Remembering spherical trigonometry

2016 ◽  
Vol 100 (547) ◽  
pp. 1-8 ◽  
Author(s):  
John Conway ◽  
Alex Ryba
Keyword(s):  
High School ◽  
20Th Century ◽  
Great Circle ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
The Subject ◽  
Image Position ◽  
Two Sides

Although high school textbooks from early in the 20th century show that spherical trigonometry was still widely taught then, today very few mathematicians have any familiarity with the subject. The first thing to understand is that all six parts of a spherical triangle are really angles — see Figure 1.This shows a spherical triangle ABC on a sphere centred at O. The typical side is a = BC is a great circle arc from to that lies in the plane OBC; its length is the angle subtended at O. Similarly, the typical angle between the two sides AB and AC is the angle between the planes OAB and OAC.

New Formulae for Combined Spherical Triangles

2018 ◽  
Vol 72 (2) ◽  
pp. 503-512
Author(s):  
Tsung-Hsuan Hsieh ◽  
Shengzheng Wang ◽  
Wei Liu ◽  
Jiansen Zhao
Keyword(s):  
Research Field ◽  
Great Circle ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
Calculation Process ◽  
Different Types ◽  
Celestial Bodies ◽  
Spherical Triangles

Spherical trigonometry formulae are widely adopted to solve various navigation problems. However, these formulae only express the relationships between the sides and angles of a single spherical triangle. In fact, many problems may involve different types of spherical shapes. If we can develop the different formulae for specific spherical shapes, it will help us solve these problems directly. Thus, we propose two types of formulae for combined spherical triangles. The first set are the formulae of the divided spherical triangle, and the second set are the formulae of the spherical quadrilateral. By applying the formulae of the divided spherical triangle, waypoints on a great circle track can be obtained directly without finding the initial great circle course angle in advance. By applying the formulae of the spherical quadrilateral, the astronomical vessel position can be yielded directly from two celestial bodies, and the calculation process concept is easier to comprehend. The formulae we propose can not only be directly used to solve corresponding problems, but also expand the spherical trigonometry research field.

“Western” Histories of World Literature

2018 ◽  
Vol 3 (4) ◽  
pp. 524-551
Author(s):  
Matthias Buschmeier
Keyword(s):  
20Th Century ◽  
World Literature ◽  
National Identities ◽  
The West ◽  
The Subject ◽  
To Come ◽  
The Many ◽  
Two Sides ◽  
Western Historiography

Abstract This article reviews attempts to define histories of world literature during the late 19th and first half of the 20th century. It submits that “World Literature” and national philology are two sides of the same coin, in that they serve to produce specific national identities and legitimize colonial hegemonic practices. Astonishingly, some patterns of these early histories of world literature can still be observed in contemporary theoretical debates on the subject. Thus, it is argued that, rather than dismissing this heritage of Western historiography (with or without condemnation), we should strive seriously to come up with alternative histories, wherein “West” is no longer treated as synonymous with “world,” and vice versa. The West should be seen as just one form of society and culture among the many others, all of which are due consideration when invoking the term “world.”

The Cartographic Solution of Great Circle Problems

1942 ◽  
Vol 46 (373) ◽  
pp. 4-31 ◽  
Author(s):  
A. J. Dilloway
Keyword(s):  
Special Needs ◽  
Great Circle ◽  
Spherical Trigonometry ◽  
Long Distance ◽  
Traditional Practice ◽  
Map Projection ◽  
Map Projections ◽  
The Earth ◽  
The Subject

During the last twenty years a revival of interest in the subject of map projections has taken the form of a renewed attack upon certain basic limitations which face the cartographer. A necessary stimulus has been provided by the growth of a technique of air navigation, which in many ways has departed from the traditional practice of the marine navigator. From this and other directions emphasis has been laid on the need for precise cartographic representation of distance, direction and position on the earth. But although certain problems may be treated by devising new projections to meet special needs, it is obvious that there is a limit to any purely cartographic approach.The requirements of the new navigation are, briefly, that its methods should be rapid, convenient to use, and of an accuracy consistent with the limitations imposed. The traditional solution by means of spherical trigonometry is thus ruled out on at least two of these counts. In its place, there are now available a number of graphical, mechanical and simplified tabular methods, many of which have no cartographic basis. Since a map projection is an obvious medium for the measurement of spherical relations, the needs of long-distance air navigation have encouraged the adaptation of certain projections in the form of special instruments and devices.

