting the surface separating two different bodies will not pass onward, but suffer total reflection back into the first body, if the.
Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wave-lengths X, IX, 3A, 4A...
Let us suppose that two trains of sine waves of length A and amplitude a are travelling in opposite directions with velocity U.
Briggs pointed out in his lectures at Gresham College that it would be more convenient that o should stand for the logarithm of the whole sine as in the Descriptio, but that the logarithm of the tenth part of the whole sine should be Io,000,000,000.
The minor axis, on the other hand, is not constant, but, as we have already seen, depends on the latitude, being the product of the major axis into the sine of the latitude.
Aricia was one of the oldest cities of Latium, and appears as a serious opponent of Rome at the end of the period of the kings and beginning of the republic. In 338 B.C. it was conquered by C. Maenius and became a civitas sine suffragio, but was soon given full rights.
This view has the merit of giving the book a practical religious aim - a sine qua non to any theory of an early Christian writing.
that is to say, the angular velocity about each axis is proportional to the sine of the angle between the other two.
C2P2=IC2cos obliquity (The obliquity which is found to answer best in practice is about 143/4; its cosine is about i3/4, and its sine about 3/4.
Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI bye, c=2r sin 0; and therefore For the flank, p C=2r sine ~~_r} (29) For the face, pC2r sin 8 ~-~
But the quarrel, known as the controversy de auxiliis gratiae, was brought to an end by Pope Paul V., who closed the debates and adjourned his decision sine die.
Pouillet in 1837 contributed the sine and tangent compass, and W.
And the third, where the cause is really ended, and it is sine fraude without relation to any precedent promise....
If 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 .
Both the aberration of axis points, and the deviation from the sine condition, rapidly increase in most (uncorrected) systems with the aperture.
The aberrations of the third order are: (1) aberration of the axis point; (2) aberration of points whose distance from the Aberra- axis is very small, less than of the third order - the tions of deviation from the sine condition and coma here fall together in one class; (3) astigmatism; (4) curvature of the field; (5) distortion.
In order to render spherical aberration and the deviation from the sine condition small throughout the whole aperture, there is given to a ray with a finite angle of aperture u* (with infinitely distant objects: with a finite height of incidence the same distance of intersection, and the same sine ratio as to one neighbouring the axis (u* or h* may not be much smaller than the largest aperture U or H to be used in the system).
The rays with an angle of aperture smaller than u* would not have the same distance of intersection and the same sine ratio; these deviations are called "zones," and the constructor endeavours to reduce these to a minimum.
Spherical aberration and changes of the sine ratios are often represented graphically as functions of the aperture, in the same way as the deviations of two astigmatic image surfaces of the image plane of the axis point are represented as functions of the angles of the field of view.
The following may be regarded as typical: - (t) Largest aperture; necessary corrections are - for the axis point, and sine condition; errors of the field of view are almost disregarded; example - highpower microscope objectives.
The condition for the reproduction of a surface element in the place of a sharply reproduced point - the constant of the sine relation - must also be fulfilled with large apertures for several colours.
Abbe succeeded in computing microscope objectives free from error of the axis point and satisfying the sine condition for several colours, which therefore, according to his definition, were " aplanatic for several colours "; such systems he termed " apochromatic."
To integrate this equation for a solid of given form is probably difficult, but it is easy to see that at some distance on either side of the body, where the liquid is sensibly at rest, the crest of the wave will approximate to an asymptote inclined to the path of the body at an angle whose sine is w/V, where w is the velocity of the wave and V is that of the body.
He also reduced the revolted Sabines to submission; a large portion of their territory was distributed among the Roman citizens, and the most important towns received the citizenship without the right of voting for magistrates (civitas sine sufJragio).
On this occasion he insisted on the re-establishment of the constitution in its integrity as a sine qua non.
In this connexion it might also be well to mention the remarkable satires on the papal court, included in the Epistolae sine titulo.
He prided himself on his motto, Nulla dies sine linea.
While, therefore, there is apparently no correspondence between the area of the wing and the animal to be raised, there is, except in the case of sailing insects, birds and bats, an unvarying relation as to the weight and number of oscillations; so that the problem of flight would seem to resolve itself into one of weight, power, velocity and small surfaces, versus buoyancy, debility, diminished speed and extensive surfaces - weight in either case being a sine qua non.
