The rules then are sine of the middle part = product of tangents of adjacent parts = product of cosines of opposite parts.
"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.
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.
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.
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.
® 6.80470 Sine 0 's eq.
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.
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.
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.
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 ?
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.
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.
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.
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.
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.
Any periodic curve may be resolved into sine or harmonic curves by Fourier's theorem.
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.
(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.
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.
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.
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.
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.
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.
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."
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.
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.
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.
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.