Vector Solutions for Great Circle Navigation

2005 ◽  
Vol 58 (3) ◽  
pp. 451-457 ◽  
Author(s):  
Michael A. Earle
Keyword(s):  
Great Circle ◽  
North Pole ◽  
Spherical Triangle ◽  
Vector Analysis ◽  
Small Computer ◽  
Spherical Trigonometry ◽  
The North ◽  
Document Vector ◽  
Spherical Triangles ◽  
Vector Approach

Traditionally, navigation has been taught with methods employing Napier's rules for spherical triangles while methods derived from vector analysis and calculus appear to have been avoided in the teaching environment. In this document, vector methods are described that allow distance and azimuth at any point on a great circle to be determined. These methods are direct and avoid reliance on the formulae of spherical trigonometry. The vector approach presented here allows waypoints to be established without the need to either ascertain the position of the vertex or select the nearest pole; the method discussed here requires only one spherical triangle having an apex at the North Pole and is also easy to implement on a small computer.

Direct Methods of Sight Reduction: An Historical Review

1982 ◽  
Vol 35 (2) ◽  
pp. 260-273 ◽  
Author(s):  
Charles H. Cotter
Keyword(s):  
Direct Method ◽  
Historical Review ◽  
Direct Methods ◽  
The Other ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
General Terms ◽  
Spherical Triangles ◽  
Two Sides ◽  
Fundamental Formula

In general terms the principal problem in astronomical navigation is the solving of a spherical triangle - the PZX-triangle. The fundamental formula of spherical trigonometry for finding an angle given the three sides of a spherical triangle is the cosine formula. By transposition this formula can be used for finding a side given the opposite angle and the other two sides. Because the cosine formula is not suitable for use with logarithms numerous formulae have been derived from it with the aim of simplifying logarithmic computation. The term ‘direct method’ applies to a method the basis of which is generally the cosine formula or any of its derivatives although some direct methods are based on Napier's Rules for right-angled spherical triangles.

Immigrant Arab Women in Chile during the Renaissance (Nahda), 1910s-1920s

1970 ◽  
pp. 47-55
Author(s):  
Sarah Limorté
Keyword(s):  
20Th Century ◽  
19Th Century ◽  
Thematic Analysis ◽  
New World ◽  
Arab Women ◽  
Arab Woman ◽  
The Subject ◽  
The 19Th Century

Levantine immigration to Chile started during the last quarter of the 19th century. This immigration, almost exclusively male at the outset, changed at the beginning of the 20th century when women started following their fathers, brothers, and husbands to the New World. Defining the role and status of the Arab woman within her community in Chile has never before been tackled in a detailed study. This article attempts to broach the subject by looking at Arabic newspapers published in Chile between 1912 and the end of the 1920s. A thematic analysis of articles dealing with the question of women or written by women, appearing in publications such as Al-Murshid, Asch-Schabibat, Al-Watan, and Oriente, will be discussed.

Planification: Potential and role in XXI century market economy

2016 ◽  
pp. 63-80 ◽  
Author(s):  
A. Buzgalin ◽  
A. Kolganov
Keyword(s):  
Economic Development ◽  
20Th Century ◽  
Market Economy ◽  
Critical Analysis ◽  
New Economy ◽  
Real Socialism ◽  
The Subject ◽  
Planning Institutions

The authors, basing on a critical analysis of the experience of planning during the 20th century in a number of countries of Europe and Asia, and also on the lessons from the economics of "real socialism", set out to substantiate their conclusions on the advisability of "reloading" this institution. The aim is to create planning mechanisms, suited to the new economy, that incorporate forecasting, projections, direct and indirect selective regulation and so forth into integral programs of economic development and that set a vector of development for particular limited spheres of what remains on the whole a market economy. New planning institutions presuppose a supersession of the forms of bureaucratic centralism and a reliance on network forms of organization of the subject and process of planning.

