Talk:Indian mathematics: Difference between revisions

Browse history interactively ← Previous editContent deleted Content addedVisual WikitextInline

Revision as of 03:25, 2 July 2006 edit212.199.22.40 (talk)No edit summary← Previous edit		Latest revision as of 14:32, 24 December 2024 edit undoMerlin.anthwares (talk \| contribs)86 edits →Aryabhata I's contributions - Trigonometry: new sectionTag: New topic
(449 intermediate revisions by 94 users not shown)
Line 1:		Line 1:
			{{WikiProject banner shell\|class=B\|1=
	This is a nationalist orgy. A good article could be written on this subject; it might even use the list of boasts in the middle section as a framework - but to ascribe the invention of trigonometry to someone who lived two centuries ''after'' ] is nonsense. ] 19:49, 7 September 2005 (UTC)
			{{WikiProject India\|importance=mid\|history=yes\|history-importance=mid\|assess-date=May 2012}}
			{{WikiProject Mathematics\|importance=high}}
			{{WikiProject Pakistan\|History=yes\|importance=mid}}
			{{WikiProject Anthropology\|oral-tradition=yes
			}}
			}}

			{{archivebox\|
	Many of the claims on this page are patently false. I will do what I can, but this page desperately needs an expert in this field.
			] 07 Sep 2005 - 31 Dec 2006<br>
			] 01 Jan 2007 - 23 Feb 2007<br>
			] 23 Feb 2007 - 25 June 2010<br>
			}}

	== ~~Pythagorean~~ ~~Theorem~~ ==		== Vaishali Ganit(a)? ==

			This page doesn't cite a source for it, and ] simply redirects back to this page. It isn't mentioned in the main academic publications (e.g. by G.G. Joseph, Kim Plofker, or Takao Hayashi), so I had to do some detective-work...
	I don't think there is anyone who thinks the Pythagorus was the first to state the ]. Indeed it was likely used well before even 800 BC (look at the article). The fact that the Indians were the first to use a "proof with specific numbers" is nonsense. Using specific numbers is not a proof at all, it merely shows one special case. The first real proof we have is due to Euclid. A "proof with specific numbers" does not imply any deeper understanding of geometry, or an understanding of what constitutes a proof. Thus, the pythagorean theorem was used well before Indian mathematicians, and it was first proved by Greek mathematicians.
	] 16:13, 15 December 2005 (UTC)

			"Vaishali Ganit" was first mentioned on this page in December 2005; no source was ever provided. There were more details about the supposed contents of this text, all uncited, prior to when the Jain math section was shortened in May 2007. The "]" page formerly had an uncited claim that "Vaishali Ganit" contained the first-ever use of this mathematical method. The claim was inserted in 2006, and deleted in 2013 by a user who noticed the suspiciousness. However, the claim had already been credulously carried over into other secondary sources about math history, which still appear in a Google search. I used Google Search's time range tool and found three other mentions of it on pages that appeared to be older than 2005, but Google was wrong: these pages all were created or updated between 2010 and 2016, and appear to have derived their information from Misplaced Pages.
	: First of all, calm down. This discussion - what consitutes ''real'' proof and what is just silly amatuers dabbling with numbers and ''have no deep understanding of geometry'' is not for this page. After reading through the ] page, I am understanding that Indians were the first to state the theorum and first to give a numerical proof (one that uses specific numbers but in such a way that it can be generalized - not ''one case'' as you mentioned). I will change this page to reflect that. If you disagree with wording or timeline please discuss on the ] page. If I have understood it incorrectly - Indians first to stating theorum and numerical proof - please discuss on this page. --] 01:14, 16 December 2005 (UTC)

			In conclusion, I can't find even a single reliable source that verifies the existence of a text with a name remotely like "Vaishali Ganit(a)". In fact, it appears that every single mention of it is either from websites that simply copy Misplaced Pages, or from pages that got their information from Misplaced Pages. Hence I am deleting the mention of it here. I have the strong suspicion that it is nothing more than a decade-long hoax. ] (]) 04:42, 26 May 2018 (UTC)
	:: I'm not so sure that there is any consensus that Indians were the first to state the theorem. There is strong evidence that it was used as far back as 2000 BC, by the monolith builders in ancient Britain and elsewhere. Indeed, there is a specific example of a solution of a problem involving the pythagorean theorem on a Bablyonian tablet which I believe is circa 1200 BC. Furthermore, "numerical proof" is nonsensical. "Proving" the pythagorean with specific numbers, but in such a way that you could also use other specific numbers and prove it for them, as well, is no proof at all.

