C. Mailer's Bavaria, both based on trigonometrical surveys.
A trigonometrical survey of British India was begun in 1800 and the country can now boast of a survey which in most respects is equal to those of most European states.
In the states of Australia cadastral surveys conducted by surveyors-general have been in progress for many years, as also trigonometrical surveys (Western Australia excepted), Australia.
A trigonometrical survey was given up and only details of immediate practical use are required.
Among the contents of this book we simply mention a trigonometrical chapter, in which the words sinus versus arcus occur, the approximate extraction of cube roots shown more at large than in the Liber abaci, and a very curious problem, which nobody would search for in a geometrical work, viz.
The trigonometrical section of the book had been issued as a separate treatise (Wittenberg, 1542) under the care of Rheticus.
The subject he had chosen for his Habilitationsschrift was the "Representation of a Function by Means of a Trigonometrical Series," a subject which Dirichlet had made his own by a now well-known series of researches.
Hero's expressions for the areas of regular polygons of from 5 to 12 sides in terms of the squares of the sides show interesting approximations to the values of trigonometrical ratios.
There has also been a certain amount of geographical sketching combined with trigonometrical observations; and there are the route surveys of native explorers.
In 1784 the base-line of the first trigonometrical survey in England was laid down on the heath.
The 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.
A collection of formulae relating to the circle, for instance, would comprise not only geometrical and trigonometrical formulae, but also approximate formulae, such as Huygens's rule (§ 91), which are the result of advanced analysis.
This is the case, for instance, with formulae which involve 7r or trigonometrical ratios.
The length of the arc of a circle, for instance, is known if the length of the chord and its distance from the middle point of the arc are known; but it may be more convenient in such a case to use a formula such as Huygens's rule than to obtain a more accurate result by means of trigonometrical tables.
If the data for any of these figures are other than those given above, trigonometrical ratios will usually be involved.
- Besides the work upon the geodetical operations connecting Paris and Greenwich, of which Legendre was one of the authors, he published in the Memoires de l'Academie for 1787 two papers on trigonometrical operations depending upon the figure of the earth, containing many theorems relating to this subject.
This triangulation was the beginning of the great trigonometrical survey which has since been extended all over the country.
It should be mentioned that in most tables of trigonometrical functions, the number io is added to all the logarithms in the table in order to avoid the use of negative characteristics, so that the characteristic 9 denotes in reality 1, 8 denotes a, io denotes o, &c. Logarithms thus increased are frequently referred to for the sake of distinction as tabular logarithms, so that the tabular logarithm =the true logarithm -IIo.
The first calculation or publication of Briggian or common logarithms of trigonometrical functions was made in 1620 by Edmund Gunter, who was Briggs's colleague as professor of 1 It was certainly published after Napier's death, as Briggs mentions his " librum posthumum."
The calculation of the logarithms not only of numbers but also of the trigonometrical functions is therefore due to Briggs and Vlacq; and the results contained in their four fundamental works - A rithmetica logarithmica (Briggs), 1624; Arithmetica logarithmica (Vlacq), 1628; Trigonometria Britannica (Briggs), 1633; Trigonometria artificialis (Vlacq), 1633 - have not been superseded by any subsequent calculations.
The calculation of tables of the natural trigonometrical functions may be said to have formed the work of the last half of the 16th century, and the great canon of natural sines for every 10 seconds to 15 places which had been calculated by Rheticus was published by Pitiscus only in 1613, the year before that in which the Descriptio appeared.
In the construction of the natural trigonometrical tables Great Britain had taken no part, and it is remarkable that the discovery of the principles and the formation of the tables that were to revolutionize or supersede all the methods of calculation then in use should have been so rapidly effected and developed in a country in which so little attention had been previously devoted to such questions.
In 1705 appeared the original edition of Sherwin's tables, the first of the series of ordinary seven-figure tables of logarithms of numbers and trigonometrical functions such as are in general use now.
Among the most useful and accessible of modern ordinary sevenfigure tables of logarithms of numbers and trigonometrical functions may be mentioned those of Bremiker, Schriin and Bruhns.
I „ Logarithms of the ratios of arcs to sines from 04 00000 to 0 4.05000, and log sines throughout the quadrant 4 „ Logarithms of the ratios of arcs to tangents from 0 4 00000 to 0 4.05000, and log tangents throughout the quadrant 4 The trigonometrical results are given for every hundred-thousandth of the quadrant (to" centesimal or 3" 24 sexagesimal).
8 4 In modern trigonometrical notation, I +sec 0: tan 0 :: I : tan Z0.