4to); Lettres a sine Princesse d'Allemagne sur quelques sujets de physique et de philosophie (St Petersburg, 1768-1772, in 3 vols.
Nomentum received the civitas sine suffragio after the last war of the Latins against Rome (338 B.C.); in its municipal constitution the chief magistrate even in imperial times bore the title of dictator.
Most of the elements are small numerical fractions: e, the eccentricity of the moon's orbit, about 0.055; e', the eccentricity of the earth's orbit, about o 017: y, the sine of half the inclination of the moon's orbit, about 0.046; m, the ratio of the mean motions of the moon and earth, about 0.075.
' - sine 2p When the primitive light is white, this expression must be summed for the different monochromatic constituents.
As spring merges into summer, sunny days become more frequent; the ever-increasing breadth of beeforage yields still more abundantly, and the excitement among the labourers crowding the hives increases, rendering room in advance, shade and ventilation, a sine qua non.
The value of the clearness of an image-point in a median section is the sine of the semi-aperture of the pencil multiplied with the refractive index of the medium.
In dry-systems only the sine of the semi-aperture is concerned; in immersion-systems it is the product of the refractive index of the immersion-liquid and the sine of the object-side semi-aperture.
if the ratio of the sine of the aperture u on the object-side to the sine of the corresponding aperture u' on the image-side is constant, i.e.
The sine of this angle is the numerical aperture for dry lenses.
p. 42, 1604 fol.), " Officinis non sine risu eruditorum, Graeco articulo adjecto, Olibanus vocatur."
adjourned sine die.
The sine qua non is to decide whether treaties require popular approval.
returns: the sine of the argument. cos public static double cos (double a) Returns the trigonometric cosine of an angle.
Power series, radius of convergence, important examples including exponential, sine and cosine series.
A tuning fork produces a single note - a pure sine wave.
imperium cui est semper sine fine eleyson.
Laus virtus pax et imperium cui est semper sine fine eleyson.
FSINCOS ANS r1 - r2 r3 r2 is the sine of the radian angle r1. r3 is the cosine of the radian angle r1. r3 is the cosine of the radian angle r1.
FSINCOS ANS r1 - r2 r3 r2 is the sine of the radian angle r1. r3 is the cosine of the radian angle r1.
See Also: Constant Field Values Method Detail sin public static double sin (double a) returns the trigonometric sine of an angle.
Belief in natural selection has become the sine qua non of entry to much of polite society.
They use sine, cosine and tangent in right-angled triangles when solving problems in two dimensions.
I have sine reordered a couple of times and have been delighted.
ASN A function giving the arc sine of its argument in radians.
The inverse sine is defined only in the domain from ` -1 ' to ` 1 ' .
Parameters: a - the value whose arc sine is to be returned.
sine wave input signal should be plotted in the upper graph.
sine curve: easier topics offer relief from more difficult topics.
sine function for these angles?
sine rule is simpler to remember but not always applicable.
Such vigilant monitoring is a conditio sine qua non for any physician who wishes to cure the patient of her malady.
Using trigonometry: angles and the trigonometric functions sine, cosine and tangent; usage in the Earth and the science of surveying.
sine qua non condition for the existence of the two identities.
sine of an angle appears in the work of the Hindus.
sine of the latitude of the place = the secant of the suns latitude.
sine waves of different frequencies.
The spectrum contains just one component, at 256 Hz, because the pure tone is a single sine wave.
The Spectrum Any sound can be constructed by adding up enough sine waves of differing intensity levels, frequencies and starting times.
sine wave supply, we cannot be held responsible for any damage caused to computer equipment and personal appliances.
The short vertical line above the sine theta axis marks the position of the Ti K absorption edge.
Stockhausen used close analysis of sound to compose with sine waves to create timbres.
Don't think I'll ever get that toaster, I can't seem to sine anybody up.
Relationships and common formulae from elementary trigonometry, including the basic properties of sine and cosine.
undulate in a slow sine wave.
Racing staff are the sine qua non of racing: at present they are merely unsung and undervalued heroes.
virtus pax et imperium cui est semper sine fine eleyson.
By making the reference bias voltages applied to the diodes equal, an approximate square wave output is obtained from a sine wave input.