			== Invented or recorded? ==
	::I did not at any point imply that the Indian mathematicians were "silly amatuers" or anything of the like. They certainly had many important contributions to mathematics and science.

			This article had said that the decimal system was ''invented'' by Indian mathematicians and recently {{u\|Deacon Vorbis}} changed that to say it was ''recorded'' by ancient Indians. There followed a bit of back and forth by various editors and I was involved with part of that. Even though it seems to fly in the face of "common knowledge", I do believe that Deacon Vorbis' intent was correct. The statement was cited to Kim Plofker's ''Mathematics in India'', a work in which Plofker tries to set the record straight and point out what is historically known and what is speculation about the history of Indian mathematics. <s>He</s> She does not, as far as I can tell, use the term "invented", instead <s>he</s> she talks about the concept being "developed" and being "recorded" for the first time. In an earlier work <s>he</s> she pointedly remarks that the origin of the concept can not be determined from the historical record. It seems to me that using the word "invented" and citing Plofker for it is very inappropriate. --] (]) 19:36, 26 June 2018 (UTC)
	::I have changed the wording to what I hope is an agreeable compromise. As I do not dispute the accuracy of anything written on the ] page, only on this page, I do believe that this is the appropriate place for this discussion. - Grokmoo
			:You are right. I wrote the article long time ago, or wrote most of the readable sections of the article. It was "recorded." That's what Kim Plofker says. I will restore it if it hasn't been done already. ]] 17:35, 27 June 2018 (UTC)
			::Plofker is a she. "Recorded" is no put down. There is no evidence that the Indians got the ideas from any other place or civilization. What they first recorded in an orally transmitted Buddhist text is a piece of beauty. Other civilizations had bits and pieces of the puzzle. The Chinese had a place value system. But what is in use today is the Indian system, more importantly the arithmetic in use today, addition, subtraction, multiplication, long division, ..., the bread and butter of primary school math the world over, is the work of unknown mathematicians of the subcontinent. ]] 18:01, 27 June 2018 (UTC)
			My deepest apologies to Kim. I thought that I had read something about her that used the masculine pronoun and was surprised by that, having initially assumed she was female. I agree that there is nothing wrong with the use of "recorded", but another editor claimed that this might imply that the origins of the concept came from elsewhere. I don't see it that way, but I would be willing to add a short clarification to thwart that implication–not quite sure how to do that though.--] (]) 21:26, 27 June 2018 (UTC)

			== On vacation ==
	::: The page says ''Circa 2500 BCE, Megalithic monuments on the British Isles incorporate right triangles with integer sides. B.L. van der Waerden conjectures that these Pythagorean triples were discovered algebraically.'' I am reading that not as strong evidence but as ''conjecture'' by one person. Pythogorean triples were known by many civilizations but not their algebraic nature. To tell you the truth, I am becoming very weary of modern interpretations of mathematical history. If the same monuments were found anywhere other than Europe, there is not much chance anyone would have cared to attribute more knowledge than is directly visible - knowledge of Pythogorean triples.
	::: In any case, the current wording is agreeable. --] 16:27, 17 December 2005 (UTC)

			I am on vacation through September. I'd be grateful if {{u\|David Eppstein}}, {{u\|Johnuniq}} and {{u\|RegentsPark}}, {{u\|Abecedare}} could keep an eye on this article. I mean, I know you do in any case, but the times are such that more vigilance might be required. Best regards, ]] 17:01, 16 July 2020 (UTC)
	It is very interesting to see Westerners not accept the fact that in the ancient world, Eastern civilizations were more advanced than the European civilization. Nobody questions the Eurocentrism in science and technology in the modern times, but it is high time to accept the contribtuions in the ancient world from not just India (which undoubtedly has the maximum contribution to the foundations of mathematics as we know it today), but from other parts of the world, specifically Asian countries, as well, besides Greece and Egypt (which, surprisingly enough, has been given credit despite not being in Europe, perhaps due to its proximity to Europe and the fact that a lot of 'European' ideas were adopted directly from there). While earliest records of trigonometry as a studied discipline exist perhaps from Greece, no doubt exists about the fact that there are much older allusions to geometry and trigonometry in the Vedas and Hindu scriptures dating earlier than 1000 BCE. Moreover, what is found in Greece is only the elements of trigonometry. Trigonometry developed as a well-explored science in India later on, and this is exactly what the text on Indian mathematics implies. ] 19:31, 2 January 2006 (UTC)apalaria

