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==References==
{{Reflist|refs=
<ref name="Cajori_1929">{{cite book |author-first=Florian |author-last=Cajori |author-link=Florian Cajori |title=A History of Mathematical Notations |chapter=§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions |volume=2 |orig-year=March 1929 |publisher=[[Open court publishing company]] |___location=Chicago, USA |date=1952 |edition=3rd corrected printing of 1929 issue, 2nd |pages=108, 176–179, 336, 346 |isbn=978-1-60206-714-1 |url=https://books.google.com/books?id=bT5suOONXlgC |access-date=2016-01-18 |quote=[…] §473. ''Iterated logarithms'' […] We note here the symbolism used by [[Alfred Pringsheim|Pringsheim]] and [[Jules Molk|Molk]] in their joint ''Encyclopédie'' article: "<sup>2</sup>log<sub>''b''</sub> ''a'' = log<sub>''b''</sub> (log<sub>''b''</sub> ''a''), …, <sup>''k''+1</sup>log<sub>''b''</sub> ''a'' = log<sub>''b''</sub> (<sup>''k''</sup>log<sub>''b''</sub> ''a'')."{{citeref|Pringsheim|Molk|1907|a<!-- [10] -->}} […] §533. ''[[John Frederick William Herschel|John Herschel]]'s notation for inverse functions,'' sin<sup>−1</sup> ''x'', tan<sup>−1</sup> ''x'', etc., was published by him in the ''[[Philosophical Transactions of London]]'', for the year 1813. He says ({{citeref|Herschel|1813|p. 10|style=plain}}): "This notation cos.<sup>−1</sup> ''e'' must not be understood to signify 1/cos. ''e'', but what is usually written thus, arc (cos.=''e'')." He admits that some authors use cos.<sup>''m''</sup> ''A'' for (cos. ''A'')<sup>''m''</sup>, but he justifies his own notation by pointing out that since ''d''<sup>2</sup> ''x'', Δ<sup>3</sup> ''x'', Σ<sup>2</sup> ''x'' mean ''dd'' ''x'', ΔΔΔ ''x'', ΣΣ ''x'', we ought to write sin.<sup>2</sup> ''x'' for sin. sin. ''x'', log.<sup>3</sup> ''x'' for log. log. log. ''x''. Just as we write ''d''<sup>−''n''</sup> V=∫<sup>''n''</sup> V, we may write similarly sin.<sup>−1</sup> ''x''=arc (sin.=''x''), log.<sup>−1</sup> ''x''.=c<sup>''x''</sup>. Some years later Herschel explained that in 1813 he used ''f''<sup>''n''</sup>(''x''), ''f''<sup>−''n''</sup>(''x''), sin.<sup>−1</sup> ''x'', etc., "as he then supposed for the first time. The work of a German Analyst, [[Hans Heinrich Bürmann|Burmann]], has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He[Burmann], however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan<sup>−1</sup>, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption."{{citeref|Herschel|1820|b<!-- [4] -->}} […] §535. ''Persistence of rival notations for inverse function.''— […] The use of Herschel's notation underwent a slight change in [[Benjamin Peirce]]'s books, to remove the chief objection to them; Peirce wrote: "cos<sup>[−1]</sup> ''x''," "log<sup>[−1]</sup> ''x''."{{citeref|Peirce|1852|c<!-- [1] -->}} […] §537. ''Powers of trigonometric functions.''—Three principal notations have been used to denote, say, the square of sin ''x'', namely, (sin ''x'')<sup>2</sup>, sin ''x''<sup>2</sup>, sin<sup>2</sup> ''x''. The prevailing notation at present is sin<sup>2</sup> ''x'', though the first is least likely to be misinterpreted. In the case of sin<sup>2</sup> ''x'' two interpretations suggest themselves; first, sin ''x'' &
<ref name="Herschel_1813">{{cite journal |author-first=John Frederick William |author-last=Herschel |author-link=John Frederick William Herschel |title=On a Remarkable Application of Cotes's Theorem |journal=[[Philosophical Transactions of the Royal Society of London]] |publisher=[[Royal Society of London]], printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall |___location=London |volume=103 |number=Part 1 |date=1813 |orig-year=1812-11-12 |jstor=107384 |pages=8–26 [10]|doi=10.1098/rstl.1813.0005 |s2cid=118124706 |doi-access=free }}</ref>
<ref name="Herschel_1820">{{cite book |author-first=John Frederick William |author-last=Herschel |author-link=John Frederick William Herschel |title=A Collection of Examples of the Applications of the Calculus of Finite Differences |chapter=Part III. Section I. Examples of the Direct Method of Differences |___location=Cambridge, UK |publisher=Printed by J. Smith, sold by J. Deighton & sons |date=1820 |pages=1–13 [5–6] |chapter-url=https://books.google.com/books?id=PWcSAAAAIAAJ&pg=PA5 |access-date=2020-08-04 |url-status=live |archive-url=https://web.archive.org/web/20200804031020/https://books.google.de/books?hl=de&id=PWcSAAAAIAAJ&jtp=5 |archive-date=2020-08-04}} [https://archive.org/details/acollectionexam00lacrgoog] (NB. Inhere, Herschel refers to his {{citeref|Herschel|1813|1813 work|style=plain}} and mentions [[Hans Heinrich Bürmann]]'s older work.)</ref>
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