In that case a sine wave type vibration did show up on the photographs.
"Napier's analogies" are the four formulae - tan (A +B) cos = cos l(a - b) cotzC, tan2(A - B) - sin(a - b)cot2C; a (a+b) sine (a+b) tan a(a +b) = cos(A - B) tan1c, tan2(a - b) = s in(A - B) tan2c.
cos z (A+B) sine (A+B) They were first published after his death in the Constructio among the formulae in spherical trigonometry, which were the results of his latest work.
® 6.80470 Sine 0 's eq.
ZI /t = - (a - s) M'Q 2 sine cos Ã‚° - EQ sin() =[ - (a - (3)M'U+E]V (8) Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid - aM' (Ã‡_vR), - (3M' (-- E -FUR), - EC' dR, (9) so that its equations of motion are M (Ã‡ - vR) _ - aM' (_vR) - (a - $) M'VR, (io) M (Ã‡+uR) = - OM' (dV+U R) - (a - ()M'UR - R, '(II) C dR = dR + (a - Q)M'UV+0V; (12) and putting as before M+aM'=ci, M+13M' = c2, C+EC'=C3, ci dU - c2VR=o, dV +(c1U+E)R=o, c 3 dR - (c 1 U+ - c 2 U)V =o; showing the modification of the equations of plane motion, due to the component E of the circulation.
Any periodic curve may be resolved into sine or harmonic curves by Fourier's theorem.
But Cyprian of Carthage said long ago, Consuetudo sine veritate vetustas erroris est; and the bare fact of previous existence is no argument for the re-introduction of obsolete and antiquated institutions and theories.
Thus if x= log x =1 - where the path of integration in the plane of the complex variable t is any curve which does not pass through the origin; but now log x is not a uniform function, that is to say, if x describes a closed curve it does not follow that log x also describes a closed curve: in fact we have log (E +in) = log,/ Q 2 +7)2) t i (a + 2 n 7r), where a is the numerically least angle whose cosine and sine are (2 +n 2) and 71/,i (t 2 +n 2), and n denotes any integer.
The lemma is enunciated as follows: " Quaestiones omnes, quae per sinus, tangentes, atque secantes absolvi solent, per solam prosthaphaeresim, id est, per solam additionem, subtractionem, sine laboriosa numerorum multiplicatione divisioneque expedire."
The extension of the image away from the axis or size of field available for covering a photographic plate with fair definition is a function in the first place of the ratio between focal length and aperture, the longer focus having the greater relative or angular covering power, and in the second a function of the curvatures of the lenses, in the sense that the objective must be free from coma at the foci of oblique pencils or must fulfil the sine condition (see Aberration).
The moment of the resultant force R of the wrench about this line is Rr sine, and that of the couple G is G cos 9.
These aberrations are avoided if, according to Abbe, the " sine condition," sin u'1/sin u, = sin u' 2 /sin u 2, holds for all rays reproducing the point O.
Discussing the question whether a prince who becomes a cliens of another loses his majestas, he concludes that, unlike the tru i vassal, the cliens is not deprived of sovereignty: " Nihilominus in foederibus et pacis actionibus, quae inter principes aut populos societate et amicitia conjunctissimos sancientur; earn vim habet ut nec alter alteri pareat, nec imperet: sed ut alter alterius majestatem observare, sine ulla majestatis minutione teneatur.
SIN A function giving the sine of its radian argument.
Prove that twice the sine of the latitude of the place = the secant of the suns latitude.
See Also: Constant Field Values Method Detail sin public static double sin (double a) Returns the trigonometric sine of an angle.
The inverse sine is defined only in the domain from ` -1 ' to ` 1 '.
The sampled sine wave input signal should be plotted in the upper graph.
It goes like a sine curve: easier topics offer relief from more difficult topics.
Now what about the sine function for these angles?
In an amplitude plot, you see the actual amplitude of sine components.
The sine rule is simpler to remember but not always applicable.
This would abolish the distinction which is the sine qua non condition for the existence of the two identities.
The first actual appearance of the sine of an angle appears in the work of the Hindus.
When you perform Fourier analysis, you assume that the signal is composed of sine waves of different frequencies.
This is not a pure sine wave supply, we cannot be held responsible for any damage caused to computer equipment and personal appliances.