		⚫	== Kerala mathematics ==
	:: If you can, you should try to get a copy of some of Ptolemy's work. If you read the Almagest, you will find a quite complete and comprehensive development of trigonometry. It is certain that many of these ideas were well known to the Greeks well before Ptolemy, but his work is sufficient to place the development of trigonometry at least as early as about 150 CE. Vague references in scripture are not development of mathematics. If you know of even a relatively complete treatment that was written before this time, I would love to see it, but I do not think any such treatise exists. Unless you can prove otherwise, the statement that the Indians developed trigonometry is patently false. ] 15:02, 5 January 2006 (UTC)

			In kerala mathematics section, it is said that the geometric series formula discovered by kerala mathrmaticians were discovered earlier in 10th century by arab mathrmatician ibn al haytham. And i checked the link and no where in the book it is mentioned that alhazen were aware about this geometric series formula. It seems like kerala mathematicians were the first to discover it. So i want you guys to remove that comment. ] (]) 11:52, 15 January 2021 (UTC)
	:: I removed the reference to trigonometry. However, I do think this Aryabhata should probably be mentioned on this page, but I am not sure where and how, so I'll let someone else decide. ] 15:06, 5 January 2006 (UTC)
	:::The problem is that most (a lot, rather) of the works of Indian mathematicians/astronomers/etc. were destroyed in the course of the several invasions that happened. Hence, today there is not enough paper (or whatever) evidence of the original works simply because they no longer exist. It is indeed unfortunate that the works have been destroyed. As for westerners, a very few of them actually even know that science and mathematics ever existed in ancient India. And telling them so usually evokes skepticism. ] 14:36, 1 March 2006 (UTC)
	:::: If the evidence is destroyed, why are you sure the groundbreaking works ever existed? {{unsigned\|212.199.22.40}} 02:53, July 2, 2006

			: Without thinking too carefully, I reverted your change. I've now reverted myself. Because (a) I don't know for myself; (b) the impressive-looking Edwards ref was , and I have no means of checking it; (c) the ] doesn't have anything similar ] (]) 19:05, 15 January 2021 (UTC)
⚫	== ~~Vedic~~ mathematics ==
			::This "we invented it first" sports-spectator view is a bad way to understand the history of mathematics. And the formula for a finite geometric series is in Euclid (Book IX, prop. 35). It's an easy step from there to the infinite one but it is plausible to me that the Kerala mathematicians stated that step before others, because the concept of infinite limits may not have been current elsewhere (but see Zeno). It's that concept, not the specific formula, that is the interesting part. —] (]) 19:11, 15 January 2021 (UTC)

			== Narayana pandit ==
	{{main\|Vedic mathematics}}
	Propose a name change for the article from ] - to something like ] or ].

			Narayana pandit is not a kerala mathematician. He lived and worked in north india . So i have decided to remove that part ] (]) 17:38, 15 January 2021 (UTC)
	- OR -
			:This change is in agreement with Plofker's '']'', which puts Narayana in an earlier chapter than Kerala. —] (]) 19:12, 15 January 2021 (UTC)

			== Sum of integral powers of numbers ==
	Change the section title of this article to "Mathematics of/from Vedas" or something like that. --<FONT style="color:#5A3696">]</FONT>ː<FONT style="color:#008000">]</FONT> 08:15, 17 January 2006 (UTC)

