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| '''Prosthaphaeresis''' was an [[algorithm]] used in the late 16th century and early 17th century for approximate [[multiplication]] and [[Division (mathematics)|division]] using formulas from [[trigonometry]]. For the 25 years preceding the invention of the [[logarithm]] in 1614, it was the only known generally-applicable way of approximating products quickly. Its name comes from the [[Greek language|Greek]] ''prosthesis'' and ''aphaeresis'', meaning ''addition'' and ''subtraction'', two steps in the process.<ref>{{cite journal | author = Pierce, R. C., Jr. |date=January 1977 | title = A Brief History of Logarithms | journal = The Two-Year College Mathematics Journal | volume = 8 | issue = 1 | pages = 22–26 | publisher = Mathematical Association of America| doi = 10.2307/3026878 | jstor = 3026878 }}</ref><ref>[http://www.nmt.edu/~borchers/prost.pdf Prosthaphaeresis], by Brian Borchers</ref>
| | == '秦ゆう三兄弟これは単に直接の休暇です == |
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| == History and motivation == | | 中庭の外では、中庭のドアが広く開い土地です。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 東京] 3村の病院である中年男が笑って話しています。<br><br>盛夏フラグは中年の男が言っている最も高いに向かって行きました：「ファン徐村、あなたが通知を受け取っているはず、これは今日から、高騰3年以上の最新アレンジして、彼らは意志村の感触にあなたのキャラクターです。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_7.php クリスチャンルブタン通販] '<br><br>ファンが自慢、少なくとも2メートル背が高く、ひょろっとし、ルックスが、それは静かです [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン サイズ]。<br><br>手ルピナス。ファン徐秦ゆうが3つに微笑んだ：「私は村のファン徐福睡眠のチーフだった、あなたは秦ゆう、侯飛、村にある黒い羽トリオバーはいくつかのことに注意を払う必要がある、私はあなたのための準備ができている。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン 店舗] 「ファン徐はヒスイジェーンを手渡した。<br><br>秦ゆうの魂がこのヒスイジェーンを一掃してから、引き渡す、言った： [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_11.php クリスチャンルブタン ブーツ] '村は、その後私たちは、3人の兄弟を邪魔しませんが、盛夏フラグの弟を残し、これに私たちを送ってくれてありがとう、のまま。'秦ゆう三兄弟これは単に直接の休暇です。<br><br>「ねえ、私のゆうジェーンが自分の税金を得るために二度書いた、彼はワイヤーべき参照 |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.baimusic.cn/forum.php?mod=viewthread&tid=163627 http://www.baimusic.cn/forum.php?mod=viewthread&tid=163627]</li> |
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| | <li>[http://bbs.zuoyou.mobi/home.php?mod=space&uid=7006 http://bbs.zuoyou.mobi/home.php?mod=space&uid=7006]</li> |
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| | <li>[http://www.yueyelianmeng.com/forum.php?mod=viewthread&tid=74715 http://www.yueyelianmeng.com/forum.php?mod=viewthread&tid=74715]</li> |
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| | </ul> |
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| [[Image:RechtwKugeldreieck.svg|right|thumb|250px|A spherical triangle]]
| | == 'ディロンの顔が変更されました == |
| In sixteenth century Europe, [[celestial navigation]] of ships on long voyages relied heavily on [[ephemerides]] to determine their position and course. These voluminous charts prepared by [[astronomer]]s detailed the position of stars and planets at various points in time. The models used to compute these were based on [[spherical trigonometry]], which relates the angles and arc lengths of spherical triangles (see diagram, right) using formulas such as:
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| * cos ''a'' = cos ''b'' cos ''c'' + sin ''b'' sin ''c'' cos α
| | Yuは、あなたはそれで彼をとりこにしますが、完全にダウンに彼の記憶を奪われなければならない [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン メンズ]。殿下ディロン道路に対する昆虫カストディアンのexhortationsの「大覚醒 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 東京]。<br><br>Jingzhe法執行はすぐに答えた： 'の下に......」<br><br>「悪い、誰かが私の9邪悪寺院の基盤を壊した！ [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン ブーツ] 'ディロンの顔が変更されました。スピリチュアルな知識はすぐにその後、流され、叫んだ「チベットは宮殿の位置、誰かが私と一緒に来てすぐに、法律八卦ジェン、紳士兄弟を破っている。「ディロンは既にホールを貫通した。<br><br>ヤンディ、ディ徐、ディ風、ディアロー、ジョセフ·ディ、ディ熊6殿下はまたすぐにチベットの宮殿に向かって突進 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン パンプス]。<br>最も重要な寺院の中でチベットの宮殿が、9悪ホール12ホール<br> [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 東京]。その他の再構築することができ宮殿が、宮殿の貴重な所有物は、建物ではなく、不死の内部は、不死は、これらの邪悪な捕食寺院の9、及び一定の検索半径内の9000マイルは、千年以上はただ......そのような蓄積を持たないようにすることができます、<br><br>'破る！'<br><br>物質エグゼクティブヤンチー剣、ダウン、緑内障が点滅アレイベース刺すの「ハム」位置に対する秦ゆう傍若無人剣 |
| * sin ''b'' sin α = sin ''a'' sin β
| | 相关的主题文章： |
| | | <ul> |
| where ''a'', ''b'' and ''c'' are the angles subtended at the centre of the sphere by the corresponding arcs.