Fit Details The Add Sine Component Linear Fit Details adds a statistical breakdown of each sinusoid in the fit.
Do n't think I'll ever get that toaster, I ca n't seem to sine anybody up.
The keys on the keyboard undulate in a slow sine wave.
Three basic waves, sine, triangles and square, generated by a locked waveform generator, are applied to shape the display.
The Christmas Guitar Chord Songbook - Over 40 songs are available in this publication including Auld Lang Sine, Deck the Halls and Coventry Carol.
Sine the early nineteen hundreds, Gabrielle 'Coco' Chanel has been one of the world's fashion darlings.
Sine Qua Non Salo: A salon in the runnings for City's Best, this salon promises a creative environment that is attentive to the client's needs.
In Squier and Crehore's " Synchronograph " system " sine waves of current, instead of sharp " makes and breaks," or sharp reversals, are employed for transmitting signals, the waves being produced by an alternating-current dynamo, and regulated by means of a perforated paper ribbon, as in the Wheatstone automatic system.
Much that had been done by bishops, sine strepitu forensi et figura judicii, is now done in the course of regular judicial procedure.
This process is called conceptual synthesis, the possibility of which is a sine qua non for the exchange of information by speech and writing.
In the parable of the rich man and Lazarus, Basilides saw the proof of naturam sine radice et sine loco rebus supervenientem (Acta Archelai).
" Intuitio," " speculatio," " visio sine comprehensione," " comprehensio incomprehensibilis," " mystica theologia," " tertius caelus," are some of the terms he applies to this knowledge above knowledge; but in the working out of his system he is remarkably free from extravagance.
" Miracula sine doctrina nihil valent " is the principle now generally recognized.
The conferences were opened at the close of July in the camp of the grand vizier, who was pressing Belgrade hard and demanded the surrender of the city as a sine qua non.
Other views of the original aerarii are that they were: - artisans and freedmen (Niebuhr); inhabitants of towns united with Rome by a hospitium publicum, who had become domiciled on Roman territory (Lange); only a class of degraded citizens, including neither the cives sine sufJragio nor the artisans (Madvig); identical with the capite censi of the Servian constitution (Belot, Greenidge).
The improvement of the telescope was justly regarded as a sine qua non for the advancement of astronomical knowledge.
(25) The sum of the disturbance is obtained by adding (24) and (25) y = y l +y 2 = 2a cos Ut s i n 57 x, (26) At any given instant t this is a sine curve of amplitude 2a cos (27r/A)Ut, and of wave-length A, and with nodes at x = o, a A, A, ..., that is, there is no displacement at these nodes whatever the value of t, and between them the displacement is always a sine curve, but of amplitude varying between +2a and - 2a.
Formerly it was generally supposed that the Tartini tone was due to the beats themselves, that the mere variation in the amplitude was equivalent, as far as the ear is concerned, to a superposition on the two original tones of a smooth sine displacement of the same periodicity as that variation.
It was already in the hands of the Romans in 306 B.C., and since in the 3rd century B.C. it issued copper coins with a Latin legend it must have had the civitas sine suffragio.
12), begun in 1897 and opened for traffic in 1903, has a span of 1600 ft., a versed sine of 176 ft., and a width of 118 ft.
The only two points on which he departed from the orthodox Lutheran faith of his day were the requirement of regeneration as the sine qua non of the true theologian, and the expectation of the conversion of the Jews and the fall of Papacy as the prelude of the triumph of the church.
Of his very numerous literary productions the following may be mentioned: Beitrage zur Kenntnis der nati rlichen and politischen Verfassung des oettingischen Vaterlandes (1786); Ein Votum fiber den Wucher von einem Manne sine voto (1791); Historische Entwicklung der deutschen Steuerverfassungen (1793); Historische Priifung des vermeintlichen Alters der deutschen Landstande (1796); Neuere Geschichte des Fi rstentums Bayreuth (1486-1603) (1798-1811); Tabellen fiber Flacheninhalt &c. and bevorstehende Verluste der deutschen Reichsstande.
An intimate collaboration with England and France was a conditio sine qua non for Czechoslovakia.