			Why is the equation
	: I've written a note in the ''Vedic Mathematics'' section of this article to inform the reader that the article ] is based on a system developed by Shri Bharati Krishna Tirtha. - Jagged 85, 18/01/2006
			1^k+2^k+3^k .... for large k /n^k+1 is equal to 1/k+1 is attributed to alhazen ? First of all, he only found out the sum of 2nd and 4th powers of finite numbers.
	::Thanks Jagged. Though I think it doesn't help much in absence of some stronger disambig or unless the other article is renamed more descriptively. Because, a user querying for Vedic + Mathematics would always land up on that page - where there are 3 disambig notices already! --<FONT style="color:#5A3696">]</FONT>ː<FONT style="color:#008000">]</FONT> 11:41, 19 January 2006 (UTC)
			It was kerala school who gave a general formula for any integral powers of finite numbers and also made the realization that for infinite n , lower order terms can be neglected and can be written as n^k+1/k+1. It was kerala school who explicitly stated this result. So it should be credited to kerala school only. So i am editing that paragraph. ] (]) 17:33, 24 January 2021 (UTC)
			:The general formula for finite geometric series is in Euclid, long before Kerala. As I already clearly stated two sections up, but apparently you didn't read that. —] (]) 19:23, 24 January 2021 (UTC)

			Again read the comment carefully. I am not talking about geometric formula . We already discussed about that. In this, i am talking the about sum of integral powers which should be credited to kerala school and alhazen should not be mentioned ] (]) 05:41, 25 January 2021 (UTC)
	==Article structure==
			:You are of course correct re the formula, sorry. I have no useful knowledge of Alhazen's accomplishments here but we should certainly not be mentioning him without a source. —] (]) 06:00, 25 January 2021 (UTC)
	Shouldn't this article talk about Indian Mathematics rather than Indian Mathematicians? There should be a ] article covering this material, and ] should be about Indian mathematics in general. In short, there are probably too much lists. There's a lot of great material in here that should be written in prose. ] 13:04, 18 April 2006 (UTC)

	== ~~Hic~~ ~~Rhodus, hic salta~~ ==		== Article's title ==

			If this article showed that in India there is no such thing as counting, or that south of the Himalayas multiplication has no effect, or perhaps that dividing twelve by three gives sixty-five in India, then the current title would certainly be justified.
	Anybody claiming Indian priority in inventing calculus please quote the ancient sources. Don't tell me the dog ate them or the invading Turks burnt them. I want to see your proof!

			Isn't it all just mathematics? Where is the article on "Argentinian gravity"? ] (]) 18:13, 11 September 2024 (UTC)
			:A title suggestion accompanying your criticism would have been a good idea. History of Mathematics in India? Indian contributions to mathematics? Something else?] <small>(])</small> 20:23, 11 September 2024 (UTC)
			::I think both of your possibilities are good. There's also already a redirect from "Mathematics in India". Any of those seems (to me) accurate enough and suitable enough, and I wouldn't argue against any of them. People with a stronger connection to the article than I have, might prefer one of them over another, or might have more ideas; I'd be fine with any consensus that prevents the kind of obvious wrong interpretations I complained about. ] (]) 21:43, 11 September 2024 (UTC)

			== Aryabhata I's contributions - Trigonometry ==

			Not sure if "Contains the earliest tables ...", "Contains the trigonometric formula ..." is the best verbiage. Would it be better to use "Created" instead of "Contains"? ] (]) 14:32, 24 December 2024 (UTC)

Latest revision as of 14:32, 24 December 2024

This article is rated B-class on Misplaced Pages's content assessment scale.
It is of interest to the following WikiProjects:

India: History Mid‑importance

	India portal This article is within the scope of WikiProject India, which aims to improve Misplaced Pages's coverage of India-related topics. If you would like to participate, please visit the project page.IndiaWikipedia:WikiProject IndiaTemplate:WikiProject IndiaIndia
Mid	This article has been rated as Mid-importance on the project's importance scale.
	This article is supported by the Indian history workgroup (assessed as Mid-importance).
	This article was last assessed in May 2012.

Mathematics High‑priority

	Mathematics portal This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics
High	This article has been rated as High-priority on the project's priority scale.

Pakistan Mid‑importance

	Pakistan portal This article is within the scope of WikiProject Pakistan, a collaborative effort to improve the coverage of Pakistan on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.PakistanWikipedia:WikiProject PakistanTemplate:WikiProject PakistanPakistan
Mid	This article has been rated as Mid-importance on the project's importance scale.
	This article is supported by WikiProject Pakistani history.