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| | | <li>[http://caishen588.net/home.php?mod=space&uid=24677 http://caishen588.net/home.php?mod=space&uid=24677]</li> |
| When one quantity in such a formula is unknown but the others are known, the unknown quantity can be computed using a series of multiplications, divisions, and trigonometric table lookups. Astronomers had to make thousands of such calculations, and because the best method of multiplication available was [[Multiplication algorithm#Long multiplication|long multiplication]], most of this time was spent taxingly multiplying out products.
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| | | <li>[http://0851gz.com/home.php?mod=space&uid=148414 http://0851gz.com/home.php?mod=space&uid=148414]</li> |
| Mathematicians, particularly those who were also astronomers, were looking for an easier way, and trigonometry was one of the most advanced and familiar fields to these people. Prosthaphaeresis appeared in the 1580s, but its originator is not known for certain; its contributors included the mathematicians [[Paul Wittich]], [[Ibn Yunis]], [[Joost Bürgi]], [[Johannes Werner]], [[Christopher Clavius]], and [[François Viète]]. Wittich, Yunis, and Clavius were all astronomers and have all been credited by various sources with discovering the method. Its most well-known proponent was [[Tycho Brahe]], who used it extensively for astronomical calculations such as those described above. It was also used by [[John Napier]], who is credited with inventing the logarithms that would supplant it.
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| (Additional information: Nicholas Copernicus mentions 'prosthaphaeresis' several times in his work De Revolutionibus Orbium Coelestium, published in 1543, meaning the "great parallax" caused by the displacement of the observer due to the Earth's annual motion.)
| | <li>[http://www.gpxzw.com/home.php?mod=space&uid=135163 http://www.gpxzw.com/home.php?mod=space&uid=135163]</li> |
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| == The identities ==
| | </ul> |
| The [[trigonometric identity|trigonometric identities]] exploited by prosthaphaeresis relate products of [[trigonometric function]]s to sums. They include the following:
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| * sin ''a'' sin ''b'' = ½[cos(''a'' − ''b'') − cos(''a'' + ''b'')]
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| * cos ''a'' cos ''b'' = ½[cos(''a'' − ''b'') + cos(''a'' + ''b'')]
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| * sin ''a'' cos ''b'' = ½[sin(''a'' + ''b'') + sin(''a'' − ''b'')]
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| * cos ''a'' sin ''b'' = ½[sin(''a'' + ''b'') − sin(''a'' − ''b'')].
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| The first two of these are believed to have been derived by [[Jost Bürgi|Bürgi]], who related them to Brahe; the others follow easily from these two. If both sides are multiplied by 2, these formulas are also called the ''Werner formulas''.
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| == The algorithm ==
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| Using the second formula above, the technique for multiplication works as follows:
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| # '''Scale down''': By shifting the decimal point to the left or right, scale both numbers to a value between −1 and 1.
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| # '''Inverse cosine''': Using an inverse cosine table, find two angles whose cosines are our two values.
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| # '''Sum and difference''': Find the sum and difference of the two angles.
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| # '''Average the cosines''': Find the cosines of the sum and difference angles using a cosine table and average them.
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| # '''Scale up''': Shift the decimal place in the answer to the right (or left) as many places as you shifted the decimal place to the left (or right) in the first step, for each input.
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| For example, say we want to multiply 105 and 720. Following the steps:
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| # '''Scale down''': Shift the decimal point three places to the left in each. We get: 0.105, 0.720
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| # '''Inverse cosine''': cos(84°) is about 0.105, cos(44°) is about 0.720
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| # '''Sum and difference''': 84 + 44 = 128, 84 − 44 = 40
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| # '''Average the cosines''': ½[cos(128°) + cos(40°)] is about ½[−0.616 + 0.766], or 0.075
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| # '''Scale up''': For each of 105 and 720 we shifted the decimal point three places to the left, so in the answer we shift six places to the right. The result is 75,000. This is very close to the actual product, 75,600.
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| If we want the product of the cosines of the two initial values, which is useful in some of the astronomical calculations mentioned above, this is surprisingly even easier: only steps 3 and 4 above are necessary.
