# Talk:Angle of parallelism

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Field: Geometry

## Bonola and Halsted

In 1912 Roberto Bonola published a textbook with Open Court in Chicago on the topic of non-Euclidean geometry. When Dover re-issued the book half a century later, it included work by G.B. Halsted that significantly improved coverage of the subject, particularly a translation of Lobachevski. It is that appendix that is cited for an English reference for the topic "angle of parallelism". Recently the Bonola text has been scanned into Archive.org. One can now find the link at the end of Non-Euclidean geometry. Since the text scanned was the original 1912 edition it does not include Halsted's translation of Lobachevski. Hence I have removed the link as inappropriate for this article.Rgdboer (talk) 22:45, 19 February 2008 (UTC)

The Halsted manuscript has become available Google Books so the old print source in the back of the Dover edition of Bonola is unnecessary.
The text is available by template:Geometrical Researches on the Theory of Parallels, p. 13, at Google Books.
Bonola's original edition did not include Halsted and the paging is unclear in editions that include the translation.Rgdboer (talk) 01:24, 10 February 2015 (UTC)

## Napier ?

The second of the equivalent descriptions of angle of parallelism is

$\tan(\tfrac{1}{2}\Pi(a)) = e^{-a} .$

Here the e is the base of natural logarithms. The historic English reference cited is Halsted 1891, now on GB:

Geometrical Researches on the Theory of Parallels, p. 41, at Google Books.

Unfortunately Halsted considers unit of length in connection with selection of a base for logarithm, and in doing so refers to Napierian logarithm, a topic frequently confused with natural logarithm, but is in fact a rather different function. All reference to Napierian logarithm should be avoided; better references might be found to complete the thought Halsted meant to express before this misdirection.Rgdboer (talk) 02:04, 11 February 2015 (UTC) Correct date 1891.Rgdboer (talk) 02:07, 11 February 2015 (UTC)

Consequently, the following quotation from Halsted was removed:

...$\tan(\tfrac{1}{2} \Pi(x)) = \theta ^{-x}$ where $\theta$ may be any arbitrary number, which is geater than unity, since $\Pi(x) = 0$ for $x = \infty$.
Since the unit by which lines are measured are arbitrary , so we may also understand by $\theta$ the base of Napierian logarithms.

The quotation is found on page 41 of Halsted's translation of Lobachevsky.Rgdboer (talk) 02:21, 20 February 2015 (UTC)

## Negative angle

The following confusing contribution was removed:

By definition for a negative angle p:
$\Pi(p) + \Pi(-p) = \pi$

Before the text was made consistent with a as variable segment length, some uses were p. It does not make sense to then call p an angle. Negative segment length doesn't make sense either. Perhaps a discussion here can clarify the idea, or a reference can be produced to justify this addition.Rgdboer (talk) 20:08, 7 March 2015 (UTC)

Reference: Halsteds translation of Lobachevsky's "Geometrische Untersuchungen zur Theory der Parallellinien" page 20-21, end of paragraph 23 (taken from Bonola):
Since we are wholy at liberty we will understand by the symbol $\Pi(p)$ when the line p is expressed by a negative number we will assume
$\Pi(p) + \Pi(-p) = \pi$,
as equation which shall hold for all values of p, positive as well as negative and for p = 0.
Maybe my shortening of this was not really clear, or maybe it is later given another meaning
WillemienH (talk) 20:51, 9 March 2015 (UTC)