IMPELEMENTATION OF SELF-REGULATED LEARNING ON JUNIOR HIGH SCHOOL STUDENT OF LOW SOCIOECONOMIC STATUS

2020 ◽  
Vol 2 (1) ◽  
pp. 1-15
Author(s):  
Fatin Rohmah Wahidah ◽  
Farida Kurniawati
Keyword(s):  
High School ◽  
Junior High School ◽  
Junior High ◽  
Learning Motivation ◽  
Learning Abilities ◽  
Low Socioeconomic ◽  
Self Regulated Learning ◽  
Regulated Learning ◽  
The Subject ◽  
Motivation And Learning

The purpose of this study was to determine the application of self-regulated learning interventions to increase learning motivation and learning abilities in junior high school students who come from families with low socioeconomic levels. The intervention was given for five sessions through psychoeducation and assignments. The subject of this study is a male, 15 years old. This study uses a qualitative method with a case study approach. Data analysis using triangulation techniques. The results of the analysis of observations, interviews, and assignment documents show that there are differences before and after the subject was given intervention. In general, self-regulated learning intervention has a positive impact on learning motivation and learning abilities of the subject. Several factors that support and inhibit intervention were explained. Some suggestions are given to develop the further intervention.

John of Kronshtadt and Leo Tolstoy

2020 ◽  
Vol 102 ◽  
pp. 471-478
Author(s):  
Peter A. Shevchenko
Keyword(s):  
Comparative Analysis ◽  
20Th Century ◽  
Real Life ◽  
Leo Tolstoy ◽  
Russian Society ◽  
Common People ◽  
Religious Consciousness ◽  
The Subject ◽  
Main Components

The article provides a comparative analysis of the influence of L.N. Tolstoy and I.I. Sergiev (John of Kronstadt) on the formation of personal worldview in Russian society. The analysis is based on the testimonies of the contemporaries and the previously not reissued publication of “Novy Put” (“New Way”) journal on the subject. In the context of the declared problematics, special attention is paid to the question of transformation of religious consciousness in the course of the personality formation in relation to the period under consideration (the beginning of the 20th century). The author reveals and analyzes the main components of the life stand of Tolstoy and Father John of Kronstadt in the context of their influence on contemporaries. The results of the study allow to reveal the following antitheses that characterize Tolstoy and John of Kronstadt, respectively: doubt - faith, search for oneself – following the once chosen path, preaching of non-resistance as part of the philosophy of not-doing (not doing evil) – preaching of active upholding of faith (doing good), “simple living” – real life with and for common people.

THE EFFECT OF APPLYING DEMONSTRATION METHOD ON STUDENTS’ ACHIEVEMENT IN SPEAKING SKILL AT THE THIRD GRADE OF SMK PAB 3 MEDAN ESTATE

2015 ◽  
Vol 4 (2) ◽  
Author(s):  
Meryanti Napitupulu And Anni Holila Pulungan
Keyword(s):  
High School ◽  
Control Group ◽  
Vocational High School ◽  
Control Groups ◽  
Speaking Test ◽  
The Third ◽  
Speaking Skill ◽  
The Subject ◽  
And Control ◽  
Experimental Group

This study was conducted as an attempt to discover the effect of applying Demonstration Method on students’ achievement in speaking skill. It was an experimental research. The subject was students of Grade XII, Vocational High School (Sekolah Menengah Kejuruan: SMK), which consisted of 79 students. The research was divided into two groups: experimental and control groups. The instrument used to collect the data was speaking test. To obtain the reliability of the test, the writer applied Kuder Richardson 21 formula. The result of the reliability was 0.7, and it was found that the test was reliable. The data were analyzed by using t-test formula. The analysis showed that the scores of the students in the experimental group were significantly higher than the scores of the students in the control group at the level of significant m = 0.05 with the degree of freedom (df) 77, t-observed value 8.9 > t-table value 1.99. The findings indicate that using Demonstration Method significantly affected the students’ achievement in speaking skill. So, English teachers are suggested to use Demonstration Method in order to improve students’ achievement in speaking skill.

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