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| To divide, we exploit the definition of the secant as the reciprocal of the cosine. To divide 3500 by 70, we scale the numbers to 0.35 and 7.0. The cosine of 69.5 degrees is 0.35. Then use a table of [[Trigonometric function|secant]]s to find out that 7.0 is the secant of 81.8 degrees. This means that 1/7.0 is the cosine of 81.8 degrees, and so we can multiply 0.35 by 1/7.0 using the above procedure. Average the cosine of the sum of the angles, 81.8+69.5=151.3, with the cosine of their difference, 81.8-69.5=12.3
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| :½[cos(151°) + cos(−15°)] is about ½[−0.877 + 0.977], or 0.050
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| Scaling up to locate the decimal point gives the approximate answer, 50
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| Algorithms using the other formulas are similar, but each using different tables (sine, inverse sine, cosine, and inverse cosine) in different places. The first two are the easiest because they each only require two tables. Using the second formula, however, has the unique advantage that if only a cosine table is available, it can be used to estimate inverse cosines by searching for the angle with the nearest cosine value.
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| Notice how similar the above algorithm is to the process for multiplying using logarithms, which follows the steps: scale down, take logarithms, add, take inverse logarithm, scale up. It's no surprise that the originators of logarithms had used prosthaphaeresis.
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| Indeed the two are closely related mathematically. In modern terms, prosthaphaeresis can be viewed as relying on the logarithm of complex numbers, in particular on the identity <math>e^{ix}=\cos x + i \sin x</math>.
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| == Decreasing the error ==
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| If all the operations are performed with high precision, the product can be as accurate as desired. Although sums, differences, and averages are easy to compute with high precision, even by hand, trigonometric functions and especially inverse trigonometric functions are not. For this reason, the accuracy of the method depends to a large extent on the accuracy and detail of the trigonometric tables used.
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| For example, a sine table with an entry for each degree can be off by as much as 0.0087 if we just [[Nearest-neighbor interpolation|choose the closest number]]; each time we double the size of the table we halve this error. Tables were painstakingly constructed for prosthaphaeresis with values for every second, or 3600th of a degree.
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| Inverse sine and cosine functions are particularly troublesome, because they become steep near −1 and 1. One solution is to include more table values in this area. Another is to scale the inputs to numbers between −0.9 and 0.9. For example, 950 would become 0.095 instead of 0.950.
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| Another effective approach to enhancing the accuracy is [[linear interpolation]], which chooses a value between two adjacent table values. For example, if we know the sine of 45° is about 0.707 and the sine of 46° is about 0.719, we can estimate the sine of 45.7° as:
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| : 0.707 × (1 − 0.7) + 0.719 × 0.7 = 0.7154.
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| The actual sine is 0.7157. A table of cosines with only 180 entries combined with linear interpolation is as accurate as a table with about 45000 entries without it. Even a quick estimate of the interpolated value is often much closer than the nearest table value. See [[lookup table]] for more details.
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| == Reverse identities ==
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| The product formulas can also be manipulated to obtain formulas that express addition in terms of multiplication. Although less useful for computing products, these are still useful for deriving trigonometric results:
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| * sin ''a'' + sin ''b'' = 2sin[½(''a'' + ''b'')]cos[½(''a'' − ''b'')]
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| * sin ''a'' − sin ''b'' = 2cos[½(''a'' + ''b'')]sin[½(''a'' − ''b'')]
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| * cos ''a'' + cos ''b'' = 2cos[½(''a'' + ''b'')]cos[½(''a'' − ''b'')]
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| * cos ''a'' − cos ''b'' = −2sin[½(''a'' + ''b'')]sin[½(''a'' − ''b'')]
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| ==References==
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| {{reflist}}
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| == External links ==
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| * [http://planetmath.org/encyclopedia/ProsthaphaeresisFormulas.html PlanetMath: Prosthaphaeresis formulas]
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| * Daniel E. Otero [http://cerebro.xu.edu/math/math147/02f/briggs/briggsintro.html Henry Briggs]. Introduction: the need for speed in calculation.
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| * [http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html Mathworld: Prosthaphaeresis formulas]
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| * Adam Mosley. [http://www.hps.cam.ac.uk/starry/tychomaths.html Tycho Brahe and Mathematical Techniques]. University of Cambridge.
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| * IEEE Computer Society. [http://pages.cpsc.ucalgary.ca/~williams/History_web_site/time%201500_1800/John%20Napier%20and%20invention%20of%20logs.htm History of computing: John Napier and the invention of logarithms].(doesn't open)
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| * [http://www.pballew.net/arithm18.html#Prostha Math Words: Prosthaphaeresis]
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| * Beatrice Lumpkin. ''[http://www.pps.k12.or.us/depts-c/mc-me/be-af-ma.pdf African and African-American Contributions to Mathematics]''. Discusses Ibn Yunis's contribution to prosthaphaeresis.
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| * [http://www4.ncsu.edu/~njrose/pdfFiles/Prostha.pdf Prosthaphaeresis] and beat phenomenon in the theory of vibrations, by Nicholas J. Rose
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| [[Category:Trigonometry]]
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| [[Category:Arithmetic]]
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