Anthropology: Oral tradition

	This article is within the scope of WikiProject Anthropology, a collaborative effort to improve the coverage of Anthropology on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.AnthropologyWikipedia:WikiProject AnthropologyTemplate:WikiProject AnthropologyAnthropology
???	This article has not yet received a rating on the importance scale.
	This article is supported by Oral tradition taskforce.

Archives

/Archive 1 07 Sep 2005 - 31 Dec 2006
/Archive 2 01 Jan 2007 - 23 Feb 2007
/Archive 3 23 Feb 2007 - 25 June 2010

Vaishali Ganit(a)?

This page doesn't cite a source for it, and Vaishali Ganit simply redirects back to this page. It isn't mentioned in the main academic publications (e.g. by G.G. Joseph, Kim Plofker, or Takao Hayashi), so I had to do some detective-work...

"Vaishali Ganit" was first mentioned on this page in December 2005; no source was ever provided. There were more details about the supposed contents of this text, all uncited, prior to when the Jain math section was shortened in May 2007. The "false position method" page formerly had an uncited claim that "Vaishali Ganit" contained the first-ever use of this mathematical method. The claim was inserted in 2006, and deleted in 2013 by a user who noticed the suspiciousness. However, the claim had already been credulously carried over into other secondary sources about math history, which still appear in a Google search. I used Google Search's time range tool and found three other mentions of it on pages that appeared to be older than 2005, but Google was wrong: these pages all were created or updated between 2010 and 2016, and appear to have derived their information from Misplaced Pages.

In conclusion, I can't find even a single reliable source that verifies the existence of a text with a name remotely like "Vaishali Ganit(a)". In fact, it appears that every single mention of it is either from websites that simply copy Misplaced Pages, or from pages that got their information from Misplaced Pages. Hence I am deleting the mention of it here. I have the strong suspicion that it is nothing more than a decade-long hoax. Avantiputra7 (talk) 04:42, 26 May 2018 (UTC)

Invented or recorded?

This article had said that the decimal system was invented by Indian mathematicians and recently Deacon Vorbis changed that to say it was recorded by ancient Indians. There followed a bit of back and forth by various editors and I was involved with part of that. Even though it seems to fly in the face of "common knowledge", I do believe that Deacon Vorbis' intent was correct. The statement was cited to Kim Plofker's Mathematics in India, a work in which Plofker tries to set the record straight and point out what is historically known and what is speculation about the history of Indian mathematics. He She does not, as far as I can tell, use the term "invented", instead he she talks about the concept being "developed" and being "recorded" for the first time. In an earlier work he she pointedly remarks that the origin of the concept can not be determined from the historical record. It seems to me that using the word "invented" and citing Plofker for it is very inappropriate. --Bill Cherowitzo (talk) 19:36, 26 June 2018 (UTC)

You are right. I wrote the article long time ago, or wrote most of the readable sections of the article. It was "recorded." That's what Kim Plofker says. I will restore it if it hasn't been done already. Fowler&fowler «Talk» 17:35, 27 June 2018 (UTC)

Plofker is a she. "Recorded" is no put down. There is no evidence that the Indians got the ideas from any other place or civilization. What they first recorded in an orally transmitted Buddhist text is a piece of beauty. Other civilizations had bits and pieces of the puzzle. The Chinese had a place value system. But what is in use today is the Indian system, more importantly the arithmetic in use today, addition, subtraction, multiplication, long division, ..., the bread and butter of primary school math the world over, is the work of unknown mathematicians of the subcontinent. Fowler&fowler «Talk» 18:01, 27 June 2018 (UTC)

My deepest apologies to Kim. I thought that I had read something about her that used the masculine pronoun and was surprised by that, having initially assumed she was female. I agree that there is nothing wrong with the use of "recorded", but another editor claimed that this might imply that the origins of the concept came from elsewhere. I don't see it that way, but I would be willing to add a short clarification to thwart that implication–not quite sure how to do that though.--Bill Cherowitzo (talk) 21:26, 27 June 2018 (UTC)