Bruckner (Leipzig, 1901; also written in Polish); Chmielowski, History of Polish Literature (in Polish, 3 vols.); Stanislaus Tarnowski, History of Polish Literature (in Polish); Grabowski, Poezya Polska po roku 1863 (Cracow, 1903); Heinrich Nitschmann, Geschichte der polnischen Literatur (Leipzig; sine anno).
The ancient city of Fundi in 338 B.C. (or 332) received (with Formiae) the civitas sine suffragio, because it had always secured the Romans safe passage through its territory; the people as a whole did not join Privernum in its war against Rome three years later, though Vitruvius Vacca, the leader, was a native of Fundi.
The legislatures of Connecticut and Rhode Island, and town meetings in Cheshire and Grafton counties (New Hampshire) and in Windham county (Vermont) accepted the invitation, and the convention, composed of 12 delegates from Massachusetts, 7 from Connecticut, 4 from Rhode Island, 2 from New Hampshire and 1 from Vermont, all Federalists, met on the 15th of December 1814, chose George Cabot of Massachusetts president and Theodore Dwight of Connecticut secretary, and remained in secret session until the 5th of January 1815, when it adjourned sine die.
The commissioners feared that, so long as Greek was a sine qua non at the universities, these schools would be cut off from direct connexion with the universities, while the universities would in some degree lose their control over a portion of the higher culture of the nation.
As far as the circlesquaring functions are concerned, it would seem that Gregory was the first (in 1670) to make known the series for the arc in terms of the tangent, the series for the tangent in terms of the arc, and the secant in terms of the arc; and in 1669 Newton showed to Isaac Barrow a little treatise in manuscript containing the series for the arc in terms of the sine, for the sine in terms of the arc, and for the cosine in terms of the arc. These discoveries 1 See Euler, ” Annotationes in locum quendam Cartesii," in Nov.
A spectroscope may be compared to a mechanical harmonic analyser which when fed with an irregular function of one variable represented by a curve supplies us with the sine curves into which the original function may be resolved.
Not so; like Kant himself, Hodgson supposes something beyond; not, however, an unknown thing in itself causing sensations, but a condition, or sine qua non, of their existence, without being a cause of their nature.
What is known as the method of sines is used, for since the axes of the two magnets are always at right angles when the mirror magnet is in its zero position, the ratio M/H is proportional to the sine of the angle between the magnetic axis of the mirror magnet and the magnetic - = meridian.
ii.) that the abnormal values of the variation observed in the wood-built ships of his day was due to deviation of the compass caused by the iron in the ship; that the deviation was zero when the ship's head was near the north and south points; that it attained its maximum on the east and west points, and varied as the sine of the azimuth of the ship's head reckoned from the zero points.
non tamen sine epistopi auctoritate."
According to another point of view, an arboretum should be constructed with regard to picturesque beauty rather than systematically, although it is admitted that for scientific purposes a systematic arrangement is a sine qua non.
A Simple-Harmonic or Sine Wave is the only kind which is propagated without change of form.
The Democratic National Convention adopted (August 29, 1864) a resolution (drafted by Vallandigham) declaring the war a failure, and demanding a cessation of hostilities; it nominated M'Clellan for president, and instead of adjourning sine die as usual, remained organized, and subject to be convened at any time and place by the executive national committee.
The possession of the doctorate is a sine qua non for eligibility to a university chair, and to a lectureship in the university of Paris.
Regarded as a linear velocity, the parallactic motion is the same for all stars, being exactly equal and opposite to the solar motion; but its amount, as measured by the corresponding angular displacement of the star, is inversely proportional to the distance of the star from the earth, and foreshortening causes it to vary as the sine of the angular distance from the apex.
These currents are furnished by an alternator which transmits sine currents over the line and operates a motor at the distant end of the line, both machines running in synchronism.
The integral of (14) and (15) may be written ciU+E=Fcoso, c 2 V= - Fsino, dx F cost o F sinz o 71 = U cos o - V sin o = cl + c c ic os o, chi = U sine +V coso= (F - F) sin cos o - l sino, (19) c i 2 2 2 sin o cos o - l ?
Nor do we know anything of its history between 334 (when it probably became a civitas sine suffragio under Roman domination, shortly afterwards receiving, in 318, a praefectus iure dicundo) and 215, when the Romans introduced a garrison of 6000 men to protect the town from Hannibal, who besieged it in vain for three days in 214.
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