Comparison Between Rhumb-Line and Great-Circle Courses
IT is well known that, while the shortest course between two points on a sphere is a great circle, the course which is easiest for navigation is a rhumb line, i.e . a course which cuts all the successive meridians at the same angle, so that the ship always steers in a direction making an angle of th...
Published in: | The Mathematical Gazette |
---|---|
Main Author: | |
Format: | Article in Journal/Newspaper |
Language: | English |
Published: |
Cambridge University Press (CUP)
1926
|
Subjects: | |
Online Access: | http://dx.doi.org/10.2307/3603804 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0025557200093098 |
id |
crcambridgeupr:10.2307/3603804 |
---|---|
record_format |
openpolar |
spelling |
crcambridgeupr:10.2307/3603804 2023-05-15T18:25:59+02:00 Comparison Between Rhumb-Line and Great-Circle Courses Heawood, P. J. 1926 http://dx.doi.org/10.2307/3603804 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0025557200093098 en eng Cambridge University Press (CUP) https://www.cambridge.org/core/terms The Mathematical Gazette volume 13, issue 183, page 137-145 ISSN 0025-5572 2056-6328 General Mathematics journal-article 1926 crcambridgeupr https://doi.org/10.2307/3603804 2022-04-07T08:02:46Z IT is well known that, while the shortest course between two points on a sphere is a great circle, the course which is easiest for navigation is a rhumb line, i.e . a course which cuts all the successive meridians at the same angle, so that the ship always steers in a direction making an angle of the same number of degrees with due north or due south. Such a course is represented by a straight line on a Mercator’s chart, which thus makes it appear more direct than it really is. On a great circle, though the course is the shortest, you are (except on the equator or a meridian) continually altering your direction as measured by the angle which it makes with the north. Between any two points there is thus a difference in favour of great-circle sailing; but to demonstrate rigorously the greatest value which this difference can possibly take is a somewhat difficult and attractive mathematical problem. I have heard it stated that it is not susceptible of rigorous mathematical treatment. The following is suggested as giving a complete solution of the question: it would be interesting to learn whether any shorter or more direct method is available, which makes no unwarranted assumption. For the abstract treatment of the problem (though indeed in the Pacific or the Southern Ocean fairly extended courses are practically available) we suppose the globe cleared of all such inconvenient encumbrances as continents or islands: moreover, we treat it as a perfect sphere; and to word the question more exactly we should say: Find two points such that the difference between the shortest great-circle course and the shortest rhumb-line course joining them is a maximum; for the great circle through two points will (in general) consist of a major and a minor arc, and on the other hand a rhumb line might start the longer way round the earth, or indeed make any number of revolutions about the pole before reaching its objective. In either case it is the shortest course of the kind with which we deal. Article in Journal/Newspaper Southern Ocean Cambridge University Press (via Crossref) Southern Ocean Pacific The Mathematical Gazette 13 183 137 145 |
institution |
Open Polar |
collection |
Cambridge University Press (via Crossref) |
op_collection_id |
crcambridgeupr |
language |
English |
topic |
General Mathematics |
spellingShingle |
General Mathematics Heawood, P. J. Comparison Between Rhumb-Line and Great-Circle Courses |
topic_facet |
General Mathematics |
description |
IT is well known that, while the shortest course between two points on a sphere is a great circle, the course which is easiest for navigation is a rhumb line, i.e . a course which cuts all the successive meridians at the same angle, so that the ship always steers in a direction making an angle of the same number of degrees with due north or due south. Such a course is represented by a straight line on a Mercator’s chart, which thus makes it appear more direct than it really is. On a great circle, though the course is the shortest, you are (except on the equator or a meridian) continually altering your direction as measured by the angle which it makes with the north. Between any two points there is thus a difference in favour of great-circle sailing; but to demonstrate rigorously the greatest value which this difference can possibly take is a somewhat difficult and attractive mathematical problem. I have heard it stated that it is not susceptible of rigorous mathematical treatment. The following is suggested as giving a complete solution of the question: it would be interesting to learn whether any shorter or more direct method is available, which makes no unwarranted assumption. For the abstract treatment of the problem (though indeed in the Pacific or the Southern Ocean fairly extended courses are practically available) we suppose the globe cleared of all such inconvenient encumbrances as continents or islands: moreover, we treat it as a perfect sphere; and to word the question more exactly we should say: Find two points such that the difference between the shortest great-circle course and the shortest rhumb-line course joining them is a maximum; for the great circle through two points will (in general) consist of a major and a minor arc, and on the other hand a rhumb line might start the longer way round the earth, or indeed make any number of revolutions about the pole before reaching its objective. In either case it is the shortest course of the kind with which we deal. |
format |
Article in Journal/Newspaper |
author |
Heawood, P. J. |
author_facet |
Heawood, P. J. |
author_sort |
Heawood, P. J. |
title |
Comparison Between Rhumb-Line and Great-Circle Courses |
title_short |
Comparison Between Rhumb-Line and Great-Circle Courses |
title_full |
Comparison Between Rhumb-Line and Great-Circle Courses |
title_fullStr |
Comparison Between Rhumb-Line and Great-Circle Courses |
title_full_unstemmed |
Comparison Between Rhumb-Line and Great-Circle Courses |
title_sort |
comparison between rhumb-line and great-circle courses |
publisher |
Cambridge University Press (CUP) |
publishDate |
1926 |
url |
http://dx.doi.org/10.2307/3603804 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0025557200093098 |
geographic |
Southern Ocean Pacific |
geographic_facet |
Southern Ocean Pacific |
genre |
Southern Ocean |
genre_facet |
Southern Ocean |
op_source |
The Mathematical Gazette volume 13, issue 183, page 137-145 ISSN 0025-5572 2056-6328 |
op_rights |
https://www.cambridge.org/core/terms |
op_doi |
https://doi.org/10.2307/3603804 |
container_title |
The Mathematical Gazette |
container_volume |
13 |
container_issue |
183 |
container_start_page |
137 |
op_container_end_page |
145 |
_version_ |
1766207757358202880 |