With Vieta, by reason of the advance in arithmetic, the style of treatment becomes more strictly trigonometrical; indeed, the Universales Inspectiones, in which the calculation occurs, would now be called plane and spherical trigonometry, and the accompanying Canon mathematicus a table of sines, tangents and secants.'
II) whose centre is 0, AC its chord, and HK the tangent drawn at the middle point of the arc and bounded by OA, OC produced, then, according to Archimedes, AMC< HK, but > AC. In modern trigonometrical notation the propositions to be compared stand as follows: 2 tan 20 >2 sin 28 (Archimedes); tan 10+2 sin 3B>0> 3 sin B (Snell).
2-}-cos B It is readily shown that the latter gives the best approximation to 0; but, while the former requires for its application a knowledge of the trigonometrical ratios of only one angle (in other words, the ratios of the sides of only one right-angled triangle), the latter requires the same for two angles, 0 and 3B.
The point separating the integers from the decimal fractions seems to be the invention of Bartholomaeus Pitiscus, in whose trigonometrical tables (1612) it occurs and it was accepted by John Napier in his logarithmic papers (1614 and 1619).
Russian explorers and natives of India trained for geographical reconnaissance, and employed in connexion with the great trigonometrical survey of India, had done so much towards clearing away the mists which enveloped the actual course of the river, that all the primary affluents were known, although their relative value was misunderstood, but the nature of the districts which bordered the river in Afghan Turkestan was so imperfectly mapped as to give rise to considerable political complication in framing the boundary agreement between Great Britain and Russia.
The French academician Bouger, who was chief of the scientific commission sent to Ecuador in 1736 to measure a degree of the meridian on the equator, made a trigonometrical measurement of Iliniza, and Wagner ascended to within 800 ft.
In applying tests of memory, it may be legitimate to allow a candidate to pass who answers correctly from 30 to 50% of the questions; such an allowance if applied to a test of capacity, such as the performance of a sum in addition, the solution of triangles by means of trigonometrical tables, or the translation of an easy passage from a foreign language, appears to be irrational.
To avoid the intricate trigonometrical calculations which would often be necessary, C graphical devices have been introduced by H.
If the quadratic (38) has a negative root, the trigonometrical functions in (36) are to be replaced by real exponentials, and the position x=o, y=o is unstable.
In 1885 a trigonometrical survey and a map on the scale of r in.
At the present time constructors almost always employ the inverse method: they compose a system from certain, often quite personal experiences, and test, by the trigonometrical calculation of the paths of several rays, whether the system gives the desired reproduction (examples are given in A.
Dehra is the headquarters of the Trigonometrical Survey and of the Forest Department, besides being a cantonment for a Gurkha force.
(Russian trigonometrical survey), and ending in Khorasan, the great Elburz range presents on its southern, or inward, face a more or less abrupt scarp rising above immense gravel slopes, and reaches in some of its summits a height of nearly 13,000 ft.; and the peak of Demavend, north-west of Teheran, has a height of at least 18,000 ft.
The heights of peaks determined by exact processes of trigonometrical observation are bound to be more or less in error for three reasons: (1) the extraordinary geoidal deformation of the level surface at the observing stations in submontane regions; (2) ignorance of the laws of refraction when rays traverse rarefied air in snow-covered regions; (3) ignorance of the variations in the actual height of peaks due to the increase, or decrease, of snow.
Trigonometrical Surveys in India, to be in probable error to the following extent: These determinations have the effect of placing Kinchinjunga second and K 2 third on the list.
He co-operated in the Danish and Hanoverian measurements of an arc and trigonometrical operations (1821-1848), and wrote (1843, 1846) the two memoirs Ãœber GegenstÃnde der hÃ¶heren GeodÃsie.
At first he purposed to study law; but this he abandoned on his father's death in 1794, and entered the army, obtaining a commission in the 12th regiment of foot, then stationed in India, where he rendered valuable assistance in the great trigonometrical survey.
De Morgan's other principal mathematical works were The Elements of Algebra (1835), a valuable but somewhat dry elementary treatise; the [[Essay]] on Probabilities (1838), forming the 107th volume of Lardner's Cyclopaedia, which forms a valuable introduction to the subject; and The Elements of Trigonometry and Trigonometrical Analysis, preliminary to the Differential Calculus (1837).
Its elevation as at present determined by trigonometrical observation is 2 9, 002 ft., but it is possible that further investigation into the value of refraction at such altitudes will result in placing the summit even higher.
The peak possesses no distinctive native name and has been called Everest after Sir George Everest, who completed the trigonometrical survey of the Himalayas in 1841 and first fixed its position and altitude.