On vacation

I am on vacation through September. I'd be grateful if David Eppstein, Johnuniq and RegentsPark, Abecedare could keep an eye on this article. I mean, I know you do in any case, but the times are such that more vigilance might be required. Best regards, Fowler&fowler «Talk» 17:01, 16 July 2020 (UTC)

Kerala mathematics

In kerala mathematics section, it is said that the geometric series formula discovered by kerala mathrmaticians were discovered earlier in 10th century by arab mathrmatician ibn al haytham. And i checked the link and no where in the book it is mentioned that alhazen were aware about this geometric series formula. It seems like kerala mathematicians were the first to discover it. So i want you guys to remove that comment. Jino john1996 (talk) 11:52, 15 January 2021 (UTC)

Without thinking too carefully, I reverted your change. I've now reverted myself. Because (a) I don't know for myself; (b) the impressive-looking Edwards ref was added by an anon in 2006, and I have no means of checking it; (c) the Ibn al-Haytham doesn't have anything similar William M. Connolley (talk) 19:05, 15 January 2021 (UTC)

This "we invented it first" sports-spectator view is a bad way to understand the history of mathematics. And the formula for a finite geometric series is in Euclid (Book IX, prop. 35). It's an easy step from there to the infinite one but it is plausible to me that the Kerala mathematicians stated that step before others, because the concept of infinite limits may not have been current elsewhere (but see Zeno). It's that concept, not the specific formula, that is the interesting part. —David Eppstein (talk) 19:11, 15 January 2021 (UTC)

Narayana pandit

Narayana pandit is not a kerala mathematician. He lived and worked in north india . So i have decided to remove that part Jino john1996 (talk) 17:38, 15 January 2021 (UTC)

This change is in agreement with Plofker's Mathematics in India, which puts Narayana in an earlier chapter than Kerala. —David Eppstein (talk) 19:12, 15 January 2021 (UTC)

Sum of integral powers of numbers

Why is the equation 1^k+2^k+3^k .... for large k /n^k+1 is equal to 1/k+1 is attributed to alhazen ? First of all, he only found out the sum of 2nd and 4th powers of finite numbers. It was kerala school who gave a general formula for any integral powers of finite numbers and also made the realization that for infinite n , lower order terms can be neglected and can be written as n^k+1/k+1. It was kerala school who explicitly stated this result. So it should be credited to kerala school only. So i am editing that paragraph. Jino john1996 (talk) 17:33, 24 January 2021 (UTC)

The general formula for finite geometric series is in Euclid, long before Kerala. As I already clearly stated two sections up, but apparently you didn't read that. —David Eppstein (talk) 19:23, 24 January 2021 (UTC)

Again read the comment carefully. I am not talking about geometric formula . We already discussed about that. In this, i am talking the about sum of integral powers which should be credited to kerala school and alhazen should not be mentioned Jino john1996 (talk) 05:41, 25 January 2021 (UTC)

You are of course correct re the formula, sorry. I have no useful knowledge of Alhazen's accomplishments here but we should certainly not be mentioning him without a source. —David Eppstein (talk) 06:00, 25 January 2021 (UTC)

Article's title

If this article showed that in India there is no such thing as counting, or that south of the Himalayas multiplication has no effect, or perhaps that dividing twelve by three gives sixty-five in India, then the current title would certainly be justified.

Isn't it all just mathematics? Where is the article on "Argentinian gravity"? TooManyFingers (talk) 18:13, 11 September 2024 (UTC)

A title suggestion accompanying your criticism would have been a good idea. History of Mathematics in India? Indian contributions to mathematics? Something else?RegentsPark (comment) 20:23, 11 September 2024 (UTC)

I think both of your possibilities are good. There's also already a redirect from "Mathematics in India". Any of those seems (to me) accurate enough and suitable enough, and I wouldn't argue against any of them. People with a stronger connection to the article than I have, might prefer one of them over another, or might have more ideas; I'd be fine with any consensus that prevents the kind of obvious wrong interpretations I complained about. TooManyFingers (talk) 21:43, 11 September 2024 (UTC)

Aryabhata I's contributions - Trigonometry

Not sure if "Contains the earliest tables ...", "Contains the trigonometric formula ..." is the best verbiage. Would it be better to use "Created" instead of "Contains"? Merlin.anthwares (talk) 14:32, 24 December 2024 (UTC)

Categories: